Subgroups — C₆₇×#60944 #30740195

Order 17152.

1846 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3444	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3250	C₂	1	4	yes	maximal, complemented
3	C₆₇×#3231	C₂	1	2	yes	maximal, complemented
4	C₆₇×#3228	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3210	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3207	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3191	C₂	1	4	yes	maximal, complemented
8	C₆₇×#3170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3169	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#3167	C₂	1	2	yes	maximal, complemented
11	C₆₇×#3102	C₂	1	8	yes	maximal, complemented
12	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	#533×C₆₇	C₂²	1	2	yes	—
14	#533×C₆₇	C₂²	1	2	yes	—
15	#533×C₆₇	C₂²	1	2	yes	—
16	#532×C₆₇	C₂²	1	2	yes	—
17	#532×C₆₇	C₂²	1	2	yes	—
18	#532×C₆₇	C₂²	1	2	yes	—
19	#529×C₆₇	—	2	2	no	—
20	#529×C₆₇	—	2	2	no	—
21	#529×C₆₇	—	2	2	no	—
22	#529×C₆₇	—	2	2	no	—
23	#529×C₆₇	C₂²	1	4	yes	—
24	#529×C₆₇	C₂²	1	4	yes	—
25	#525×C₆₇	—	2	2	no	—
26	#525×C₆₇	—	2	2	no	—
27	#525×C₆₇	C₂²	1	4	yes	—
28	C₄×#41×C₆₇	—	2	2	no	—
29	C₄×#41×C₆₇	—	2	2	no	—
30	C₄×#41×C₆₇	C₂²	1	4	yes	—
31	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
32	#500×C₆₇	C₂²	1	1	yes	characteristic, complemented
33	#500×C₆₇	C₂²	1	1	yes	characteristic, complemented
34	#498×C₆₇	C₂²	1	2	yes	complemented
35	#494×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	#493×C₆₇	C₂²	1	1	yes	characteristic, complemented
37	#492×C₆₇	C₂²	1	2	yes	complemented
38	#487×C₆₇	C₂²	1	2	yes	complemented
39	#487×C₆₇	C₂²	1	2	yes	complemented
40	#486×C₆₇	C₂²	1	2	yes	complemented
41	#486×C₆₇	C₂²	1	2	yes	complemented
42	#485×C₆₇	C₂²	1	8	yes	—
43	#482×C₆₇	C₂²	1	8	yes	—
44	#479×C₆₇	C₂²	1	8	yes	complemented
45	#478×C₆₇	C₂²	1	8	yes	complemented
46	#467×C₆₇	C₂²	1	4	yes	—
47	#466×C₆₇	C₂²	1	4	yes	—
48	#465×C₆₇	C₂²	1	8	yes	—
49	#464×C₆₇	C₂²	1	8	yes	complemented
50	#463×C₆₇	C₂²	1	8	yes	complemented
51	#452×C₆₇	C₂²	1	8	yes	—
52	#449×C₆₇	C₂²	1	8	yes	—
53	C₄×Q₁₆×C₆₇	C₂²	1	4	yes	complemented
54	C₄×SD₁₆×C₆₇	C₂²	1	8	yes	complemented
55	C₄×D₈×C₆₇	C₂²	1	4	yes	complemented
56	#431×C₆₇	C₂²	1	4	yes	—
57	#431×C₆₇	C₂²	1	2	yes	—
58	#431×C₆₇	C₂²	1	2	yes	—
59	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
60	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
61	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
62	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
63	C₂×#41×C₆₇	—	2	2	no	—
64	C₂×#41×C₆₇	—	2	2	no	—
65	C₂×#41×C₆₇	C₂³	1	4	yes	—
66	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
67	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
68	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
70	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
71	C₂×D₈×C₆₇	C₂³	1	4	yes	—
72	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
73	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
74	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
75	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
76	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
77	#127×C₆₇	C₂³	1	2	yes	—
78	#126×C₆₇	—	2	4	no	—
79	#126×C₆₇	—	2	4	no	—
80	#125×C₆₇	—	2	2	no	—
81	#125×C₆₇	—	2	2	no	—
82	#125×C₆₇	C₂³	1	4	yes	—
83	#124×C₆₇	—	2	2	no	—
84	#124×C₆₇	—	2	2	no	—
85	#124×C₆₇	C₂³	1	4	yes	—
86	#123×C₆₇	—	2	4	no	—
87	#123×C₆₇	—	2	4	no	—
88	#123×C₆₇	—	2	4	no	—
89	C₂²⋊Q₈×C₆₇	—	2	2	no	—
90	C₂²⋊Q₈×C₆₇	—	2	2	no	—
91	C₂²⋊Q₈×C₆₇	—	2	2	no	—
92	C₂²⋊Q₈×C₆₇	—	2	2	no	—
93	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
94	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
95	#121×C₆₇	—	2	2	no	—
96	#121×C₆₇	—	2	2	no	—
97	#121×C₆₇	—	2	2	no	—
98	#121×C₆₇	—	2	2	no	—
99	#121×C₆₇	C₂³	1	4	yes	—
100	#121×C₆₇	C₂³	1	4	yes	—
101	#120×C₆₇	—	2	4	no	—
102	C₄×Q₈×C₆₇	—	2	2	no	—
103	C₄×Q₈×C₆₇	—	2	2	no	—
104	C₄×Q₈×C₆₇	—	2	2	no	—
105	C₄×Q₈×C₆₇	—	2	2	no	—
106	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
107	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
108	C₄×D₄×C₆₇	—	2	4	no	—
109	C₄×D₄×C₆₇	—	2	4	no	—
110	C₄×D₄×C₆₇	—	2	4	no	—
111	C₄×D₄×C₆₇	—	2	4	no	—
112	C₄×D₄×C₆₇	—	2	4	no	—
113	C₄×D₄×C₆₇	—	2	4	no	—
114	C₄×D₄×C₆₇	—	2	2	no	—
115	C₄×D₄×C₆₇	—	2	2	no	—
116	C₄×D₄×C₆₇	—	2	2	no	—
117	C₄×D₄×C₆₇	—	2	2	no	—
118	C₄×D₄×C₆₇	C₂³	1	4	yes	—
119	C₄×D₄×C₆₇	C₂³	1	4	yes	—
120	#117×C₆₇	—	2	2	no	—
121	#117×C₆₇	—	2	2	no	—
122	#117×C₆₇	—	2	4	no	—
123	#117×C₆₇	—	2	4	no	—
124	#117×C₆₇	—	2	4	no	—
125	#117×C₆₇	C₂³	1	4	yes	—
126	C₂×C₄²×C₆₇	—	2	4	no	—
127	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
128	C₂×C₄²×C₆₇	D₄	1	2	yes	—
129	C₂×C₄²×C₆₇	D₄	1	2	yes	—
130	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
131	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
132	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
133	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
135	#107×C₆₇	C₂³	1	4	yes	—
136	#107×C₆₇	C₂³	1	4	yes	—
137	#106×C₆₇	C₂³	1	8	yes	—
138	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
139	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
140	#103×C₆₇	C₂³	1	8	yes	—
141	#103×C₆₇	C₂³	1	8	yes	—
142	#102×C₆₇	C₂³	1	8	yes	—
143	#102×C₆₇	C₂³	1	8	yes	—
144	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
145	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
146	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
147	C₄×C₈×C₆₇	C₂³	1	2	yes	—
148	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
149	C₄×C₈×C₆₇	C₂³	1	2	yes	—
150	C₄×C₈×C₆₇	C₂³	1	2	yes	—
151	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
152	#41×C₆₇	—	4	8	no	—
153	#41×C₆₇	—	4	8	no	—
154	#41×C₆₇	—	4	8	no	—
155	C₂×Q₈×C₆₇	—	2	2	no	—
156	C₂×Q₈×C₆₇	—	2	2	no	—
157	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
158	C₂×D₄×C₆₇	—	2	4	no	—
159	C₂×D₄×C₆₇	—	2	4	no	—
160	C₂×D₄×C₆₇	—	2	4	no	—
161	C₂×D₄×C₆₇	—	2	2	no	—
162	C₂×D₄×C₆₇	—	2	2	no	—
163	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₆₇	—	2	4	no	—
165	C₂²×C₄×C₆₇	—	2	4	no	—
166	C₂²×C₄×C₆₇	—	2	4	no	—
167	C₂²×C₄×C₆₇	—	2	4	no	—
168	C₂²×C₄×C₆₇	—	4	2	no	—
169	C₂²×C₄×C₆₇	—	4	2	no	—
170	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
177	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
179	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
182	Q₁₆×C₆₇	—	4	4	no	—
183	SD₁₆×C₆₇	—	4	8	no	—
184	D₈×C₆₇	—	4	4	no	—
185	C₂×C₈×C₆₇	—	4	2	no	—
186	C₂×C₈×C₆₇	—	4	2	no	—
187	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
188	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
189	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
190	C₄⋊C₄×C₆₇	—	2	2	no	—
191	C₄⋊C₄×C₆₇	—	2	2	no	—
192	C₄⋊C₄×C₆₇	—	2	2	no	—
193	C₄⋊C₄×C₆₇	—	2	2	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	4	no	—
200	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
201	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
202	C₂²⋊C₄×C₆₇	—	2	4	no	—
203	C₂²⋊C₄×C₆₇	—	2	4	no	—
204	C₂²⋊C₄×C₆₇	—	2	4	no	—
205	C₂²⋊C₄×C₆₇	—	2	4	no	—
206	C₂²⋊C₄×C₆₇	—	2	4	no	—
207	C₂²⋊C₄×C₆₇	—	2	4	no	—
208	C₂²⋊C₄×C₆₇	—	2	4	no	—
209	C₂²⋊C₄×C₆₇	—	2	4	no	—
210	C₂²⋊C₄×C₆₇	—	2	4	no	—
211	C₄²×C₆₇	—	2	4	no	—
212	C₄²×C₆₇	—	2	4	no	—
213	C₄²×C₆₇	—	2	4	no	—
214	C₄²×C₆₇	C₂×D₄	1	2	yes	—
215	C₄²×C₆₇	C₂×D₄	1	2	yes	—
216	C₄²×C₆₇	C₂×D₄	1	2	yes	—
217	C₄²×C₆₇	C₂×D₄	1	2	yes	—
218	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
219	C₄²×C₆₇	C₂×D₄	1	2	yes	—
220	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
221	C₄²×C₆₇	C₂⁴	1	2	yes	—
222	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
223	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
224	C₂³×C₆₇	—	2	2	no	—
225	C₂³×C₆₇	—	2	4	no	—
226	C₂³×C₆₇	—	4	1	no	—
227	C₂³×C₆₇	—	4	1	no	—
228	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
229	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
230	Q₈×C₆₇	—	2	8	no	—
231	Q₈×C₆₇	—	4	4	no	—
232	Q₈×C₆₇	—	4	4	no	—
233	D₄×C₆₇	—	2	8	no	—
234	D₄×C₆₇	—	4	8	no	—
235	D₄×C₆₇	—	4	8	no	—
236	D₄×C₆₇	—	4	8	no	—
237	D₄×C₆₇	—	4	4	no	—
238	D₄×C₆₇	—	4	4	no	—
239	C₂×C₄×C₆₇	—	2	2	no	—
240	C₂×C₄×C₆₇	—	2	4	no	—
241	C₂×C₄×C₆₇	—	2	4	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	4	no	—
244	C₂×C₄×C₆₇	—	2	2	no	—
245	C₂×C₄×C₆₇	—	2	4	no	—
246	C₂×C₄×C₆₇	—	2	4	no	—
247	C₂×C₄×C₆₇	—	2	4	no	—
248	C₂×C₄×C₆₇	—	2	4	no	—
249	C₂×C₄×C₆₇	—	4	1	no	—
250	C₂×C₄×C₆₇	—	4	2	no	—
251	C₂×C₄×C₆₇	—	4	1	no	—
252	C₂×C₄×C₆₇	—	4	2	no	—
253	C₂×C₄×C₆₇	—	4	8	no	—
254	C₂×C₄×C₆₇	—	4	8	no	—
255	C₂×C₄×C₆₇	—	4	8	no	—
256	C₂×C₄×C₆₇	#135	1	4	yes	—
257	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
258	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
259	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
260	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
261	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
262	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
263	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
264	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
265	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
266	C₈×C₆₇	—	2	2	no	—
267	C₈×C₆₇	—	2	4	no	—
268	C₈×C₆₇	—	2	2	no	—
269	C₂²×C₆₇	—	2	1	no	—
270	C₂²×C₆₇	—	2	1	no	—
271	C₂²×C₆₇	—	2	2	no	—
272	C₂²×C₆₇	—	2	2	no	—
273	C₂²×C₆₇	—	2	2	no	—
274	C₂²×C₆₇	—	4	4	no	—
275	C₂²×C₆₇	—	4	4	no	—
276	C₂²×C₆₇	—	4	4	no	—
277	C₂²×C₆₇	—	4	4	no	—
278	C₂²×C₆₇	C₂×#135	1	2	yes	—
279	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
280	C₄×C₆₇	—	2	2	no	—
281	C₄×C₆₇	—	2	1	no	—
282	C₄×C₆₇	—	2	2	no	—
283	C₄×C₆₇	—	2	2	no	—
284	C₄×C₆₇	—	2	1	no	—
285	C₄×C₆₇	—	4	4	no	—
286	C₄×C₆₇	—	4	4	no	—
287	C₄×C₆₇	—	4	4	no	—
288	C₄×C₆₇	C₂×#135	1	2	yes	—
289	C₄×C₆₇	C₂×#135	1	2	yes	—
290	C₄×C₆₇	#575	1	2	yes	—
291	C₄×C₆₇	#585	1	2	yes	—
292	#60944	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
293	C₂×C₆₇	—	2	2	no	—
294	C₂×C₆₇	—	4	2	no	—
295	C₂×C₆₇	—	8	4	no	—
296	C₂×C₆₇	#3449	1	1	yes	characteristic
297	C₂×C₆₇	C₂²×#135	1	1	yes	characteristic
298	C₂×C₆₇	C₂×#575	1	1	yes	characteristic
299	#3444	C₂×C₆₇	1	2	yes	complemented
300	#3250	C₂×C₆₇	1	4	yes	complemented
301	#3231	C₂×C₆₇	1	2	yes	complemented
302	#3228	C₂×C₆₇	1	2	yes	complemented
303	#3210	C₂×C₆₇	1	2	yes	complemented
304	#3207	C₂×C₆₇	1	2	yes	complemented
305	#3191	C₂×C₆₇	1	4	yes	complemented
306	#3170	C₂×C₆₇	1	1	yes	characteristic, complemented
307	#3169	C₂×C₆₇	1	1	yes	characteristic, complemented
308	#3167	C₂×C₆₇	1	2	yes	complemented
309	#3102	C₂×C₆₇	1	8	yes	complemented
310	C₂×#431	C₂×C₆₇	1	1	yes	characteristic, complemented
311	C₆₇	#60944	1	1	yes	characteristic, direct product factor, Sylow
312	#533	C₂²×C₆₇	1	2	yes	—
313	#533	C₂²×C₆₇	1	2	yes	—
314	#533	C₂²×C₆₇	1	2	yes	—
315	#532	C₂²×C₆₇	1	2	yes	—
316	#532	C₂²×C₆₇	1	2	yes	—
317	#532	C₂²×C₆₇	1	2	yes	—
318	#529	—	2	2	no	—
319	#529	—	2	2	no	—
320	#529	—	2	2	no	—
321	#529	—	2	2	no	—
322	#529	C₂²×C₆₇	1	4	yes	—
323	#529	C₂²×C₆₇	1	4	yes	—
324	#525	—	2	2	no	—
325	#525	—	2	2	no	—
326	#525	C₂²×C₆₇	1	4	yes	—
327	C₄×#41	—	2	2	no	—
328	C₄×#41	—	2	2	no	—
329	C₄×#41	C₂²×C₆₇	1	4	yes	—
330	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
331	#500	C₂²×C₆₇	1	1	yes	characteristic, complemented
332	#500	C₂²×C₆₇	1	1	yes	characteristic, complemented
333	#498	C₂²×C₆₇	1	2	yes	complemented
334	#494	C₂²×C₆₇	1	1	yes	characteristic, complemented
335	#493	C₂²×C₆₇	1	1	yes	characteristic, complemented
336	#492	C₂²×C₆₇	1	2	yes	complemented
337	#487	C₂²×C₆₇	1	2	yes	complemented
338	#487	C₂²×C₆₇	1	2	yes	complemented
339	#486	C₂²×C₆₇	1	2	yes	complemented
340	#486	C₂²×C₆₇	1	2	yes	complemented
341	#485	C₂²×C₆₇	1	8	yes	—
342	#482	C₂²×C₆₇	1	8	yes	—
343	#479	C₂²×C₆₇	1	8	yes	complemented
344	#478	C₂²×C₆₇	1	8	yes	complemented
345	#467	C₂²×C₆₇	1	4	yes	—
346	#466	C₂²×C₆₇	1	4	yes	—
347	#465	C₂²×C₆₇	1	8	yes	—
348	#464	C₂²×C₆₇	1	8	yes	complemented
349	#463	C₂²×C₆₇	1	8	yes	complemented
350	#452	C₂²×C₆₇	1	8	yes	—
351	#449	C₂²×C₆₇	1	8	yes	—
352	C₄×Q₁₆	C₂²×C₆₇	1	4	yes	complemented
353	C₄×SD₁₆	C₂²×C₆₇	1	8	yes	complemented
354	C₄×D₈	C₂²×C₆₇	1	4	yes	complemented
355	#431	C₂²×C₆₇	1	4	yes	—
356	#431	C₂²×C₆₇	1	2	yes	—
357	#431	C₂²×C₆₇	1	2	yes	—
358	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
359	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
360	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
361	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
362	C₂×#41	—	2	2	no	—
363	C₂×#41	—	2	2	no	—
364	C₂×#41	C₂³×C₆₇	1	4	yes	—
365	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
366	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
367	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
368	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
369	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
370	C₂×D₈	C₂³×C₆₇	1	4	yes	—
371	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
372	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
373	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
374	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
375	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
376	#127	C₂³×C₆₇	1	2	yes	—
377	#126	—	2	4	no	—
378	#126	—	2	4	no	—
379	#125	—	2	2	no	—
380	#125	—	2	2	no	—
381	#125	C₂³×C₆₇	1	4	yes	—
382	#124	—	2	2	no	—
383	#124	—	2	2	no	—
384	#124	C₂³×C₆₇	1	4	yes	—
385	#123	—	2	4	no	—
386	#123	—	2	4	no	—
387	#123	—	2	4	no	—
388	C₂²⋊Q₈	—	2	2	no	—
389	C₂²⋊Q₈	—	2	2	no	—
390	C₂²⋊Q₈	—	2	2	no	—
391	C₂²⋊Q₈	—	2	2	no	—
392	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
393	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
394	#121	—	2	2	no	—
395	#121	—	2	2	no	—
396	#121	—	2	2	no	—
397	#121	—	2	2	no	—
398	#121	C₂³×C₆₇	1	4	yes	—
399	#121	C₂³×C₆₇	1	4	yes	—
400	#120	—	2	4	no	—
401	C₄×Q₈	—	2	2	no	—
402	C₄×Q₈	—	2	2	no	—
403	C₄×Q₈	—	2	2	no	—
404	C₄×Q₈	—	2	2	no	—
405	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
406	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
407	C₄×D₄	—	2	4	no	—
408	C₄×D₄	—	2	4	no	—
409	C₄×D₄	—	2	4	no	—
410	C₄×D₄	—	2	4	no	—
411	C₄×D₄	—	2	4	no	—
412	C₄×D₄	—	2	4	no	—
413	C₄×D₄	—	2	2	no	—
414	C₄×D₄	—	2	2	no	—
415	C₄×D₄	—	2	2	no	—
416	C₄×D₄	—	2	2	no	—
417	C₄×D₄	C₂³×C₆₇	1	4	yes	—
418	C₄×D₄	C₂³×C₆₇	1	4	yes	—
419	#117	—	2	2	no	—
420	#117	—	2	2	no	—
421	#117	—	2	4	no	—
422	#117	—	2	4	no	—
423	#117	—	2	4	no	—
424	#117	C₂³×C₆₇	1	4	yes	—
425	C₂×C₄²	—	2	4	no	—
426	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
427	C₂×C₄²	D₄×C₆₇	1	2	yes	—
428	C₂×C₄²	D₄×C₆₇	1	2	yes	—
429	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
430	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
431	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
432	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
433	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
434	#107	C₂³×C₆₇	1	4	yes	—
435	#107	C₂³×C₆₇	1	4	yes	—
436	#106	C₂³×C₆₇	1	8	yes	—
437	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
438	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
439	#103	C₂³×C₆₇	1	8	yes	—
440	#103	C₂³×C₆₇	1	8	yes	—
441	#102	C₂³×C₆₇	1	8	yes	—
442	#102	C₂³×C₆₇	1	8	yes	—
443	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
444	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
445	C₄×C₈	C₂³×C₆₇	1	2	yes	—
446	C₄×C₈	C₂³×C₆₇	1	2	yes	—
447	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
448	C₄×C₈	C₂³×C₆₇	1	2	yes	—
449	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
450	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
451	#41	—	4	8	no	—
452	#41	—	4	8	no	—
453	#41	—	4	8	no	—
454	C₂×Q₈	—	2	2	no	—
455	C₂×Q₈	—	2	2	no	—
456	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
457	C₂×D₄	—	2	4	no	—
458	C₂×D₄	—	2	4	no	—
459	C₂×D₄	—	2	4	no	—
460	C₂×D₄	—	2	2	no	—
461	C₂×D₄	—	2	2	no	—
462	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	—	2	4	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	4	no	—
467	C₂²×C₄	—	4	2	no	—
468	C₂²×C₄	—	4	2	no	—
469	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
470	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
471	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
472	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
473	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
474	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
475	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
476	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
478	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
479	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
480	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
481	Q₁₆	—	4	4	no	—
482	SD₁₆	—	4	8	no	—
483	D₈	—	4	4	no	—
484	C₂×C₈	—	4	2	no	—
485	C₂×C₈	—	4	2	no	—
486	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
487	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
488	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	2	no	—
491	C₄⋊C₄	—	2	2	no	—
492	C₄⋊C₄	—	2	2	no	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	4	no	—
495	C₄⋊C₄	—	2	4	no	—
496	C₄⋊C₄	—	2	4	no	—
497	C₄⋊C₄	—	2	4	no	—
498	C₄⋊C₄	—	2	4	no	—
499	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
500	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₂²⋊C₄	—	2	4	no	—
503	C₂²⋊C₄	—	2	4	no	—
504	C₂²⋊C₄	—	2	4	no	—
505	C₂²⋊C₄	—	2	4	no	—
506	C₂²⋊C₄	—	2	4	no	—
507	C₂²⋊C₄	—	2	4	no	—
508	C₂²⋊C₄	—	2	4	no	—
509	C₂²⋊C₄	—	2	4	no	—
510	C₄²	—	2	4	no	—
511	C₄²	—	2	4	no	—
512	C₄²	—	2	4	no	—
513	C₄²	C₂×D₄×C₆₇	1	2	yes	—
514	C₄²	C₂×D₄×C₆₇	1	2	yes	—
515	C₄²	C₂×D₄×C₆₇	1	2	yes	—
516	C₄²	C₂×D₄×C₆₇	1	2	yes	—
517	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
518	C₄²	C₂×D₄×C₆₇	1	2	yes	—
519	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
520	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
521	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
522	C₄²	C₂⁴×C₆₇	1	2	yes	—
523	C₂³	—	2	2	no	—
524	C₂³	—	2	4	no	—
525	C₂³	—	4	1	no	—
526	C₂³	—	4	1	no	—
527	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
528	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
529	Q₈	—	2	8	no	—
530	Q₈	—	4	4	no	—
531	Q₈	—	4	4	no	—
532	D₄	—	2	8	no	—
533	D₄	—	4	8	no	—
534	D₄	—	4	8	no	—
535	D₄	—	4	8	no	—
536	D₄	—	4	4	no	—
537	D₄	—	4	4	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	4	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	2	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	2	no	—
552	C₂×C₄	—	4	8	no	—
553	C₂×C₄	—	4	8	no	—
554	C₂×C₄	—	4	8	no	—
555	C₂×C₄	#135×C₆₇	1	4	yes	—
556	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
557	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
558	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
559	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
560	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
561	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
562	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
563	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
564	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
565	C₈	—	2	2	no	—
566	C₈	—	2	4	no	—
567	C₈	—	2	2	no	—
568	C₂²	—	2	1	no	—
569	C₂²	—	2	1	no	—
570	C₂²	—	2	2	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	4	4	no	—
574	C₂²	—	4	4	no	—
575	C₂²	—	4	4	no	—
576	C₂²	—	4	4	no	—
577	C₂²	C₂×#135×C₆₇	1	2	yes	—
578	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
579	C₄	—	2	2	no	—
580	C₄	—	2	2	no	—
581	C₄	—	2	1	no	—
582	C₄	—	2	2	no	—
583	C₄	—	2	1	no	—
584	C₄	—	4	4	no	—
585	C₄	—	4	4	no	—
586	C₄	—	4	4	no	—
587	C₄	C₂×#135×C₆₇	1	2	yes	—
588	C₄	C₂×#135×C₆₇	1	2	yes	—
589	C₄	#575×C₆₇	1	2	yes	—
590	C₄	#585×C₆₇	1	2	yes	—
591	C₂	—	2	2	no	—
592	C₂	—	4	2	no	—
593	C₂	—	8	4	no	—
594	C₂	C₆₇×#3449	1	1	yes	characteristic
595	C₂	C₂²×#135×C₆₇	1	1	yes	characteristic
596	C₂	C₂×#575×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂²×C₆₇	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3444	C₂	semidirect
2. C₆₇×#3250	C₂	semidirect
3. C₆₇×#3231	C₂	semidirect
4. C₆₇×#3228	C₂	semidirect
5. C₆₇×#3210	C₂	semidirect
6. C₆₇×#3207	C₂	semidirect
7. C₆₇×#3191	C₂	semidirect
8. C₆₇×#3170	C₂	semidirect
9. C₆₇×#3169	C₂	semidirect
10. C₆₇×#3167	C₂	semidirect
11. C₆₇×#3102	C₂	semidirect
12. C₂×#431×C₆₇	C₂	semidirect
32. #500×C₆₇	C₂²	semidirect
33. #500×C₆₇	C₂²	semidirect
34. #498×C₆₇	C₂²	semidirect
35. #494×C₆₇	C₂²	semidirect
36. #493×C₆₇	C₂²	semidirect
37. #492×C₆₇	C₂²	semidirect
38. #487×C₆₇	C₂²	semidirect
39. #487×C₆₇	C₂²	semidirect
40. #486×C₆₇	C₂²	semidirect
41. #486×C₆₇	C₂²	semidirect
44. #479×C₆₇	C₂²	semidirect
45. #478×C₆₇	C₂²	semidirect
49. #464×C₆₇	C₂²	semidirect
50. #463×C₆₇	C₂²	semidirect
53. C₄×Q₁₆×C₆₇	C₂²	semidirect
54. C₄×SD₁₆×C₆₇	C₂²	semidirect
55. C₄×D₈×C₆₇	C₂²	semidirect
292. #60944	C₆₇	direct (N × H)
299. #3444	C₂×C₆₇	semidirect
300. #3250	C₂×C₆₇	semidirect
301. #3231	C₂×C₆₇	semidirect
302. #3228	C₂×C₆₇	semidirect
303. #3210	C₂×C₆₇	semidirect
304. #3207	C₂×C₆₇	semidirect
305. #3191	C₂×C₆₇	semidirect
306. #3170	C₂×C₆₇	semidirect
307. #3169	C₂×C₆₇	semidirect
308. #3167	C₂×C₆₇	semidirect
309. #3102	C₂×C₆₇	semidirect
310. C₂×#431	C₂×C₆₇	semidirect
311. C₆₇	#60944	direct (N × H)
331. #500	C₂²×C₆₇	semidirect
332. #500	C₂²×C₆₇	semidirect
333. #498	C₂²×C₆₇	semidirect
334. #494	C₂²×C₆₇	semidirect
335. #493	C₂²×C₆₇	semidirect
336. #492	C₂²×C₆₇	semidirect
337. #487	C₂²×C₆₇	semidirect
338. #487	C₂²×C₆₇	semidirect
339. #486	C₂²×C₆₇	semidirect
340. #486	C₂²×C₆₇	semidirect
343. #479	C₂²×C₆₇	semidirect
344. #478	C₂²×C₆₇	semidirect
348. #464	C₂²×C₆₇	semidirect
349. #463	C₂²×C₆₇	semidirect
352. C₄×Q₁₆	C₂²×C₆₇	semidirect
353. C₄×SD₁₆	C₂²×C₆₇	semidirect
354. C₄×D₈	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×#3444	2
C₆₇×#3250	2
C₆₇×#3231	2
C₆₇×#3228	2
C₆₇×#3210	2
C₆₇×#3207	2
C₆₇×#3191	2
C₆₇×#3170	2
C₆₇×#3169	2
C₆₇×#3167	2
C₆₇×#3102	2
C₂×#431×C₆₇	2
#60944	67

Subgroups — C67×#60944 #30740195

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60944 #30740195