Subgroups — C₆₇×#61708 #30740959

Order 17152.

1806 conjugacy classes of subgroups.

The subgroup lattice has 622 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×#553×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3346	C₂	1	8	yes	maximal, complemented
4	C₆₇×#3320	C₂	1	8	yes	maximal, complemented
5	C₆₇×#3236	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3193	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3187	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3110	C₂	1	2	yes	maximal, complemented
9	C₆₇×#3095	C₂	1	2	yes	maximal, complemented
10	C₂×#484×C₆₇	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2973	C₂	1	1	yes	maximal, characteristic, complemented
12	#554×C₆₇	—	2	4	no	—
13	#554×C₆₇	C₂²	1	4	yes	complemented
14	#553×C₆₇	—	2	4	no	—
15	#553×C₆₇	C₂²	1	8	yes	—
16	D₄×Q₈×C₆₇	C₂²	1	4	yes	complemented
17	#544×C₆₇	—	2	1	no	—
18	#543×C₆₇	—	2	1	no	—
19	#543×C₆₇	—	2	1	no	—
20	#533×C₆₇	C₂²	1	2	yes	—
21	C₂×#126×C₆₇	—	2	1	no	—
22	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#123×C₆₇	—	2	1	no	—
24	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
26	C₂×#121×C₆₇	—	2	1	no	—
27	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
29	#518×C₆₇	—	2	1	no	—
30	C₂×C₄×D₄×C₆₇	—	2	1	no	—
31	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
34	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
35	#501×C₆₇	C₂²	1	4	yes	complemented
36	#491×C₆₇	C₂²	1	4	yes	complemented
37	#485×C₆₇	C₂²	1	8	yes	complemented
38	#484×C₆₇	C₂²	1	4	yes	—
39	#484×C₆₇	C₂²	1	4	yes	—
40	#484×C₆₇	C₂²	1	8	yes	complemented
41	#483×C₆₇	C₂²	1	4	yes	—
42	#481×C₆₇	C₂²	1	4	yes	complemented
43	#480×C₆₇	C₂²	1	4	yes	complemented
44	#479×C₆₇	C₂²	1	8	yes	complemented
45	#474×C₆₇	C₂²	1	4	yes	complemented
46	#468×C₆₇	C₂²	1	8	yes	complemented
47	#465×C₆₇	C₂²	1	8	yes	complemented
48	#461×C₆₇	C₂²	1	4	yes	complemented
49	#450×C₆₇	C₂²	1	4	yes	—
50	#448×C₆₇	C₂²	1	4	yes	complemented
51	#448×C₆₇	C₂²	1	4	yes	complemented
52	#440×C₆₇	C₂²	1	8	yes	complemented
53	#435×C₆₇	C₂²	1	8	yes	complemented
54	#432×C₆₇	C₂²	1	2	yes	—
55	#428×C₆₇	C₂²	1	4	yes	complemented
56	C₂×#107×C₆₇	C₂²	1	4	yes	complemented
57	#419×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	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
62	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
63	C₂²×D₄×C₆₇	—	2	1	no	—
64	C₂²×D₄×C₆₇	—	2	1	no	—
65	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
66	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₆₇	—	2	1	no	—
68	C₂³×C₄×C₆₇	—	2	1	no	—
69	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
72	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
73	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
74	C₄⋊Q₈×C₆₇	—	2	4	no	—
75	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
76	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
77	#126×C₆₇	—	2	8	no	—
78	#125×C₆₇	—	2	4	no	—
79	#125×C₆₇	—	2	2	no	—
80	#125×C₆₇	—	2	2	no	—
81	#125×C₆₇	C₂³	1	4	yes	—
82	#123×C₆₇	—	2	8	no	—
83	C₂²⋊Q₈×C₆₇	—	2	2	no	—
84	C₂²⋊Q₈×C₆₇	—	2	2	no	—
85	C₂²⋊Q₈×C₆₇	—	2	2	no	—
86	C₂²⋊Q₈×C₆₇	—	2	2	no	—
87	C₂²⋊Q₈×C₆₇	—	2	2	no	—
88	C₂²⋊Q₈×C₆₇	—	2	4	no	—
89	C₂²⋊Q₈×C₆₇	—	2	4	no	—
90	C₂²⋊Q₈×C₆₇	—	2	4	no	—
91	C₂²⋊Q₈×C₆₇	—	2	4	no	—
92	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
93	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
94	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
95	#121×C₆₇	—	2	8	no	—
96	#121×C₆₇	—	2	2	no	—
97	#121×C₆₇	—	2	2	no	—
98	#121×C₆₇	C₂³	1	4	yes	—
99	#121×C₆₇	C₂³	1	4	yes	—
100	C₄×Q₈×C₆₇	—	2	2	no	—
101	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
102	C₄×D₄×C₆₇	—	2	4	no	—
103	C₄×D₄×C₆₇	—	2	4	no	—
104	C₄×D₄×C₆₇	—	2	8	no	—
105	C₄×D₄×C₆₇	—	2	2	no	—
106	C₄×D₄×C₆₇	D₄	1	4	yes	—
107	C₄×D₄×C₆₇	C₂³	1	4	yes	—
108	C₄×D₄×C₆₇	C₂³	1	4	yes	—
109	#117×C₆₇	—	2	2	no	—
110	#117×C₆₇	—	2	2	no	—
111	#117×C₆₇	C₂³	1	4	yes	—
112	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
113	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
114	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
115	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
116	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
118	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
120	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
121	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
122	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
123	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
124	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
125	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
126	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
127	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
128	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
129	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
130	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
131	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
132	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
133	C₂×C₄²×C₆₇	—	2	1	no	—
134	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
135	#107×C₆₇	C₂³	1	8	yes	—
136	#107×C₆₇	C₂³	1	8	yes	—
137	#106×C₆₇	C₂³	1	8	yes	—
138	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
139	#103×C₆₇	C₂³	1	8	yes	—
140	#103×C₆₇	C₂³	1	8	yes	—
141	#103×C₆₇	C₂³	1	8	yes	—
142	#102×C₆₇	C₂³	1	8	yes	—
143	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
144	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
145	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
146	#97×C₆₇	C₂³	1	4	yes	—
147	C₂⁴×C₆₇	—	2	1	no	—
148	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
149	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
150	C₂×Q₈×C₆₇	—	2	2	no	—
151	C₂×Q₈×C₆₇	—	2	2	no	—
152	C₂×Q₈×C₆₇	—	2	2	no	—
153	C₂×Q₈×C₆₇	—	4	4	no	—
154	C₂×Q₈×C₆₇	—	4	4	no	—
155	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
156	C₂×D₄×C₆₇	—	2	4	no	—
157	C₂×D₄×C₆₇	—	2	4	no	—
158	C₂×D₄×C₆₇	—	2	4	no	—
159	C₂×D₄×C₆₇	—	2	1	no	—
160	C₂×D₄×C₆₇	—	4	2	no	—
161	C₂×D₄×C₆₇	—	4	4	no	—
162	C₂×D₄×C₆₇	—	8	2	no	—
163	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
164	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₆₇	—	2	1	no	—
166	C₂²×C₄×C₆₇	—	2	1	no	—
167	C₂²×C₄×C₆₇	—	2	1	no	—
168	C₂²×C₄×C₆₇	—	2	2	no	—
169	C₂²×C₄×C₆₇	—	2	2	no	—
170	C₂²×C₄×C₆₇	—	2	2	no	—
171	C₂²×C₄×C₆₇	—	2	2	no	—
172	C₂²×C₄×C₆₇	—	2	4	no	—
173	C₂²×C₄×C₆₇	—	2	4	no	—
174	C₂²×C₄×C₆₇	—	4	2	no	—
175	C₂²×C₄×C₆₇	—	4	2	no	—
176	C₂²×C₄×C₆₇	—	8	2	no	—
177	C₂²×C₄×C₆₇	—	8	2	no	—
178	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
183	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
184	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
185	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
186	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
187	SD₁₆×C₆₇	—	4	8	no	—
188	M₁₆×C₆₇	—	4	4	no	—
189	C₂×C₈×C₆₇	—	4	4	no	—
190	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
191	C₄⋊C₄×C₆₇	—	2	1	no	—
192	C₄⋊C₄×C₆₇	—	2	1	no	—
193	C₄⋊C₄×C₆₇	—	2	4	no	—
194	C₄⋊C₄×C₆₇	—	2	4	no	—
195	C₄⋊C₄×C₆₇	—	2	4	no	—
196	C₄⋊C₄×C₆₇	—	2	4	no	—
197	C₄⋊C₄×C₆₇	—	2	4	no	—
198	C₄⋊C₄×C₆₇	—	2	4	no	—
199	C₄⋊C₄×C₆₇	—	2	1	no	—
200	C₄⋊C₄×C₆₇	—	2	2	no	—
201	C₄⋊C₄×C₆₇	—	2	2	no	—
202	C₄⋊C₄×C₆₇	—	2	2	no	—
203	C₄⋊C₄×C₆₇	—	2	2	no	—
204	C₄⋊C₄×C₆₇	—	2	2	no	—
205	C₄⋊C₄×C₆₇	—	2	4	no	—
206	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
207	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
208	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
209	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
210	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
211	C₂²⋊C₄×C₆₇	—	2	4	no	—
212	C₂²⋊C₄×C₆₇	—	2	4	no	—
213	C₂²⋊C₄×C₆₇	—	2	4	no	—
214	C₂²⋊C₄×C₆₇	—	2	4	no	—
215	C₂²⋊C₄×C₆₇	—	2	4	no	—
216	C₂²⋊C₄×C₆₇	—	2	4	no	—
217	C₂²⋊C₄×C₆₇	—	2	4	no	—
218	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
219	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
220	C₄²×C₆₇	—	2	4	no	—
221	C₄²×C₆₇	—	2	4	no	—
222	C₄²×C₆₇	C₂×D₄	1	2	yes	—
223	C₄²×C₆₇	C₂⁴	1	2	yes	—
224	C₂³×C₆₇	—	2	2	no	—
225	C₂³×C₆₇	—	2	2	no	—
226	C₂³×C₆₇	—	4	1	no	—
227	C₂³×C₆₇	—	4	1	no	—
228	C₂³×C₆₇	—	4	1	no	—
229	C₂³×C₆₇	—	4	1	no	—
230	C₂³×C₆₇	—	4	2	no	—
231	C₂³×C₆₇	—	4	2	no	—
232	C₂³×C₆₇	—	4	2	no	—
233	C₂³×C₆₇	#137	1	4	yes	—
234	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
235	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
236	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
237	Q₈×C₆₇	—	2	8	no	—
238	Q₈×C₆₇	—	8	2	no	—
239	Q₈×C₆₇	—	8	2	no	—
240	Q₈×C₆₇	—	8	2	no	—
241	D₄×C₆₇	—	2	8	no	—
242	D₄×C₆₇	—	2	4	no	—
243	D₄×C₆₇	—	4	2	no	—
244	D₄×C₆₇	—	4	8	no	—
245	D₄×C₆₇	—	8	4	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	4	no	—
248	C₂×C₄×C₆₇	—	2	4	no	—
249	C₂×C₄×C₆₇	—	2	2	no	—
250	C₂×C₄×C₆₇	—	2	2	no	—
251	C₂×C₄×C₆₇	—	2	2	no	—
252	C₂×C₄×C₆₇	—	2	4	no	—
253	C₂×C₄×C₆₇	—	2	4	no	—
254	C₂×C₄×C₆₇	—	2	2	no	—
255	C₂×C₄×C₆₇	—	4	2	no	—
256	C₂×C₄×C₆₇	—	4	2	no	—
257	C₂×C₄×C₆₇	—	4	1	no	—
258	C₂×C₄×C₆₇	—	4	1	no	—
259	C₂×C₄×C₆₇	—	4	1	no	—
260	C₂×C₄×C₆₇	—	4	2	no	—
261	C₂×C₄×C₆₇	—	4	2	no	—
262	C₂×C₄×C₆₇	—	4	2	no	—
263	C₂×C₄×C₆₇	—	8	4	no	—
264	C₂×C₄×C₆₇	—	8	2	no	—
265	C₂×C₄×C₆₇	—	8	2	no	—
266	C₂×C₄×C₆₇	—	8	4	no	—
267	C₂×C₄×C₆₇	—	8	2	no	—
268	C₂×C₄×C₆₇	—	8	2	no	—
269	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
270	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
271	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
272	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
273	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
274	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
275	C₈×C₆₇	—	2	8	no	—
276	C₂²×C₆₇	—	2	1	no	—
277	C₂²×C₆₇	—	2	2	no	—
278	C₂²×C₆₇	—	2	1	no	—
279	C₂²×C₆₇	—	2	2	no	—
280	C₂²×C₆₇	—	2	2	no	—
281	C₂²×C₆₇	—	2	2	no	—
282	C₂²×C₆₇	—	2	2	no	—
283	C₂²×C₆₇	—	4	4	no	—
284	C₂²×C₆₇	—	4	2	no	—
285	C₂²×C₆₇	—	4	2	no	—
286	C₂²×C₆₇	—	4	2	no	—
287	C₂²×C₆₇	—	8	2	no	—
288	C₂²×C₆₇	—	8	4	no	—
289	C₂²×C₆₇	C₂×#137	1	2	yes	—
290	C₂²×C₆₇	C₂×#137	1	4	yes	—
291	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
292	C₄×C₆₇	—	2	2	no	—
293	C₄×C₆₇	—	2	1	no	—
294	C₄×C₆₇	—	2	2	no	—
295	C₄×C₆₇	—	2	2	no	—
296	C₄×C₆₇	—	2	2	no	—
297	C₄×C₆₇	—	4	2	no	—
298	C₄×C₆₇	—	4	2	no	—
299	C₄×C₆₇	—	4	4	no	—
300	C₄×C₆₇	—	4	4	no	—
301	C₄×C₆₇	—	4	2	no	—
302	C₄×C₆₇	#585	1	2	yes	—
303	#61708	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
304	C₂×C₆₇	—	2	2	no	—
305	C₂×C₆₇	—	2	4	no	—
306	C₂×C₆₇	—	4	2	no	—
307	C₂×C₆₇	—	8	2	no	—
308	C₂×C₆₇	#3485	1	1	yes	characteristic
309	C₂×C₆₇	C₂²×#137	1	1	yes	characteristic
310	C₂×C₆₇	#3585	1	1	yes	characteristic
311	#3501	C₂×C₆₇	1	1	yes	characteristic, complemented
312	C₂×#553	C₂×C₆₇	1	1	yes	characteristic
313	#3346	C₂×C₆₇	1	8	yes	complemented
314	#3320	C₂×C₆₇	1	8	yes	complemented
315	#3236	C₂×C₆₇	1	2	yes	complemented
316	#3193	C₂×C₆₇	1	2	yes	complemented
317	#3187	C₂×C₆₇	1	2	yes	complemented
318	#3110	C₂×C₆₇	1	2	yes	complemented
319	#3095	C₂×C₆₇	1	2	yes	complemented
320	C₂×#484	C₂×C₆₇	1	2	yes	complemented
321	#2973	C₂×C₆₇	1	1	yes	characteristic, complemented
322	C₆₇	#61708	1	1	yes	characteristic, direct product factor, Sylow
323	#554	—	2	4	no	—
324	#554	C₂²×C₆₇	1	4	yes	complemented
325	#553	—	2	4	no	—
326	#553	C₂²×C₆₇	1	8	yes	—
327	D₄×Q₈	C₂²×C₆₇	1	4	yes	complemented
328	#544	—	2	1	no	—
329	#543	—	2	1	no	—
330	#543	—	2	1	no	—
331	#533	C₂²×C₆₇	1	2	yes	—
332	C₂×#126	—	2	1	no	—
333	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
334	C₂×#123	—	2	1	no	—
335	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
336	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
337	C₂×#121	—	2	1	no	—
338	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
339	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
340	#518	—	2	1	no	—
341	C₂×C₄×D₄	—	2	1	no	—
342	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
343	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
344	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
345	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
346	#501	C₂²×C₆₇	1	4	yes	complemented
347	#491	C₂²×C₆₇	1	4	yes	complemented
348	#485	C₂²×C₆₇	1	8	yes	complemented
349	#484	C₂²×C₆₇	1	4	yes	—
350	#484	C₂²×C₆₇	1	4	yes	—
351	#484	C₂²×C₆₇	1	8	yes	complemented
352	#483	C₂²×C₆₇	1	4	yes	—
353	#481	C₂²×C₆₇	1	4	yes	complemented
354	#480	C₂²×C₆₇	1	4	yes	complemented
355	#479	C₂²×C₆₇	1	8	yes	complemented
356	#474	C₂²×C₆₇	1	4	yes	complemented
357	#468	C₂²×C₆₇	1	8	yes	complemented
358	#465	C₂²×C₆₇	1	8	yes	complemented
359	#461	C₂²×C₆₇	1	4	yes	complemented
360	#450	C₂²×C₆₇	1	4	yes	—
361	#448	C₂²×C₆₇	1	4	yes	complemented
362	#448	C₂²×C₆₇	1	4	yes	complemented
363	#440	C₂²×C₆₇	1	8	yes	complemented
364	#435	C₂²×C₆₇	1	8	yes	complemented
365	#432	C₂²×C₆₇	1	2	yes	—
366	#428	C₂²×C₆₇	1	4	yes	complemented
367	C₂×#107	C₂²×C₆₇	1	4	yes	complemented
368	#419	C₂²×C₆₇	1	4	yes	complemented
369	C₂×#103	C₂²×C₆₇	1	4	yes	complemented
370	#407	C₂²×C₆₇	1	2	yes	—
371	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
372	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
373	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
374	C₂²×D₄	—	2	1	no	—
375	C₂²×D₄	—	2	1	no	—
376	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
377	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
378	C₂³×C₄	—	2	1	no	—
379	C₂³×C₄	—	2	1	no	—
380	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
381	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
382	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
383	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
384	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
385	C₄⋊Q₈	—	2	4	no	—
386	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
387	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
388	#126	—	2	8	no	—
389	#125	—	2	4	no	—
390	#125	—	2	2	no	—
391	#125	—	2	2	no	—
392	#125	C₂³×C₆₇	1	4	yes	—
393	#123	—	2	8	no	—
394	C₂²⋊Q₈	—	2	2	no	—
395	C₂²⋊Q₈	—	2	2	no	—
396	C₂²⋊Q₈	—	2	2	no	—
397	C₂²⋊Q₈	—	2	2	no	—
398	C₂²⋊Q₈	—	2	2	no	—
399	C₂²⋊Q₈	—	2	4	no	—
400	C₂²⋊Q₈	—	2	4	no	—
401	C₂²⋊Q₈	—	2	4	no	—
402	C₂²⋊Q₈	—	2	4	no	—
403	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
404	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
405	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
406	#121	—	2	8	no	—
407	#121	—	2	2	no	—
408	#121	—	2	2	no	—
409	#121	C₂³×C₆₇	1	4	yes	—
410	#121	C₂³×C₆₇	1	4	yes	—
411	C₄×Q₈	—	2	2	no	—
412	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
413	C₄×D₄	—	2	4	no	—
414	C₄×D₄	—	2	4	no	—
415	C₄×D₄	—	2	8	no	—
416	C₄×D₄	—	2	2	no	—
417	C₄×D₄	D₄×C₆₇	1	4	yes	—
418	C₄×D₄	C₂³×C₆₇	1	4	yes	—
419	C₄×D₄	C₂³×C₆₇	1	4	yes	—
420	#117	—	2	2	no	—
421	#117	—	2	2	no	—
422	#117	C₂³×C₆₇	1	4	yes	—
423	C₂×C₄⋊C₄	—	2	1	no	—
424	C₂×C₄⋊C₄	—	2	2	no	—
425	C₂×C₄⋊C₄	—	2	1	no	—
426	C₂×C₄⋊C₄	—	2	1	no	—
427	C₂×C₄⋊C₄	—	2	2	no	—
428	C₂×C₄⋊C₄	—	2	2	no	—
429	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
430	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
431	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
432	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
433	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
434	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
435	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
436	C₂×C₂²⋊C₄	—	2	1	no	—
437	C₂×C₂²⋊C₄	—	2	2	no	—
438	C₂×C₂²⋊C₄	—	2	2	no	—
439	C₂×C₂²⋊C₄	—	2	1	no	—
440	C₂×C₂²⋊C₄	—	2	1	no	—
441	C₂×C₂²⋊C₄	—	2	1	no	—
442	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
443	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
444	C₂×C₄²	—	2	1	no	—
445	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
446	#107	C₂³×C₆₇	1	8	yes	—
447	#107	C₂³×C₆₇	1	8	yes	—
448	#106	C₂³×C₆₇	1	8	yes	—
449	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
450	#103	C₂³×C₆₇	1	8	yes	—
451	#103	C₂³×C₆₇	1	8	yes	—
452	#103	C₂³×C₆₇	1	8	yes	—
453	#102	C₂³×C₆₇	1	8	yes	—
454	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
455	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
456	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
457	#97	C₂³×C₆₇	1	4	yes	—
458	C₂⁴	—	2	1	no	—
459	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
460	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
461	C₂×Q₈	—	2	2	no	—
462	C₂×Q₈	—	2	2	no	—
463	C₂×Q₈	—	2	2	no	—
464	C₂×Q₈	—	4	4	no	—
465	C₂×Q₈	—	4	4	no	—
466	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
467	C₂×D₄	—	2	4	no	—
468	C₂×D₄	—	2	4	no	—
469	C₂×D₄	—	2	4	no	—
470	C₂×D₄	—	2	1	no	—
471	C₂×D₄	—	4	2	no	—
472	C₂×D₄	—	4	4	no	—
473	C₂×D₄	—	8	2	no	—
474	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
475	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
476	C₂²×C₄	—	2	1	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	2	1	no	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	—	2	2	no	—
482	C₂²×C₄	—	2	2	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	2	4	no	—
485	C₂²×C₄	—	4	2	no	—
486	C₂²×C₄	—	4	2	no	—
487	C₂²×C₄	—	8	2	no	—
488	C₂²×C₄	—	8	2	no	—
489	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
490	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
491	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
492	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
493	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
494	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
495	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
496	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
497	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
498	SD₁₆	—	4	8	no	—
499	M₁₆	—	4	4	no	—
500	C₂×C₈	—	4	4	no	—
501	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	—	2	1	no	—
504	C₄⋊C₄	—	2	4	no	—
505	C₄⋊C₄	—	2	4	no	—
506	C₄⋊C₄	—	2	4	no	—
507	C₄⋊C₄	—	2	4	no	—
508	C₄⋊C₄	—	2	4	no	—
509	C₄⋊C₄	—	2	4	no	—
510	C₄⋊C₄	—	2	1	no	—
511	C₄⋊C₄	—	2	2	no	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	—	2	2	no	—
514	C₄⋊C₄	—	2	2	no	—
515	C₄⋊C₄	—	2	2	no	—
516	C₄⋊C₄	—	2	4	no	—
517	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
518	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
519	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
520	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
521	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
522	C₂²⋊C₄	—	2	4	no	—
523	C₂²⋊C₄	—	2	4	no	—
524	C₂²⋊C₄	—	2	4	no	—
525	C₂²⋊C₄	—	2	4	no	—
526	C₂²⋊C₄	—	2	4	no	—
527	C₂²⋊C₄	—	2	4	no	—
528	C₂²⋊C₄	—	2	4	no	—
529	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
530	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
531	C₄²	—	2	4	no	—
532	C₄²	—	2	4	no	—
533	C₄²	C₂×D₄×C₆₇	1	2	yes	—
534	C₄²	C₂⁴×C₆₇	1	2	yes	—
535	C₂³	—	2	2	no	—
536	C₂³	—	2	2	no	—
537	C₂³	—	4	1	no	—
538	C₂³	—	4	1	no	—
539	C₂³	—	4	1	no	—
540	C₂³	—	4	1	no	—
541	C₂³	—	4	2	no	—
542	C₂³	—	4	2	no	—
543	C₂³	—	4	2	no	—
544	C₂³	#137×C₆₇	1	4	yes	—
545	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
546	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
547	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
548	Q₈	—	2	8	no	—
549	Q₈	—	8	2	no	—
550	Q₈	—	8	2	no	—
551	Q₈	—	8	2	no	—
552	D₄	—	2	8	no	—
553	D₄	—	2	4	no	—
554	D₄	—	4	2	no	—
555	D₄	—	4	8	no	—
556	D₄	—	8	4	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	4	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	4	no	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	4	2	no	—
567	C₂×C₄	—	4	2	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	2	no	—
572	C₂×C₄	—	4	2	no	—
573	C₂×C₄	—	4	2	no	—
574	C₂×C₄	—	8	4	no	—
575	C₂×C₄	—	8	4	no	—
576	C₂×C₄	—	8	2	no	—
577	C₂×C₄	—	8	2	no	—
578	C₂×C₄	—	8	2	no	—
579	C₂×C₄	—	8	2	no	—
580	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
581	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
582	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
583	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
584	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
585	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
586	C₈	—	2	8	no	—
587	C₂²	—	2	1	no	—
588	C₂²	—	2	2	no	—
589	C₂²	—	2	1	no	—
590	C₂²	—	2	2	no	—
591	C₂²	—	2	2	no	—
592	C₂²	—	2	2	no	—
593	C₂²	—	2	2	no	—
594	C₂²	—	4	4	no	—
595	C₂²	—	4	2	no	—
596	C₂²	—	4	2	no	—
597	C₂²	—	4	2	no	—
598	C₂²	—	8	2	no	—
599	C₂²	—	8	4	no	—
600	C₂²	C₂×#137×C₆₇	1	2	yes	—
601	C₂²	C₂×#137×C₆₇	1	4	yes	—
602	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
603	C₄	—	2	2	no	—
604	C₄	—	2	2	no	—
605	C₄	—	2	1	no	—
606	C₄	—	2	2	no	—
607	C₄	—	2	2	no	—
608	C₄	—	4	2	no	—
609	C₄	—	4	2	no	—
610	C₄	—	4	2	no	—
611	C₄	—	4	4	no	—
612	C₄	—	4	4	no	—
613	C₄	#585×C₆₇	1	2	yes	—
614	C₂	—	2	2	no	—
615	C₂	—	2	4	no	—
616	C₂	—	4	2	no	—
617	C₂	—	8	2	no	—
618	C₂	C₆₇×#3485	1	1	yes	characteristic
619	C₂	C₂²×#137×C₆₇	1	1	yes	characteristic
620	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×#553×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×#3346	C₂	semidirect
4. C₆₇×#3320	C₂	semidirect
5. C₆₇×#3236	C₂	semidirect
6. C₆₇×#3193	C₂	semidirect
7. C₆₇×#3187	C₂	semidirect
8. C₆₇×#3110	C₂	semidirect
9. C₆₇×#3095	C₂	semidirect
10. C₂×#484×C₆₇	C₂	semidirect
11. C₆₇×#2973	C₂	semidirect
13. #554×C₆₇	C₂²	semidirect
16. D₄×Q₈×C₆₇	C₂²	semidirect
35. #501×C₆₇	C₂²	semidirect
36. #491×C₆₇	C₂²	semidirect
37. #485×C₆₇	C₂²	semidirect
40. #484×C₆₇	C₂²	semidirect
42. #481×C₆₇	C₂²	semidirect
43. #480×C₆₇	C₂²	semidirect
44. #479×C₆₇	C₂²	semidirect
45. #474×C₆₇	C₂²	semidirect
46. #468×C₆₇	C₂²	semidirect
47. #465×C₆₇	C₂²	semidirect
48. #461×C₆₇	C₂²	semidirect
50. #448×C₆₇	C₂²	semidirect
51. #448×C₆₇	C₂²	semidirect
52. #440×C₆₇	C₂²	semidirect
53. #435×C₆₇	C₂²	semidirect
55. #428×C₆₇	C₂²	semidirect
56. C₂×#107×C₆₇	C₂²	semidirect
57. #419×C₆₇	C₂²	semidirect
58. C₂×#103×C₆₇	C₂²	semidirect
303. #61708	C₆₇	direct (N × H)
311. #3501	C₂×C₆₇	semidirect
313. #3346	C₂×C₆₇	semidirect
314. #3320	C₂×C₆₇	semidirect
315. #3236	C₂×C₆₇	semidirect
316. #3193	C₂×C₆₇	semidirect
317. #3187	C₂×C₆₇	semidirect
318. #3110	C₂×C₆₇	semidirect
319. #3095	C₂×C₆₇	semidirect
320. C₂×#484	C₂×C₆₇	semidirect
321. #2973	C₂×C₆₇	semidirect
322. C₆₇	#61708	direct (N × H)
324. #554	C₂²×C₆₇	semidirect
327. D₄×Q₈	C₂²×C₆₇	semidirect
346. #501	C₂²×C₆₇	semidirect
347. #491	C₂²×C₆₇	semidirect
348. #485	C₂²×C₆₇	semidirect
351. #484	C₂²×C₆₇	semidirect
353. #481	C₂²×C₆₇	semidirect
354. #480	C₂²×C₆₇	semidirect
355. #479	C₂²×C₆₇	semidirect
356. #474	C₂²×C₆₇	semidirect
357. #468	C₂²×C₆₇	semidirect
358. #465	C₂²×C₆₇	semidirect
359. #461	C₂²×C₆₇	semidirect
361. #448	C₂²×C₆₇	semidirect
362. #448	C₂²×C₆₇	semidirect
363. #440	C₂²×C₆₇	semidirect
364. #435	C₂²×C₆₇	semidirect
366. #428	C₂²×C₆₇	semidirect
367. C₂×#107	C₂²×C₆₇	semidirect
368. #419	C₂²×C₆₇	semidirect
369. C₂×#103	C₂²×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×#553×C₆₇	2
C₆₇×#3346	2
C₆₇×#3320	2
C₆₇×#3236	2
C₆₇×#3193	2
C₆₇×#3187	2
C₆₇×#3110	2
C₆₇×#3095	2
C₂×#484×C₆₇	2
C₆₇×#2973	2
#61708	67

Subgroups — C67×#61708 #30740959

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61708 #30740959