Subgroups — C₆₇×#61560 #30740811

Order 17152.

2574 conjugacy classes of subgroups.

The subgroup lattice has 516 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×D₄²×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3295	C₂	1	8	yes	maximal, complemented
4	D₄×SD₁₆×C₆₇	C₂	1	8	yes	maximal, complemented
5	C₆₇×#3208	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3187	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3101	C₂	1	2	yes	maximal, complemented
8	C₂×#450×C₆₇	C₂	1	4	yes	maximal, complemented
9	C₆₇×#2972	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₄×C₆₇	—	2	1	no	—
11	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
12	C₂²×SD₁₆×C₆₇	C₂²	1	8	yes	complemented
13	#554×C₆₇	—	2	4	no	—
14	#554×C₆₇	C₂²	1	4	yes	complemented
15	D₄×Q₈×C₆₇	C₂²	1	4	yes	complemented
16	D₄²×C₆₇	—	2	4	no	—
17	D₄²×C₆₇	C₂²	1	8	yes	—
18	#544×C₆₇	—	2	1	no	—
19	#543×C₆₇	—	2	2	no	—
20	#533×C₆₇	C₂²	1	2	yes	complemented
21	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
23	C₂×#121×C₆₇	—	2	1	no	—
24	C₂×#121×C₆₇	C₂²	1	2	yes	—
25	C₂×#120×C₆₇	—	2	2	no	—
26	#518×C₆₇	—	2	1	no	—
27	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
29	#492×C₆₇	C₂²	1	4	yes	complemented
30	#486×C₆₇	C₂²	1	4	yes	complemented
31	#483×C₆₇	C₂²	1	4	yes	complemented
32	#481×C₆₇	C₂²	1	8	yes	complemented
33	#476×C₆₇	C₂²	1	4	yes	complemented
34	#465×C₆₇	C₂²	1	16	yes	complemented
35	#461×C₆₇	C₂²	1	4	yes	complemented
36	#460×C₆₇	C₂²	1	8	yes	complemented
37	#450×C₆₇	C₂²	1	4	yes	—
38	#450×C₆₇	C₂²	1	16	yes	complemented
39	#450×C₆₇	C₂²	1	8	yes	complemented
40	#448×C₆₇	C₂²	1	8	yes	complemented
41	C₄×SD₁₆×C₆₇	C₂²	1	8	yes	complemented
42	C₈×D₄×C₆₇	C₂²	1	8	yes	complemented
43	#431×C₆₇	C₂²	1	2	yes	complemented
44	C₂×#102×C₆₇	C₂²	1	8	yes	complemented
45	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	complemented
46	C₂⁵×C₆₇	—	2	2	no	—
47	C₂²×Q₈×C₆₇	C₂³	1	4	yes	complemented
48	C₂²×D₄×C₆₇	—	2	2	no	—
49	C₂²×D₄×C₆₇	—	2	8	no	—
50	C₂²×D₄×C₆₇	—	2	2	no	—
51	C₂²×D₄×C₆₇	—	2	1	no	—
52	C₂²×D₄×C₆₇	—	2	4	no	—
53	C₂²×D₄×C₆₇	—	2	1	no	—
54	C₂²×D₄×C₆₇	—	4	8	no	—
55	C₂²×D₄×C₆₇	—	4	8	no	—
56	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
57	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
59	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
60	C₂³×C₄×C₆₇	—	2	1	no	—
61	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
62	C₂×SD₁₆×C₆₇	—	4	16	no	—
63	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
64	C₂×SD₁₆×C₆₇	C₂³	1	16	yes	complemented
65	C₂²×C₈×C₆₇	C₂³	1	8	yes	complemented
66	C₄⋊Q₈×C₆₇	—	2	4	no	—
67	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	complemented
68	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	complemented
69	#127×C₆₇	—	2	2	no	—
70	#127×C₆₇	C₂³	1	4	yes	—
71	#125×C₆₇	—	2	4	no	—
72	#125×C₆₇	—	2	2	no	—
73	C₂²⋊Q₈×C₆₇	—	2	4	no	—
74	C₂²⋊Q₈×C₆₇	—	2	4	no	—
75	C₂²⋊Q₈×C₆₇	—	2	2	no	—
76	C₂²⋊Q₈×C₆₇	—	2	8	no	—
77	C₂²⋊Q₈×C₆₇	—	2	8	no	—
78	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
79	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	complemented
80	#121×C₆₇	—	2	4	no	—
81	#121×C₆₇	—	2	8	no	—
82	#121×C₆₇	C₂³	1	8	yes	—
83	#120×C₆₇	—	2	16	no	—
84	C₄×Q₈×C₆₇	—	2	2	no	—
85	C₄×Q₈×C₆₇	C₂³	1	4	yes	complemented
86	C₄×D₄×C₆₇	—	2	4	no	—
87	C₄×D₄×C₆₇	—	2	4	no	—
88	C₄×D₄×C₆₇	—	2	2	no	—
89	C₄×D₄×C₆₇	D₄	1	4	yes	—
90	C₄×D₄×C₆₇	C₂³	1	4	yes	—
91	C₄×D₄×C₆₇	C₂³	1	4	yes	—
92	#117×C₆₇	—	2	2	no	—
93	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
94	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	complemented
97	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
98	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
99	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
100	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
101	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
102	#106×C₆₇	C₂³	1	8	yes	complemented
103	C₄⋊C₈×C₆₇	C₂³	1	4	yes	complemented
104	#103×C₆₇	C₂³	1	8	yes	complemented
105	#102×C₆₇	C₂³	1	8	yes	—
106	#102×C₆₇	C₂³	1	16	yes	complemented
107	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
108	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	complemented
109	C₄×C₈×C₆₇	C₂³	1	4	yes	complemented
110	C₂⁴×C₆₇	—	2	1	no	—
111	C₂⁴×C₆₇	—	2	2	no	—
112	C₂⁴×C₆₇	—	2	8	no	—
113	C₂⁴×C₆₇	—	4	8	no	—
114	C₂⁴×C₆₇	—	4	8	no	—
115	C₂⁴×C₆₇	—	8	4	no	—
116	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
117	C₂×Q₈×C₆₇	—	2	2	no	—
118	C₂×Q₈×C₆₇	—	2	4	no	—
119	C₂×Q₈×C₆₇	—	4	8	no	—
120	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
121	C₂×D₄×C₆₇	—	2	4	no	—
122	C₂×D₄×C₆₇	—	2	16	no	—
123	C₂×D₄×C₆₇	—	2	8	no	—
124	C₂×D₄×C₆₇	—	2	1	no	—
125	C₂×D₄×C₆₇	—	2	8	no	—
126	C₂×D₄×C₆₇	—	2	2	no	—
127	C₂×D₄×C₆₇	—	2	4	no	—
128	C₂×D₄×C₆₇	—	2	4	no	—
129	C₂×D₄×C₆₇	—	4	8	no	—
130	C₂×D₄×C₆₇	—	4	4	no	—
131	C₂×D₄×C₆₇	—	4	16	no	—
132	C₂×D₄×C₆₇	—	4	8	no	—
133	C₂×D₄×C₆₇	—	4	8	no	—
134	C₂×D₄×C₆₇	—	4	32	no	—
135	C₂×D₄×C₆₇	—	4	4	no	—
136	C₂×D₄×C₆₇	—	8	2	no	—
137	C₂×D₄×C₆₇	SD₁₆	1	8	yes	complemented
138	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
139	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
140	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
141	C₂²×C₄×C₆₇	—	2	1	no	—
142	C₂²×C₄×C₆₇	—	2	4	no	—
143	C₂²×C₄×C₆₇	—	2	4	no	—
144	C₂²×C₄×C₆₇	—	2	4	no	—
145	C₂²×C₄×C₆₇	—	4	4	no	—
146	C₂²×C₄×C₆₇	—	4	4	no	—
147	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
149	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
150	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
153	SD₁₆×C₆₇	—	2	16	no	—
154	SD₁₆×C₆₇	—	4	16	no	—
155	C₂×C₈×C₆₇	—	4	8	no	—
156	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
157	C₄⋊C₄×C₆₇	—	2	4	no	—
158	C₄⋊C₄×C₆₇	—	2	8	no	—
159	C₄⋊C₄×C₆₇	—	2	1	no	—
160	C₄⋊C₄×C₆₇	—	2	4	no	—
161	C₄⋊C₄×C₆₇	—	2	2	no	—
162	C₄⋊C₄×C₆₇	—	2	2	no	—
163	C₄⋊C₄×C₆₇	—	2	2	no	—
164	C₄⋊C₄×C₆₇	—	2	4	no	—
165	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
166	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
167	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
168	C₂²⋊C₄×C₆₇	—	2	4	no	—
169	C₂²⋊C₄×C₆₇	—	2	8	no	—
170	C₂²⋊C₄×C₆₇	—	2	4	no	—
171	C₂²⋊C₄×C₆₇	—	2	8	no	—
172	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
173	C₄²×C₆₇	—	2	4	no	—
174	C₄²×C₆₇	C₂×D₄	1	2	yes	—
175	C₄²×C₆₇	C₂⁴	1	2	yes	—
176	C₂³×C₆₇	—	2	4	no	—
177	C₂³×C₆₇	—	2	2	no	—
178	C₂³×C₆₇	—	2	4	no	—
179	C₂³×C₆₇	—	4	2	no	—
180	C₂³×C₆₇	—	4	2	no	—
181	C₂³×C₆₇	—	4	16	no	—
182	C₂³×C₆₇	—	4	2	no	—
183	C₂³×C₆₇	—	4	4	no	—
184	C₂³×C₆₇	—	4	4	no	—
185	C₂³×C₆₇	—	4	2	no	—
186	C₂³×C₆₇	—	4	16	no	—
187	C₂³×C₆₇	—	4	4	no	—
188	C₂³×C₆₇	—	8	8	no	—
189	C₂³×C₆₇	—	8	32	no	—
190	C₂³×C₆₇	C₂×SD₁₆	1	8	yes	complemented
191	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
192	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
193	Q₈×C₆₇	—	2	8	no	—
194	Q₈×C₆₇	—	8	2	no	—
195	Q₈×C₆₇	—	8	4	no	—
196	D₄×C₆₇	—	2	4	no	—
197	D₄×C₆₇	—	2	2	no	—
198	D₄×C₆₇	—	2	8	no	—
199	D₄×C₆₇	—	4	16	no	—
200	D₄×C₆₇	—	4	4	no	—
201	D₄×C₆₇	—	4	16	no	—
202	D₄×C₆₇	—	4	8	no	—
203	D₄×C₆₇	—	8	4	no	—
204	D₄×C₆₇	C₂×SD₁₆	1	16	yes	complemented
205	D₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
206	C₂×C₄×C₆₇	—	2	8	no	—
207	C₂×C₄×C₆₇	—	2	2	no	—
208	C₂×C₄×C₆₇	—	2	4	no	—
209	C₂×C₄×C₆₇	—	2	4	no	—
210	C₂×C₄×C₆₇	—	2	2	no	—
211	C₂×C₄×C₆₇	—	4	4	no	—
212	C₂×C₄×C₆₇	—	4	2	no	—
213	C₂×C₄×C₆₇	—	4	1	no	—
214	C₂×C₄×C₆₇	—	4	2	no	—
215	C₂×C₄×C₆₇	—	4	8	no	—
216	C₂×C₄×C₆₇	—	8	4	no	—
217	C₂×C₄×C₆₇	—	8	4	no	—
218	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	complemented
219	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
221	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
222	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
223	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
224	C₈×C₆₇	—	2	8	no	—
225	C₂²×C₆₇	—	2	1	no	—
226	C₂²×C₆₇	—	2	4	no	—
227	C₂²×C₆₇	—	2	1	no	—
228	C₂²×C₆₇	—	2	4	no	—
229	C₂²×C₆₇	—	2	4	no	—
230	C₂²×C₆₇	—	4	8	no	—
231	C₂²×C₆₇	—	4	8	no	—
232	C₂²×C₆₇	—	4	2	no	—
233	C₂²×C₆₇	—	4	2	no	—
234	C₂²×C₆₇	—	4	4	no	—
235	C₂²×C₆₇	—	4	4	no	—
236	C₂²×C₆₇	—	4	4	no	—
237	C₂²×C₆₇	—	8	32	no	—
238	C₂²×C₆₇	—	8	8	no	—
239	C₂²×C₆₇	C₂²×SD₁₆	1	2	yes	—
240	C₂²×C₆₇	C₂²×SD₁₆	1	8	yes	—
241	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
242	C₄×C₆₇	—	2	2	no	—
243	C₄×C₆₇	—	2	1	no	—
244	C₄×C₆₇	—	2	4	no	—
245	C₄×C₆₇	—	4	4	no	—
246	C₄×C₆₇	—	4	4	no	—
247	C₄×C₆₇	—	4	2	no	—
248	C₄×C₆₇	C₂²×SD₁₆	1	4	yes	—
249	C₄×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
250	#61560	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
251	C₂×C₆₇	—	2	2	no	—
252	C₂×C₆₇	—	2	8	no	—
253	C₂×C₆₇	—	4	4	no	—
254	C₂×C₆₇	—	8	4	no	—
255	C₂×C₆₇	#3484	1	1	yes	characteristic
256	C₂×C₆₇	C₂³×SD₁₆	1	1	yes	characteristic
257	C₂×C₆₇	#3585	1	1	yes	characteristic
258	#3501	C₂×C₆₇	1	1	yes	characteristic, complemented
259	C₂×D₄²	C₂×C₆₇	1	1	yes	characteristic
260	#3295	C₂×C₆₇	1	8	yes	complemented
261	D₄×SD₁₆	C₂×C₆₇	1	8	yes	complemented
262	#3208	C₂×C₆₇	1	2	yes	complemented
263	#3187	C₂×C₆₇	1	4	yes	complemented
264	#3101	C₂×C₆₇	1	2	yes	complemented
265	C₂×#450	C₂×C₆₇	1	4	yes	complemented
266	#2972	C₂×C₆₇	1	1	yes	characteristic, complemented
267	C₆₇	#61560	1	1	yes	characteristic, direct product factor, Sylow
268	C₂³×D₄	—	2	1	no	—
269	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
270	C₂²×SD₁₆	C₂²×C₆₇	1	8	yes	complemented
271	#554	—	2	4	no	—
272	#554	C₂²×C₆₇	1	4	yes	complemented
273	D₄×Q₈	C₂²×C₆₇	1	4	yes	complemented
274	D₄²	—	2	4	no	—
275	D₄²	C₂²×C₆₇	1	8	yes	—
276	#544	—	2	1	no	—
277	#543	—	2	2	no	—
278	#533	C₂²×C₆₇	1	2	yes	complemented
279	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
280	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	complemented
281	C₂×#121	—	2	1	no	—
282	C₂×#121	C₂²×C₆₇	1	2	yes	—
283	C₂×#120	—	2	2	no	—
284	#518	—	2	1	no	—
285	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
286	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
287	#492	C₂²×C₆₇	1	4	yes	complemented
288	#486	C₂²×C₆₇	1	4	yes	complemented
289	#483	C₂²×C₆₇	1	4	yes	complemented
290	#481	C₂²×C₆₇	1	8	yes	complemented
291	#476	C₂²×C₆₇	1	4	yes	complemented
292	#465	C₂²×C₆₇	1	16	yes	complemented
293	#461	C₂²×C₆₇	1	4	yes	complemented
294	#460	C₂²×C₆₇	1	8	yes	complemented
295	#450	C₂²×C₆₇	1	4	yes	—
296	#450	C₂²×C₆₇	1	8	yes	complemented
297	#450	C₂²×C₆₇	1	16	yes	complemented
298	#448	C₂²×C₆₇	1	8	yes	complemented
299	C₄×SD₁₆	C₂²×C₆₇	1	8	yes	complemented
300	C₈×D₄	C₂²×C₆₇	1	8	yes	complemented
301	#431	C₂²×C₆₇	1	2	yes	complemented
302	C₂×#102	C₂²×C₆₇	1	8	yes	complemented
303	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	complemented
304	C₂⁵	—	2	2	no	—
305	C₂²×Q₈	C₂³×C₆₇	1	4	yes	complemented
306	C₂²×D₄	—	2	2	no	—
307	C₂²×D₄	—	2	8	no	—
308	C₂²×D₄	—	2	2	no	—
309	C₂²×D₄	—	2	1	no	—
310	C₂²×D₄	—	2	4	no	—
311	C₂²×D₄	—	2	1	no	—
312	C₂²×D₄	—	4	8	no	—
313	C₂²×D₄	—	4	8	no	—
314	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
315	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
316	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
317	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
318	C₂³×C₄	—	2	1	no	—
319	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
320	C₂×SD₁₆	—	4	16	no	—
321	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
322	C₂×SD₁₆	C₂³×C₆₇	1	16	yes	complemented
323	C₂²×C₈	C₂³×C₆₇	1	8	yes	complemented
324	C₄⋊Q₈	—	2	4	no	—
325	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	complemented
326	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	complemented
327	#127	—	2	2	no	—
328	#127	C₂³×C₆₇	1	4	yes	—
329	#125	—	2	4	no	—
330	#125	—	2	2	no	—
331	C₂²⋊Q₈	—	2	4	no	—
332	C₂²⋊Q₈	—	2	4	no	—
333	C₂²⋊Q₈	—	2	2	no	—
334	C₂²⋊Q₈	—	2	8	no	—
335	C₂²⋊Q₈	—	2	8	no	—
336	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
337	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	complemented
338	#121	—	2	4	no	—
339	#121	—	2	8	no	—
340	#121	C₂³×C₆₇	1	8	yes	—
341	#120	—	2	16	no	—
342	C₄×Q₈	—	2	2	no	—
343	C₄×Q₈	C₂³×C₆₇	1	4	yes	complemented
344	C₄×D₄	—	2	4	no	—
345	C₄×D₄	—	2	4	no	—
346	C₄×D₄	—	2	2	no	—
347	C₄×D₄	D₄×C₆₇	1	4	yes	—
348	C₄×D₄	C₂³×C₆₇	1	4	yes	—
349	C₄×D₄	C₂³×C₆₇	1	4	yes	—
350	#117	—	2	2	no	—
351	C₂×C₄⋊C₄	—	2	4	no	—
352	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
353	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
354	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	complemented
355	C₂×C₂²⋊C₄	—	2	4	no	—
356	C₂×C₂²⋊C₄	—	2	1	no	—
357	C₂×C₂²⋊C₄	—	2	2	no	—
358	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
359	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
360	#106	C₂³×C₆₇	1	8	yes	complemented
361	C₄⋊C₈	C₂³×C₆₇	1	4	yes	complemented
362	#103	C₂³×C₆₇	1	8	yes	complemented
363	#102	C₂³×C₆₇	1	8	yes	—
364	#102	C₂³×C₆₇	1	16	yes	complemented
365	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
366	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	complemented
367	C₄×C₈	C₂³×C₆₇	1	4	yes	complemented
368	C₂⁴	—	2	1	no	—
369	C₂⁴	—	2	2	no	—
370	C₂⁴	—	2	8	no	—
371	C₂⁴	—	4	8	no	—
372	C₂⁴	—	4	8	no	—
373	C₂⁴	—	8	4	no	—
374	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
375	C₂×Q₈	—	2	2	no	—
376	C₂×Q₈	—	2	4	no	—
377	C₂×Q₈	—	4	8	no	—
378	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
379	C₂×D₄	—	2	4	no	—
380	C₂×D₄	—	2	16	no	—
381	C₂×D₄	—	2	8	no	—
382	C₂×D₄	—	2	1	no	—
383	C₂×D₄	—	2	8	no	—
384	C₂×D₄	—	2	2	no	—
385	C₂×D₄	—	2	4	no	—
386	C₂×D₄	—	2	4	no	—
387	C₂×D₄	—	4	8	no	—
388	C₂×D₄	—	4	4	no	—
389	C₂×D₄	—	4	16	no	—
390	C₂×D₄	—	4	8	no	—
391	C₂×D₄	—	4	8	no	—
392	C₂×D₄	—	4	32	no	—
393	C₂×D₄	—	4	4	no	—
394	C₂×D₄	—	8	2	no	—
395	C₂×D₄	SD₁₆×C₆₇	1	8	yes	complemented
396	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
397	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
398	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	4	no	—
401	C₂²×C₄	—	2	4	no	—
402	C₂²×C₄	—	2	4	no	—
403	C₂²×C₄	—	4	4	no	—
404	C₂²×C₄	—	4	4	no	—
405	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
406	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
407	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
408	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
409	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
410	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
411	SD₁₆	—	2	16	no	—
412	SD₁₆	—	4	16	no	—
413	C₂×C₈	—	4	8	no	—
414	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
415	C₄⋊C₄	—	2	4	no	—
416	C₄⋊C₄	—	2	8	no	—
417	C₄⋊C₄	—	2	1	no	—
418	C₄⋊C₄	—	2	4	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	—	2	2	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	2	4	no	—
423	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
424	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
425	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
426	C₂²⋊C₄	—	2	4	no	—
427	C₂²⋊C₄	—	2	8	no	—
428	C₂²⋊C₄	—	2	4	no	—
429	C₂²⋊C₄	—	2	8	no	—
430	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
431	C₄²	—	2	4	no	—
432	C₄²	C₂×D₄×C₆₇	1	2	yes	—
433	C₄²	C₂⁴×C₆₇	1	2	yes	—
434	C₂³	—	2	4	no	—
435	C₂³	—	2	2	no	—
436	C₂³	—	2	4	no	—
437	C₂³	—	4	2	no	—
438	C₂³	—	4	2	no	—
439	C₂³	—	4	16	no	—
440	C₂³	—	4	2	no	—
441	C₂³	—	4	4	no	—
442	C₂³	—	4	4	no	—
443	C₂³	—	4	2	no	—
444	C₂³	—	4	16	no	—
445	C₂³	—	4	4	no	—
446	C₂³	—	8	8	no	—
447	C₂³	—	8	32	no	—
448	C₂³	C₂×SD₁₆×C₆₇	1	8	yes	complemented
449	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
450	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
451	Q₈	—	2	8	no	—
452	Q₈	—	8	4	no	—
453	Q₈	—	8	2	no	—
454	D₄	—	2	4	no	—
455	D₄	—	2	2	no	—
456	D₄	—	2	8	no	—
457	D₄	—	4	16	no	—
458	D₄	—	4	4	no	—
459	D₄	—	4	16	no	—
460	D₄	—	4	8	no	—
461	D₄	—	8	4	no	—
462	D₄	C₂×SD₁₆×C₆₇	1	16	yes	complemented
463	D₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
464	C₂×C₄	—	2	8	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	4	4	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	8	no	—
474	C₂×C₄	—	8	4	no	—
475	C₂×C₄	—	8	4	no	—
476	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	complemented
477	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
478	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
479	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
480	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
481	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
482	C₈	—	2	8	no	—
483	C₂²	—	2	1	no	—
484	C₂²	—	2	4	no	—
485	C₂²	—	2	1	no	—
486	C₂²	—	2	4	no	—
487	C₂²	—	2	4	no	—
488	C₂²	—	4	8	no	—
489	C₂²	—	4	8	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	4	2	no	—
492	C₂²	—	4	4	no	—
493	C₂²	—	4	4	no	—
494	C₂²	—	4	4	no	—
495	C₂²	—	8	32	no	—
496	C₂²	—	8	8	no	—
497	C₂²	C₂²×SD₁₆×C₆₇	1	2	yes	—
498	C₂²	C₂²×SD₁₆×C₆₇	1	8	yes	—
499	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
500	C₄	—	2	2	no	—
501	C₄	—	2	1	no	—
502	C₄	—	2	4	no	—
503	C₄	—	4	2	no	—
504	C₄	—	4	4	no	—
505	C₄	—	4	4	no	—
506	C₄	C₂²×SD₁₆×C₆₇	1	4	yes	—
507	C₄	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
508	C₂	—	2	2	no	—
509	C₂	—	2	8	no	—
510	C₂	—	4	4	no	—
511	C₂	—	8	4	no	—
512	C₂	C₆₇×#3484	1	1	yes	characteristic
513	C₂	C₂³×SD₁₆×C₆₇	1	1	yes	characteristic
514	C₂	C₆₇×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂×D₄²×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×#3295	C₂	semidirect
4. D₄×SD₁₆×C₆₇	C₂	semidirect
5. C₆₇×#3208	C₂	semidirect
6. C₆₇×#3187	C₂	semidirect
7. C₆₇×#3101	C₂	semidirect
8. C₂×#450×C₆₇	C₂	semidirect
9. C₆₇×#2972	C₂	semidirect
12. C₂²×SD₁₆×C₆₇	C₂²	semidirect
14. #554×C₆₇	C₂²	semidirect
15. D₄×Q₈×C₆₇	C₂²	semidirect
20. #533×C₆₇	C₂²	semidirect
22. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
29. #492×C₆₇	C₂²	semidirect
30. #486×C₆₇	C₂²	semidirect
31. #483×C₆₇	C₂²	semidirect
32. #481×C₆₇	C₂²	semidirect
33. #476×C₆₇	C₂²	semidirect
34. #465×C₆₇	C₂²	semidirect
35. #461×C₆₇	C₂²	semidirect
36. #460×C₆₇	C₂²	semidirect
38. #450×C₆₇	C₂²	semidirect
39. #450×C₆₇	C₂²	semidirect
40. #448×C₆₇	C₂²	semidirect
41. C₄×SD₁₆×C₆₇	C₂²	semidirect
42. C₈×D₄×C₆₇	C₂²	semidirect
43. #431×C₆₇	C₂²	semidirect
44. C₂×#102×C₆₇	C₂²	semidirect
45. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
47. C₂²×Q₈×C₆₇	C₂³	semidirect
64. C₂×SD₁₆×C₆₇	C₂³	semidirect
65. C₂²×C₈×C₆₇	C₂³	semidirect
67. C₄⋊Q₈×C₆₇	C₂³	semidirect
68. C₄⋊Q₈×C₆₇	C₂³	semidirect
79. C₂²⋊Q₈×C₆₇	C₂³	semidirect
85. C₄×Q₈×C₆₇	C₂³	semidirect
96. C₂×C₄⋊C₄×C₆₇	C₂³	semidirect
102. #106×C₆₇	C₂³	semidirect
103. C₄⋊C₈×C₆₇	C₂³	semidirect
104. #103×C₆₇	C₂³	semidirect
106. #102×C₆₇	C₂³	semidirect
108. C₂²⋊C₈×C₆₇	C₂³	semidirect
109. C₄×C₈×C₆₇	C₂³	semidirect
137. C₂×D₄×C₆₇	SD₁₆	semidirect
190. C₂³×C₆₇	C₂×SD₁₆	semidirect
204. D₄×C₆₇	C₂×SD₁₆	semidirect
218. C₂×C₄×C₆₇	C₂×SD₁₆	semidirect
250. #61560	C₆₇	direct (N × H)
258. #3501	C₂×C₆₇	semidirect
260. #3295	C₂×C₆₇	semidirect
261. D₄×SD₁₆	C₂×C₆₇	semidirect
262. #3208	C₂×C₆₇	semidirect
263. #3187	C₂×C₆₇	semidirect
264. #3101	C₂×C₆₇	semidirect
265. C₂×#450	C₂×C₆₇	semidirect
266. #2972	C₂×C₆₇	semidirect
267. C₆₇	#61560	direct (N × H)
270. C₂²×SD₁₆	C₂²×C₆₇	semidirect
272. #554	C₂²×C₆₇	semidirect
273. D₄×Q₈	C₂²×C₆₇	semidirect
278. #533	C₂²×C₆₇	semidirect
280. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
287. #492	C₂²×C₆₇	semidirect
288. #486	C₂²×C₆₇	semidirect
289. #483	C₂²×C₆₇	semidirect
290. #481	C₂²×C₆₇	semidirect
291. #476	C₂²×C₆₇	semidirect
292. #465	C₂²×C₆₇	semidirect
293. #461	C₂²×C₆₇	semidirect
294. #460	C₂²×C₆₇	semidirect
296. #450	C₂²×C₆₇	semidirect
297. #450	C₂²×C₆₇	semidirect
298. #448	C₂²×C₆₇	semidirect
299. C₄×SD₁₆	C₂²×C₆₇	semidirect
300. C₈×D₄	C₂²×C₆₇	semidirect
301. #431	C₂²×C₆₇	semidirect
302. C₂×#102	C₂²×C₆₇	semidirect
303. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
305. C₂²×Q₈	C₂³×C₆₇	semidirect
322. C₂×SD₁₆	C₂³×C₆₇	semidirect
323. C₂²×C₈	C₂³×C₆₇	semidirect
325. C₄⋊Q₈	C₂³×C₆₇	semidirect
326. C₄⋊Q₈	C₂³×C₆₇	semidirect
337. C₂²⋊Q₈	C₂³×C₆₇	semidirect
343. C₄×Q₈	C₂³×C₆₇	semidirect
354. C₂×C₄⋊C₄	C₂³×C₆₇	semidirect
360. #106	C₂³×C₆₇	semidirect
361. C₄⋊C₈	C₂³×C₆₇	semidirect
362. #103	C₂³×C₆₇	semidirect
364. #102	C₂³×C₆₇	semidirect
366. C₂²⋊C₈	C₂³×C₆₇	semidirect
367. C₄×C₈	C₂³×C₆₇	semidirect
395. C₂×D₄	SD₁₆×C₆₇	semidirect
448. C₂³	C₂×SD₁₆×C₆₇	semidirect
462. D₄	C₂×SD₁₆×C₆₇	semidirect
476. C₂×C₄	C₂×SD₁₆×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×D₄²×C₆₇	2
C₆₇×#3295	2
D₄×SD₁₆×C₆₇	2
C₆₇×#3208	2
C₆₇×#3187	2
C₆₇×#3101	2
C₂×#450×C₆₇	2
C₆₇×#2972	2
#61560	67

Subgroups — C67×#61560 #30740811

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61560 #30740811