Subgroups — #34081754

Order 19328.

350 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752931	C₂	1	1	yes	maximal, characteristic, complemented
2	#23752921	C₂	1	1	yes	maximal, characteristic, complemented
3	#23752566	C₂	1	1	yes	maximal, characteristic, complemented
4	#23752565	C₂	1	1	yes	maximal, characteristic, complemented
5	#23752561	C₂	1	1	yes	maximal, characteristic, complemented
6	#23752521	C₂	1	1	yes	maximal, characteristic, complemented
7	#23752520	C₂	1	1	yes	maximal, characteristic, complemented
8	#23752500	C₂	1	1	yes	maximal, characteristic, complemented
9	#23752425	C₂	1	1	yes	maximal, characteristic
10	#23752415	C₂	1	1	yes	maximal, characteristic, complemented
11	#23752336	C₂	1	1	yes	maximal, characteristic, complemented
12	SD₁₆×Dic₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#14558992	C₂	1	1	yes	maximal, characteristic, complemented
14	D₈×Dic₁₅₁	C₂	1	1	yes	maximal, characteristic
15	#463×C₁₅₁	C₂	1	1	yes	maximal, characteristic
16	C₂×#14558992	C₂²	1	1	yes	characteristic
17	#18012738	C₂²	1	1	yes	characteristic
18	#18012737	C₂²	1	1	yes	characteristic
19	#18012732	—	2	1	no	—
20	#18012732	—	2	1	no	—
21	#18012731	C₂²	1	1	yes	characteristic
22	#18012730	C₂²	1	1	yes	characteristic
23	#18012726	C₂²	1	1	yes	characteristic
24	#18012718	—	2	1	no	—
25	#18012716	—	2	1	no	—
26	#18012716	—	2	1	no	—
27	#18012716	—	2	1	no	—
28	Q₈×Dic₁₅₁	C₂²	1	1	yes	characteristic
29	C₄×Dic₃₀₂	C₂²	1	1	yes	characteristic
30	C₄×Dic₃₀₂	C₂²	1	1	yes	characteristic
31	D₄×Dic₁₅₁	C₂²	1	1	yes	characteristic
32	D₄×Dic₁₅₁	C₂²	1	1	yes	characteristic
33	D₄×Dic₁₅₁	C₂²	1	1	yes	characteristic
34	#18012695	—	2	1	no	—
35	C₂×Dic₁₅₁⋊C₄	—	2	1	no	—
36	#18012673	C₂²	1	1	yes	characteristic
37	#18012671	C₂²	1	1	yes	characteristic
38	#18012670	C₂²	1	1	yes	characteristic
39	Dic₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
41	#18012664	C₂²	1	1	yes	characteristic
42	#18012664	C₂²	1	1	yes	characteristic
43	#18012663	C₂²	1	1	yes	characteristic
44	#18012663	C₂²	1	1	yes	characteristic
45	#18012662	C₂²	1	1	yes	characteristic
46	#18012660	C₂²	1	1	yes	characteristic
47	#18012660	C₂²	1	1	yes	characteristic
48	#18012660	C₂²	1	1	yes	characteristic
49	C₄×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₅₁	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₅₁	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
53	C₂×D₈×C₁₅₁	C₂²	1	1	yes	characteristic
54	#124×C₁₅₁	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₁₅₁	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₅₁	C₂²	1	1	yes	characteristic
57	#103×C₁₅₁	C₂²	1	1	yes	characteristic
58	#102×C₁₅₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₀₂	—	2	1	no	—
60	C₂×Dic₃₀₂	D₄	1	1	yes	characteristic
61	C₂×Dic₃₀₂	C₂³	1	1	yes	characteristic
62	C₂²×Dic₁₅₁	—	2	1	no	—
63	C₂²×Dic₁₅₁	—	2	1	no	—
64	C₂²×Dic₁₅₁	—	2	1	no	—
65	#14558992	—	2	2	no	—
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	D₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
82	Dic₁₅₁⋊C₄	D₄	1	1	yes	characteristic
83	C₂²⋊Dic₁₅₁	—	2	1	no	—
84	C₂²⋊Dic₁₅₁	—	2	1	no	—
85	C₂²⋊Dic₁₅₁	—	2	1	no	—
86	C₂²⋊Dic₁₅₁	—	2	1	no	—
87	C₂²⋊Dic₁₅₁	—	2	1	no	—
88	C₄×Dic₁₅₁	—	2	1	no	—
89	C₄×Dic₁₅₁	—	2	1	no	—
90	C₄×Dic₁₅₁	C₂³	1	1	yes	characteristic
91	C₄×Dic₁₅₁	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₁₅₁	C₂³	1	1	yes	characteristic
93	C₂×D₄×C₁₅₁	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₁₅₁	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₁₅₁	—	2	1	no	—
96	SD₁₆×C₁₅₁	—	2	2	no	—
97	D₈×C₁₅₁	—	2	2	no	—
98	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
101	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
102	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
103	C₂²⋊C₄×C₁₅₁	—	2	1	no	—
104	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
105	Dic₃₀₂	—	2	2	no	—
106	Dic₃₀₂	—	2	1	no	—
107	Dic₃₀₂	—	2	1	no	—
108	Dic₃₀₂	—	8	1	no	—
109	C₂×Dic₁₅₁	—	2	1	no	—
110	C₂×Dic₁₅₁	—	2	1	no	—
111	C₂×Dic₁₅₁	—	2	1	no	—
112	C₂×Dic₁₅₁	—	2	1	no	—
113	C₂×Dic₁₅₁	—	4	2	no	—
114	C₂×Dic₁₅₁	—	4	2	no	—
115	C₂×Dic₁₅₁	—	8	1	no	—
116	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
120	C₁₅₁⋊C₈	—	2	1	no	—
121	C₁₅₁⋊C₈	—	2	1	no	—
122	C₂³×C₁₅₁	—	2	1	no	—
123	C₂³×C₁₅₁	—	2	1	no	—
124	Q₈×C₁₅₁	—	2	1	no	—
125	Q₈×C₁₅₁	—	2	1	no	—
126	D₄×C₁₅₁	—	2	1	no	—
127	D₄×C₁₅₁	—	2	1	no	—
128	D₄×C₁₅₁	—	2	1	no	—
129	D₄×C₁₅₁	#41	1	2	yes	—
130	C₂×C₄×C₁₅₁	—	2	1	no	—
131	C₂×C₄×C₁₅₁	—	2	1	no	—
132	C₂×C₄×C₁₅₁	—	8	1	no	—
133	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
136	C₈×C₁₅₁	—	2	1	no	—
137	C₈×C₁₅₁	—	2	1	no	—
138	C₂²×C₁₅₁	—	2	2	no	—
139	C₂²×C₁₅₁	—	4	1	no	—
140	C₂²×C₁₅₁	—	4	1	no	—
141	C₂²×C₁₅₁	—	4	1	no	—
142	C₂²×C₁₅₁	—	4	1	no	—
143	C₂²×C₁₅₁	—	4	1	no	—
144	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
145	C₄×C₁₅₁	—	2	1	no	—
146	C₄×C₁₅₁	—	2	1	no	—
147	C₄×C₁₅₁	—	4	1	no	—
148	C₄×C₁₅₁	—	4	1	no	—
149	C₄×C₁₅₁	C₂×#41	1	1	yes	characteristic
150	C₄×C₁₅₁	2₋¹⁺⁴	1	1	yes	characteristic
151	Dic₁₅₁	—	2	1	no	—
152	Dic₁₅₁	—	2	1	no	—
153	Dic₁₅₁	—	2	1	no	—
154	Dic₁₅₁	—	4	1	no	—
155	Dic₁₅₁	—	4	1	no	—
156	Dic₁₅₁	—	4	1	no	—
157	Dic₁₅₁	—	4	1	no	—
158	Dic₁₅₁	#135	1	2	yes	—
159	C₂×C₁₅₁	—	4	2	no	—
160	C₂×C₁₅₁	—	8	1	no	—
161	C₂×C₁₅₁	#548	1	1	yes	characteristic
162	C₂×C₁₅₁	C₂×#135	1	1	yes	characteristic
163	C₂×C₁₅₁	#577	1	1	yes	characteristic
164	C₁₅₁	#3360	1	1	yes	characteristic, complemented, Sylow
165	#3360	—	151	1	no	maximal, Sylow
166	#556	—	151	1	no	—
167	#554	—	151	1	no	—
168	#488	—	151	1	no	—
169	#488	—	151	1	no	—
170	#487	—	151	1	no	—
171	#479	—	151	1	no	—
172	#479	—	151	1	no	—
173	#476	—	151	1	no	—
174	#463	—	151	1	no	—
175	#463	D₁₅₁	1	1	yes	characteristic
176	#461	—	151	1	no	—
177	C₈×Q₈	—	151	1	no	—
178	C₄×SD₁₆	—	151	1	no	—
179	C₄×SD₁₆	—	151	1	no	—
180	C₄×D₈	—	151	1	no	—
181	C₂×SD₁₆	—	151	1	no	—
182	C₂×SD₁₆	C₂×D₁₅₁	1	1	yes	characteristic
183	C₂×D₈	C₂×D₁₅₁	1	1	yes	characteristic
184	C₄⋊Q₈	—	151	1	no	—
185	C₄⋊Q₈	—	151	1	no	—
186	#126	—	302	1	no	—
187	#126	—	302	1	no	—
188	#125	—	151	1	no	—
189	#125	—	151	1	no	—
190	#124	—	151	1	no	—
191	#124	C₂×D₁₅₁	1	1	yes	characteristic
192	C₂²⋊Q₈	—	302	1	no	—
193	C₂²⋊Q₈	—	302	1	no	—
194	C₂²⋊Q₈	—	302	1	no	—
195	C₂²⋊Q₈	—	302	1	no	—
196	C₄×Q₈	—	151	1	no	—
197	C₄×Q₈	—	151	1	no	—
198	C₄×Q₈	—	151	1	no	—
199	C₄×D₄	—	151	1	no	—
200	C₄×D₄	—	151	1	no	—
201	C₄×D₄	—	151	1	no	—
202	C₄×D₄	C₂×D₁₅₁	1	1	yes	characteristic
203	#117	—	302	1	no	—
204	C₂×C₄⋊C₄	—	302	1	no	—
205	#107	—	151	1	no	—
206	#106	—	151	1	no	—
207	#106	—	151	1	no	—
208	C₄⋊C₈	—	151	1	no	—
209	C₄⋊C₈	—	151	1	no	—
210	C₄⋊C₈	C₂×D₁₅₁	1	1	yes	characteristic
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	#102	—	151	1	no	—
218	#102	—	151	1	no	—
219	#102	—	151	1	no	—
220	#102	C₂×D₁₅₁	1	1	yes	characteristic
221	C₄×C₈	—	151	1	no	—
222	C₄×C₈	—	151	1	no	—
223	C₄×C₈	—	151	1	no	—
224	C₂×Q₈	—	302	1	no	—
225	C₂×Q₈	—	151	1	no	—
226	C₂×Q₈	—	151	1	no	—
227	C₂×Q₈	C₂²×D₁₅₁	1	1	yes	characteristic
228	C₂×D₄	C₂²×D₁₅₁	1	1	yes	characteristic
229	C₂×D₄	C₂²×D₁₅₁	1	1	yes	characteristic
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	302	1	no	—
232	C₂²×C₄	—	302	1	no	—
233	C₂²×C₄	—	302	1	no	—
234	SD₁₆	—	2	2	no	—
235	SD₁₆	—	302	2	no	—
236	D₈	—	2	2	no	—
237	C₂×C₈	—	151	1	no	—
238	C₂×C₈	—	151	1	no	—
239	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
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	—	151	1	no	—
251	C₄⋊C₄	—	151	1	no	—
252	C₄⋊C₄	—	151	1	no	—
253	C₄⋊C₄	—	151	1	no	—
254	C₄⋊C₄	—	151	1	no	—
255	C₄⋊C₄	—	151	1	no	—
256	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	302	1	no	—
261	C₂²⋊C₄	—	302	1	no	—
262	C₂²⋊C₄	—	302	1	no	—
263	C₂²⋊C₄	—	302	1	no	—
264	C₂²⋊C₄	—	302	1	no	—
265	C₄²	—	302	1	no	—
266	C₄²	—	302	1	no	—
267	C₄²	—	151	1	no	—
268	C₄²	—	151	1	no	—
269	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
270	C₂³	—	2	1	no	—
271	C₂³	—	2	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	302	2	no	—
275	Q₈	—	302	1	no	—
276	Q₈	—	302	1	no	—
277	Q₈	—	1208	1	no	—
278	D₄	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	2	1	no	—
281	D₄	#14559003	1	2	yes	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	8	1	no	—
285	C₂×C₄	—	302	1	no	—
286	C₂×C₄	—	302	1	no	—
287	C₂×C₄	—	302	1	no	—
288	C₂×C₄	—	302	1	no	—
289	C₂×C₄	—	604	2	no	—
290	C₂×C₄	—	604	2	no	—
291	C₂×C₄	—	151	1	no	—
292	C₂×C₄	—	151	1	no	—
293	C₂×C₄	—	151	1	no	—
294	C₂×C₄	—	151	1	no	—
295	C₂×C₄	—	1208	1	no	—
296	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
297	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	302	1	no	—
302	C₈	—	302	1	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	C₂×D₄×D₁₅₁	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	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	—	604	1	no	—
320	C₄	—	604	1	no	—
321	C₄	—	151	2	no	—
322	C₄	C₂×#14559003	1	1	yes	characteristic
323	C₄	#18012786	1	1	yes	characteristic
324	C₂	—	4	2	no	—
325	C₂	—	8	1	no	—
326	C₂	#23752868	1	1	yes	characteristic
327	C₂	#135×D₁₅₁	1	1	yes	characteristic
328	C₂	#23753022	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₅₁
commutator, cosocle	C₂×C₄×C₁₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	#463×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂²×D₄
prime radical	C₂×D₈×C₁₅₁	C₂²
hypercenter	#463	D₁₅₁
hypocenter	C₁₅₁	#3360
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #463

Complements

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

N	H ≅ G/N	type
1. #23752931	C₂	semidirect
2. #23752921	C₂	semidirect
3. #23752566	C₂	semidirect
4. #23752565	C₂	semidirect
5. #23752561	C₂	semidirect
6. #23752521	C₂	semidirect
7. #23752520	C₂	semidirect
8. #23752500	C₂	semidirect
10. #23752415	C₂	semidirect
11. #23752336	C₂	semidirect
12. SD₁₆×Dic₁₅₁	C₂	semidirect
13. C₄×#14558992	C₂	semidirect
164. C₁₅₁	#3360	semidirect

Maximal subgroups

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

subgroup	index
#23752931	2
#23752921	2
#23752566	2
#23752565	2
#23752561	2
#23752521	2
#23752520	2
#23752500	2
#23752425	2
#23752415	2
#23752336	2
SD₁₆×Dic₁₅₁	2
C₄×#14558992	2
D₈×Dic₁₅₁	2
#463×C₁₅₁	2
#3360	151