Subgroups — #34080314

Order 19328.

324 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752766	C₂	1	1	yes	maximal, characteristic
2	#125×D₁₅₁	C₂	1	1	yes	maximal, characteristic
3	#23752620	C₂	1	1	yes	maximal, characteristic, complemented
4	#23752619	C₂	1	1	yes	maximal, characteristic, complemented
5	#23752579	C₂	1	1	yes	maximal, characteristic, complemented
6	#23752579	C₂	1	1	yes	maximal, characteristic, complemented
7	#23752549	C₂	1	1	yes	maximal, characteristic
8	#23752546	C₂	1	1	yes	maximal, characteristic
9	#23752544	C₂	1	1	yes	maximal, characteristic
10	#23752541	C₂	1	1	yes	maximal, characteristic
11	#23752520	C₂	1	1	yes	maximal, characteristic, complemented
12	#23752519	C₂	1	1	yes	maximal, characteristic, complemented
13	#23752516	C₂	1	1	yes	maximal, characteristic, complemented
14	#23752268	C₂	1	1	yes	maximal, characteristic
15	#479×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
16	#18012738	C₂²	1	1	yes	characteristic
17	#18012737	C₂²	1	1	yes	characteristic
18	#18012731	C₂²	1	1	yes	characteristic
19	#18012730	C₂²	1	1	yes	characteristic
20	#18012730	C₂²	1	1	yes	characteristic
21	#18012729	—	2	1	no	—
22	#18012729	—	2	1	no	—
23	#18012729	—	2	1	no	—
24	#18012719	C₂²	1	1	yes	characteristic
25	#18012717	—	2	1	no	—
26	#18012715	C₂²	1	1	yes	characteristic
27	C₄×Dic₃₀₂	—	2	1	no	—
28	C₄×Dic₃₀₂	C₂²	1	1	yes	characteristic
29	#18012696	—	2	1	no	—
30	C₄⋊C₄×D₁₅₁	—	2	1	no	—
31	C₄⋊C₄×D₁₅₁	—	2	1	no	—
32	C₄⋊C₄×D₁₅₁	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₁₅₁	C₂²	1	1	yes	characteristic
34	C₄²×D₁₅₁	C₂²	1	1	yes	characteristic
35	#18012673	C₂²	1	1	yes	characteristic
36	#18012672	C₂²	1	1	yes	characteristic
37	#18012672	C₂²	1	1	yes	characteristic
38	#18012671	C₂²	1	1	yes	characteristic
39	#18012670	C₂²	1	1	yes	characteristic
40	#18012670	C₂²	1	1	yes	characteristic
41	C₄⋊(C₁₅₁⋊C₈)	C₂²	1	1	yes	characteristic
42	#18012664	C₂²	1	1	yes	characteristic
43	#18012664	C₂²	1	1	yes	characteristic
44	#18012663	C₂²	1	1	yes	characteristic
45	#18012663	C₂²	1	1	yes	characteristic
46	#18012662	C₂²	1	1	yes	characteristic
47	#18012662	C₂²	1	1	yes	characteristic
48	(C₂×D₁₅₁)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₁₅₁)⋊C₈	C₂²	1	1	yes	characteristic
50	#18012649	C₂²	1	1	yes	characteristic
51	#18012649	C₂²	1	1	yes	characteristic
52	#125×C₁₅₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
54	#107×C₁₅₁	C₂²	1	1	yes	characteristic
55	#106×C₁₅₁	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₅₁	C₂²	1	1	yes	characteristic
57	#103×C₁₅₁	C₂²	1	1	yes	characteristic
58	#103×C₁₅₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₀₂	—	2	1	no	—
60	C₂×Dic₃₀₂	C₂³	1	1	yes	characteristic
61	C₂×C₄×D₁₅₁	—	2	1	no	—
62	C₂×C₄×D₁₅₁	—	2	1	no	—
63	C₂×C₄×D₁₅₁	D₄	1	1	yes	characteristic
64	C₂×C₄×D₁₅₁	D₄	1	1	yes	characteristic
65	C₂×C₄×D₁₅₁	C₂³	1	1	yes	characteristic
66	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₁₅₁	—	2	1	no	—
69	C₄⋊Dic₁₅₁	—	2	1	no	—
70	C₄⋊Dic₁₅₁	—	2	1	no	—
71	C₄⋊Dic₁₅₁	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₁₅₁	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₅₁	C₂³	1	1	yes	characteristic
74	Dic₁₅₁⋊C₄	—	2	1	no	—
75	Dic₁₅₁⋊C₄	—	2	1	no	—
76	Dic₁₅₁⋊C₄	—	2	1	no	—
77	Dic₁₅₁⋊C₄	—	2	1	no	—
78	Dic₁₅₁⋊C₄	—	2	1	no	—
79	Dic₁₅₁⋊C₄	—	2	1	no	—
80	Dic₁₅₁⋊C₄	—	2	1	no	—
81	Dic₁₅₁⋊C₄	—	2	1	no	—
82	Dic₁₅₁⋊C₄	—	2	1	no	—
83	Dic₁₅₁⋊C₄	—	2	1	no	—
84	Dic₁₅₁⋊C₄	—	2	1	no	—
85	(C₂×D₁₅₁)⋊C₄	—	2	1	no	—
86	(C₂×D₁₅₁)⋊C₄	—	2	1	no	—
87	C₄×Dic₁₅₁	—	2	1	no	—
88	C₄×Dic₁₅₁	D₄	1	1	yes	characteristic
89	C₄×Dic₁₅₁	D₄	1	1	yes	characteristic
90	C₄×Dic₁₅₁	C₂³	1	1	yes	characteristic
91	C₂×Q₈×C₁₅₁	C₂³	1	1	yes	characteristic
92	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
93	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
94	C₄⋊C₄×C₁₅₁	—	2	1	no	—
95	C₄⋊C₄×C₁₅₁	—	2	1	no	—
96	C₄⋊C₄×C₁₅₁	—	2	1	no	—
97	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
100	C₄²×C₁₅₁	—	2	1	no	—
101	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
102	C₂²×D₁₅₁	C₂×D₄	1	1	yes	characteristic
103	Dic₃₀₂	—	2	1	no	—
104	Dic₃₀₂	—	2	1	no	—
105	Dic₃₀₂	—	4	2	no	—
106	C₂×Dic₁₅₁	—	2	1	no	—
107	C₂×Dic₁₅₁	—	2	1	no	—
108	C₂×Dic₁₅₁	—	2	1	no	—
109	C₂×Dic₁₅₁	—	2	1	no	—
110	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
111	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
112	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
113	C₄×D₁₅₁	—	2	2	no	—
114	C₄×D₁₅₁	—	2	1	no	—
115	C₄×D₁₅₁	—	2	1	no	—
116	C₄×D₁₅₁	—	4	1	no	—
117	C₄×D₁₅₁	—	4	2	no	—
118	C₄×D₁₅₁	—	4	2	no	—
119	C₁₅₁⋊C₈	—	2	1	no	—
120	C₁₅₁⋊C₈	—	2	1	no	—
121	Q₈×C₁₅₁	—	2	1	no	—
122	Q₈×C₁₅₁	—	2	1	no	—
123	C₂×C₄×C₁₅₁	—	2	1	no	—
124	C₂×C₄×C₁₅₁	—	2	1	no	—
125	C₂×C₄×C₁₅₁	—	2	1	no	—
126	C₂×C₄×C₁₅₁	—	2	1	no	—
127	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
130	C₈×C₁₅₁	—	2	1	no	—
131	C₈×C₁₅₁	—	2	1	no	—
132	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
133	C₂×D₁₅₁	—	2	1	no	—
134	C₂×D₁₅₁	—	2	1	no	—
135	C₂×D₁₅₁	#137	1	2	yes	—
136	C₄×C₁₅₁	—	2	1	no	—
137	C₄×C₁₅₁	—	2	1	no	—
138	C₄×C₁₅₁	—	4	1	no	—
139	C₄×C₁₅₁	—	4	1	no	—
140	C₄×C₁₅₁	—	4	1	no	—
141	C₄×C₁₅₁	—	4	1	no	—
142	C₄×C₁₅₁	2₋¹⁺⁴	1	1	yes	characteristic
143	C₄×C₁₅₁	2₋¹⁺⁴	1	1	yes	characteristic
144	Dic₁₅₁	—	2	1	no	—
145	Dic₁₅₁	—	2	1	no	—
146	Dic₁₅₁	—	2	1	no	—
147	Dic₁₅₁	—	4	1	no	—
148	Dic₁₅₁	—	4	1	no	—
149	Dic₁₅₁	—	4	1	no	—
150	Dic₁₅₁	—	4	1	no	—
151	C₂×C₁₅₁	#536	1	1	yes	characteristic
152	C₂×C₁₅₁	C₂×#137	1	1	yes	characteristic
153	C₂×C₁₅₁	#575	1	1	yes	characteristic
154	D₁₅₁	—	2	2	no	—
155	C₁₅₁	#3236	1	1	yes	characteristic, complemented, Sylow
156	#3236	—	151	1	no	maximal, Sylow
157	#533	—	151	1	no	—
158	C₂×#125	—	151	1	no	—
159	#501	—	151	1	no	—
160	#501	—	151	1	no	—
161	#491	—	151	1	no	—
162	#491	—	151	1	no	—
163	#484	—	151	1	no	—
164	#484	—	151	1	no	—
165	#483	—	151	1	no	—
166	#483	—	151	1	no	—
167	#479	—	151	1	no	—
168	#479	—	151	1	no	—
169	#479	—	151	1	no	—
170	#479	D₁₅₁	1	1	yes	characteristic, complemented
171	#432	—	151	1	no	—
172	C₄⋊Q₈	—	151	1	no	—
173	C₄⋊Q₈	—	151	1	no	—
174	#125	—	302	1	no	—
175	#125	—	302	1	no	—
176	#125	—	302	1	no	—
177	#125	—	151	1	no	—
178	#125	—	151	1	no	—
179	#125	—	151	1	no	—
180	#125	C₂×D₁₅₁	1	1	yes	characteristic
181	C₂²⋊Q₈	—	302	1	no	—
182	C₂²⋊Q₈	—	151	1	no	—
183	C₂²⋊Q₈	—	151	1	no	—
184	C₄×Q₈	—	302	1	no	—
185	C₄×Q₈	—	151	1	no	—
186	C₄×Q₈	C₂×D₁₅₁	1	1	yes	characteristic
187	#117	—	302	1	no	—
188	C₂×C₄⋊C₄	—	302	1	no	—
189	C₂×C₄⋊C₄	—	302	1	no	—
190	C₂×C₄⋊C₄	—	151	1	no	—
191	C₂×C₄⋊C₄	—	151	1	no	—
192	C₂×C₄²	—	151	1	no	—
193	#107	—	151	1	no	—
194	#107	—	151	1	no	—
195	#107	—	151	1	no	—
196	#107	C₂×D₁₅₁	1	1	yes	characteristic
197	#106	—	151	1	no	—
198	#106	—	151	1	no	—
199	#106	—	151	1	no	—
200	#106	C₂×D₁₅₁	1	1	yes	characteristic
201	C₄⋊C₈	—	151	1	no	—
202	C₄⋊C₈	C₂×D₁₅₁	1	1	yes	characteristic
203	#103	—	151	1	no	—
204	#103	—	151	1	no	—
205	#103	—	151	1	no	—
206	#103	—	151	1	no	—
207	#103	—	151	1	no	—
208	#103	—	151	1	no	—
209	#103	C₂×D₁₅₁	1	1	yes	characteristic
210	#103	C₂×D₁₅₁	1	1	yes	characteristic
211	C₂²⋊C₈	—	151	1	no	—
212	C₂²⋊C₈	—	151	1	no	—
213	#97	—	151	1	no	—
214	#97	—	151	1	no	—
215	C₂×Q₈	—	302	1	no	—
216	C₂×Q₈	—	151	1	no	—
217	C₂×Q₈	C₂²×D₁₅₁	1	1	yes	characteristic
218	C₂²×C₄	—	302	1	no	—
219	C₂²×C₄	—	302	1	no	—
220	C₂²×C₄	—	151	1	no	—
221	C₂²×C₄	—	151	1	no	—
222	C₂²×C₄	—	151	1	no	—
223	C₂×C₈	—	151	1	no	—
224	C₂×C₈	—	151	1	no	—
225	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
226	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	302	1	no	—
231	C₄⋊C₄	—	302	1	no	—
232	C₄⋊C₄	—	302	1	no	—
233	C₄⋊C₄	—	302	1	no	—
234	C₄⋊C₄	—	302	1	no	—
235	C₄⋊C₄	—	302	1	no	—
236	C₄⋊C₄	—	302	1	no	—
237	C₄⋊C₄	—	302	1	no	—
238	C₄⋊C₄	—	302	1	no	—
239	C₄⋊C₄	—	302	1	no	—
240	C₄⋊C₄	—	302	1	no	—
241	C₄⋊C₄	—	302	1	no	—
242	C₄⋊C₄	—	302	1	no	—
243	C₄⋊C₄	—	302	1	no	—
244	C₄⋊C₄	—	151	1	no	—
245	C₄⋊C₄	—	151	1	no	—
246	C₄⋊C₄	—	151	1	no	—
247	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
248	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
250	C₂²⋊C₄	—	302	1	no	—
251	C₂²⋊C₄	—	302	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	302	1	no	—
254	C₄²	—	151	1	no	—
255	C₄²	—	151	1	no	—
256	C₄²	—	151	1	no	—
257	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
258	C₂³	—	151	1	no	—
259	Q₈	—	2	1	no	—
260	Q₈	—	2	1	no	—
261	Q₈	—	302	1	no	—
262	Q₈	—	302	1	no	—
263	Q₈	—	604	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	302	2	no	—
269	C₂×C₄	—	302	1	no	—
270	C₂×C₄	—	302	1	no	—
271	C₂×C₄	—	302	1	no	—
272	C₂×C₄	—	302	1	no	—
273	C₂×C₄	—	302	1	no	—
274	C₂×C₄	—	302	1	no	—
275	C₂×C₄	—	604	1	no	—
276	C₂×C₄	—	604	2	no	—
277	C₂×C₄	—	604	2	no	—
278	C₂×C₄	—	151	1	no	—
279	C₂×C₄	—	151	1	no	—
280	C₂×C₄	—	151	1	no	—
281	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
282	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
283	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	302	1	no	—
287	C₈	—	302	1	no	—
288	C₂²	—	302	1	no	—
289	C₂²	—	302	1	no	—
290	C₂²	—	151	2	no	—
291	C₂²	C₂×D₄×D₁₅₁	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	—	4	1	no	—
298	C₄	—	302	1	no	—
299	C₄	—	302	1	no	—
300	C₄	—	302	1	no	—
301	C₄	—	604	1	no	—
302	C₄	—	604	1	no	—
303	C₄	—	604	1	no	—
304	C₄	—	604	1	no	—
305	C₄	#18012786	1	1	yes	characteristic
306	C₄	#18012787	1	1	yes	characteristic
307	C₂	—	302	2	no	—
308	C₂	#23752785	1	1	yes	characteristic
309	C₂	#137×D₁₅₁	1	1	yes	characteristic
310	C₂	#23753009	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₅₁
commutator, cosocle	C₂×C₄×C₁₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	#479×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂²×D₄
prime radical	C₂²×D₁₅₁	C₂×D₄
hypercenter	#479	D₁₅₁
hypocenter	C₁₅₁	#3236
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
3. #23752620	C₂	semidirect
4. #23752619	C₂	semidirect
5. #23752579	C₂	semidirect
6. #23752579	C₂	semidirect
11. #23752520	C₂	semidirect
12. #23752519	C₂	semidirect
13. #23752516	C₂	semidirect
15. #479×C₁₅₁	C₂	semidirect
155. C₁₅₁	#3236	semidirect
170. #479	D₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#23752766	2
#125×D₁₅₁	2
#23752620	2
#23752619	2
#23752579	2
#23752549	2
#23752546	2
#23752544	2
#23752541	2
#23752520	2
#23752519	2
#23752516	2
#23752268	2
#479×C₁₅₁	2
#3236	151