Subgroups — #30749015

Order 17152.

408 conjugacy classes of subgroups.

The subgroup lattice has 316 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	#22558965	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	#16863147	—	2	1	no	—
11	#16863147	—	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	2	no	—
115	C₂×C₄×C₆₇	—	2	1	no	—
116	C₂×C₄×C₆₇	—	2	1	no	—
117	C₂×C₄×C₆₇	—	4	1	no	—
118	C₂×C₄×C₆₇	—	4	1	no	—
119	C₂×C₄×C₆₇	—	4	2	no	—
120	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
123	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
124	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
126	C₈×C₆₇	—	2	1	no	—
127	C₈×C₆₇	—	2	1	no	—
128	C₈×C₆₇	—	2	1	no	—
129	C₈×C₆₇	—	4	1	no	—
130	C₈×C₆₇	—	4	1	no	—
131	C₈×C₆₇	#104	1	2	yes	—
132	C₆₇⋊C₈	—	4	4	no	—
133	C₆₇⋊C₈	—	8	2	no	—
134	C₂²×C₆₇	—	2	1	no	—
135	C₂²×C₆₇	—	2	1	no	—
136	C₂²×C₆₇	—	2	2	no	—
137	C₂²×C₆₇	—	2	1	no	—
138	C₂²×C₆₇	—	4	1	no	—
139	C₂²×C₆₇	—	4	1	no	—
140	C₂²×C₆₇	#380	1	1	yes	characteristic
141	C₄×C₆₇	—	2	1	no	—
142	C₄×C₆₇	—	2	1	no	—
143	C₄×C₆₇	—	2	2	no	—
144	C₄×C₆₇	—	2	1	no	—
145	C₄×C₆₇	—	4	1	no	—
146	C₄×C₆₇	—	4	1	no	—
147	C₄×C₆₇	C₂×#104	1	1	yes	characteristic
148	C₄×C₆₇	#421	1	1	yes	characteristic
149	C₄×C₆₇	C₂×#108	1	1	yes	characteristic
150	C₄×C₆₇	C₂×#108	1	1	yes	characteristic
151	C₄×C₆₇	#471	1	2	yes	—
152	#8670	—	67	1	no	maximal, Sylow
153	C₂×C₆₇	—	4	1	no	—
154	C₂×C₆₇	—	4	2	no	—
155	C₂×C₆₇	#1806	1	1	yes	characteristic
156	C₂×C₆₇	#1810	1	1	yes	characteristic
157	C₂×C₆₇	#1815	1	1	yes	characteristic
158	C₈×#41	D₆₇	1	1	yes	characteristic
159	C₄×#108	—	67	1	no	—
160	#1566	—	67	1	no	—
161	#1489	—	67	2	no	—
162	#1299	—	67	2	no	—
163	C₆₇	#8670	1	1	yes	characteristic, complemented, Sylow
164	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
165	C₈×Q₈	Dic₆₇	1	1	yes	characteristic
166	C₈×Q₈	Dic₆₇	1	1	yes	characteristic
167	C₈×D₄	—	2	1	no	—
168	C₈×D₄	—	2	1	no	—
169	C₈×D₄	Dic₆₇	1	1	yes	characteristic
170	C₈×D₄	Dic₆₇	1	1	yes	characteristic
171	#431	—	2	1	no	—
172	#431	C₂×D₆₇	1	1	yes	characteristic
173	C₂×#108	—	134	1	no	—
174	#423	—	67	2	no	—
175	C₄×M₁₆	—	67	2	no	—
176	C₂×C₄×C₈	—	2	1	no	—
177	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
178	#341	—	134	1	no	—
179	#334	—	134	1	no	—
180	#334	—	134	1	no	—
181	Q₈⋊C₈	—	134	2	no	—
182	D₄⋊C₈	—	134	2	no	—
183	C₂×#41	Dic₁₃₄	1	1	yes	characteristic
184	C₂×M₁₆	—	134	2	no	—
185	C₂×M₁₆	—	134	2	no	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
189	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
190	C₄×Q₈	D₂₆₈	1	1	yes	characteristic
191	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
192	C₄×D₄	—	2	1	no	—
193	C₄×D₄	—	2	1	no	—
194	C₄×D₄	D₂₆₈	1	1	yes	characteristic
195	C₄×D₄	C₂×Dic₆₇	1	1	yes	characteristic
196	#117	—	2	1	no	—
197	#117	Dic₁₃₄	1	1	yes	characteristic
198	C₂×C₄²	—	2	1	no	—
199	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
200	#108	—	134	4	no	—
201	C₄⋊C₈	—	2	1	no	—
202	C₄⋊C₈	—	2	1	no	—
203	C₄⋊C₈	—	134	2	no	—
204	C₄⋊C₈	—	134	2	no	—
205	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
206	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
207	C₂²⋊C₈	—	2	1	no	—
208	C₂²⋊C₈	—	2	1	no	—
209	C₂²⋊C₈	—	2	1	no	—
210	#97	—	134	2	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	134	2	no	—
215	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
216	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
217	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
218	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
219	#41	—	2	2	no	—
220	#41	—	2	2	no	—
221	C₂×Q₈	C₄⋊Dic₆₇	1	1	yes	characteristic
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	C₄⋊Dic₆₇	1	1	yes	characteristic
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
228	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
229	C₂²×C₄	#11761628	1	1	yes	characteristic
230	M₁₆	—	268	2	no	—
231	M₁₆	—	536	2	no	—
232	M₁₆	—	134	4	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	268	2	no	—
240	C₂×C₈	—	268	2	no	—
241	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
242	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
243	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
244	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
248	C₄⋊C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
249	C₂²⋊C₄	—	2	1	no	—
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
256	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
257	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
258	C₄²	#11761628	1	1	yes	characteristic
259	C₂³	—	2	1	no	—
260	C₂³	#14524022	1	1	yes	characteristic
261	Q₈	—	2	1	no	—
262	Q₈	#14523979	1	2	yes	complemented
263	D₄	—	2	4	no	—
264	D₄	—	2	1	no	—
265	D₄	#14523979	1	2	yes	complemented
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	1	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	—	4	2	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
280	C₂×C₄	#14524002	1	1	yes	characteristic
281	C₂×C₄	#14524017	1	1	yes	characteristic
282	C₂×C₄	#14524017	1	1	yes	characteristic
283	C₂×C₄	#14524022	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	—	4	1	no	—
288	C₈	—	4	1	no	—
289	C₈	—	268	4	no	—
290	C₈	—	536	2	no	—
291	C₈	#14523970	1	2	yes	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	#16863256	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	C₂×#14523970	1	1	yes	characteristic
306	C₄	#16863413	1	1	yes	characteristic
307	C₄	C₂×#14523979	1	1	yes	characteristic
308	C₄	C₂×#14523979	1	1	yes	characteristic
309	C₄	#16863659	1	2	yes	—
310	C₂	—	4	1	no	—
311	C₂	—	4	2	no	—
312	C₂	#22560229	1	1	yes	characteristic
313	C₂	#22560247	1	1	yes	characteristic
314	C₂	#22560268	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	#8670
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. #22558965	C₂	semidirect
4. #22558100	C₂	semidirect
104. Q₈×C₆₇	#108	semidirect
107. D₄×C₆₇	#108	semidirect
163. C₆₇	#8670	semidirect
262. Q₈	#14523979	semidirect
265. 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
#22558965	2
#22558648	2
#22558100	2
C₈×#41×C₆₇	2
#8670	67