Subgroups — C₆₇×#11422 #30690673

Order 17152.

528 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#408×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2112	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1315	C₂	1	4	yes	maximal, complemented
5	C₂²×#41×C₆₇	C₄	1	1	yes	characteristic
6	C₂²×M₁₆×C₆₇	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
8	C₂×M₃₂×C₆₇	—	2	1	no	—
9	C₂²×C₁₆×C₆₇	—	2	1	no	—
10	#408×C₆₇	—	2	1	no	—
11	#408×C₆₇	—	2	1	no	—
12	#408×C₆₇	C₄	1	2	yes	—
13	#408×C₆₇	C₂²	1	2	yes	—
14	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
15	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
16	C₂²⋊C₁₆×C₆₇	—	2	1	no	—
17	C₂²⋊C₁₆×C₆₇	—	2	1	no	—
18	C₂²⋊C₁₆×C₆₇	C₂²	1	2	yes	complemented
19	C₂²⋊C₁₆×C₆₇	C₂²	1	2	yes	complemented
20	C₂×#41×C₆₇	—	2	1	no	—
21	C₂×#41×C₆₇	—	2	1	no	—
22	C₂×#41×C₆₇	—	4	2	no	—
23	C₂×#41×C₆₇	—	4	2	no	—
24	C₂×#41×C₆₇	C₈	1	2	yes	—
25	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
26	C₂²×Q₈×C₆₇	C₈	1	1	yes	characteristic
27	C₂²×D₄×C₆₇	—	2	1	no	—
28	C₂²×D₄×C₆₇	C₈	1	1	yes	characteristic
29	C₂³×C₄×C₆₇	—	2	1	no	—
30	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂×M₁₆×C₆₇	—	2	1	no	—
32	C₂×M₁₆×C₆₇	—	4	2	no	—
33	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₆₇	—	2	1	no	—
35	C₂²×C₈×C₆₇	—	4	2	no	—
36	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
42	M₃₂×C₆₇	—	4	2	no	—
43	C₂×C₁₆×C₆₇	—	2	2	no	—
44	C₂×C₁₆×C₆₇	—	2	2	no	—
45	C₂×C₁₆×C₆₇	—	4	2	no	—
46	C₂²⋊C₈×C₆₇	—	2	2	no	—
47	C₂²⋊C₈×C₆₇	—	2	2	no	—
48	C₂²⋊C₈×C₆₇	—	2	2	no	—
49	C₂²⋊C₈×C₆₇	—	2	2	no	—
50	C₂⁴×C₆₇	—	2	1	no	—
51	C₂⁴×C₆₇	M₁₆	1	1	yes	characteristic
52	#41×C₆₇	—	4	4	no	—
53	#41×C₆₇	—	4	4	no	—
54	#41×C₆₇	—	4	2	no	—
55	#41×C₆₇	—	4	2	no	—
56	#41×C₆₇	—	4	2	no	—
57	#41×C₆₇	—	4	2	no	—
58	C₂×Q₈×C₆₇	—	2	1	no	—
59	C₂×Q₈×C₆₇	—	4	2	no	—
60	C₂×Q₈×C₆₇	C₂×C₈	1	2	yes	—
61	C₂×D₄×C₆₇	—	2	2	no	—
62	C₂×D₄×C₆₇	—	2	2	no	—
63	C₂×D₄×C₆₇	—	2	1	no	—
64	C₂×D₄×C₆₇	—	4	2	no	—
65	C₂×D₄×C₆₇	—	4	4	no	—
66	C₂×D₄×C₆₇	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₆₇	—	2	2	no	—
68	C₂²×C₄×C₆₇	—	2	2	no	—
69	C₂²×C₄×C₆₇	—	2	1	no	—
70	C₂²×C₄×C₆₇	—	2	2	no	—
71	C₂²×C₄×C₆₇	—	2	2	no	—
72	C₂²×C₄×C₆₇	—	4	2	no	—
73	C₂²×C₄×C₆₇	—	4	2	no	—
74	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	M₁₆×C₆₇	—	4	2	no	—
81	M₁₆×C₆₇	—	4	2	no	—
82	C₂×C₈×C₆₇	—	2	2	no	—
83	C₂×C₈×C₆₇	—	2	2	no	—
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	—	4	1	no	—
86	C₂×C₈×C₆₇	—	4	1	no	—
87	C₂×C₈×C₆₇	—	4	1	no	—
88	C₂×C₈×C₆₇	—	4	1	no	—
89	C₂×C₈×C₆₇	—	4	2	no	—
90	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₁₆×C₆₇	—	4	2	no	—
95	C₁₆×C₆₇	—	4	2	no	—
96	C₂³×C₆₇	—	2	2	no	—
97	C₂³×C₆₇	—	2	2	no	—
98	C₂³×C₆₇	—	2	2	no	—
99	C₂³×C₆₇	—	4	1	no	—
100	C₂³×C₆₇	—	4	2	no	—
101	C₂³×C₆₇	—	4	2	no	—
102	C₂³×C₆₇	—	4	2	no	—
103	C₂³×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
104	C₂³×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
105	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
106	Q₈×C₆₇	—	4	2	no	—
107	Q₈×C₆₇	—	4	2	no	—
108	D₄×C₆₇	—	4	2	no	—
109	D₄×C₆₇	—	4	4	no	—
110	D₄×C₆₇	—	4	4	no	—
111	D₄×C₆₇	—	4	2	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	2	1	no	—
116	C₂×C₄×C₆₇	—	2	1	no	—
117	C₂×C₄×C₆₇	—	2	1	no	—
118	C₂×C₄×C₆₇	—	2	1	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	4	4	no	—
121	C₂×C₄×C₆₇	—	4	2	no	—
122	C₂×C₄×C₆₇	—	4	2	no	—
123	C₂×C₄×C₆₇	—	4	2	no	—
124	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
126	C₂×C₄×C₆₇	#99	1	2	yes	—
127	C₂×C₄×C₆₇	#100	1	2	yes	—
128	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
130	C₈×C₆₇	—	2	1	no	—
131	C₈×C₆₇	—	2	1	no	—
132	C₈×C₆₇	—	2	1	no	—
133	C₈×C₆₇	—	2	1	no	—
134	C₈×C₆₇	—	4	2	no	—
135	C₂²×C₆₇	—	2	1	no	—
136	C₂²×C₆₇	—	2	1	no	—
137	C₂²×C₆₇	—	2	1	no	—
138	C₂²×C₆₇	—	2	1	no	—
139	C₂²×C₆₇	—	2	1	no	—
140	C₂²×C₆₇	—	2	1	no	—
141	C₂²×C₆₇	—	2	1	no	—
142	C₂²×C₆₇	—	4	4	no	—
143	C₂²×C₆₇	—	4	2	no	—
144	C₂²×C₆₇	—	4	2	no	—
145	C₂²×C₆₇	—	4	2	no	—
146	C₂²×C₆₇	—	8	4	no	—
147	C₂²×C₆₇	#323	1	2	yes	—
148	C₂²×C₆₇	#350	1	2	yes	—
149	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
150	C₄×C₆₇	—	2	1	no	—
151	C₄×C₆₇	—	2	1	no	—
152	C₄×C₆₇	—	4	2	no	—
153	C₄×C₆₇	#323	1	2	yes	—
154	C₄×C₆₇	C₂×#99	1	1	yes	characteristic
155	C₄×C₆₇	C₂×#100	1	1	yes	characteristic
156	#11422	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂×C₆₇	—	2	2	no	—
158	C₂×C₆₇	—	2	2	no	—
159	C₂×C₆₇	—	4	1	no	—
160	C₂×C₆₇	—	8	2	no	—
161	C₂×C₆₇	C₂×#323	1	1	yes	characteristic
162	C₂×C₆₇	C₂×#350	1	1	yes	characteristic
163	C₂×C₆₇	#2117	1	1	yes	characteristic
164	C₂×#408	C₂×C₆₇	1	1	yes	characteristic
165	#2112	C₂×C₆₇	1	1	yes	characteristic, complemented
166	C₂×C₂²⋊C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
167	#1315	C₂×C₆₇	1	4	yes	complemented
168	C₆₇	#11422	1	1	yes	characteristic, direct product factor, Sylow
169	C₂²×#41	C₄×C₆₇	1	1	yes	characteristic
170	C₂²×M₁₆	C₄×C₆₇	1	1	yes	characteristic
171	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
172	C₂×M₃₂	—	2	1	no	—
173	C₂²×C₁₆	—	2	1	no	—
174	#408	—	2	1	no	—
175	#408	—	2	1	no	—
176	#408	C₄×C₆₇	1	2	yes	—
177	#408	C₂²×C₆₇	1	2	yes	—
178	C₂×C₂²⋊C₈	—	2	1	no	—
179	C₂×C₂²⋊C₈	—	2	1	no	—
180	C₂²⋊C₁₆	—	2	1	no	—
181	C₂²⋊C₁₆	—	2	1	no	—
182	C₂²⋊C₁₆	C₂²×C₆₇	1	2	yes	complemented
183	C₂²⋊C₁₆	C₂²×C₆₇	1	2	yes	complemented
184	C₂×#41	—	2	1	no	—
185	C₂×#41	—	2	1	no	—
186	C₂×#41	—	4	2	no	—
187	C₂×#41	—	4	2	no	—
188	C₂×#41	C₈×C₆₇	1	2	yes	—
189	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
190	C₂²×Q₈	C₈×C₆₇	1	1	yes	characteristic
191	C₂²×D₄	—	2	1	no	—
192	C₂²×D₄	C₈×C₆₇	1	1	yes	characteristic
193	C₂³×C₄	—	2	1	no	—
194	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
195	C₂×M₁₆	—	2	1	no	—
196	C₂×M₁₆	—	4	2	no	—
197	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	4	2	no	—
200	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
201	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
202	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
203	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
204	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
205	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
206	M₃₂	—	4	2	no	—
207	C₂×C₁₆	—	2	2	no	—
208	C₂×C₁₆	—	2	2	no	—
209	C₂×C₁₆	—	4	2	no	—
210	C₂²⋊C₈	—	2	2	no	—
211	C₂²⋊C₈	—	2	2	no	—
212	C₂²⋊C₈	—	2	2	no	—
213	C₂²⋊C₈	—	2	2	no	—
214	C₂⁴	—	2	1	no	—
215	C₂⁴	M₁₆×C₆₇	1	1	yes	characteristic
216	#41	—	4	4	no	—
217	#41	—	4	4	no	—
218	#41	—	4	2	no	—
219	#41	—	4	2	no	—
220	#41	—	4	2	no	—
221	#41	—	4	2	no	—
222	C₂×Q₈	—	2	1	no	—
223	C₂×Q₈	—	4	2	no	—
224	C₂×Q₈	C₂×C₈×C₆₇	1	2	yes	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	4	4	no	—
230	C₂×D₄	C₂×C₈×C₆₇	1	2	yes	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	—	4	2	no	—
238	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
239	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₈×C₆₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₈×C₆₇	1	1	yes	characteristic
242	C₂²×C₄	M₁₆×C₆₇	1	1	yes	characteristic
243	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
244	M₁₆	—	4	2	no	—
245	M₁₆	—	4	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	4	1	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
255	C₂×C₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
256	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
257	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
258	C₁₆	—	4	2	no	—
259	C₁₆	—	4	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	4	1	no	—
264	C₂³	—	4	2	no	—
265	C₂³	—	4	2	no	—
266	C₂³	—	4	2	no	—
267	C₂³	C₂²⋊C₈×C₆₇	1	1	yes	characteristic
268	C₂³	C₂²⋊C₈×C₆₇	1	1	yes	characteristic
269	C₂³	C₂×M₁₆×C₆₇	1	1	yes	characteristic
270	Q₈	—	4	2	no	—
271	Q₈	—	4	2	no	—
272	D₄	—	4	2	no	—
273	D₄	—	4	4	no	—
274	D₄	—	4	4	no	—
275	D₄	—	4	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	C₂²⋊C₈×C₆₇	1	1	yes	characteristic
289	C₂×C₄	C₂²⋊C₈×C₆₇	1	1	yes	characteristic
290	C₂×C₄	#99×C₆₇	1	2	yes	—
291	C₂×C₄	#100×C₆₇	1	2	yes	—
292	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	4	2	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	4	4	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	4	2	no	—
310	C₂²	—	8	4	no	—
311	C₂²	#323×C₆₇	1	2	yes	—
312	C₂²	#350×C₆₇	1	2	yes	—
313	C₂²	C₂×C₂²⋊C₈×C₆₇	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	2	no	—
317	C₄	#323×C₆₇	1	2	yes	—
318	C₄	C₂×#99×C₆₇	1	1	yes	characteristic
319	C₄	C₂×#100×C₆₇	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	2	2	no	—
322	C₂	—	4	1	no	—
323	C₂	—	8	2	no	—
324	C₂	C₂×#323×C₆₇	1	1	yes	characteristic
325	C₂	C₂×#350×C₆₇	1	1	yes	characteristic
326	C₂	C₆₇×#2117	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₆₇
Frattini, cosocle	C₂²×C₈	C₂³×C₆₇
socle	C₂²×C₆₇	C₂×C₂²⋊C₈
prime radical	C₂²×#41×C₆₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂³×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2112	C₂	semidirect
3. C₂×C₂²⋊C₁₆×C₆₇	C₂	semidirect
4. C₆₇×#1315	C₂	semidirect
18. C₂²⋊C₁₆×C₆₇	C₂²	semidirect
19. C₂²⋊C₁₆×C₆₇	C₂²	semidirect
156. #11422	C₆₇	direct (N × H)
165. #2112	C₂×C₆₇	semidirect
166. C₂×C₂²⋊C₁₆	C₂×C₆₇	semidirect
167. #1315	C₂×C₆₇	semidirect
168. C₆₇	#11422	direct (N × H)
182. C₂²⋊C₁₆	C₂²×C₆₇	semidirect
183. C₂²⋊C₁₆	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#408×C₆₇	2
C₆₇×#2112	2
C₂×C₂²⋊C₁₆×C₆₇	2
C₆₇×#1315	2
#11422	67

Subgroups — C67×#11422 #30690673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#11422 #30690673