Subgroups — C₆₇×#61573 #30740824

Order 17152.

2094 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3501	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#548×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3316	C₂	1	8	yes	maximal, complemented
4	C₆₇×#3270	C₂	1	8	yes	maximal, complemented
5	C₆₇×#3215	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3195	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3187	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3113	C₂	1	2	yes	maximal, complemented
9	C₆₇×#3007	C₂	1	2	yes	maximal, complemented
10	C₂×#451×C₆₇	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2973	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#137×C₆₇	C₂²	1	4	yes	complemented
15	C₂²×Q₁₆×C₆₇	C₂²	1	4	yes	complemented
16	#554×C₆₇	—	2	4	no	—
17	#554×C₆₇	C₂²	1	4	yes	complemented
18	D₄×Q₈×C₆₇	C₂²	1	4	yes	complemented
19	#548×C₆₇	—	2	4	no	—
20	#548×C₆₇	C₂²	1	8	yes	—
21	#544×C₆₇	—	2	1	no	—
22	#543×C₆₇	—	2	1	no	—
23	#543×C₆₇	—	2	1	no	—
24	#533×C₆₇	C₂²	1	2	yes	—
25	C₂×#124×C₆₇	—	2	1	no	—
26	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
28	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
29	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
31	C₂×#121×C₆₇	—	2	1	no	—
32	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
33	#518×C₆₇	—	2	1	no	—
34	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×C₄×D₄×C₆₇	—	2	1	no	—
36	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
37	#497×C₆₇	C₂²	1	4	yes	complemented
38	#489×C₆₇	C₂²	1	4	yes	complemented
39	#483×C₆₇	C₂²	1	4	yes	—
40	#483×C₆₇	C₂²	1	4	yes	complemented
41	#481×C₆₇	C₂²	1	4	yes	complemented
42	#475×C₆₇	C₂²	1	4	yes	complemented
43	#470×C₆₇	C₂²	1	8	yes	complemented
44	#465×C₆₇	C₂²	1	8	yes	complemented
45	#464×C₆₇	C₂²	1	8	yes	complemented
46	#462×C₆₇	C₂²	1	4	yes	complemented
47	#452×C₆₇	C₂²	1	8	yes	complemented
48	#451×C₆₇	C₂²	1	4	yes	—
49	#451×C₆₇	C₂²	1	4	yes	—
50	#451×C₆₇	C₂²	1	4	yes	complemented
51	#451×C₆₇	C₂²	1	8	yes	complemented
52	#450×C₆₇	C₂²	1	4	yes	—
53	#448×C₆₇	C₂²	1	4	yes	complemented
54	#441×C₆₇	C₂²	1	8	yes	complemented
55	#435×C₆₇	C₂²	1	8	yes	complemented
56	#432×C₆₇	C₂²	1	2	yes	—
57	#417×C₆₇	C₂²	1	4	yes	complemented
58	C₂×#103×C₆₇	C₂²	1	4	yes	complemented
59	#407×C₆₇	C₂²	1	2	yes	—
60	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
61	C₂×#41×C₆₇	—	2	2	no	—
62	C₂×#41×C₆₇	—	4	4	no	—
63	C₂×#41×C₆₇	C₂³	1	4	yes	—
64	C₂²×Q₈×C₆₇	—	2	1	no	—
65	C₂²×Q₈×C₆₇	—	2	2	no	—
66	C₂²×Q₈×C₆₇	—	4	4	no	—
67	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
69	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
70	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
72	C₂²×D₄×C₆₇	—	2	1	no	—
73	C₂²×D₄×C₆₇	—	2	1	no	—
74	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
75	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₆₇	—	2	1	no	—
77	C₂³×C₄×C₆₇	—	2	1	no	—
78	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
80	#137×C₆₇	—	4	8	no	—
81	C₂×Q₁₆×C₆₇	—	4	8	no	—
82	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	—
83	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	—
84	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
85	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
86	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
87	C₄⋊Q₈×C₆₇	—	2	4	no	—
88	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
89	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
90	#125×C₆₇	—	2	4	no	—
91	#125×C₆₇	—	2	2	no	—
92	#124×C₆₇	—	2	8	no	—
93	#124×C₆₇	—	2	2	no	—
94	#124×C₆₇	C₂³	1	4	yes	—
95	C₂²⋊Q₈×C₆₇	—	2	2	no	—
96	C₂²⋊Q₈×C₆₇	—	2	2	no	—
97	C₂²⋊Q₈×C₆₇	—	2	2	no	—
98	C₂²⋊Q₈×C₆₇	—	2	2	no	—
99	C₂²⋊Q₈×C₆₇	—	2	2	no	—
100	C₂²⋊Q₈×C₆₇	—	2	2	no	—
101	C₂²⋊Q₈×C₆₇	—	2	8	no	—
102	C₂²⋊Q₈×C₆₇	—	2	4	no	—
103	C₂²⋊Q₈×C₆₇	—	2	4	no	—
104	C₂²⋊Q₈×C₆₇	—	2	4	no	—
105	C₂²⋊Q₈×C₆₇	—	2	4	no	—
106	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
107	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
108	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
109	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
110	#121×C₆₇	—	2	8	no	—
111	#121×C₆₇	—	2	2	no	—
112	#121×C₆₇	C₂³	1	4	yes	—
113	C₄×Q₈×C₆₇	—	2	2	no	—
114	C₄×Q₈×C₆₇	—	2	2	no	—
115	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
116	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
117	C₄×D₄×C₆₇	—	2	4	no	—
118	C₄×D₄×C₆₇	—	2	4	no	—
119	C₄×D₄×C₆₇	—	2	8	no	—
120	C₄×D₄×C₆₇	D₄	1	4	yes	—
121	C₄×D₄×C₆₇	C₂³	1	4	yes	—
122	#117×C₆₇	—	2	2	no	—
123	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
124	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
125	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
127	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
128	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
129	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
130	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
131	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
132	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
133	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
134	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
135	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
136	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
137	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
138	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
139	C₂×C₄²×C₆₇	—	2	1	no	—
140	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
141	#106×C₆₇	C₂³	1	8	yes	—
142	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
143	#103×C₆₇	C₂³	1	8	yes	—
144	#103×C₆₇	C₂³	1	8	yes	—
145	#103×C₆₇	C₂³	1	8	yes	—
146	#102×C₆₇	C₂³	1	8	yes	—
147	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
148	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
149	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
150	#97×C₆₇	C₂³	1	4	yes	—
151	C₂⁴×C₆₇	—	2	1	no	—
152	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
153	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
154	#41×C₆₇	—	2	8	no	—
155	#41×C₆₇	—	4	4	no	—
156	#41×C₆₇	—	4	8	no	—
157	C₂×Q₈×C₆₇	—	2	4	no	—
158	C₂×Q₈×C₆₇	—	2	1	no	—
159	C₂×Q₈×C₆₇	—	2	4	no	—
160	C₂×Q₈×C₆₇	—	2	1	no	—
161	C₂×Q₈×C₆₇	—	2	2	no	—
162	C₂×Q₈×C₆₇	—	2	2	no	—
163	C₂×Q₈×C₆₇	—	2	2	no	—
164	C₂×Q₈×C₆₇	—	2	8	no	—
165	C₂×Q₈×C₆₇	—	4	4	no	—
166	C₂×Q₈×C₆₇	—	4	2	no	—
167	C₂×Q₈×C₆₇	—	4	4	no	—
168	C₂×Q₈×C₆₇	—	4	4	no	—
169	C₂×Q₈×C₆₇	—	4	4	no	—
170	C₂×Q₈×C₆₇	—	4	8	no	—
171	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
172	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
173	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
174	C₂×D₄×C₆₇	—	2	4	no	—
175	C₂×D₄×C₆₇	—	2	4	no	—
176	C₂×D₄×C₆₇	—	2	4	no	—
177	C₂×D₄×C₆₇	—	2	1	no	—
178	C₂×D₄×C₆₇	—	4	2	no	—
179	C₂×D₄×C₆₇	—	4	4	no	—
180	C₂×D₄×C₆₇	—	4	4	no	—
181	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
182	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
183	C₂²×C₄×C₆₇	—	2	1	no	—
184	C₂²×C₄×C₆₇	—	2	1	no	—
185	C₂²×C₄×C₆₇	—	2	1	no	—
186	C₂²×C₄×C₆₇	—	2	4	no	—
187	C₂²×C₄×C₆₇	—	2	2	no	—
188	C₂²×C₄×C₆₇	—	2	2	no	—
189	C₂²×C₄×C₆₇	—	2	2	no	—
190	C₂²×C₄×C₆₇	—	2	2	no	—
191	C₂²×C₄×C₆₇	—	2	4	no	—
192	C₂²×C₄×C₆₇	—	4	2	no	—
193	C₂²×C₄×C₆₇	—	4	2	no	—
194	C₂²×C₄×C₆₇	—	4	4	no	—
195	C₂²×C₄×C₆₇	—	4	4	no	—
196	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
197	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
198	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
199	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
200	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
201	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
202	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
203	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
204	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
205	Q₁₆×C₆₇	—	2	16	no	—
206	Q₁₆×C₆₇	—	4	8	no	—
207	SD₁₆×C₆₇	—	4	8	no	—
208	M₁₆×C₆₇	—	4	4	no	—
209	C₂×C₈×C₆₇	—	4	4	no	—
210	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
211	C₄⋊C₄×C₆₇	—	2	4	no	—
212	C₄⋊C₄×C₆₇	—	2	4	no	—
213	C₄⋊C₄×C₆₇	—	2	4	no	—
214	C₄⋊C₄×C₆₇	—	2	4	no	—
215	C₄⋊C₄×C₆₇	—	2	4	no	—
216	C₄⋊C₄×C₆₇	—	2	1	no	—
217	C₄⋊C₄×C₆₇	—	2	2	no	—
218	C₄⋊C₄×C₆₇	—	2	2	no	—
219	C₄⋊C₄×C₆₇	—	2	2	no	—
220	C₄⋊C₄×C₆₇	—	2	2	no	—
221	C₄⋊C₄×C₆₇	—	2	2	no	—
222	C₄⋊C₄×C₆₇	—	2	4	no	—
223	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
224	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
225	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
226	C₂²⋊C₄×C₆₇	—	2	4	no	—
227	C₂²⋊C₄×C₆₇	—	2	4	no	—
228	C₂²⋊C₄×C₆₇	—	2	4	no	—
229	C₂²⋊C₄×C₆₇	—	2	4	no	—
230	C₂²⋊C₄×C₆₇	—	2	4	no	—
231	C₂²⋊C₄×C₆₇	—	2	4	no	—
232	C₂²⋊C₄×C₆₇	—	2	4	no	—
233	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
234	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
235	C₄²×C₆₇	—	2	4	no	—
236	C₄²×C₆₇	—	2	4	no	—
237	C₄²×C₆₇	C₂×D₄	1	2	yes	—
238	C₄²×C₆₇	C₂⁴	1	2	yes	—
239	C₂³×C₆₇	—	2	2	no	—
240	C₂³×C₆₇	—	2	2	no	—
241	C₂³×C₆₇	—	4	1	no	—
242	C₂³×C₆₇	—	4	1	no	—
243	C₂³×C₆₇	—	4	1	no	—
244	C₂³×C₆₇	—	4	1	no	—
245	C₂³×C₆₇	—	4	2	no	—
246	C₂³×C₆₇	—	4	2	no	—
247	C₂³×C₆₇	—	4	2	no	—
248	C₂³×C₆₇	#137	1	4	yes	complemented
249	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
250	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
251	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
252	Q₈×C₆₇	—	2	4	no	—
253	Q₈×C₆₇	—	2	2	no	—
254	Q₈×C₆₇	—	2	8	no	—
255	Q₈×C₆₇	—	2	4	no	—
256	Q₈×C₆₇	—	4	8	no	—
257	Q₈×C₆₇	—	4	2	no	—
258	Q₈×C₆₇	—	8	2	no	—
259	Q₈×C₆₇	—	8	2	no	—
260	Q₈×C₆₇	—	8	2	no	—
261	Q₈×C₆₇	H(5,𝔽₂)	1	4	yes	—
262	D₄×C₆₇	—	2	8	no	—
263	D₄×C₆₇	—	2	4	no	—
264	D₄×C₆₇	—	4	8	no	—
265	D₄×C₆₇	—	4	2	no	—
266	D₄×C₆₇	—	4	8	no	—
267	C₂×C₄×C₆₇	—	2	4	no	—
268	C₂×C₄×C₆₇	—	2	2	no	—
269	C₂×C₄×C₆₇	—	2	2	no	—
270	C₂×C₄×C₆₇	—	2	4	no	—
271	C₂×C₄×C₆₇	—	2	4	no	—
272	C₂×C₄×C₆₇	—	2	2	no	—
273	C₂×C₄×C₆₇	—	2	2	no	—
274	C₂×C₄×C₆₇	—	2	4	no	—
275	C₂×C₄×C₆₇	—	2	4	no	—
276	C₂×C₄×C₆₇	—	2	2	no	—
277	C₂×C₄×C₆₇	—	4	1	no	—
278	C₂×C₄×C₆₇	—	4	2	no	—
279	C₂×C₄×C₆₇	—	4	1	no	—
280	C₂×C₄×C₆₇	—	4	2	no	—
281	C₂×C₄×C₆₇	—	4	1	no	—
282	C₂×C₄×C₆₇	—	4	8	no	—
283	C₂×C₄×C₆₇	—	4	8	no	—
284	C₂×C₄×C₆₇	—	4	2	no	—
285	C₂×C₄×C₆₇	—	4	2	no	—
286	C₂×C₄×C₆₇	—	8	2	no	—
287	C₂×C₄×C₆₇	—	8	2	no	—
288	C₂×C₄×C₆₇	—	8	2	no	—
289	C₂×C₄×C₆₇	—	8	2	no	—
290	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
291	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
292	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
293	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
294	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
295	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
296	C₈×C₆₇	—	2	8	no	—
297	C₂²×C₆₇	—	2	1	no	—
298	C₂²×C₆₇	—	2	2	no	—
299	C₂²×C₆₇	—	2	1	no	—
300	C₂²×C₆₇	—	2	2	no	—
301	C₂²×C₆₇	—	2	2	no	—
302	C₂²×C₆₇	—	2	2	no	—
303	C₂²×C₆₇	—	2	2	no	—
304	C₂²×C₆₇	—	4	4	no	—
305	C₂²×C₆₇	—	4	2	no	—
306	C₂²×C₆₇	—	4	2	no	—
307	C₂²×C₆₇	—	4	2	no	—
308	C₂²×C₆₇	—	8	2	no	—
309	C₂²×C₆₇	—	8	4	no	—
310	C₂²×C₆₇	C₂×#137	1	2	yes	—
311	C₂²×C₆₇	C₂×#137	1	4	yes	—
312	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
313	C₄×C₆₇	—	2	2	no	—
314	C₄×C₆₇	—	2	4	no	—
315	C₄×C₆₇	—	2	2	no	—
316	C₄×C₆₇	—	2	1	no	—
317	C₄×C₆₇	—	2	2	no	—
318	C₄×C₆₇	—	2	2	no	—
319	C₄×C₆₇	—	4	4	no	—
320	C₄×C₆₇	—	4	2	no	—
321	C₄×C₆₇	—	4	4	no	—
322	C₄×C₆₇	—	4	2	no	—
323	C₄×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
324	#61573	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
325	C₂×C₆₇	—	2	2	no	—
326	C₂×C₆₇	—	2	4	no	—
327	C₂×C₆₇	—	4	2	no	—
328	C₂×C₆₇	—	8	2	no	—
329	C₂×C₆₇	#3484	1	1	yes	characteristic
330	C₂×C₆₇	C₂²×#137	1	1	yes	characteristic
331	C₂×C₆₇	#3586	1	1	yes	characteristic
332	#3501	C₂×C₆₇	1	1	yes	characteristic, complemented
333	C₂×#548	C₂×C₆₇	1	1	yes	characteristic
334	#3316	C₂×C₆₇	1	8	yes	complemented
335	#3270	C₂×C₆₇	1	8	yes	complemented
336	#3215	C₂×C₆₇	1	2	yes	complemented
337	#3195	C₂×C₆₇	1	2	yes	complemented
338	#3187	C₂×C₆₇	1	2	yes	complemented
339	#3113	C₂×C₆₇	1	2	yes	complemented
340	#3007	C₂×C₆₇	1	2	yes	complemented
341	C₂×#451	C₂×C₆₇	1	2	yes	complemented
342	#2973	C₂×C₆₇	1	1	yes	characteristic, complemented
343	C₆₇	#61573	1	1	yes	characteristic, direct product factor, Sylow
344	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
345	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
346	C₂×#137	C₂²×C₆₇	1	4	yes	complemented
347	C₂²×Q₁₆	C₂²×C₆₇	1	4	yes	complemented
348	#554	—	2	4	no	—
349	#554	C₂²×C₆₇	1	4	yes	complemented
350	D₄×Q₈	C₂²×C₆₇	1	4	yes	complemented
351	#548	—	2	4	no	—
352	#548	C₂²×C₆₇	1	8	yes	—
353	#544	—	2	1	no	—
354	#543	—	2	1	no	—
355	#543	—	2	1	no	—
356	#533	C₂²×C₆₇	1	2	yes	—
357	C₂×#124	—	2	1	no	—
358	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
359	C₂×C₂²⋊Q₈	—	2	1	no	—
360	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
361	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
362	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
363	C₂×#121	—	2	1	no	—
364	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
365	#518	—	2	1	no	—
366	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
367	C₂×C₄×D₄	—	2	1	no	—
368	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
369	#497	C₂²×C₆₇	1	4	yes	complemented
370	#489	C₂²×C₆₇	1	4	yes	complemented
371	#483	C₂²×C₆₇	1	4	yes	—
372	#483	C₂²×C₆₇	1	4	yes	complemented
373	#481	C₂²×C₆₇	1	4	yes	complemented
374	#475	C₂²×C₆₇	1	4	yes	complemented
375	#470	C₂²×C₆₇	1	8	yes	complemented
376	#465	C₂²×C₆₇	1	8	yes	complemented
377	#464	C₂²×C₆₇	1	8	yes	complemented
378	#462	C₂²×C₆₇	1	4	yes	complemented
379	#452	C₂²×C₆₇	1	8	yes	complemented
380	#451	C₂²×C₆₇	1	4	yes	—
381	#451	C₂²×C₆₇	1	4	yes	—
382	#451	C₂²×C₆₇	1	4	yes	complemented
383	#451	C₂²×C₆₇	1	8	yes	complemented
384	#450	C₂²×C₆₇	1	4	yes	—
385	#448	C₂²×C₆₇	1	4	yes	complemented
386	#441	C₂²×C₆₇	1	8	yes	complemented
387	#435	C₂²×C₆₇	1	8	yes	complemented
388	#432	C₂²×C₆₇	1	2	yes	—
389	#417	C₂²×C₆₇	1	4	yes	complemented
390	C₂×#103	C₂²×C₆₇	1	4	yes	complemented
391	#407	C₂²×C₆₇	1	2	yes	—
392	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
393	C₂×#41	—	2	2	no	—
394	C₂×#41	—	4	4	no	—
395	C₂×#41	C₂³×C₆₇	1	4	yes	—
396	C₂²×Q₈	—	2	1	no	—
397	C₂²×Q₈	—	2	2	no	—
398	C₂²×Q₈	—	4	4	no	—
399	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
400	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
401	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
402	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
403	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
404	C₂²×D₄	—	2	1	no	—
405	C₂²×D₄	—	2	1	no	—
406	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
407	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
408	C₂³×C₄	—	2	1	no	—
409	C₂³×C₄	—	2	1	no	—
410	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
411	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
412	#137	—	4	8	no	—
413	C₂×Q₁₆	—	4	8	no	—
414	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
415	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
416	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
417	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
418	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
419	C₄⋊Q₈	—	2	4	no	—
420	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
421	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
422	#125	—	2	4	no	—
423	#125	—	2	2	no	—
424	#124	—	2	8	no	—
425	#124	—	2	2	no	—
426	#124	C₂³×C₆₇	1	4	yes	—
427	C₂²⋊Q₈	—	2	2	no	—
428	C₂²⋊Q₈	—	2	2	no	—
429	C₂²⋊Q₈	—	2	2	no	—
430	C₂²⋊Q₈	—	2	2	no	—
431	C₂²⋊Q₈	—	2	2	no	—
432	C₂²⋊Q₈	—	2	2	no	—
433	C₂²⋊Q₈	—	2	8	no	—
434	C₂²⋊Q₈	—	2	4	no	—
435	C₂²⋊Q₈	—	2	4	no	—
436	C₂²⋊Q₈	—	2	4	no	—
437	C₂²⋊Q₈	—	2	4	no	—
438	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
439	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
440	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
441	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
442	#121	—	2	8	no	—
443	#121	—	2	2	no	—
444	#121	C₂³×C₆₇	1	4	yes	—
445	C₄×Q₈	—	2	2	no	—
446	C₄×Q₈	—	2	2	no	—
447	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
448	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
449	C₄×D₄	—	2	4	no	—
450	C₄×D₄	—	2	4	no	—
451	C₄×D₄	—	2	8	no	—
452	C₄×D₄	D₄×C₆₇	1	4	yes	—
453	C₄×D₄	C₂³×C₆₇	1	4	yes	—
454	#117	—	2	2	no	—
455	C₂×C₄⋊C₄	—	2	1	no	—
456	C₂×C₄⋊C₄	—	2	1	no	—
457	C₂×C₄⋊C₄	—	2	2	no	—
458	C₂×C₄⋊C₄	—	2	2	no	—
459	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
460	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
461	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
462	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
463	C₂×C₂²⋊C₄	—	2	2	no	—
464	C₂×C₂²⋊C₄	—	2	2	no	—
465	C₂×C₂²⋊C₄	—	2	1	no	—
466	C₂×C₂²⋊C₄	—	2	1	no	—
467	C₂×C₂²⋊C₄	—	2	1	no	—
468	C₂×C₂²⋊C₄	—	2	1	no	—
469	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
470	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
471	C₂×C₄²	—	2	1	no	—
472	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
473	#106	C₂³×C₆₇	1	8	yes	—
474	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
475	#103	C₂³×C₆₇	1	8	yes	—
476	#103	C₂³×C₆₇	1	8	yes	—
477	#103	C₂³×C₆₇	1	8	yes	—
478	#102	C₂³×C₆₇	1	8	yes	—
479	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
480	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
481	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
482	#97	C₂³×C₆₇	1	4	yes	—
483	C₂⁴	—	2	1	no	—
484	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
485	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
486	#41	—	2	8	no	—
487	#41	—	4	4	no	—
488	#41	—	4	8	no	—
489	C₂×Q₈	—	2	4	no	—
490	C₂×Q₈	—	2	1	no	—
491	C₂×Q₈	—	2	4	no	—
492	C₂×Q₈	—	2	1	no	—
493	C₂×Q₈	—	2	2	no	—
494	C₂×Q₈	—	2	2	no	—
495	C₂×Q₈	—	2	2	no	—
496	C₂×Q₈	—	2	8	no	—
497	C₂×Q₈	—	4	4	no	—
498	C₂×Q₈	—	4	2	no	—
499	C₂×Q₈	—	4	4	no	—
500	C₂×Q₈	—	4	4	no	—
501	C₂×Q₈	—	4	4	no	—
502	C₂×Q₈	—	4	8	no	—
503	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
504	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
505	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
506	C₂×D₄	—	2	4	no	—
507	C₂×D₄	—	2	4	no	—
508	C₂×D₄	—	2	4	no	—
509	C₂×D₄	—	2	1	no	—
510	C₂×D₄	—	4	2	no	—
511	C₂×D₄	—	4	4	no	—
512	C₂×D₄	—	4	4	no	—
513	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
514	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
515	C₂²×C₄	—	2	1	no	—
516	C₂²×C₄	—	2	1	no	—
517	C₂²×C₄	—	2	1	no	—
518	C₂²×C₄	—	2	4	no	—
519	C₂²×C₄	—	2	2	no	—
520	C₂²×C₄	—	2	2	no	—
521	C₂²×C₄	—	2	2	no	—
522	C₂²×C₄	—	2	2	no	—
523	C₂²×C₄	—	2	4	no	—
524	C₂²×C₄	—	4	2	no	—
525	C₂²×C₄	—	4	2	no	—
526	C₂²×C₄	—	4	4	no	—
527	C₂²×C₄	—	4	4	no	—
528	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
529	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
530	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
531	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
532	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
533	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
534	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
535	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
536	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
537	Q₁₆	—	2	16	no	—
538	Q₁₆	—	4	8	no	—
539	SD₁₆	—	4	8	no	—
540	M₁₆	—	4	4	no	—
541	C₂×C₈	—	4	4	no	—
542	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
543	C₄⋊C₄	—	2	4	no	—
544	C₄⋊C₄	—	2	4	no	—
545	C₄⋊C₄	—	2	4	no	—
546	C₄⋊C₄	—	2	4	no	—
547	C₄⋊C₄	—	2	4	no	—
548	C₄⋊C₄	—	2	1	no	—
549	C₄⋊C₄	—	2	2	no	—
550	C₄⋊C₄	—	2	2	no	—
551	C₄⋊C₄	—	2	2	no	—
552	C₄⋊C₄	—	2	2	no	—
553	C₄⋊C₄	—	2	2	no	—
554	C₄⋊C₄	—	2	4	no	—
555	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
556	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
557	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
558	C₂²⋊C₄	—	2	4	no	—
559	C₂²⋊C₄	—	2	4	no	—
560	C₂²⋊C₄	—	2	4	no	—
561	C₂²⋊C₄	—	2	4	no	—
562	C₂²⋊C₄	—	2	4	no	—
563	C₂²⋊C₄	—	2	4	no	—
564	C₂²⋊C₄	—	2	4	no	—
565	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
566	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
567	C₄²	—	2	4	no	—
568	C₄²	—	2	4	no	—
569	C₄²	C₂×D₄×C₆₇	1	2	yes	—
570	C₄²	C₂⁴×C₆₇	1	2	yes	—
571	C₂³	—	2	2	no	—
572	C₂³	—	2	2	no	—
573	C₂³	—	4	1	no	—
574	C₂³	—	4	1	no	—
575	C₂³	—	4	1	no	—
576	C₂³	—	4	1	no	—
577	C₂³	—	4	2	no	—
578	C₂³	—	4	2	no	—
579	C₂³	—	4	2	no	—
580	C₂³	#137×C₆₇	1	4	yes	complemented
581	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
582	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
583	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
584	Q₈	—	2	4	no	—
585	Q₈	—	2	2	no	—
586	Q₈	—	2	8	no	—
587	Q₈	—	2	4	no	—
588	Q₈	—	4	8	no	—
589	Q₈	—	4	2	no	—
590	Q₈	—	8	2	no	—
591	Q₈	—	8	2	no	—
592	Q₈	—	8	2	no	—
593	Q₈	H(5,𝔽₂)×C₆₇	1	4	yes	—
594	D₄	—	2	8	no	—
595	D₄	—	2	4	no	—
596	D₄	—	4	8	no	—
597	D₄	—	4	2	no	—
598	D₄	—	4	8	no	—
599	C₂×C₄	—	2	4	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	2	4	no	—
603	C₂×C₄	—	2	4	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	4	no	—
607	C₂×C₄	—	2	4	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	2	no	—
611	C₂×C₄	—	4	1	no	—
612	C₂×C₄	—	4	2	no	—
613	C₂×C₄	—	4	1	no	—
614	C₂×C₄	—	4	8	no	—
615	C₂×C₄	—	4	8	no	—
616	C₂×C₄	—	4	2	no	—
617	C₂×C₄	—	4	2	no	—
618	C₂×C₄	—	8	2	no	—
619	C₂×C₄	—	8	2	no	—
620	C₂×C₄	—	8	2	no	—
621	C₂×C₄	—	8	2	no	—
622	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
623	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
624	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
625	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
626	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
627	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
628	C₈	—	2	8	no	—
629	C₂²	—	2	1	no	—
630	C₂²	—	2	2	no	—
631	C₂²	—	2	1	no	—
632	C₂²	—	2	2	no	—
633	C₂²	—	2	2	no	—
634	C₂²	—	2	2	no	—
635	C₂²	—	2	2	no	—
636	C₂²	—	4	4	no	—
637	C₂²	—	4	2	no	—
638	C₂²	—	4	2	no	—
639	C₂²	—	4	2	no	—
640	C₂²	—	8	2	no	—
641	C₂²	—	8	4	no	—
642	C₂²	C₂×#137×C₆₇	1	2	yes	—
643	C₂²	C₂×#137×C₆₇	1	4	yes	—
644	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
645	C₄	—	2	2	no	—
646	C₄	—	2	2	no	—
647	C₄	—	2	4	no	—
648	C₄	—	2	2	no	—
649	C₄	—	2	1	no	—
650	C₄	—	2	2	no	—
651	C₄	—	4	2	no	—
652	C₄	—	4	4	no	—
653	C₄	—	4	2	no	—
654	C₄	—	4	4	no	—
655	C₄	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
656	C₂	—	2	2	no	—
657	C₂	—	2	4	no	—
658	C₂	—	4	2	no	—
659	C₂	—	8	2	no	—
660	C₂	C₆₇×#3484	1	1	yes	characteristic
661	C₂	C₂²×#137×C₆₇	1	1	yes	characteristic
662	C₂	C₆₇×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂×#548×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₆₇×#3501	C₂	semidirect
3. C₆₇×#3316	C₂	semidirect
4. C₆₇×#3270	C₂	semidirect
5. C₆₇×#3215	C₂	semidirect
6. C₆₇×#3195	C₂	semidirect
7. C₆₇×#3187	C₂	semidirect
8. C₆₇×#3113	C₂	semidirect
9. C₆₇×#3007	C₂	semidirect
10. C₂×#451×C₆₇	C₂	semidirect
11. C₆₇×#2973	C₂	semidirect
14. C₂×#137×C₆₇	C₂²	semidirect
15. C₂²×Q₁₆×C₆₇	C₂²	semidirect
17. #554×C₆₇	C₂²	semidirect
18. D₄×Q₈×C₆₇	C₂²	semidirect
37. #497×C₆₇	C₂²	semidirect
38. #489×C₆₇	C₂²	semidirect
40. #483×C₆₇	C₂²	semidirect
41. #481×C₆₇	C₂²	semidirect
42. #475×C₆₇	C₂²	semidirect
43. #470×C₆₇	C₂²	semidirect
44. #465×C₆₇	C₂²	semidirect
45. #464×C₆₇	C₂²	semidirect
46. #462×C₆₇	C₂²	semidirect
47. #452×C₆₇	C₂²	semidirect
50. #451×C₆₇	C₂²	semidirect
51. #451×C₆₇	C₂²	semidirect
53. #448×C₆₇	C₂²	semidirect
54. #441×C₆₇	C₂²	semidirect
55. #435×C₆₇	C₂²	semidirect
57. #417×C₆₇	C₂²	semidirect
58. C₂×#103×C₆₇	C₂²	semidirect
248. C₂³×C₆₇	#137	semidirect
324. #61573	C₆₇	direct (N × H)
332. #3501	C₂×C₆₇	semidirect
334. #3316	C₂×C₆₇	semidirect
335. #3270	C₂×C₆₇	semidirect
336. #3215	C₂×C₆₇	semidirect
337. #3195	C₂×C₆₇	semidirect
338. #3187	C₂×C₆₇	semidirect
339. #3113	C₂×C₆₇	semidirect
340. #3007	C₂×C₆₇	semidirect
341. C₂×#451	C₂×C₆₇	semidirect
342. #2973	C₂×C₆₇	semidirect
343. C₆₇	#61573	direct (N × H)
346. C₂×#137	C₂²×C₆₇	semidirect
347. C₂²×Q₁₆	C₂²×C₆₇	semidirect
349. #554	C₂²×C₆₇	semidirect
350. D₄×Q₈	C₂²×C₆₇	semidirect
369. #497	C₂²×C₆₇	semidirect
370. #489	C₂²×C₆₇	semidirect
372. #483	C₂²×C₆₇	semidirect
373. #481	C₂²×C₆₇	semidirect
374. #475	C₂²×C₆₇	semidirect
375. #470	C₂²×C₆₇	semidirect
376. #465	C₂²×C₆₇	semidirect
377. #464	C₂²×C₆₇	semidirect
378. #462	C₂²×C₆₇	semidirect
379. #452	C₂²×C₆₇	semidirect
382. #451	C₂²×C₆₇	semidirect
383. #451	C₂²×C₆₇	semidirect
385. #448	C₂²×C₆₇	semidirect
386. #441	C₂²×C₆₇	semidirect
387. #435	C₂²×C₆₇	semidirect
389. #417	C₂²×C₆₇	semidirect
390. C₂×#103	C₂²×C₆₇	semidirect
580. C₂³	#137×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₆₇×#3501	2
C₂×#548×C₆₇	2
C₆₇×#3316	2
C₆₇×#3270	2
C₆₇×#3215	2
C₆₇×#3195	2
C₆₇×#3187	2
C₆₇×#3113	2
C₆₇×#3007	2
C₂×#451×C₆₇	2
C₆₇×#2973	2
#61573	67

Subgroups — C67×#61573 #30740824

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61573 #30740824