Subgroups — C₆₇×#25926 #30705177

Order 17152.

1012 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3503	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3140	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3040	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2640	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2028	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2027	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1997	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1995	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1883	C₂	1	2	yes	maximal
10	C₆₇×#1882	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1759	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	1	yes	characteristic
14	Q₈²×C₆₇	—	2	1	no	—
15	#557×C₆₇	—	2	1	no	—
16	#554×C₆₇	—	2	1	no	—
17	#554×C₆₇	—	2	2	no	—
18	D₄×Q₈×C₆₇	—	2	2	no	—
19	D₄×Q₈×C₆₇	—	2	1	no	—
20	#544×C₆₇	—	2	2	no	—
21	#544×C₆₇	C₂²	1	2	yes	—
22	#533×C₆₇	—	2	1	no	—
23	#533×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
25	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
27	#520×C₆₇	C₂²	1	2	yes	—
28	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
29	#489×C₆₇	—	2	1	no	—
30	#489×C₆₇	—	2	1	no	—
31	#488×C₆₇	—	2	2	no	—
32	#464×C₆₇	—	2	1	no	—
33	#464×C₆₇	—	2	1	no	—
34	#463×C₆₇	—	2	1	no	—
35	#463×C₆₇	—	2	1	no	—
36	#423×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×#104×C₆₇	C₂²	1	2	yes	—
38	#419×C₆₇	C₂²	1	1	yes	characteristic
39	#419×C₆₇	C₂²	1	2	yes	—
40	#418×C₆₇	C₂²	1	1	yes	characteristic
41	#418×C₆₇	C₂²	1	2	yes	—
42	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
43	#397×C₆₇	—	2	2	no	—
44	#393×C₆₇	—	2	2	no	—
45	#391×C₆₇	C₂²	1	2	yes	—
46	#390×C₆₇	C₂²	1	2	yes	—
47	#376×C₆₇	C₂²	1	1	yes	characteristic
48	#343×C₆₇	C₂²	1	2	yes	—
49	#343×C₆₇	C₂²	1	2	yes	—
50	#339×C₆₇	C₂²	1	2	yes	—
51	#339×C₆₇	C₂²	1	2	yes	—
52	#339×C₆₇	C₂²	1	2	yes	—
53	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×Q₈×C₆₇	—	2	2	no	—
55	C₂²×Q₈×C₆₇	D₄	1	2	yes	—
56	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
58	#137×C₆₇	—	2	4	no	—
59	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
60	C₂×Q₁₆×C₆₇	—	2	1	no	—
61	C₂×SD₁₆×C₆₇	—	2	1	no	—
62	C₂×SD₁₆×C₆₇	—	2	1	no	—
63	C₂×D₈×C₆₇	—	2	1	no	—
64	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
67	C₄⋊Q₈×C₆₇	—	2	2	no	—
68	C₄⋊Q₈×C₆₇	—	2	2	no	—
69	C₄⋊Q₈×C₆₇	—	2	4	no	—
70	C₄⋊Q₈×C₆₇	—	2	2	no	—
71	C₄⋊Q₈×C₆₇	—	2	2	no	—
72	C₄⋊Q₈×C₆₇	—	2	1	no	—
73	C₄⋊Q₈×C₆₇	—	2	1	no	—
74	C₄⋊Q₈×C₆₇	—	2	1	no	—
75	C₄⋊Q₈×C₆₇	—	2	1	no	—
76	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
77	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
78	#125×C₆₇	—	2	4	no	—
79	#125×C₆₇	—	2	2	no	—
80	#125×C₆₇	—	2	2	no	—
81	#125×C₆₇	D₄	1	2	yes	—
82	#124×C₆₇	—	2	1	no	—
83	#124×C₆₇	—	2	1	no	—
84	#124×C₆₇	—	2	1	no	—
85	#124×C₆₇	—	2	1	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	1	no	—
91	C₂²⋊Q₈×C₆₇	—	2	1	no	—
92	C₂²⋊Q₈×C₆₇	—	2	4	no	—
93	C₄×Q₈×C₆₇	—	2	2	no	—
94	C₄×Q₈×C₆₇	—	2	2	no	—
95	C₄×Q₈×C₆₇	—	2	2	no	—
96	C₄×Q₈×C₆₇	—	2	1	no	—
97	C₄×Q₈×C₆₇	—	2	1	no	—
98	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
99	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
100	C₄×D₄×C₆₇	—	2	2	no	—
101	C₄×D₄×C₆₇	—	2	1	no	—
102	C₄×D₄×C₆₇	—	2	1	no	—
103	C₄×D₄×C₆₇	—	2	4	no	—
104	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
105	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
106	#117×C₆₇	—	2	2	no	—
107	#117×C₆₇	—	2	1	no	—
108	#117×C₆₇	C₂³	1	1	yes	characteristic
109	#117×C₆₇	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
111	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
112	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
113	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
115	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
116	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
117	C₂×C₄²×C₆₇	—	2	1	no	—
118	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
119	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
120	C₄⋊C₈×C₆₇	—	2	1	no	—
121	C₄⋊C₈×C₆₇	—	2	1	no	—
122	#104×C₆₇	—	2	4	no	—
123	#103×C₆₇	—	2	1	no	—
124	#103×C₆₇	—	2	1	no	—
125	#103×C₆₇	—	2	2	no	—
126	#103×C₆₇	—	2	2	no	—
127	#102×C₆₇	—	2	1	no	—
128	#102×C₆₇	—	2	1	no	—
129	#102×C₆₇	—	2	2	no	—
130	#102×C₆₇	—	2	2	no	—
131	#97×C₆₇	—	2	1	no	—
132	C₄×C₈×C₆₇	—	2	1	no	—
133	#95×C₆₇	—	2	1	no	—
134	#95×C₆₇	—	2	1	no	—
135	#95×C₆₇	—	2	4	no	—
136	C₂⁴×C₆₇	—	2	1	no	—
137	#41×C₆₇	—	2	2	no	—
138	#41×C₆₇	#41	1	4	yes	—
139	C₂×Q₈×C₆₇	—	2	4	no	—
140	C₂×Q₈×C₆₇	—	2	1	no	—
141	C₂×Q₈×C₆₇	—	2	2	no	—
142	C₂×Q₈×C₆₇	—	2	1	no	—
143	C₂×Q₈×C₆₇	—	2	2	no	—
144	C₂×Q₈×C₆₇	—	2	1	no	—
145	C₂×Q₈×C₆₇	—	2	1	no	—
146	C₂×Q₈×C₆₇	—	2	2	no	—
147	C₂×Q₈×C₆₇	—	2	2	no	—
148	C₂×Q₈×C₆₇	—	8	2	no	—
149	C₂×Q₈×C₆₇	—	8	4	no	—
150	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
151	C₂×D₄×C₆₇	—	2	1	no	—
152	C₂×D₄×C₆₇	—	2	1	no	—
153	C₂×D₄×C₆₇	—	2	1	no	—
154	C₂×D₄×C₆₇	—	2	2	no	—
155	C₂×D₄×C₆₇	—	2	2	no	—
156	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	—	2	1	no	—
158	C₂²×C₄×C₆₇	—	2	4	no	—
159	C₂²×C₄×C₆₇	—	2	1	no	—
160	C₂²×C₄×C₆₇	—	2	1	no	—
161	C₂²×C₄×C₆₇	—	2	2	no	—
162	C₂²×C₄×C₆₇	—	2	1	no	—
163	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
164	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
167	Q₁₆×C₆₇	—	4	2	no	—
168	SD₁₆×C₆₇	—	4	2	no	—
169	SD₁₆×C₆₇	—	4	2	no	—
170	D₈×C₆₇	—	4	2	no	—
171	M₁₆×C₆₇	—	2	2	no	—
172	M₁₆×C₆₇	—	2	2	no	—
173	M₁₆×C₆₇	—	2	2	no	—
174	M₁₆×C₆₇	—	2	2	no	—
175	C₂×C₈×C₆₇	—	2	2	no	—
176	C₂×C₈×C₆₇	—	2	1	no	—
177	C₂×C₈×C₆₇	—	2	1	no	—
178	C₄⋊C₄×C₆₇	—	2	4	no	—
179	C₄⋊C₄×C₆₇	—	2	4	no	—
180	C₄⋊C₄×C₆₇	—	2	4	no	—
181	C₄⋊C₄×C₆₇	—	2	4	no	—
182	C₄⋊C₄×C₆₇	—	2	1	no	—
183	C₄⋊C₄×C₆₇	—	2	2	no	—
184	C₄⋊C₄×C₆₇	—	2	2	no	—
185	C₄⋊C₄×C₆₇	—	2	1	no	—
186	C₄⋊C₄×C₆₇	—	2	1	no	—
187	C₄⋊C₄×C₆₇	—	2	1	no	—
188	C₄⋊C₄×C₆₇	—	2	2	no	—
189	C₄⋊C₄×C₆₇	—	2	2	no	—
190	C₄⋊C₄×C₆₇	—	2	2	no	—
191	C₄⋊C₄×C₆₇	—	2	2	no	—
192	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
193	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
194	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
195	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
196	C₂²⋊C₄×C₆₇	—	2	4	no	—
197	C₂²⋊C₄×C₆₇	—	2	2	no	—
198	C₂²⋊C₄×C₆₇	—	2	1	no	—
199	C₂²⋊C₄×C₆₇	—	2	1	no	—
200	C₂²⋊C₄×C₆₇	—	2	1	no	—
201	C₂²⋊C₄×C₆₇	—	4	1	no	—
202	C₂²⋊C₄×C₆₇	—	4	1	no	—
203	C₂²⋊C₄×C₆₇	—	4	1	no	—
204	C₂²⋊C₄×C₆₇	—	4	1	no	—
205	C₂²⋊C₄×C₆₇	—	8	4	no	—
206	C₄²×C₆₇	—	2	4	no	—
207	C₄²×C₆₇	—	2	1	no	—
208	C₄²×C₆₇	—	2	1	no	—
209	C₄²×C₆₇	—	2	1	no	—
210	C₄²×C₆₇	—	2	2	no	—
211	C₄²×C₆₇	—	2	2	no	—
212	C₄²×C₆₇	—	2	2	no	—
213	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
214	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
215	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
216	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
217	C₂³×C₆₇	—	2	1	no	—
218	C₂³×C₆₇	—	4	2	no	—
219	C₂³×C₆₇	—	4	2	no	—
220	C₂³×C₆₇	—	4	1	no	—
221	C₂³×C₆₇	—	4	2	no	—
222	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
223	Q₈×C₆₇	—	2	1	no	—
224	Q₈×C₆₇	—	4	2	no	—
225	Q₈×C₆₇	—	4	4	no	—
226	Q₈×C₆₇	—	4	2	no	—
227	Q₈×C₆₇	—	4	1	no	—
228	Q₈×C₆₇	—	4	1	no	—
229	Q₈×C₆₇	—	4	2	no	—
230	Q₈×C₆₇	—	8	2	no	—
231	Q₈×C₆₇	—	8	4	no	—
232	Q₈×C₆₇	#124	1	2	yes	—
233	D₄×C₆₇	—	2	4	no	—
234	D₄×C₆₇	—	2	1	no	—
235	D₄×C₆₇	—	4	1	no	—
236	D₄×C₆₇	—	4	1	no	—
237	D₄×C₆₇	—	4	2	no	—
238	D₄×C₆₇	#124	1	2	yes	—
239	C₂×C₄×C₆₇	—	2	1	no	—
240	C₂×C₄×C₆₇	—	2	4	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	1	no	—
243	C₂×C₄×C₆₇	—	2	1	no	—
244	C₂×C₄×C₆₇	—	2	4	no	—
245	C₂×C₄×C₆₇	—	2	2	no	—
246	C₂×C₄×C₆₇	—	4	1	no	—
247	C₂×C₄×C₆₇	—	4	1	no	—
248	C₂×C₄×C₆₇	—	4	2	no	—
249	C₂×C₄×C₆₇	—	4	2	no	—
250	C₂×C₄×C₆₇	—	4	4	no	—
251	C₂×C₄×C₆₇	—	4	2	no	—
252	C₂×C₄×C₆₇	—	4	2	no	—
253	C₂×C₄×C₆₇	—	4	1	no	—
254	C₂×C₄×C₆₇	—	4	2	no	—
255	C₂×C₄×C₆₇	—	4	2	no	—
256	C₂×C₄×C₆₇	—	8	4	no	—
257	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
258	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
259	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
260	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
261	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
262	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
263	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
264	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
265	C₈×C₆₇	—	4	1	no	—
266	C₈×C₆₇	—	4	1	no	—
267	C₈×C₆₇	—	4	2	no	—
268	C₂²×C₆₇	—	2	1	no	—
269	C₂²×C₆₇	—	2	1	no	—
270	C₂²×C₆₇	—	2	2	no	—
271	C₂²×C₆₇	—	4	1	no	—
272	C₂²×C₆₇	—	4	1	no	—
273	C₂²×C₆₇	—	8	4	no	—
274	C₂²×C₆₇	—	8	1	no	—
275	C₂²×C₆₇	—	8	1	no	—
276	C₂²×C₆₇	—	8	1	no	—
277	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
278	C₂²×C₆₇	#548	1	2	yes	—
279	C₄×C₆₇	—	2	2	no	—
280	C₄×C₆₇	—	2	1	no	—
281	C₄×C₆₇	—	2	1	no	—
282	C₄×C₆₇	—	2	1	no	—
283	C₄×C₆₇	—	2	1	no	—
284	C₄×C₆₇	—	4	1	no	—
285	C₄×C₆₇	—	4	4	no	—
286	C₄×C₆₇	—	4	2	no	—
287	C₄×C₆₇	—	4	1	no	—
288	C₄×C₆₇	—	4	2	no	—
289	C₄×C₆₇	—	8	2	no	—
290	C₄×C₆₇	—	8	1	no	—
291	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
292	C₄×C₆₇	#536	1	1	yes	characteristic
293	C₄×C₆₇	#548	1	2	yes	—
294	#25926	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
295	C₂×C₆₇	—	2	2	no	—
296	C₂×C₆₇	—	4	2	no	—
297	C₂×C₆₇	—	16	1	no	—
298	C₂×C₆₇	#2436	1	1	yes	characteristic
299	C₂×C₆₇	#3019	1	1	yes	characteristic
300	C₂×C₆₇	#3020	1	1	yes	characteristic
301	#3503	C₂×C₆₇	1	1	yes	characteristic, complemented
302	#3140	C₂×C₆₇	1	1	yes	characteristic, complemented
303	#3040	C₂×C₆₇	1	1	yes	characteristic
304	#2640	C₂×C₆₇	1	1	yes	characteristic, complemented
305	#2028	C₂×C₆₇	1	2	yes	complemented
306	#2027	C₂×C₆₇	1	2	yes	complemented
307	#1997	C₂×C₆₇	1	1	yes	characteristic, complemented
308	#1995	C₂×C₆₇	1	1	yes	characteristic, complemented
309	#1883	C₂×C₆₇	1	2	yes	—
310	#1882	C₂×C₆₇	1	2	yes	complemented
311	#1759	C₂×C₆₇	1	1	yes	characteristic, complemented
312	C₆₇	#25926	1	1	yes	characteristic, direct product factor, Sylow
313	C₂×#137	C₂²×C₆₇	1	1	yes	characteristic
314	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	1	yes	characteristic
315	Q₈²	—	2	1	no	—
316	#557	—	2	1	no	—
317	#554	—	2	1	no	—
318	#554	—	2	2	no	—
319	D₄×Q₈	—	2	2	no	—
320	D₄×Q₈	—	2	1	no	—
321	#544	—	2	2	no	—
322	#544	C₂²×C₆₇	1	2	yes	—
323	#533	—	2	1	no	—
324	#533	C₂²×C₆₇	1	1	yes	characteristic
325	C₂×C₄⋊Q₈	—	2	1	no	—
326	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
327	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
328	#520	C₂²×C₆₇	1	2	yes	—
329	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
330	#489	—	2	1	no	—
331	#489	—	2	1	no	—
332	#488	—	2	2	no	—
333	#464	—	2	1	no	—
334	#464	—	2	1	no	—
335	#463	—	2	1	no	—
336	#463	—	2	1	no	—
337	#423	C₂²×C₆₇	1	1	yes	characteristic
338	C₂×#104	C₂²×C₆₇	1	2	yes	—
339	#419	C₂²×C₆₇	1	2	yes	—
340	#419	C₂²×C₆₇	1	1	yes	characteristic
341	#418	C₂²×C₆₇	1	2	yes	—
342	#418	C₂²×C₆₇	1	1	yes	characteristic
343	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
344	#397	—	2	2	no	—
345	#393	—	2	2	no	—
346	#391	C₂²×C₆₇	1	2	yes	—
347	#390	C₂²×C₆₇	1	2	yes	—
348	#376	C₂²×C₆₇	1	1	yes	characteristic
349	#343	C₂²×C₆₇	1	2	yes	—
350	#343	C₂²×C₆₇	1	2	yes	—
351	#339	C₂²×C₆₇	1	2	yes	—
352	#339	C₂²×C₆₇	1	2	yes	—
353	#339	C₂²×C₆₇	1	2	yes	—
354	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
355	C₂²×Q₈	—	2	2	no	—
356	C₂²×Q₈	D₄×C₆₇	1	2	yes	—
357	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
358	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
359	#137	—	2	4	no	—
360	AGL(1,ℤ/8)	—	2	4	no	—
361	C₂×Q₁₆	—	2	1	no	—
362	C₂×SD₁₆	—	2	1	no	—
363	C₂×SD₁₆	—	2	1	no	—
364	C₂×D₈	—	2	1	no	—
365	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
366	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
367	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
368	C₄⋊Q₈	—	2	2	no	—
369	C₄⋊Q₈	—	2	2	no	—
370	C₄⋊Q₈	—	2	4	no	—
371	C₄⋊Q₈	—	2	2	no	—
372	C₄⋊Q₈	—	2	2	no	—
373	C₄⋊Q₈	—	2	1	no	—
374	C₄⋊Q₈	—	2	1	no	—
375	C₄⋊Q₈	—	2	1	no	—
376	C₄⋊Q₈	—	2	1	no	—
377	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
378	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
379	#125	—	2	4	no	—
380	#125	—	2	2	no	—
381	#125	—	2	2	no	—
382	#125	D₄×C₆₇	1	2	yes	—
383	#124	—	2	1	no	—
384	#124	—	2	1	no	—
385	#124	—	2	1	no	—
386	#124	—	2	1	no	—
387	C₂²⋊Q₈	—	2	2	no	—
388	C₂²⋊Q₈	—	2	2	no	—
389	C₂²⋊Q₈	—	2	4	no	—
390	C₂²⋊Q₈	—	2	4	no	—
391	C₂²⋊Q₈	—	2	1	no	—
392	C₂²⋊Q₈	—	2	1	no	—
393	C₂²⋊Q₈	—	2	4	no	—
394	C₄×Q₈	—	2	2	no	—
395	C₄×Q₈	—	2	2	no	—
396	C₄×Q₈	—	2	2	no	—
397	C₄×Q₈	—	2	1	no	—
398	C₄×Q₈	—	2	1	no	—
399	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
400	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
401	C₄×D₄	—	2	2	no	—
402	C₄×D₄	—	2	1	no	—
403	C₄×D₄	—	2	1	no	—
404	C₄×D₄	—	2	4	no	—
405	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
406	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
407	#117	—	2	2	no	—
408	#117	—	2	1	no	—
409	#117	C₂³×C₆₇	1	1	yes	characteristic
410	#117	C₂³×C₆₇	1	2	yes	—
411	C₂×C₄⋊C₄	—	2	2	no	—
412	C₂×C₄⋊C₄	—	2	2	no	—
413	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
414	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
415	C₂×C₂²⋊C₄	—	2	1	no	—
416	C₂×C₂²⋊C₄	—	2	1	no	—
417	C₂×C₂²⋊C₄	—	2	4	no	—
418	C₂×C₄²	—	2	1	no	—
419	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
420	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
421	C₄⋊C₈	—	2	1	no	—
422	C₄⋊C₈	—	2	1	no	—
423	#104	—	2	4	no	—
424	#103	—	2	1	no	—
425	#103	—	2	1	no	—
426	#103	—	2	2	no	—
427	#103	—	2	2	no	—
428	#102	—	2	1	no	—
429	#102	—	2	1	no	—
430	#102	—	2	2	no	—
431	#102	—	2	2	no	—
432	#97	—	2	1	no	—
433	C₄×C₈	—	2	1	no	—
434	#95	—	2	1	no	—
435	#95	—	2	1	no	—
436	#95	—	2	4	no	—
437	C₂⁴	—	2	1	no	—
438	#41	—	2	2	no	—
439	#41	#41×C₆₇	1	4	yes	—
440	C₂×Q₈	—	2	4	no	—
441	C₂×Q₈	—	2	1	no	—
442	C₂×Q₈	—	2	2	no	—
443	C₂×Q₈	—	2	1	no	—
444	C₂×Q₈	—	2	2	no	—
445	C₂×Q₈	—	2	1	no	—
446	C₂×Q₈	—	2	1	no	—
447	C₂×Q₈	—	2	2	no	—
448	C₂×Q₈	—	2	2	no	—
449	C₂×Q₈	—	8	2	no	—
450	C₂×Q₈	—	8	4	no	—
451	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
452	C₂×D₄	—	2	1	no	—
453	C₂×D₄	—	2	1	no	—
454	C₂×D₄	—	2	1	no	—
455	C₂×D₄	—	2	2	no	—
456	C₂×D₄	—	2	2	no	—
457	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	4	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
465	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
466	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
467	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
468	Q₁₆	—	4	2	no	—
469	SD₁₆	—	4	2	no	—
470	SD₁₆	—	4	2	no	—
471	D₈	—	4	2	no	—
472	M₁₆	—	2	2	no	—
473	M₁₆	—	2	2	no	—
474	M₁₆	—	2	2	no	—
475	M₁₆	—	2	2	no	—
476	C₂×C₈	—	2	2	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₄⋊C₄	—	2	4	no	—
480	C₄⋊C₄	—	2	4	no	—
481	C₄⋊C₄	—	2	4	no	—
482	C₄⋊C₄	—	2	4	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	2	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	1	no	—
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	C₂×D₄×C₆₇	1	2	yes	—
494	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
495	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
496	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
497	C₂²⋊C₄	—	2	4	no	—
498	C₂²⋊C₄	—	2	2	no	—
499	C₂²⋊C₄	—	2	1	no	—
500	C₂²⋊C₄	—	2	1	no	—
501	C₂²⋊C₄	—	2	1	no	—
502	C₂²⋊C₄	—	4	1	no	—
503	C₂²⋊C₄	—	4	1	no	—
504	C₂²⋊C₄	—	4	1	no	—
505	C₂²⋊C₄	—	4	1	no	—
506	C₂²⋊C₄	—	8	4	no	—
507	C₄²	—	2	4	no	—
508	C₄²	—	2	1	no	—
509	C₄²	—	2	1	no	—
510	C₄²	—	2	1	no	—
511	C₄²	—	2	2	no	—
512	C₄²	—	2	2	no	—
513	C₄²	—	2	2	no	—
514	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
515	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
516	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
517	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
518	C₂³	—	2	1	no	—
519	C₂³	—	4	2	no	—
520	C₂³	—	4	2	no	—
521	C₂³	—	4	1	no	—
522	C₂³	—	4	2	no	—
523	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
524	Q₈	—	2	1	no	—
525	Q₈	—	4	4	no	—
526	Q₈	—	4	2	no	—
527	Q₈	—	4	2	no	—
528	Q₈	—	4	1	no	—
529	Q₈	—	4	1	no	—
530	Q₈	—	4	2	no	—
531	Q₈	—	8	2	no	—
532	Q₈	—	8	4	no	—
533	Q₈	#124×C₆₇	1	2	yes	—
534	D₄	—	2	4	no	—
535	D₄	—	2	1	no	—
536	D₄	—	4	1	no	—
537	D₄	—	4	1	no	—
538	D₄	—	4	2	no	—
539	D₄	#124×C₆₇	1	2	yes	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	2	no	—
550	C₂×C₄	—	4	2	no	—
551	C₂×C₄	—	4	4	no	—
552	C₂×C₄	—	4	2	no	—
553	C₂×C₄	—	4	2	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	2	no	—
556	C₂×C₄	—	4	2	no	—
557	C₂×C₄	—	8	4	no	—
558	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
559	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
560	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
561	C₂×C₄	#120×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	C₂×#41×C₆₇	1	2	yes	—
565	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
566	C₈	—	4	1	no	—
567	C₈	—	4	1	no	—
568	C₈	—	4	2	no	—
569	C₂²	—	2	1	no	—
570	C₂²	—	2	1	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	4	1	no	—
573	C₂²	—	4	1	no	—
574	C₂²	—	8	4	no	—
575	C₂²	—	8	1	no	—
576	C₂²	—	8	1	no	—
577	C₂²	—	8	1	no	—
578	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
579	C₂²	#548×C₆₇	1	2	yes	—
580	C₄	—	2	1	no	—
581	C₄	—	2	1	no	—
582	C₄	—	2	1	no	—
583	C₄	—	2	1	no	—
584	C₄	—	2	2	no	—
585	C₄	—	4	1	no	—
586	C₄	—	4	4	no	—
587	C₄	—	4	2	no	—
588	C₄	—	4	1	no	—
589	C₄	—	4	2	no	—
590	C₄	—	8	2	no	—
591	C₄	—	8	1	no	—
592	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
593	C₄	#536×C₆₇	1	1	yes	characteristic
594	C₄	#548×C₆₇	1	2	yes	—
595	C₂	—	2	2	no	—
596	C₂	—	4	2	no	—
597	C₂	—	16	1	no	—
598	C₂	C₆₇×#2436	1	1	yes	characteristic
599	C₂	C₆₇×#3019	1	1	yes	characteristic
600	C₂	C₆₇×#3020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×AGL(1,ℤ/8)×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3503	C₂	semidirect
2. C₆₇×#3140	C₂	semidirect
4. C₆₇×#2640	C₂	semidirect
5. C₆₇×#2028	C₂	semidirect
6. C₆₇×#2027	C₂	semidirect
7. C₆₇×#1997	C₂	semidirect
8. C₆₇×#1995	C₂	semidirect
10. C₆₇×#1882	C₂	semidirect
11. C₆₇×#1759	C₂	semidirect
294. #25926	C₆₇	direct (N × H)
301. #3503	C₂×C₆₇	semidirect
302. #3140	C₂×C₆₇	semidirect
304. #2640	C₂×C₆₇	semidirect
305. #2028	C₂×C₆₇	semidirect
306. #2027	C₂×C₆₇	semidirect
307. #1997	C₂×C₆₇	semidirect
308. #1995	C₂×C₆₇	semidirect
310. #1882	C₂×C₆₇	semidirect
311. #1759	C₂×C₆₇	semidirect
312. C₆₇	#25926	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3503	2
C₆₇×#3140	2
C₆₇×#3040	2
C₆₇×#2640	2
C₆₇×#2028	2
C₆₇×#2027	2
C₆₇×#1997	2
C₆₇×#1995	2
C₆₇×#1883	2
C₆₇×#1882	2
C₆₇×#1759	2
#25926	67

Subgroups — C67×#25926 #30705177

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25926 #30705177