Subgroups — #30749020

Order 17152.

408 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523979	C₂	1	1	yes	maximal, characteristic, complemented
2	#22558968	C₂	1	1	yes	maximal, characteristic, complemented
3	#22558648	C₂	1	2	yes	maximal
4	#22558100	C₂	1	2	yes	maximal, complemented
5	C₈×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523979	—	2	1	no	—
7	#16863420	C₂²	1	2	yes	—
8	C₄×#11761612	C₂²	1	2	yes	—
9	#16863161	—	2	1	no	—
10	#16863149	—	2	1	no	—
11	#16863149	—	2	1	no	—
12	Q₈⋊(C₆₇⋊C₈)	—	2	2	no	—
13	D₄⋊(C₆₇⋊C₈)	—	2	2	no	—
14	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₆₇	C₄	1	1	yes	characteristic
16	C₈×Q₈×C₆₇	C₄	1	1	yes	characteristic
17	C₈×D₄×C₆₇	—	2	1	no	—
18	C₈×D₄×C₆₇	—	2	1	no	—
19	C₈×D₄×C₆₇	C₄	1	1	yes	characteristic
20	C₈×D₄×C₆₇	C₄	1	1	yes	characteristic
21	#431×C₆₇	—	2	1	no	—
22	#431×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₇	—	2	1	no	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#11761612	—	2	2	no	—
26	C₂×#11761612	—	2	2	no	—
27	#14523979	—	2	4	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
29	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
30	#14523954	—	2	2	no	—
31	C₄×C₆₇⋊C₈	—	2	2	no	—
32	C₂×#41×C₆₇	Q₈	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	—	2	1	no	—
34	C₂²×C₈×C₆₇	—	2	1	no	—
35	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
37	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
39	C₄×D₄×C₆₇	—	2	1	no	—
40	C₄×D₄×C₆₇	—	2	1	no	—
41	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
43	#117×C₆₇	—	2	1	no	—
44	#117×C₆₇	Q₈	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	—	2	1	no	—
46	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₆₇	—	2	1	no	—
48	C₄⋊C₈×C₆₇	—	2	1	no	—
49	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂²⋊C₈×C₆₇	—	2	1	no	—
52	C₂²⋊C₈×C₆₇	—	2	1	no	—
53	C₂²⋊C₈×C₆₇	—	2	1	no	—
54	C₄×C₈×C₆₇	—	2	1	no	—
55	C₄×C₈×C₆₇	—	2	1	no	—
56	C₄×C₈×C₆₇	—	2	1	no	—
57	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
60	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
61	#11761612	—	2	4	no	—
62	#11761612	—	4	2	no	—
63	#11761612	—	8	2	no	—
64	C₂×C₆₇⋊C₈	—	4	2	no	—
65	C₂×C₆₇⋊C₈	—	4	2	no	—
66	#41×C₆₇	—	2	2	no	—
67	#41×C₆₇	—	2	2	no	—
68	C₂×Q₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
69	C₂×D₄×C₆₇	—	2	1	no	—
70	C₂×D₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	—	2	1	no	—
72	C₂²×C₄×C₆₇	—	2	1	no	—
73	C₂²×C₄×C₆₇	—	2	1	no	—
74	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
77	C₂×C₈×C₆₇	—	2	2	no	—
78	C₂×C₈×C₆₇	—	2	1	no	—
79	C₂×C₈×C₆₇	—	2	1	no	—
80	C₂×C₈×C₆₇	—	4	1	no	—
81	C₂×C₈×C₆₇	—	4	2	no	—
82	C₂×C₈×C₆₇	—	4	2	no	—
83	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
87	C₄⋊C₄×C₆₇	—	2	1	no	—
88	C₄⋊C₄×C₆₇	—	2	1	no	—
89	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
90	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
91	C₂²⋊C₄×C₆₇	—	2	1	no	—
92	C₂²⋊C₄×C₆₇	—	2	1	no	—
93	C₂²⋊C₄×C₆₇	—	2	1	no	—
94	C₄²×C₆₇	—	2	1	no	—
95	C₄²×C₆₇	—	2	1	no	—
96	C₄²×C₆₇	—	2	1	no	—
97	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₆₇	#41	1	1	yes	characteristic
101	C₂³×C₆₇	—	2	1	no	—
102	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
103	Q₈×C₆₇	—	2	1	no	—
104	Q₈×C₆₇	#108	1	2	yes	complemented
105	D₄×C₆₇	—	2	4	no	—
106	D₄×C₆₇	—	2	1	no	—
107	D₄×C₆₇	#108	1	2	yes	complemented
108	C₂×C₄×C₆₇	—	2	4	no	—
109	C₂×C₄×C₆₇	—	2	1	no	—
110	C₂×C₄×C₆₇	—	2	1	no	—
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	—	2	1	no	—
113	C₂×C₄×C₆₇	—	2	1	no	—
114	C₂×C₄×C₆₇	—	2	1	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	—	4	1	no	—
119	C₂×C₄×C₆₇	—	4	1	no	—
120	C₂×C₄×C₆₇	—	4	1	no	—
121	C₂×C₄×C₆₇	—	4	1	no	—
122	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
125	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
126	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
128	C₈×C₆₇	—	2	1	no	—
129	C₈×C₆₇	—	2	1	no	—
130	C₈×C₆₇	—	2	1	no	—
131	C₈×C₆₇	—	4	1	no	—
132	C₈×C₆₇	—	4	1	no	—
133	C₈×C₆₇	#104	1	2	yes	—
134	C₆₇⋊C₈	—	4	4	no	—
135	C₆₇⋊C₈	—	8	2	no	—
136	C₂²×C₆₇	—	2	1	no	—
137	C₂²×C₆₇	—	2	1	no	—
138	C₂²×C₆₇	—	2	2	no	—
139	C₂²×C₆₇	—	2	1	no	—
140	C₂²×C₆₇	—	4	1	no	—
141	C₂²×C₆₇	—	4	1	no	—
142	C₂²×C₆₇	#380	1	1	yes	characteristic
143	C₄×C₆₇	—	2	1	no	—
144	C₄×C₆₇	—	2	1	no	—
145	C₄×C₆₇	—	2	2	no	—
146	C₄×C₆₇	—	2	1	no	—
147	C₄×C₆₇	—	4	1	no	—
148	C₄×C₆₇	—	4	1	no	—
149	C₄×C₆₇	C₂×#104	1	1	yes	characteristic
150	C₄×C₆₇	#421	1	1	yes	characteristic
151	C₄×C₆₇	C₂×#108	1	1	yes	characteristic
152	C₄×C₆₇	C₂×#108	1	1	yes	characteristic
153	C₄×C₆₇	#472	1	1	yes	characteristic
154	C₄×C₆₇	#473	1	1	yes	characteristic
155	#8671	—	67	1	no	maximal, Sylow
156	C₂×C₆₇	—	4	1	no	—
157	C₂×C₆₇	—	4	2	no	—
158	C₂×C₆₇	#1806	1	1	yes	characteristic
159	C₂×C₆₇	#1810	1	1	yes	characteristic
160	C₂×C₆₇	#1816	1	1	yes	characteristic
161	C₈×#41	D₆₇	1	1	yes	characteristic
162	C₄×#108	—	67	1	no	—
163	#1567	—	67	1	no	—
164	#1489	—	67	2	no	—
165	#1299	—	67	2	no	—
166	C₆₇	#8671	1	1	yes	characteristic, complemented, Sylow
167	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
168	C₈×Q₈	Dic₆₇	1	1	yes	characteristic
169	C₈×Q₈	Dic₆₇	1	1	yes	characteristic
170	C₈×D₄	—	2	1	no	—
171	C₈×D₄	—	2	1	no	—
172	C₈×D₄	Dic₆₇	1	1	yes	characteristic
173	C₈×D₄	Dic₆₇	1	1	yes	characteristic
174	#431	—	2	1	no	—
175	#431	C₂×D₆₇	1	1	yes	characteristic
176	C₂×#108	—	134	1	no	—
177	#423	—	67	2	no	—
178	C₄×M₁₆	—	67	2	no	—
179	C₂×C₄×C₈	—	2	1	no	—
180	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
181	#341	—	134	1	no	—
182	#335	—	134	1	no	—
183	#335	—	134	1	no	—
184	Q₈⋊C₈	—	134	2	no	—
185	D₄⋊C₈	—	134	2	no	—
186	C₂×#41	Dic₁₃₄	1	1	yes	characteristic
187	C₂×M₁₆	—	134	2	no	—
188	C₂×M₁₆	—	134	2	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
192	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
193	C₄×Q₈	D₂₆₈	1	1	yes	characteristic
194	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
195	C₄×D₄	—	2	1	no	—
196	C₄×D₄	—	2	1	no	—
197	C₄×D₄	D₂₆₈	1	1	yes	characteristic
198	C₄×D₄	C₂×Dic₆₇	1	1	yes	characteristic
199	#117	—	2	1	no	—
200	#117	Dic₁₃₄	1	1	yes	characteristic
201	C₂×C₄²	—	2	1	no	—
202	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
203	#108	—	134	4	no	—
204	C₄⋊C₈	—	2	1	no	—
205	C₄⋊C₈	—	2	1	no	—
206	C₄⋊C₈	—	134	2	no	—
207	C₄⋊C₈	—	134	2	no	—
208	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
209	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
210	C₂²⋊C₈	—	2	1	no	—
211	C₂²⋊C₈	—	2	1	no	—
212	C₂²⋊C₈	—	2	1	no	—
213	#97	—	134	2	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	2	1	no	—
216	C₄×C₈	—	2	1	no	—
217	C₄×C₈	—	134	2	no	—
218	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
219	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
220	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
221	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
222	#41	—	2	2	no	—
223	#41	—	2	2	no	—
224	C₂×Q₈	C₄⋊Dic₆₇	1	1	yes	characteristic
225	C₂×D₄	—	2	1	no	—
226	C₂×D₄	C₄⋊Dic₆₇	1	1	yes	characteristic
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
231	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
232	C₂²×C₄	#11761628	1	1	yes	characteristic
233	M₁₆	—	268	2	no	—
234	M₁₆	—	536	2	no	—
235	M₁₆	—	134	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	—	268	2	no	—
243	C₂×C₈	—	268	2	no	—
244	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
245	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
246	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
247	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
251	C₄⋊C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
259	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
260	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
261	C₄²	#11761628	1	1	yes	characteristic
262	C₂³	—	2	1	no	—
263	C₂³	#14524022	1	1	yes	characteristic
264	Q₈	—	2	1	no	—
265	Q₈	#14523979	1	2	yes	complemented
266	D₄	—	2	4	no	—
267	D₄	—	2	1	no	—
268	D₄	#14523979	1	2	yes	complemented
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
285	C₂×C₄	#14524002	1	1	yes	characteristic
286	C₂×C₄	#14524017	1	1	yes	characteristic
287	C₂×C₄	#14524017	1	1	yes	characteristic
288	C₂×C₄	#14524022	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	4	1	no	—
293	C₈	—	4	1	no	—
294	C₈	—	268	4	no	—
295	C₈	—	536	2	no	—
296	C₈	#14523970	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	1	no	—
303	C₂²	#16863256	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	2	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	C₂×#14523970	1	1	yes	characteristic
311	C₄	#16863413	1	1	yes	characteristic
312	C₄	C₂×#14523979	1	1	yes	characteristic
313	C₄	C₂×#14523979	1	1	yes	characteristic
314	C₄	#16863666	1	1	yes	characteristic
315	C₄	#16863671	1	1	yes	characteristic
316	C₂	—	4	1	no	—
317	C₂	—	4	2	no	—
318	C₂	#22560229	1	1	yes	characteristic
319	C₂	#22560247	1	1	yes	characteristic
320	C₂	#22560273	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524017
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₈×#41×C₆₇	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂×#41×C₆₇	Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₈×#41	D₆₇
hypocenter	C₆₇	#8671
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. C₄×#14523979	C₂	semidirect
2. #22558968	C₂	semidirect
4. #22558100	C₂	semidirect
104. Q₈×C₆₇	#108	semidirect
107. D₄×C₆₇	#108	semidirect
166. C₆₇	#8671	semidirect
265. Q₈	#14523979	semidirect
268. D₄	#14523979	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523979	2
#22558968	2
#22558648	2
#22558100	2
C₈×#41×C₆₇	2
#8671	67