Subgroups — C₆₇×#20156 #30699407

Order 17152.

586 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3400	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3365	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1715	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1702	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1666	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1653	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1620	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1620	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1587	C₂	1	2	yes	maximal, complemented
10	#559×C₆₇	C₂²	1	1	yes	characteristic
11	#559×C₆₇	C₂²	1	1	yes	characteristic
12	#557×C₆₇	C₂²	1	2	yes	—
13	#498×C₆₇	C₂²	1	1	yes	characteristic
14	#497×C₆₇	—	2	2	no	—
15	#493×C₆₇	C₂²	1	1	yes	characteristic
16	#492×C₆₇	C₂²	1	2	yes	—
17	#488×C₆₇	—	2	2	no	—
18	#486×C₆₇	C₂²	1	2	yes	—
19	#478×C₆₇	—	2	2	no	—
20	#475×C₆₇	C₂²	1	2	yes	—
21	#464×C₆₇	—	2	2	no	—
22	#464×C₆₇	—	2	2	no	—
23	#462×C₆₇	C₂²	1	2	yes	—
24	#461×C₆₇	—	2	2	no	—
25	#460×C₆₇	C₂²	1	2	yes	—
26	#460×C₆₇	C₂²	1	2	yes	—
27	#446×C₆₇	C₂²	1	2	yes	—
28	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
29	#440×C₆₇	—	2	2	no	—
30	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
31	C₄×SD₁₆×C₆₇	—	2	2	no	—
32	#334×C₆₇	C₂²	1	2	yes	—
33	#332×C₆₇	C₂²	1	2	yes	—
34	#331×C₆₇	C₂²	1	1	yes	characteristic
35	#331×C₆₇	C₂²	1	1	yes	characteristic
36	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
37	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
38	#322×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆×C₆₇	—	2	2	no	—
40	C₂×Q₁₆×C₆₇	D₄	1	2	yes	—
41	C₂×SD₁₆×C₆₇	—	2	2	no	—
42	C₂×SD₁₆×C₆₇	—	2	2	no	—
43	C₂×SD₁₆×C₆₇	—	2	2	no	—
44	C₂×SD₁₆×C₆₇	—	2	2	no	—
45	C₂×D₈×C₆₇	—	4	1	no	—
46	C₄⋊Q₈×C₆₇	—	2	2	no	—
47	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
48	#127×C₆₇	C₂³	1	1	yes	characteristic
49	#127×C₆₇	C₂³	1	1	yes	characteristic
50	#125×C₆₇	—	2	2	no	—
51	#125×C₆₇	—	4	2	no	—
52	#124×C₆₇	—	2	2	no	—
53	#124×C₆₇	—	2	2	no	—
54	#123×C₆₇	—	4	1	no	—
55	#123×C₆₇	—	4	1	no	—
56	C₄×Q₈×C₆₇	—	2	2	no	—
57	C₄×Q₈×C₆₇	—	2	2	no	—
58	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
59	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
60	C₄×D₄×C₆₇	—	2	2	no	—
61	C₄×D₄×C₆₇	—	2	2	no	—
62	#107×C₆₇	D₄	1	2	yes	—
63	#106×C₆₇	—	2	2	no	—
64	#106×C₆₇	—	4	1	no	—
65	C₄⋊C₈×C₆₇	—	2	2	no	—
66	C₄⋊C₈×C₆₇	—	2	2	no	—
67	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
68	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
69	#103×C₆₇	—	2	2	no	—
70	#103×C₆₇	—	2	2	no	—
71	#103×C₆₇	D₄	1	2	yes	—
72	#103×C₆₇	D₄	1	2	yes	—
73	#102×C₆₇	—	2	2	no	—
74	#102×C₆₇	—	2	2	no	—
75	#102×C₆₇	—	2	2	no	—
76	#102×C₆₇	—	2	2	no	—
77	#97×C₆₇	—	2	2	no	—
78	C₄×C₈×C₆₇	—	2	2	no	—
79	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₆₇	C₂³	1	2	yes	—
81	C₂×Q₈×C₆₇	—	2	2	no	—
82	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
83	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
84	C₂×D₄×C₆₇	—	2	2	no	—
85	C₂×D₄×C₆₇	—	2	2	no	—
86	C₂×D₄×C₆₇	—	2	1	no	—
87	C₂×D₄×C₆₇	—	2	1	no	—
88	C₂²×C₄×C₆₇	—	4	1	no	—
89	C₂²×C₄×C₆₇	—	4	1	no	—
90	Q₁₆×C₆₇	—	4	2	no	—
91	Q₁₆×C₆₇	—	8	2	no	—
92	SD₁₆×C₆₇	—	4	4	no	—
93	SD₁₆×C₆₇	—	4	4	no	—
94	SD₁₆×C₆₇	—	8	2	no	—
95	SD₁₆×C₆₇	—	8	2	no	—
96	D₈×C₆₇	—	8	2	no	—
97	C₂×C₈×C₆₇	—	2	1	no	—
98	C₂×C₈×C₆₇	—	2	2	no	—
99	C₂×C₈×C₆₇	—	2	2	no	—
100	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
102	C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₄⋊C₄×C₆₇	—	2	2	no	—
104	C₄⋊C₄×C₆₇	—	2	2	no	—
105	C₄⋊C₄×C₆₇	—	2	2	no	—
106	C₄⋊C₄×C₆₇	—	4	2	no	—
107	C₄⋊C₄×C₆₇	—	4	2	no	—
108	C₄⋊C₄×C₆₇	—	4	2	no	—
109	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
110	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
111	C₂²⋊C₄×C₆₇	—	4	2	no	—
112	C₂²⋊C₄×C₆₇	—	4	2	no	—
113	C₂²⋊C₄×C₆₇	—	4	1	no	—
114	C₂²⋊C₄×C₆₇	—	4	1	no	—
115	C₄²×C₆₇	—	2	2	no	—
116	C₄²×C₆₇	—	2	2	no	—
117	C₄²×C₆₇	—	4	2	no	—
118	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
119	C₂³×C₆₇	—	4	1	no	—
120	C₂³×C₆₇	—	4	1	no	—
121	Q₈×C₆₇	—	2	2	no	—
122	Q₈×C₆₇	—	2	2	no	—
123	Q₈×C₆₇	—	2	2	no	—
124	Q₈×C₆₇	—	2	2	no	—
125	Q₈×C₆₇	—	4	2	no	—
126	Q₈×C₆₇	—	4	2	no	—
127	D₄×C₆₇	—	4	2	no	—
128	D₄×C₆₇	—	4	2	no	—
129	D₄×C₆₇	—	4	2	no	—
130	D₄×C₆₇	—	4	2	no	—
131	D₄×C₆₇	—	8	1	no	—
132	D₄×C₆₇	—	8	1	no	—
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	2	2	no	—
137	C₂×C₄×C₆₇	—	4	2	no	—
138	C₂×C₄×C₆₇	—	16	1	no	—
139	C₂×C₄×C₆₇	—	16	1	no	—
140	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
141	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
142	C₈×C₆₇	—	2	2	no	—
143	C₈×C₆₇	—	2	2	no	—
144	C₈×C₆₇	—	2	2	no	—
145	C₈×C₆₇	—	4	2	no	—
146	C₈×C₆₇	—	4	2	no	—
147	C₂²×C₆₇	—	8	2	no	—
148	C₂²×C₆₇	—	8	2	no	—
149	C₂²×C₆₇	—	8	1	no	—
150	C₂²×C₆₇	—	8	1	no	—
151	C₂²×C₆₇	D₄²	1	1	yes	characteristic
152	C₄×C₆₇	—	2	1	no	—
153	C₄×C₆₇	—	4	2	no	—
154	C₄×C₆₇	—	4	2	no	—
155	C₄×C₆₇	—	4	2	no	—
156	C₄×C₆₇	—	4	2	no	—
157	C₄×C₆₇	—	8	2	no	—
158	C₄×C₆₇	#575	1	2	yes	—
159	C₄×C₆₇	#577	1	2	yes	—
160	#20156	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂×C₆₇	—	16	1	no	—
162	C₂×C₆₇	—	16	1	no	—
163	C₂×C₆₇	#3271	1	2	yes	—
164	C₂×C₆₇	#3291	1	1	yes	characteristic
165	#3400	C₂×C₆₇	1	2	yes	complemented
166	#3365	C₂×C₆₇	1	2	yes	complemented
167	#1715	C₂×C₆₇	1	1	yes	characteristic
168	#1702	C₂×C₆₇	1	2	yes	complemented
169	#1666	C₂×C₆₇	1	2	yes	complemented
170	#1653	C₂×C₆₇	1	2	yes	complemented
171	#1620	C₂×C₆₇	1	1	yes	characteristic
172	#1620	C₂×C₆₇	1	1	yes	characteristic
173	#1587	C₂×C₆₇	1	2	yes	complemented
174	C₆₇	#20156	1	1	yes	characteristic, direct product factor, Sylow
175	#559	C₂²×C₆₇	1	1	yes	characteristic
176	#559	C₂²×C₆₇	1	1	yes	characteristic
177	#557	C₂²×C₆₇	1	2	yes	—
178	#498	C₂²×C₆₇	1	1	yes	characteristic
179	#497	—	2	2	no	—
180	#493	C₂²×C₆₇	1	1	yes	characteristic
181	#492	C₂²×C₆₇	1	2	yes	—
182	#488	—	2	2	no	—
183	#486	C₂²×C₆₇	1	2	yes	—
184	#478	—	2	2	no	—
185	#475	C₂²×C₆₇	1	2	yes	—
186	#464	—	2	2	no	—
187	#464	—	2	2	no	—
188	#462	C₂²×C₆₇	1	2	yes	—
189	#461	—	2	2	no	—
190	#460	C₂²×C₆₇	1	2	yes	—
191	#460	C₂²×C₆₇	1	2	yes	—
192	#446	C₂²×C₆₇	1	2	yes	—
193	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
194	#440	—	2	2	no	—
195	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
196	C₄×SD₁₆	—	2	2	no	—
197	#334	C₂²×C₆₇	1	2	yes	—
198	#332	C₂²×C₆₇	1	2	yes	—
199	#331	C₂²×C₆₇	1	1	yes	characteristic
200	#331	C₂²×C₆₇	1	1	yes	characteristic
201	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
202	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
203	#322	C₂²×C₆₇	1	1	yes	characteristic
204	C₂×Q₁₆	—	2	2	no	—
205	C₂×Q₁₆	D₄×C₆₇	1	2	yes	—
206	C₂×SD₁₆	—	2	2	no	—
207	C₂×SD₁₆	—	2	2	no	—
208	C₂×SD₁₆	—	2	2	no	—
209	C₂×SD₁₆	—	2	2	no	—
210	C₂×D₈	—	4	1	no	—
211	C₄⋊Q₈	—	2	2	no	—
212	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
213	#127	C₂³×C₆₇	1	1	yes	characteristic
214	#127	C₂³×C₆₇	1	1	yes	characteristic
215	#125	—	2	2	no	—
216	#125	—	4	2	no	—
217	#124	—	2	2	no	—
218	#124	—	2	2	no	—
219	#123	—	4	1	no	—
220	#123	—	4	1	no	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×Q₈	—	2	2	no	—
223	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
224	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
225	C₄×D₄	—	2	2	no	—
226	C₄×D₄	—	2	2	no	—
227	#107	D₄×C₆₇	1	2	yes	—
228	#106	—	2	2	no	—
229	#106	—	4	1	no	—
230	C₄⋊C₈	—	2	2	no	—
231	C₄⋊C₈	—	2	2	no	—
232	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
233	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
234	#103	—	2	2	no	—
235	#103	—	2	2	no	—
236	#103	D₄×C₆₇	1	2	yes	—
237	#103	D₄×C₆₇	1	2	yes	—
238	#102	—	2	2	no	—
239	#102	—	2	2	no	—
240	#102	—	2	2	no	—
241	#102	—	2	2	no	—
242	#97	—	2	2	no	—
243	C₄×C₈	—	2	2	no	—
244	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
245	C₄×C₈	C₂³×C₆₇	1	2	yes	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
248	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	Q₁₆	—	4	2	no	—
256	Q₁₆	—	8	2	no	—
257	SD₁₆	—	4	4	no	—
258	SD₁₆	—	4	4	no	—
259	SD₁₆	—	8	2	no	—
260	SD₁₆	—	8	2	no	—
261	D₈	—	8	2	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
266	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
275	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	1	no	—
279	C₂²⋊C₄	—	4	1	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	4	2	no	—
283	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
284	C₂³	—	4	1	no	—
285	C₂³	—	4	1	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	2	2	no	—
289	Q₈	—	2	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	8	1	no	—
297	D₄	—	8	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	16	1	no	—
304	C₂×C₄	—	16	1	no	—
305	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
306	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
307	C₈	—	2	2	no	—
308	C₈	—	2	2	no	—
309	C₈	—	2	2	no	—
310	C₈	—	4	2	no	—
311	C₈	—	4	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	1	no	—
315	C₂²	—	8	1	no	—
316	C₂²	D₄²×C₆₇	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	4	2	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	2	no	—
321	C₄	—	4	2	no	—
322	C₄	—	8	2	no	—
323	C₄	#575×C₆₇	1	2	yes	—
324	C₄	#577×C₆₇	1	2	yes	—
325	C₂	—	16	1	no	—
326	C₂	—	16	1	no	—
327	C₂	C₆₇×#3271	1	2	yes	—
328	C₂	C₆₇×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	#493×C₆₇	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3400	C₂	semidirect
2. C₆₇×#3365	C₂	semidirect
4. C₆₇×#1702	C₂	semidirect
5. C₆₇×#1666	C₂	semidirect
6. C₆₇×#1653	C₂	semidirect
9. C₆₇×#1587	C₂	semidirect
160. #20156	C₆₇	direct (N × H)
165. #3400	C₂×C₆₇	semidirect
166. #3365	C₂×C₆₇	semidirect
168. #1702	C₂×C₆₇	semidirect
169. #1666	C₂×C₆₇	semidirect
170. #1653	C₂×C₆₇	semidirect
173. #1587	C₂×C₆₇	semidirect
174. C₆₇	#20156	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×#3400	2
C₆₇×#3365	2
C₆₇×#1715	2
C₆₇×#1702	2
C₆₇×#1666	2
C₆₇×#1653	2
C₆₇×#1620	2
C₆₇×#1587	2
#20156	67

Subgroups — C67×#20156 #30699407

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20156 #30699407