Subgroups — C₆₇×#20150 #30699401

Order 17152.

604 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3396	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3375	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1721	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1688	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1678	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1630	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1623	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1620	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1587	C₂	1	2	yes	maximal, complemented
10	#559×C₆₇	C₂²	1	2	yes	—
11	#559×C₆₇	C₂²	1	1	yes	characteristic
12	Q₈²×C₆₇	C₂²	1	1	yes	characteristic
13	#495×C₆₇	—	2	2	no	—
14	#493×C₆₇	C₂²	1	2	yes	—
15	#493×C₆₇	C₂²	1	1	yes	characteristic
16	#492×C₆₇	C₂²	1	1	yes	characteristic
17	#489×C₆₇	—	2	2	no	—
18	#486×C₆₇	C₂²	1	2	yes	—
19	#477×C₆₇	C₂²	1	2	yes	—
20	#474×C₆₇	—	2	2	no	—
21	#464×C₆₇	—	2	2	no	—
22	#463×C₆₇	—	2	2	no	—
23	#463×C₆₇	—	2	2	no	—
24	#460×C₆₇	C₂²	1	2	yes	—
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×D₈×C₆₇	—	2	2	no	—
32	#334×C₆₇	C₂²	1	2	yes	—
33	#333×C₆₇	C₂²	1	1	yes	characteristic
34	#331×C₆₇	C₂²	1	2	yes	—
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	D₄	1	2	yes	—
40	C₂×SD₁₆×C₆₇	—	2	2	no	—
41	C₂×SD₁₆×C₆₇	—	2	2	no	—
42	C₂×SD₁₆×C₆₇	—	2	2	no	—
43	C₂×SD₁₆×C₆₇	—	4	1	no	—
44	C₂×D₈×C₆₇	—	2	2	no	—
45	C₂×D₈×C₆₇	—	2	2	no	—
46	C₂×D₈×C₆₇	—	4	1	no	—
47	C₄⋊Q₈×C₆₇	—	2	2	no	—
48	C₄⋊Q₈×C₆₇	—	4	1	no	—
49	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
50	#127×C₆₇	C₂³	1	2	yes	—
51	#127×C₆₇	C₂³	1	1	yes	characteristic
52	#124×C₆₇	—	2	2	no	—
53	#124×C₆₇	—	2	2	no	—
54	#124×C₆₇	—	2	2	no	—
55	#123×C₆₇	—	4	1	no	—
56	#123×C₆₇	—	4	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	C₄×D₄×C₆₇	—	2	2	no	—
63	#107×C₆₇	—	2	2	no	—
64	#107×C₆₇	D₄	1	2	yes	—
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	D₄	1	2	yes	—
71	#103×C₆₇	D₄	1	2	yes	—
72	#102×C₆₇	—	2	2	no	—
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	1	no	—
82	C₂×Q₈×C₆₇	—	4	1	no	—
83	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
84	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
85	C₂×D₄×C₆₇	—	2	2	no	—
86	C₂×D₄×C₆₇	—	2	2	no	—
87	C₂×D₄×C₆₇	—	2	2	no	—
88	C₂×D₄×C₆₇	—	2	2	no	—
89	C₂×D₄×C₆₇	—	2	1	no	—
90	C₂²×C₄×C₆₇	—	4	1	no	—
91	C₂²×C₄×C₆₇	—	4	2	no	—
92	Q₁₆×C₆₇	—	4	2	no	—
93	SD₁₆×C₆₇	—	4	4	no	—
94	SD₁₆×C₆₇	—	8	2	no	—
95	SD₁₆×C₆₇	—	8	2	no	—
96	SD₁₆×C₆₇	—	8	2	no	—
97	D₈×C₆₇	—	4	4	no	—
98	D₈×C₆₇	—	8	2	no	—
99	D₈×C₆₇	—	8	2	no	—
100	C₂×C₈×C₆₇	—	2	1	no	—
101	C₂×C₈×C₆₇	—	2	2	no	—
102	C₂×C₈×C₆₇	—	2	2	no	—
103	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
105	C₄⋊C₄×C₆₇	—	2	2	no	—
106	C₄⋊C₄×C₆₇	—	2	2	no	—
107	C₄⋊C₄×C₆₇	—	2	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	2	no	—
114	C₂²⋊C₄×C₆₇	—	4	2	no	—
115	C₂²⋊C₄×C₆₇	—	4	1	no	—
116	C₄²×C₆₇	—	2	2	no	—
117	C₄²×C₆₇	—	2	2	no	—
118	C₄²×C₆₇	—	4	1	no	—
119	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
120	C₂³×C₆₇	—	4	2	no	—
121	C₂³×C₆₇	—	4	1	no	—
122	Q₈×C₆₇	—	2	2	no	—
123	Q₈×C₆₇	—	2	2	no	—
124	Q₈×C₆₇	—	2	2	no	—
125	Q₈×C₆₇	—	2	2	no	—
126	Q₈×C₆₇	—	8	1	no	—
127	Q₈×C₆₇	—	16	1	no	—
128	D₄×C₆₇	—	4	2	no	—
129	D₄×C₆₇	—	4	2	no	—
130	D₄×C₆₇	—	4	2	no	—
131	D₄×C₆₇	—	4	2	no	—
132	D₄×C₆₇	—	4	2	no	—
133	D₄×C₆₇	—	4	2	no	—
134	D₄×C₆₇	—	8	2	no	—
135	D₄×C₆₇	—	8	1	no	—
136	C₂×C₄×C₆₇	—	2	2	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	4	1	no	—
141	C₂×C₄×C₆₇	—	16	1	no	—
142	C₂×C₄×C₆₇	—	16	2	no	—
143	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
144	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
145	C₈×C₆₇	—	2	2	no	—
146	C₈×C₆₇	—	2	2	no	—
147	C₈×C₆₇	—	2	2	no	—
148	C₈×C₆₇	—	4	2	no	—
149	C₈×C₆₇	—	4	2	no	—
150	C₂²×C₆₇	—	8	2	no	—
151	C₂²×C₆₇	—	8	2	no	—
152	C₂²×C₆₇	—	8	2	no	—
153	C₂²×C₆₇	—	8	2	no	—
154	C₂²×C₆₇	—	8	1	no	—
155	C₂²×C₆₇	D₄²	1	1	yes	characteristic
156	C₄×C₆₇	—	2	1	no	—
157	C₄×C₆₇	—	4	2	no	—
158	C₄×C₆₇	—	4	2	no	—
159	C₄×C₆₇	—	4	2	no	—
160	C₄×C₆₇	—	4	2	no	—
161	C₄×C₆₇	—	8	1	no	—
162	C₄×C₆₇	#575	1	2	yes	—
163	C₄×C₆₇	#577	1	2	yes	—
164	#20150	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂×C₆₇	—	16	1	no	—
166	C₂×C₆₇	—	16	2	no	—
167	C₂×C₆₇	#3271	1	2	yes	—
168	C₂×C₆₇	#3292	1	1	yes	characteristic
169	#3396	C₂×C₆₇	1	2	yes	complemented
170	#3375	C₂×C₆₇	1	2	yes	complemented
171	#1721	C₂×C₆₇	1	1	yes	characteristic
172	#1688	C₂×C₆₇	1	2	yes	complemented
173	#1678	C₂×C₆₇	1	2	yes	complemented
174	#1630	C₂×C₆₇	1	1	yes	characteristic, complemented
175	#1623	C₂×C₆₇	1	1	yes	characteristic, complemented
176	#1620	C₂×C₆₇	1	2	yes	complemented
177	#1587	C₂×C₆₇	1	2	yes	complemented
178	C₆₇	#20150	1	1	yes	characteristic, direct product factor, Sylow
179	#559	C₂²×C₆₇	1	2	yes	—
180	#559	C₂²×C₆₇	1	1	yes	characteristic
181	Q₈²	C₂²×C₆₇	1	1	yes	characteristic
182	#495	—	2	2	no	—
183	#493	C₂²×C₆₇	1	2	yes	—
184	#493	C₂²×C₆₇	1	1	yes	characteristic
185	#492	C₂²×C₆₇	1	1	yes	characteristic
186	#489	—	2	2	no	—
187	#486	C₂²×C₆₇	1	2	yes	—
188	#477	C₂²×C₆₇	1	2	yes	—
189	#474	—	2	2	no	—
190	#464	—	2	2	no	—
191	#463	—	2	2	no	—
192	#463	—	2	2	no	—
193	#460	C₂²×C₆₇	1	2	yes	—
194	#460	C₂²×C₆₇	1	2	yes	—
195	#460	C₂²×C₆₇	1	2	yes	—
196	#446	C₂²×C₆₇	1	2	yes	—
197	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
198	#440	—	2	2	no	—
199	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
200	C₄×D₈	—	2	2	no	—
201	#334	C₂²×C₆₇	1	2	yes	—
202	#333	C₂²×C₆₇	1	1	yes	characteristic
203	#331	C₂²×C₆₇	1	2	yes	—
204	#331	C₂²×C₆₇	1	1	yes	characteristic
205	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
206	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
207	#322	C₂²×C₆₇	1	1	yes	characteristic
208	C₂×Q₁₆	D₄×C₆₇	1	2	yes	—
209	C₂×SD₁₆	—	2	2	no	—
210	C₂×SD₁₆	—	2	2	no	—
211	C₂×SD₁₆	—	2	2	no	—
212	C₂×SD₁₆	—	4	1	no	—
213	C₂×D₈	—	2	2	no	—
214	C₂×D₈	—	2	2	no	—
215	C₂×D₈	—	4	1	no	—
216	C₄⋊Q₈	—	2	2	no	—
217	C₄⋊Q₈	—	4	1	no	—
218	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
219	#127	C₂³×C₆₇	1	2	yes	—
220	#127	C₂³×C₆₇	1	1	yes	characteristic
221	#124	—	2	2	no	—
222	#124	—	2	2	no	—
223	#124	—	2	2	no	—
224	#123	—	4	1	no	—
225	#123	—	4	2	no	—
226	C₄×Q₈	—	2	2	no	—
227	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
228	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
229	C₄×D₄	—	2	2	no	—
230	C₄×D₄	—	2	2	no	—
231	C₄×D₄	—	2	2	no	—
232	#107	—	2	2	no	—
233	#107	D₄×C₆₇	1	2	yes	—
234	C₄⋊C₈	—	2	2	no	—
235	C₄⋊C₈	—	2	2	no	—
236	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
237	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
238	#103	—	2	2	no	—
239	#103	D₄×C₆₇	1	2	yes	—
240	#103	D₄×C₆₇	1	2	yes	—
241	#102	—	2	2	no	—
242	#102	—	2	2	no	—
243	#102	—	2	2	no	—
244	#102	—	2	2	no	—
245	#102	—	2	2	no	—
246	#97	—	2	2	no	—
247	C₄×C₈	—	2	2	no	—
248	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
249	C₄×C₈	C₂³×C₆₇	1	2	yes	—
250	C₂×Q₈	—	2	1	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
253	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	1	no	—
259	C₂²×C₄	—	4	1	no	—
260	C₂²×C₄	—	4	2	no	—
261	Q₁₆	—	4	2	no	—
262	SD₁₆	—	4	4	no	—
263	SD₁₆	—	8	2	no	—
264	SD₁₆	—	8	2	no	—
265	SD₁₆	—	8	2	no	—
266	D₈	—	4	4	no	—
267	D₈	—	8	2	no	—
268	D₈	—	8	2	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
273	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
279	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	4	1	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	4	1	no	—
288	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
289	C₂³	—	4	2	no	—
290	C₂³	—	4	1	no	—
291	Q₈	—	2	2	no	—
292	Q₈	—	2	2	no	—
293	Q₈	—	2	2	no	—
294	Q₈	—	2	2	no	—
295	Q₈	—	8	1	no	—
296	Q₈	—	16	1	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	8	2	no	—
304	D₄	—	8	1	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	16	1	no	—
311	C₂×C₄	—	16	2	no	—
312	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
313	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
314	C₈	—	2	2	no	—
315	C₈	—	2	2	no	—
316	C₈	—	2	2	no	—
317	C₈	—	4	2	no	—
318	C₈	—	4	2	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	8	1	no	—
324	C₂²	D₄²×C₆₇	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	2	no	—
328	C₄	—	4	2	no	—
329	C₄	—	4	2	no	—
330	C₄	—	8	1	no	—
331	C₄	#575×C₆₇	1	2	yes	—
332	C₄	#577×C₆₇	1	2	yes	—
333	C₂	—	16	1	no	—
334	C₂	—	16	2	no	—
335	C₂	C₆₇×#3271	1	2	yes	—
336	C₂	C₆₇×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₆₇×#1721	C₂
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×#3396	C₂	semidirect
2. C₆₇×#3375	C₂	semidirect
4. C₆₇×#1688	C₂	semidirect
5. C₆₇×#1678	C₂	semidirect
6. C₆₇×#1630	C₂	semidirect
7. C₆₇×#1623	C₂	semidirect
8. C₆₇×#1620	C₂	semidirect
9. C₆₇×#1587	C₂	semidirect
164. #20150	C₆₇	direct (N × H)
169. #3396	C₂×C₆₇	semidirect
170. #3375	C₂×C₆₇	semidirect
172. #1688	C₂×C₆₇	semidirect
173. #1678	C₂×C₆₇	semidirect
174. #1630	C₂×C₆₇	semidirect
175. #1623	C₂×C₆₇	semidirect
176. #1620	C₂×C₆₇	semidirect
177. #1587	C₂×C₆₇	semidirect
178. C₆₇	#20150	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×#3396	2
C₆₇×#3375	2
C₆₇×#1721	2
C₆₇×#1688	2
C₆₇×#1678	2
C₆₇×#1630	2
C₆₇×#1623	2
C₆₇×#1620	2
C₆₇×#1587	2
#20150	67

Subgroups — C67×#20150 #30699401

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20150 #30699401