Subgroups — #34079406

Order 19328.

358 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752822	C₂	1	1	yes	maximal, characteristic
2	#23752689	C₂	1	1	yes	maximal, characteristic
3	#23752553	C₂	1	1	yes	maximal, characteristic
4	#23752522	C₂	1	1	yes	maximal, characteristic, complemented
5	#23752518	C₂	1	1	yes	maximal, characteristic, complemented
6	#23752421	C₂	1	1	yes	maximal, characteristic, complemented
7	#23752421	C₂	1	1	yes	maximal, characteristic, complemented
8	#23752370	C₂	1	1	yes	maximal, characteristic
9	#23752367	C₂	1	1	yes	maximal, characteristic
10	#23752318	C₂	1	1	yes	maximal, characteristic, complemented
11	#23752316	C₂	1	1	yes	maximal, characteristic, complemented
12	#23752306	C₂	1	1	yes	maximal, characteristic, complemented
13	#23752304	C₂	1	1	yes	maximal, characteristic, complemented
14	#23752270	C₂	1	1	yes	maximal, characteristic
15	#485×C₁₅₁	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₃₀₂	C₂²	1	1	yes	characteristic
17	C₂×Dic₆₀₄	C₂²	1	1	yes	characteristic
18	C₂×Dic₆₀₄	C₂²	1	1	yes	characteristic
19	C₂×#14558994	C₂²	1	1	yes	characteristic
20	C₂×#14558994	C₂²	1	1	yes	characteristic
21	#18012737	C₂²	1	1	yes	characteristic
22	#18012732	—	2	1	no	—
23	#18012732	—	2	1	no	—
24	#18012730	—	2	1	no	—
25	#18012730	C₂²	1	1	yes	characteristic
26	#18012729	—	2	1	no	—
27	#18012718	C₂²	1	1	yes	characteristic
28	Q₈×Dic₁₅₁	C₂²	1	1	yes	characteristic
29	C₄×Dic₃₀₂	—	2	1	no	—
30	C₄×Dic₃₀₂	—	2	1	no	—
31	#18012704	C₂²	1	1	yes	characteristic
32	#18012704	C₂²	1	1	yes	characteristic
33	#18012704	C₂²	1	1	yes	characteristic
34	#18012704	C₂²	1	1	yes	characteristic
35	#18012704	C₂²	1	1	yes	characteristic
36	#18012698	C₂²	1	1	yes	characteristic
37	#18012695	—	2	1	no	—
38	#18012695	—	2	1	no	—
39	#18012672	C₂²	1	1	yes	characteristic
40	#18012670	C₂²	1	1	yes	characteristic
41	Dic₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
43	#18012664	C₂²	1	1	yes	characteristic
44	#18012664	C₂²	1	1	yes	characteristic
45	#18012662	C₂²	1	1	yes	characteristic
46	#18012662	C₂²	1	1	yes	characteristic
47	#18012662	C₂²	1	1	yes	characteristic
48	#18012662	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₁₅₁⋊C₈)	C₂²	1	1	yes	characteristic
50	#18012649	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₅₁	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
53	#117×C₁₅₁	C₂²	1	1	yes	characteristic
54	#107×C₁₅₁	C₂²	1	1	yes	characteristic
55	#106×C₁₅₁	C₂²	1	1	yes	characteristic
56	#103×C₁₅₁	C₂²	1	1	yes	characteristic
57	#103×C₁₅₁	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₅₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₀₂	—	2	2	no	—
60	C₂×Dic₃₀₂	—	2	2	no	—
61	C₂×Dic₃₀₂	—	2	1	no	—
62	C₂×Dic₃₀₂	C₂³	1	1	yes	characteristic
63	C₂×Dic₃₀₂	C₂³	1	1	yes	characteristic
64	C₂×Dic₃₀₂	C₂³	1	1	yes	characteristic
65	C₂²×Dic₁₅₁	—	2	1	no	—
66	Dic₆₀₄	—	2	2	no	—
67	Dic₆₀₄	—	2	2	no	—
68	#14558994	—	2	2	no	—
69	#14558994	—	2	2	no	—
70	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₁₅₁	—	2	1	no	—
73	C₄⋊Dic₁₅₁	—	2	1	no	—
74	C₄⋊Dic₁₅₁	D₄	1	1	yes	characteristic
75	C₄⋊Dic₁₅₁	D₄	1	1	yes	characteristic
76	C₄⋊Dic₁₅₁	C₂³	1	1	yes	characteristic
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	Dic₁₅₁⋊C₄	—	2	1	no	—
86	C₂²⋊Dic₁₅₁	—	2	1	no	—
87	C₂²⋊Dic₁₅₁	D₄	1	1	yes	characteristic
88	C₂²⋊Dic₁₅₁	D₄	1	1	yes	characteristic
89	C₄×Dic₁₅₁	—	2	1	no	—
90	C₄×Dic₁₅₁	—	2	1	no	—
91	C₄×Dic₁₅₁	—	2	1	no	—
92	C₄×Dic₁₅₁	—	2	1	no	—
93	C₄×Dic₁₅₁	—	2	1	no	—
94	C₄×Dic₁₅₁	—	2	1	no	—
95	C₄×Dic₁₅₁	C₂³	1	1	yes	characteristic
96	C₄×Dic₁₅₁	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₅₁	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₅₁	—	2	1	no	—
102	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
106	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
107	C₄²×C₁₅₁	—	2	1	no	—
108	Dic₃₀₂	—	2	4	no	—
109	Dic₃₀₂	—	2	1	no	—
110	Dic₃₀₂	—	2	1	no	—
111	Dic₃₀₂	—	2	1	no	—
112	Dic₃₀₂	—	2	1	no	—
113	Dic₃₀₂	#41	1	2	yes	—
114	Dic₃₀₂	#41	1	2	yes	—
115	C₂×Dic₁₅₁	—	2	4	no	—
116	C₂×Dic₁₅₁	—	2	1	no	—
117	C₂×Dic₁₅₁	—	2	1	no	—
118	C₂×Dic₁₅₁	—	2	1	no	—
119	C₂×Dic₁₅₁	—	2	1	no	—
120	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
124	C₁₅₁⋊C₈	—	2	1	no	—
125	C₁₅₁⋊C₈	—	2	1	no	—
126	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
127	Q₈×C₁₅₁	—	2	1	no	—
128	Q₈×C₁₅₁	—	2	1	no	—
129	C₂×C₄×C₁₅₁	—	2	1	no	—
130	C₂×C₄×C₁₅₁	—	2	1	no	—
131	C₂×C₄×C₁₅₁	—	2	1	no	—
132	C₂×C₄×C₁₅₁	—	2	1	no	—
133	C₂×C₄×C₁₅₁	—	2	1	no	—
134	C₂×C₄×C₁₅₁	—	2	1	no	—
135	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
137	C₈×C₁₅₁	—	2	1	no	—
138	C₈×C₁₅₁	—	2	1	no	—
139	C₂²×C₁₅₁	—	2	1	no	—
140	C₂²×C₁₅₁	—	2	1	no	—
141	C₂²×C₁₅₁	—	2	1	no	—
142	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₁₅₁	—	2	1	no	—
144	C₄×C₁₅₁	—	4	1	no	—
145	C₄×C₁₅₁	—	4	1	no	—
146	C₄×C₁₅₁	—	4	1	no	—
147	C₄×C₁₅₁	—	4	1	no	—
148	C₄×C₁₅₁	C₂×#41	1	1	yes	characteristic
149	C₄×C₁₅₁	C₂×#41	1	1	yes	characteristic
150	Dic₁₅₁	—	2	2	no	—
151	Dic₁₅₁	—	2	2	no	—
152	Dic₁₅₁	—	2	1	no	—
153	Dic₁₅₁	—	2	1	no	—
154	Dic₁₅₁	—	2	1	no	—
155	Dic₁₅₁	—	2	1	no	—
156	Dic₁₅₁	—	4	1	no	—
157	Dic₁₅₁	—	4	1	no	—
158	C₂×C₁₅₁	—	4	1	no	—
159	C₂×C₁₅₁	#525	1	1	yes	characteristic
160	C₂×C₁₅₁	#578	1	1	yes	characteristic
161	C₂×C₁₅₁	#578	1	1	yes	characteristic
162	C₁₅₁	#3127	1	1	yes	characteristic, complemented, Sylow
163	#3127	—	151	1	no	maximal, Sylow
164	#541	—	151	1	no	—
165	#519	—	151	1	no	—
166	#485	—	151	1	no	—
167	#485	D₁₅₁	1	1	yes	characteristic
168	#479	—	151	1	no	—
169	#479	—	151	1	no	—
170	#462	—	151	1	no	—
171	#462	—	151	1	no	—
172	#451	—	151	1	no	—
173	#451	—	151	1	no	—
174	#441	—	151	1	no	—
175	#441	—	151	1	no	—
176	C₄×Q₁₆	—	151	1	no	—
177	C₄×Q₁₆	—	151	1	no	—
178	#433	—	151	1	no	—
179	C₂²×Q₈	—	151	1	no	—
180	C₂×Q₁₆	—	151	1	no	—
181	C₂×Q₁₆	—	151	1	no	—
182	C₂×Q₁₆	—	151	1	no	—
183	C₂×Q₁₆	—	151	1	no	—
184	C₄⋊Q₈	—	151	1	no	—
185	#126	—	302	1	no	—
186	#126	—	302	1	no	—
187	#125	—	302	1	no	—
188	#125	—	302	1	no	—
189	#125	—	151	1	no	—
190	C₂²⋊Q₈	—	151	1	no	—
191	C₂²⋊Q₈	C₂×D₁₅₁	1	1	yes	characteristic
192	C₄×Q₈	—	302	1	no	—
193	C₄×Q₈	—	302	1	no	—
194	C₄×Q₈	—	151	1	no	—
195	C₄×Q₈	—	151	1	no	—
196	C₄×Q₈	—	151	1	no	—
197	C₄×Q₈	—	151	1	no	—
198	C₄×Q₈	—	151	1	no	—
199	C₄×Q₈	—	151	1	no	—
200	#117	—	302	1	no	—
201	#117	—	302	1	no	—
202	#117	—	151	1	no	—
203	#117	C₂×D₁₅₁	1	1	yes	characteristic
204	#107	—	151	1	no	—
205	#107	C₂×D₁₅₁	1	1	yes	characteristic
206	#106	—	151	1	no	—
207	#106	C₂×D₁₅₁	1	1	yes	characteristic
208	C₄⋊C₈	—	151	1	no	—
209	C₄⋊C₈	—	151	1	no	—
210	#103	—	151	1	no	—
211	#103	—	151	1	no	—
212	#103	—	151	1	no	—
213	#103	—	151	1	no	—
214	#103	—	151	1	no	—
215	#103	—	151	1	no	—
216	#103	C₂×D₁₅₁	1	1	yes	characteristic
217	#103	C₂×D₁₅₁	1	1	yes	characteristic
218	C₂²⋊C₈	—	151	1	no	—
219	C₂²⋊C₈	C₂×D₁₅₁	1	1	yes	characteristic
220	#97	—	151	1	no	—
221	C₄×C₈	—	151	1	no	—
222	C₂×Q₈	—	302	2	no	—
223	C₂×Q₈	—	302	2	no	—
224	C₂×Q₈	—	302	1	no	—
225	C₂×Q₈	—	151	1	no	—
226	C₂×Q₈	—	151	1	no	—
227	C₂×Q₈	—	151	1	no	—
228	C₂×Q₈	C₂²×D₁₅₁	1	1	yes	characteristic
229	C₂²×C₄	—	302	1	no	—
230	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
231	Q₁₆	—	302	2	no	—
232	Q₁₆	—	302	2	no	—
233	Q₁₆	—	302	2	no	—
234	Q₁₆	—	302	2	no	—
235	C₂×C₈	—	151	1	no	—
236	C₂×C₈	—	151	1	no	—
237	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
239	C₄⋊C₄	—	2	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	—	302	1	no	—
245	C₄⋊C₄	—	302	1	no	—
246	C₄⋊C₄	—	302	1	no	—
247	C₄⋊C₄	—	302	1	no	—
248	C₄⋊C₄	—	302	1	no	—
249	C₄⋊C₄	—	302	1	no	—
250	C₄⋊C₄	—	302	1	no	—
251	C₄⋊C₄	—	151	1	no	—
252	C₄⋊C₄	—	151	1	no	—
253	C₄⋊C₄	—	151	1	no	—
254	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	302	1	no	—
260	C₂²⋊C₄	—	151	1	no	—
261	C₂²⋊C₄	—	151	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	302	1	no	—
264	C₄²	—	302	1	no	—
265	C₄²	—	302	1	no	—
266	C₄²	—	302	1	no	—
267	C₄²	—	302	1	no	—
268	C₄²	—	302	1	no	—
269	C₄²	—	151	1	no	—
270	C₄²	—	151	1	no	—
271	C₂³	D₄×D₁₅₁	1	1	yes	characteristic
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	302	4	no	—
275	Q₈	—	302	1	no	—
276	Q₈	—	302	1	no	—
277	Q₈	—	302	1	no	—
278	Q₈	—	302	1	no	—
279	Q₈	—	151	2	no	—
280	Q₈	—	151	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	302	4	no	—
288	C₂×C₄	—	302	1	no	—
289	C₂×C₄	—	302	1	no	—
290	C₂×C₄	—	302	1	no	—
291	C₂×C₄	—	302	1	no	—
292	C₂×C₄	—	151	1	no	—
293	C₂×C₄	—	151	1	no	—
294	C₂×C₄	—	151	1	no	—
295	C₂×C₄	—	151	1	no	—
296	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	302	1	no	—
301	C₈	—	302	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	C₂×D₄×D₁₅₁	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	302	2	no	—
312	C₄	—	302	2	no	—
313	C₄	—	302	1	no	—
314	C₄	—	302	1	no	—
315	C₄	—	302	1	no	—
316	C₄	—	302	1	no	—
317	C₄	—	604	1	no	—
318	C₄	—	604	1	no	—
319	C₄	#41×D₁₅₁	1	1	yes	characteristic
320	C₄	#41×D₁₅₁	1	1	yes	characteristic
321	C₂	—	4	1	no	—
322	C₂	#23752720	1	1	yes	characteristic
323	C₂	#23753026	1	1	yes	characteristic
324	C₂	#23753028	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₅₁
commutator, cosocle	C₂×C₄×C₁₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	#485×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂²×D₄
prime radical	C₂³×C₁₅₁	C₂×D₄
hypercenter	#485	D₁₅₁
hypocenter	C₁₅₁	#3127
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
4. #23752522	C₂	semidirect
5. #23752518	C₂	semidirect
6. #23752421	C₂	semidirect
7. #23752421	C₂	semidirect
10. #23752318	C₂	semidirect
11. #23752316	C₂	semidirect
12. #23752306	C₂	semidirect
13. #23752304	C₂	semidirect
162. C₁₅₁	#3127	semidirect

Maximal subgroups

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

subgroup	index
#23752822	2
#23752689	2
#23752553	2
#23752522	2
#23752518	2
#23752421	2
#23752370	2
#23752367	2
#23752318	2
#23752316	2
#23752306	2
#23752304	2
#23752270	2
#485×C₁₅₁	2
#3127	151