Subgroups — C₆₇×#17994 #30697245

Order 17152.

748 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3100	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2935	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1589	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1581	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1581	C₂	1	2	yes	maximal, complemented
7	C₈×D₈×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1571	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1492	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#131×C₆₇	—	2	1	no	—
13	C₂²×M₁₆×C₆₇	—	2	1	no	—
14	#532×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
16	#480×C₆₇	C₄	1	2	yes	—
17	#474×C₆₇	C₄	1	1	yes	characteristic
18	#474×C₆₇	C₄	1	1	yes	characteristic
19	#459×C₆₇	C₄	1	1	yes	characteristic
20	#459×C₆₇	C₄	1	1	yes	characteristic
21	#447×C₆₇	C₄	1	2	yes	—
22	C₄×D₈×C₆₇	C₂²	1	2	yes	—
23	C₄×D₈×C₆₇	C₂²	1	2	yes	—
24	#436×C₆₇	C₂²	1	2	yes	—
25	#436×C₆₇	C₂²	1	2	yes	complemented
26	#435×C₆₇	—	2	2	no	—
27	#435×C₆₇	—	2	2	no	—
28	C₈×D₄×C₆₇	C₂²	1	2	yes	—
29	C₈×D₄×C₆₇	C₂²	1	2	yes	complemented
30	#431×C₆₇	C₂²	1	1	yes	characteristic
31	#423×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
33	#408×C₆₇	—	2	1	no	—
34	#407×C₆₇	—	2	1	no	—
35	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	#335×C₆₇	C₂²	1	2	yes	complemented
38	#335×C₆₇	C₂²	1	2	yes	complemented
39	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
40	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
41	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
42	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
44	D₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
45	#322×C₆₇	C₂²	1	2	yes	complemented
46	C₈²×C₆₇	C₂²	1	2	yes	complemented
47	C₂×#41×C₆₇	—	2	1	no	—
48	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₆₇	—	2	1	no	—
50	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
52	#131×C₆₇	—	4	4	no	—
53	C₂×M₁₆×C₆₇	—	2	1	no	—
54	C₂×M₁₆×C₆₇	—	2	2	no	—
55	C₂×M₁₆×C₆₇	—	2	2	no	—
56	C₂×M₁₆×C₆₇	—	4	4	no	—
57	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
58	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₆₇	—	2	1	no	—
60	C₂²×C₈×C₆₇	—	2	1	no	—
61	C₂²×C₈×C₆₇	—	2	2	no	—
62	C₂²×C₈×C₆₇	—	2	2	no	—
63	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
65	#124×C₆₇	—	2	1	no	—
66	#121×C₆₇	—	2	1	no	—
67	#121×C₆₇	C₂×C₄	1	2	yes	—
68	C₄×D₄×C₆₇	—	2	1	no	—
69	C₄×D₄×C₆₇	—	2	1	no	—
70	C₄×D₄×C₆₇	—	2	1	no	—
71	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
74	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
75	C₄×D₄×C₆₇	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
78	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
79	#107×C₆₇	C₂×C₄	1	2	yes	—
80	#107×C₆₇	C₂×C₄	1	2	yes	—
81	C₄⋊C₈×C₆₇	—	2	1	no	—
82	C₄⋊C₈×C₆₇	—	2	1	no	—
83	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
85	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
88	#102×C₆₇	C₂×C₄	1	2	yes	—
89	#102×C₆₇	C₂×C₄	1	2	yes	—
90	#102×C₆₇	C₂×C₄	1	2	yes	—
91	#102×C₆₇	C₂×C₄	1	2	yes	—
92	C₂²⋊C₈×C₆₇	—	2	2	no	—
93	C₂²⋊C₈×C₆₇	—	2	2	no	—
94	C₂²⋊C₈×C₆₇	—	2	2	no	—
95	C₂²⋊C₈×C₆₇	—	2	2	no	—
96	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
97	#97×C₆₇	D₄	1	2	yes	—
98	C₄×C₈×C₆₇	—	2	1	no	—
99	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₆₇	C₂³	1	2	yes	—
102	C₄×C₈×C₆₇	C₂³	1	2	yes	—
103	C₂⁴×C₆₇	—	2	1	no	—
104	#41×C₆₇	—	4	2	no	—
105	#41×C₆₇	—	4	2	no	—
106	C₂×Q₈×C₆₇	—	2	1	no	—
107	C₂×D₄×C₆₇	—	2	2	no	—
108	C₂×D₄×C₆₇	—	2	2	no	—
109	C₂×D₄×C₆₇	—	2	1	no	—
110	C₂×D₄×C₆₇	—	2	1	no	—
111	C₂×D₄×C₆₇	—	2	1	no	—
112	C₂×D₄×C₆₇	—	4	1	no	—
113	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
115	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	—	2	1	no	—
117	C₂²×C₄×C₆₇	—	2	1	no	—
118	C₂²×C₄×C₆₇	—	2	1	no	—
119	C₂²×C₄×C₆₇	—	2	1	no	—
120	C₂²×C₄×C₆₇	—	2	2	no	—
121	C₂²×C₄×C₆₇	—	2	1	no	—
122	C₂²×C₄×C₆₇	—	4	2	no	—
123	C₂²×C₄×C₆₇	—	4	2	no	—
124	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
127	D₈×C₆₇	—	2	4	no	—
128	D₈×C₆₇	—	4	2	no	—
129	M₁₆×C₆₇	—	2	2	no	—
130	M₁₆×C₆₇	—	4	2	no	—
131	M₁₆×C₆₇	—	4	4	no	—
132	M₁₆×C₆₇	—	4	4	no	—
133	M₁₆×C₆₇	D₈	1	4	yes	complemented
134	C₂×C₈×C₆₇	—	2	1	no	—
135	C₂×C₈×C₆₇	—	2	1	no	—
136	C₂×C₈×C₆₇	—	2	1	no	—
137	C₂×C₈×C₆₇	—	2	2	no	—
138	C₂×C₈×C₆₇	—	2	1	no	—
139	C₂×C₈×C₆₇	—	4	2	no	—
140	C₂×C₈×C₆₇	—	4	2	no	—
141	C₂×C₈×C₆₇	—	4	4	no	—
142	C₂×C₈×C₆₇	—	4	4	no	—
143	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
144	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
145	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
146	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
147	C₄⋊C₄×C₆₇	—	2	1	no	—
148	C₄⋊C₄×C₆₇	—	2	1	no	—
149	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
150	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
151	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
152	C₂²⋊C₄×C₆₇	—	2	1	no	—
153	C₂²⋊C₄×C₆₇	—	2	1	no	—
154	C₂²⋊C₄×C₆₇	—	2	2	no	—
155	C₂²⋊C₄×C₆₇	—	2	1	no	—
156	C₂²⋊C₄×C₆₇	—	2	1	no	—
157	C₂²⋊C₄×C₆₇	—	2	2	no	—
158	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
159	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
160	C₄²×C₆₇	#41	1	1	yes	characteristic
161	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
162	C₂³×C₆₇	—	2	1	no	—
163	C₂³×C₆₇	—	2	1	no	—
164	C₂³×C₆₇	—	2	2	no	—
165	C₂³×C₆₇	—	4	2	no	—
166	C₂³×C₆₇	—	4	2	no	—
167	C₂³×C₆₇	—	8	1	no	—
168	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
169	Q₈×C₆₇	—	4	2	no	—
170	D₄×C₆₇	—	2	2	no	—
171	D₄×C₆₇	—	2	1	no	—
172	D₄×C₆₇	—	2	1	no	—
173	D₄×C₆₇	—	2	1	no	—
174	D₄×C₆₇	—	2	2	no	—
175	D₄×C₆₇	—	4	2	no	—
176	D₄×C₆₇	—	4	4	no	—
177	D₄×C₆₇	—	4	2	no	—
178	D₄×C₆₇	—	8	1	no	—
179	D₄×C₆₇	#131	1	2	yes	complemented
180	C₂×C₄×C₆₇	—	2	1	no	—
181	C₂×C₄×C₆₇	—	2	1	no	—
182	C₂×C₄×C₆₇	—	2	1	no	—
183	C₂×C₄×C₆₇	—	2	1	no	—
184	C₂×C₄×C₆₇	—	2	2	no	—
185	C₂×C₄×C₆₇	—	4	1	no	—
186	C₂×C₄×C₆₇	—	4	2	no	—
187	C₂×C₄×C₆₇	—	4	1	no	—
188	C₂×C₄×C₆₇	—	4	2	no	—
189	C₂×C₄×C₆₇	—	4	1	no	—
190	C₂×C₄×C₆₇	—	4	1	no	—
191	C₂×C₄×C₆₇	—	4	1	no	—
192	C₂×C₄×C₆₇	—	4	2	no	—
193	C₂×C₄×C₆₇	—	4	2	no	—
194	C₂×C₄×C₆₇	—	4	2	no	—
195	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
197	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
199	C₂×C₄×C₆₇	#135	1	2	yes	—
200	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
201	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
203	C₈×C₆₇	—	2	2	no	—
204	C₈×C₆₇	—	2	2	no	—
205	C₈×C₆₇	—	2	2	no	—
206	C₈×C₆₇	—	2	2	no	—
207	C₈×C₆₇	—	4	1	no	—
208	C₈×C₆₇	—	4	1	no	—
209	C₈×C₆₇	—	4	1	no	—
210	C₈×C₆₇	—	4	2	no	—
211	C₈×C₆₇	C₂×D₈	1	4	yes	—
212	C₂²×C₆₇	—	2	1	no	—
213	C₂²×C₆₇	—	2	1	no	—
214	C₂²×C₆₇	—	2	2	no	—
215	C₂²×C₆₇	—	4	1	no	—
216	C₂²×C₆₇	—	4	1	no	—
217	C₂²×C₆₇	—	4	1	no	—
218	C₂²×C₆₇	—	4	2	no	—
219	C₂²×C₆₇	—	4	1	no	—
220	C₂²×C₆₇	—	4	2	no	—
221	C₂²×C₆₇	—	4	2	no	—
222	C₂²×C₆₇	—	4	1	no	—
223	C₂²×C₆₇	—	8	4	no	—
224	C₂²×C₆₇	C₄×D₈	1	2	yes	—
225	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
226	C₄×C₆₇	—	2	1	no	—
227	C₄×C₆₇	—	2	1	no	—
228	C₄×C₆₇	—	2	1	no	—
229	C₄×C₆₇	—	4	1	no	—
230	C₄×C₆₇	—	4	1	no	—
231	C₄×C₆₇	—	4	2	no	—
232	C₄×C₆₇	C₄×D₈	1	2	yes	—
233	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
234	C₄×C₆₇	#568	1	1	yes	characteristic
235	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
236	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
237	#17994	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
238	C₂×C₆₇	—	2	2	no	—
239	C₂×C₆₇	—	4	2	no	—
240	C₂×C₆₇	—	8	2	no	—
241	C₂×C₆₇	—	8	1	no	—
242	C₂×C₆₇	#2933	1	1	yes	characteristic
243	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
244	C₂×C₆₇	#2956	1	1	yes	characteristic
245	#3100	C₂×C₆₇	1	1	yes	characteristic
246	#2935	C₂×C₆₇	1	1	yes	characteristic, complemented
247	D₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
248	#1589	C₂×C₆₇	1	2	yes	complemented
249	#1581	C₂×C₆₇	1	2	yes	complemented
250	#1581	C₂×C₆₇	1	2	yes	complemented
251	C₈×D₈	C₂×C₆₇	1	2	yes	complemented
252	#1571	C₂×C₆₇	1	1	yes	characteristic, complemented
253	#1492	C₂×C₆₇	1	1	yes	characteristic, complemented
254	C₂×D₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
255	C₈×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
256	C₆₇	#17994	1	1	yes	characteristic, direct product factor, Sylow
257	C₂×#131	—	2	1	no	—
258	C₂²×M₁₆	—	2	1	no	—
259	#532	C₂²×C₆₇	1	1	yes	characteristic
260	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
261	#480	C₄×C₆₇	1	2	yes	—
262	#474	C₄×C₆₇	1	1	yes	characteristic
263	#474	C₄×C₆₇	1	1	yes	characteristic
264	#459	C₄×C₆₇	1	1	yes	characteristic
265	#459	C₄×C₆₇	1	1	yes	characteristic
266	#447	C₄×C₆₇	1	2	yes	—
267	C₄×D₈	C₂²×C₆₇	1	2	yes	—
268	C₄×D₈	C₂²×C₆₇	1	2	yes	—
269	#436	C₂²×C₆₇	1	2	yes	—
270	#436	C₂²×C₆₇	1	2	yes	complemented
271	#435	—	2	2	no	—
272	#435	—	2	2	no	—
273	C₈×D₄	C₂²×C₆₇	1	2	yes	—
274	C₈×D₄	C₂²×C₆₇	1	2	yes	complemented
275	#431	C₂²×C₆₇	1	1	yes	characteristic
276	#423	C₂²×C₆₇	1	1	yes	characteristic
277	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
278	#408	—	2	1	no	—
279	#407	—	2	1	no	—
280	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
281	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
282	#335	C₂²×C₆₇	1	2	yes	complemented
283	#335	C₂²×C₆₇	1	2	yes	complemented
284	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
285	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
286	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
287	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
288	D₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
289	D₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
290	#322	C₂²×C₆₇	1	2	yes	complemented
291	C₈²	C₂²×C₆₇	1	2	yes	complemented
292	C₂×#41	—	2	1	no	—
293	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
294	C₂³×C₄	—	2	1	no	—
295	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
296	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
297	#131	—	4	4	no	—
298	C₂×M₁₆	—	2	1	no	—
299	C₂×M₁₆	—	2	2	no	—
300	C₂×M₁₆	—	2	2	no	—
301	C₂×M₁₆	—	4	4	no	—
302	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
303	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
304	C₂²×C₈	—	2	1	no	—
305	C₂²×C₈	—	2	1	no	—
306	C₂²×C₈	—	2	2	no	—
307	C₂²×C₈	—	2	2	no	—
308	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
309	#127	C₂×C₄×C₆₇	1	1	yes	characteristic
310	#124	—	2	1	no	—
311	#121	—	2	1	no	—
312	#121	C₂×C₄×C₆₇	1	2	yes	—
313	C₄×D₄	—	2	1	no	—
314	C₄×D₄	—	2	1	no	—
315	C₄×D₄	—	2	1	no	—
316	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
317	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
318	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
319	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
320	C₄×D₄	C₂³×C₆₇	1	2	yes	—
321	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
322	C₂×C₂²⋊C₄	—	2	1	no	—
323	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
324	#107	C₂×C₄×C₆₇	1	2	yes	—
325	#107	C₂×C₄×C₆₇	1	2	yes	—
326	C₄⋊C₈	—	2	1	no	—
327	C₄⋊C₈	—	2	1	no	—
328	C₄⋊C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
329	C₄⋊C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
330	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
331	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
332	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
333	#102	C₂×C₄×C₆₇	1	2	yes	—
334	#102	C₂×C₄×C₆₇	1	2	yes	—
335	#102	C₂×C₄×C₆₇	1	2	yes	—
336	#102	C₂×C₄×C₆₇	1	2	yes	—
337	C₂²⋊C₈	—	2	2	no	—
338	C₂²⋊C₈	—	2	2	no	—
339	C₂²⋊C₈	—	2	2	no	—
340	C₂²⋊C₈	—	2	2	no	—
341	C₂²⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
342	#97	D₄×C₆₇	1	2	yes	—
343	C₄×C₈	—	2	1	no	—
344	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
345	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
346	C₄×C₈	C₂³×C₆₇	1	2	yes	—
347	C₄×C₈	C₂³×C₆₇	1	2	yes	—
348	C₂⁴	—	2	1	no	—
349	#41	—	4	2	no	—
350	#41	—	4	2	no	—
351	C₂×Q₈	—	2	1	no	—
352	C₂×D₄	—	2	2	no	—
353	C₂×D₄	—	2	2	no	—
354	C₂×D₄	—	2	1	no	—
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	—	4	1	no	—
358	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
359	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
360	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	4	2	no	—
368	C₂²×C₄	—	4	2	no	—
369	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
370	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
371	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
372	D₈	—	2	4	no	—
373	D₈	—	4	2	no	—
374	M₁₆	—	2	2	no	—
375	M₁₆	—	4	2	no	—
376	M₁₆	—	4	4	no	—
377	M₁₆	—	4	4	no	—
378	M₁₆	D₈×C₆₇	1	4	yes	complemented
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	—	4	2	no	—
385	C₂×C₈	—	4	2	no	—
386	C₂×C₈	—	4	4	no	—
387	C₂×C₈	—	4	4	no	—
388	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
389	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
390	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
391	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
392	C₄⋊C₄	—	2	1	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
395	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
396	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
397	C₂²⋊C₄	—	2	1	no	—
398	C₂²⋊C₄	—	2	1	no	—
399	C₂²⋊C₄	—	2	2	no	—
400	C₂²⋊C₄	—	2	1	no	—
401	C₂²⋊C₄	—	2	1	no	—
402	C₂²⋊C₄	—	2	2	no	—
403	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
404	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
405	C₄²	#41×C₆₇	1	1	yes	characteristic
406	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
407	C₂³	—	2	1	no	—
408	C₂³	—	2	1	no	—
409	C₂³	—	2	2	no	—
410	C₂³	—	4	2	no	—
411	C₂³	—	4	2	no	—
412	C₂³	—	8	1	no	—
413	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
414	Q₈	—	4	2	no	—
415	D₄	—	2	2	no	—
416	D₄	—	2	1	no	—
417	D₄	—	2	1	no	—
418	D₄	—	2	1	no	—
419	D₄	—	2	2	no	—
420	D₄	—	4	2	no	—
421	D₄	—	4	4	no	—
422	D₄	—	4	2	no	—
423	D₄	—	8	1	no	—
424	D₄	#131×C₆₇	1	2	yes	complemented
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	4	1	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
441	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
442	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
443	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
444	C₂×C₄	#135×C₆₇	1	2	yes	—
445	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
446	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
447	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
448	C₈	—	2	2	no	—
449	C₈	—	2	2	no	—
450	C₈	—	2	2	no	—
451	C₈	—	2	2	no	—
452	C₈	—	4	1	no	—
453	C₈	—	4	1	no	—
454	C₈	—	4	1	no	—
455	C₈	—	4	2	no	—
456	C₈	C₂×D₈×C₆₇	1	4	yes	—
457	C₂²	—	2	1	no	—
458	C₂²	—	2	1	no	—
459	C₂²	—	2	2	no	—
460	C₂²	—	4	1	no	—
461	C₂²	—	4	1	no	—
462	C₂²	—	4	1	no	—
463	C₂²	—	4	2	no	—
464	C₂²	—	4	1	no	—
465	C₂²	—	4	2	no	—
466	C₂²	—	4	2	no	—
467	C₂²	—	4	1	no	—
468	C₂²	—	8	4	no	—
469	C₂²	C₄×D₈×C₆₇	1	2	yes	—
470	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
471	C₄	—	2	1	no	—
472	C₄	—	2	1	no	—
473	C₄	—	2	1	no	—
474	C₄	—	4	1	no	—
475	C₄	—	4	1	no	—
476	C₄	—	4	2	no	—
477	C₄	C₄×D₈×C₆₇	1	2	yes	—
478	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
479	C₄	#568×C₆₇	1	1	yes	characteristic
480	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
481	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
482	C₂	—	2	2	no	—
483	C₂	—	4	2	no	—
484	C₂	—	8	1	no	—
485	C₂	—	8	2	no	—
486	C₂	C₆₇×#2933	1	1	yes	characteristic
487	C₂	C₂×C₄×D₈×C₆₇	1	1	yes	characteristic
488	C₂	C₆₇×#2956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₆₇×#3100	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₄×M₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2935	C₂	semidirect
3. D₄×M₁₆×C₆₇	C₂	semidirect
4. C₆₇×#1589	C₂	semidirect
5. C₆₇×#1581	C₂	semidirect
6. C₆₇×#1581	C₂	semidirect
7. C₈×D₈×C₆₇	C₂	semidirect
8. C₆₇×#1571	C₂	semidirect
9. C₆₇×#1492	C₂	semidirect
10. C₂×D₄⋊C₈×C₆₇	C₂	semidirect
11. C₈×M₁₆×C₆₇	C₂	semidirect
25. #436×C₆₇	C₂²	semidirect
29. C₈×D₄×C₆₇	C₂²	semidirect
37. #335×C₆₇	C₂²	semidirect
38. #335×C₆₇	C₂²	semidirect
43. D₄⋊C₈×C₆₇	C₂²	semidirect
44. D₄⋊C₈×C₆₇	C₂²	semidirect
45. #322×C₆₇	C₂²	semidirect
46. C₈²×C₆₇	C₂²	semidirect
133. M₁₆×C₆₇	D₈	semidirect
179. D₄×C₆₇	#131	semidirect
237. #17994	C₆₇	direct (N × H)
246. #2935	C₂×C₆₇	semidirect
247. D₄×M₁₆	C₂×C₆₇	semidirect
248. #1589	C₂×C₆₇	semidirect
249. #1581	C₂×C₆₇	semidirect
250. #1581	C₂×C₆₇	semidirect
251. C₈×D₈	C₂×C₆₇	semidirect
252. #1571	C₂×C₆₇	semidirect
253. #1492	C₂×C₆₇	semidirect
254. C₂×D₄⋊C₈	C₂×C₆₇	semidirect
255. C₈×M₁₆	C₂×C₆₇	semidirect
256. C₆₇	#17994	direct (N × H)
270. #436	C₂²×C₆₇	semidirect
274. C₈×D₄	C₂²×C₆₇	semidirect
282. #335	C₂²×C₆₇	semidirect
283. #335	C₂²×C₆₇	semidirect
288. D₄⋊C₈	C₂²×C₆₇	semidirect
289. D₄⋊C₈	C₂²×C₆₇	semidirect
290. #322	C₂²×C₆₇	semidirect
291. C₈²	C₂²×C₆₇	semidirect
378. M₁₆	D₈×C₆₇	semidirect
424. D₄	#131×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3100	2
C₆₇×#2935	2
D₄×M₁₆×C₆₇	2
C₆₇×#1589	2
C₆₇×#1581	2
C₈×D₈×C₆₇	2
C₆₇×#1571	2
C₆₇×#1492	2
C₂×D₄⋊C₈×C₆₇	2
C₈×M₁₆×C₆₇	2
#17994	67

Subgroups — C67×#17994 #30697245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17994 #30697245