Subgroups — #34082407

Order 19328.

1212 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752767	C₂	1	2	yes	maximal, complemented
2	#23752763	C₂	1	2	yes	maximal, complemented
3	#23752762	C₂	1	1	yes	maximal, characteristic, complemented
4	#23752749	C₂	1	4	yes	maximal, complemented
5	#23752748	C₂	1	4	yes	maximal, complemented
6	#23752747	C₂	1	4	yes	maximal, complemented
7	#23752725	C₂	1	4	yes	maximal, complemented
8	#23752724	C₂	1	2	yes	maximal, complemented
9	C₄×#14559005	C₂	1	4	yes	maximal, complemented
10	#23752680	C₂	1	2	yes	maximal, complemented
11	C₂×C₄²×D₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#533×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#14559005	C₂²	1	4	yes	—
14	C₂×#14559004	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₅₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₅₁	C₂²	1	2	yes	—
17	#18012739	C₂²	1	1	yes	characteristic, complemented
18	#18012736	C₂²	1	1	yes	characteristic, complemented
19	#18012734	C₂²	1	8	yes	complemented
20	#18012730	C₂²	1	4	yes	complemented
21	#18012728	C₂²	1	4	yes	complemented
22	#18012727	C₂²	1	2	yes	complemented
23	#18012724	C₂²	1	8	yes	—
24	#18012723	C₂²	1	4	yes	—
25	#18012719	C₂²	1	4	yes	—
26	#18012715	C₂²	1	4	yes	—
27	#18012714	C₂²	1	4	yes	—
28	#18012714	C₂²	1	4	yes	—
29	#18012713	C₂²	1	4	yes	—
30	#18012709	C₂²	1	4	yes	—
31	Q₈×Dic₁₅₁	C₂²	1	4	yes	complemented
32	#18012704	C₂²	1	4	yes	complemented
33	C₄×D₆₀₄	C₂²	1	8	yes	—
34	C₄×D₆₀₄	C₂²	1	4	yes	complemented
35	#18012701	C₂²	1	8	yes	—
36	#18012701	C₂²	1	4	yes	complemented
37	#18012701	C₂²	1	4	yes	complemented
38	(C₂²⋊D₁₅₁)⋊C₄	C₂²	1	8	yes	—
39	#18012697	C₂²	1	4	yes	—
40	#18012697	C₂²	1	8	yes	complemented
41	#18012696	C₂²	1	4	yes	complemented
42	C₂×C₄×Dic₁₅₁	C₂²	1	2	yes	—
43	C₂×C₄×Dic₁₅₁	C₂²	1	1	yes	characteristic
44	C₄²×D₁₅₁	C₂²	1	2	yes	—
45	C₄²×D₁₅₁	C₂²	1	2	yes	—
46	C₄²×D₁₅₁	C₂²	1	2	yes	—
47	C₄²×D₁₅₁	C₂²	1	2	yes	—
48	C₄²×D₁₅₁	C₂²	1	4	yes	complemented
49	C₄⋊Q₈×C₁₅₁	C₂²	1	2	yes	complemented
50	#125×C₁₅₁	C₂²	1	2	yes	complemented
51	C₂²⋊Q₈×C₁₅₁	C₂²	1	4	yes	—
52	C₄×Q₈×C₁₅₁	C₂²	1	4	yes	complemented
53	#117×C₁₅₁	C₂²	1	2	yes	—
54	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
55	C₂³×D₁₅₁	C₂³	1	1	yes	characteristic
56	#14559005	—	2	16	no	—
57	#14559004	—	2	8	no	—
58	C₂×Dic₃₀₂	C₂³	1	2	yes	—
59	C₂×D₆₀₄	C₂³	1	8	yes	—
60	C₂×D₆₀₄	C₂³	1	2	yes	—
61	C₂×D₆₀₄	C₂³	1	4	yes	—
62	C₂×C₂²⋊D₁₅₁	C₂³	1	4	yes	—
63	C₂²×Dic₁₅₁	C₂³	1	2	yes	—
64	C₂²×Dic₁₅₁	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₅₁	—	2	8	no	—
66	C₂×C₄×D₁₅₁	—	2	4	no	—
67	C₂×C₄×D₁₅₁	C₂³	1	2	yes	—
68	C₂×C₄×D₁₅₁	C₂³	1	2	yes	—
69	C₂×C₄×D₁₅₁	C₂³	1	2	yes	—
70	C₂×C₄×D₁₅₁	C₂³	1	2	yes	—
71	C₂×C₄×D₁₅₁	C₂³	1	2	yes	—
72	C₂×C₄×D₁₅₁	C₂³	1	4	yes	—
73	C₂×C₄×D₁₅₁	C₂³	1	8	yes	—
74	C₂×C₄×D₁₅₁	C₂³	1	2	yes	—
75	C₄⋊Dic₁₅₁	C₂³	1	4	yes	—
76	C₄⋊Dic₁₅₁	C₂³	1	8	yes	—
77	Dic₁₅₁⋊C₄	C₂³	1	8	yes	—
78	(C₂×D₁₅₁)⋊C₄	C₂³	1	8	yes	—
79	(C₂×D₁₅₁)⋊C₄	C₂³	1	8	yes	—
80	(C₂×D₁₅₁)⋊C₄	C₂³	1	8	yes	—
81	(C₂×D₁₅₁)⋊C₄	C₂³	1	8	yes	—
82	C₄×Dic₁₅₁	C₂³	1	1	yes	characteristic
83	C₄×Dic₁₅₁	C₂³	1	1	yes	characteristic
84	C₄×Dic₁₅₁	C₂³	1	2	yes	—
85	C₄×Dic₁₅₁	C₂³	1	2	yes	—
86	C₄×Dic₁₅₁	C₂³	1	2	yes	—
87	C₄×Dic₁₅₁	C₂³	1	2	yes	—
88	C₄×Dic₁₅₁	C₂³	1	2	yes	—
89	C₄×Dic₁₅₁	C₂³	1	8	yes	—
90	C₂×Q₈×C₁₅₁	C₂³	1	4	yes	—
91	C₂²×C₄×C₁₅₁	C₂³	1	2	yes	—
92	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₁₅₁	C₂³	1	4	yes	—
94	C₄⋊C₄×C₁₅₁	C₂³	1	8	yes	—
95	C₄⋊C₄×C₁₅₁	C₂³	1	4	yes	—
96	C₂²⋊C₄×C₁₅₁	C₂³	1	4	yes	—
97	C₄²×C₁₅₁	C₂³	1	2	yes	—
98	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
99	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
100	C₄²×C₁₅₁	C₂³	1	4	yes	—
101	C₂²×D₁₅₁	—	2	2	no	—
102	C₂²×D₁₅₁	—	2	4	no	—
103	C₂²×D₁₅₁	C₂⁴	1	4	yes	—
104	C₂²×D₁₅₁	C₂⁴	1	2	yes	—
105	Dic₃₀₂	—	2	4	no	—
106	D₆₀₄	—	2	16	no	—
107	D₆₀₄	—	2	4	no	—
108	D₆₀₄	—	2	8	no	—
109	C₂²⋊D₁₅₁	—	2	8	no	—
110	C₂×Dic₁₅₁	—	2	2	no	—
111	C₂×Dic₁₅₁	—	2	4	no	—
112	C₂×Dic₁₅₁	C₂⁴	1	2	yes	—
113	C₂×Dic₁₅₁	C₂⁴	1	2	yes	—
114	C₂×Dic₁₅₁	C₂⁴	1	4	yes	—
115	C₂×Dic₁₅₁	C₂⁴	1	2	yes	—
116	C₄×D₁₅₁	—	2	4	no	—
117	C₄×D₁₅₁	—	2	4	no	—
118	C₄×D₁₅₁	—	2	4	no	—
119	C₄×D₁₅₁	—	2	4	no	—
120	C₄×D₁₅₁	—	2	8	no	—
121	C₄×D₁₅₁	—	2	4	no	—
122	C₄×D₁₅₁	—	2	16	no	—
123	C₄×D₁₅₁	#41	1	8	yes	complemented
124	C₂³×C₁₅₁	C₂⁴	1	1	yes	characteristic
125	Q₈×C₁₅₁	—	2	8	no	—
126	C₂×C₄×C₁₅₁	—	2	4	no	—
127	C₂×C₄×C₁₅₁	#41	1	4	yes	complemented
128	C₂×C₄×C₁₅₁	C₂⁴	1	8	yes	—
129	C₂×C₄×C₁₅₁	C₂⁴	1	2	yes	—
130	C₂×C₄×C₁₅₁	C₂⁴	1	2	yes	—
131	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
132	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
133	C₂²×C₁₅₁	—	2	2	no	—
134	C₂²×C₁₅₁	C₂×#41	1	2	yes	—
135	C₂²×C₁₅₁	C₂⁵	1	1	yes	characteristic
136	C₂×D₁₅₁	—	2	8	no	—
137	C₂×D₁₅₁	—	2	2	no	—
138	C₂×D₁₅₁	—	2	2	no	—
139	C₂×D₁₅₁	—	2	4	no	—
140	C₂×D₁₅₁	—	2	8	no	—
141	C₂×D₁₅₁	C₂×#41	1	4	yes	—
142	C₄×C₁₅₁	—	2	2	no	—
143	C₄×C₁₅₁	—	2	8	no	—
144	C₄×C₁₅₁	C₂×#41	1	4	yes	—
145	C₄×C₁₅₁	C₂×#41	1	2	yes	—
146	C₄×C₁₅₁	C₂×#41	1	2	yes	—
147	Dic₁₅₁	—	2	2	no	—
148	Dic₁₅₁	—	2	2	no	—
149	Dic₁₅₁	—	2	4	no	—
150	Dic₁₅₁	C₂×#41	1	4	yes	—
151	C₂×C₁₅₁	—	2	2	no	—
152	C₂×C₁₅₁	C₂²×#41	1	2	yes	—
153	C₂×C₁₅₁	C₂²×#41	1	1	yes	characteristic
154	D₁₅₁	—	2	4	no	—
155	D₁₅₁	—	4	4	no	—
156	C₁₅₁	#3444	1	1	yes	characteristic, complemented, Sylow
157	#3444	—	151	1	no	maximal, Sylow
158	#533	—	151	2	no	—
159	#533	D₁₅₁	1	1	yes	characteristic, complemented
160	#532	—	151	2	no	—
161	#532	—	151	1	no	—
162	#529	—	151	4	no	—
163	#529	—	151	4	no	—
164	#529	—	151	4	no	—
165	#525	—	151	4	no	—
166	#525	—	151	2	no	—
167	C₄×#41	—	151	4	no	—
168	C₄×#41	—	151	2	no	—
169	C₂²×C₄²	—	151	1	no	—
170	C₂×#41	—	151	4	no	—
171	C₂×#41	—	151	2	no	—
172	C₂³×C₄	—	151	2	no	—
173	C₂³×C₄	—	151	1	no	—
174	C₄⋊Q₈	—	151	1	no	—
175	C₄⋊Q₈	C₂×D₁₅₁	1	2	yes	complemented
176	#127	—	151	1	no	—
177	#126	—	151	8	no	—
178	#125	—	151	4	no	—
179	#125	C₂×D₁₅₁	1	2	yes	complemented
180	#124	—	151	4	no	—
181	#124	—	151	2	no	—
182	#123	—	151	8	no	—
183	#123	—	151	4	no	—
184	C₂²⋊Q₈	—	151	4	no	—
185	C₂²⋊Q₈	—	151	4	no	—
186	C₂²⋊Q₈	C₂×D₁₅₁	1	4	yes	—
187	#121	—	151	4	no	—
188	#121	—	151	4	no	—
189	#121	—	151	4	no	—
190	#120	—	151	4	no	—
191	C₄×Q₈	—	151	4	no	—
192	C₄×Q₈	—	151	4	no	—
193	C₄×Q₈	C₂×D₁₅₁	1	4	yes	complemented
194	C₄×D₄	—	151	8	no	—
195	C₄×D₄	—	151	4	no	—
196	C₄×D₄	—	151	4	no	—
197	C₄×D₄	—	151	8	no	—
198	C₄×D₄	—	151	8	no	—
199	C₄×D₄	—	151	4	no	—
200	#117	—	151	4	no	—
201	#117	—	151	8	no	—
202	#117	—	151	4	no	—
203	#117	C₂×D₁₅₁	1	2	yes	—
204	C₂×C₄²	—	151	1	no	—
205	C₂×C₄²	—	151	2	no	—
206	C₂×C₄²	—	151	4	no	—
207	C₂×C₄²	—	151	2	no	—
208	C₂×C₄²	—	151	2	no	—
209	C₂×C₄²	—	151	2	no	—
210	C₂×C₄²	—	151	2	no	—
211	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
212	C₂⁴	—	151	1	no	—
213	#41	—	302	16	no	—
214	#41	—	302	8	no	—
215	C₂×Q₈	—	151	2	no	—
216	C₂×Q₈	C₂²×D₁₅₁	1	4	yes	—
217	C₂×D₄	—	151	4	no	—
218	C₂×D₄	—	151	8	no	—
219	C₂×D₄	—	151	4	no	—
220	C₂×D₄	—	151	2	no	—
221	C₂²×C₄	—	302	8	no	—
222	C₂²×C₄	—	302	4	no	—
223	C₂²×C₄	—	151	2	no	—
224	C₂²×C₄	—	151	1	no	—
225	C₂²×C₄	—	151	2	no	—
226	C₂²×C₄	—	151	2	no	—
227	C₂²×C₄	—	151	2	no	—
228	C₂²×C₄	—	151	2	no	—
229	C₂²×C₄	—	151	8	no	—
230	C₂²×C₄	—	151	2	no	—
231	C₂²×C₄	—	151	4	no	—
232	C₂²×C₄	—	151	2	no	—
233	C₂²×C₄	C₂²×D₁₅₁	1	2	yes	—
234	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
235	C₄⋊C₄	—	151	8	no	—
236	C₄⋊C₄	—	151	4	no	—
237	C₄⋊C₄	—	151	8	no	—
238	C₄⋊C₄	C₂²×D₁₅₁	1	4	yes	—
239	C₄⋊C₄	C₂²×D₁₅₁	1	8	yes	—
240	C₄⋊C₄	C₂²×D₁₅₁	1	4	yes	—
241	C₂²⋊C₄	—	151	8	no	—
242	C₂²⋊C₄	—	151	8	no	—
243	C₂²⋊C₄	—	151	8	no	—
244	C₂²⋊C₄	—	151	8	no	—
245	C₂²⋊C₄	C₂²×D₁₅₁	1	4	yes	—
246	C₄²	—	151	2	no	—
247	C₄²	—	151	2	no	—
248	C₄²	—	151	2	no	—
249	C₄²	—	151	2	no	—
250	C₄²	—	151	2	no	—
251	C₄²	—	151	8	no	—
252	C₄²	—	151	1	no	—
253	C₄²	—	151	1	no	—
254	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
255	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
256	C₄²	C₂²×D₁₅₁	1	2	yes	—
257	C₄²	C₂²×D₁₅₁	1	4	yes	—
258	C₂³	—	302	4	no	—
259	C₂³	—	302	2	no	—
260	C₂³	—	151	4	no	—
261	C₂³	—	151	2	no	—
262	C₂³	C₂³×D₁₅₁	1	1	yes	characteristic
263	Q₈	—	2	8	no	—
264	Q₈	—	302	4	no	—
265	D₄	—	302	16	no	—
266	D₄	—	302	8	no	—
267	D₄	—	302	8	no	—
268	D₄	—	302	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	302	4	no	—
271	C₂×C₄	—	302	4	no	—
272	C₂×C₄	—	302	16	no	—
273	C₂×C₄	—	302	2	no	—
274	C₂×C₄	—	302	4	no	—
275	C₂×C₄	—	302	4	no	—
276	C₂×C₄	—	302	4	no	—
277	C₂×C₄	—	302	4	no	—
278	C₂×C₄	—	302	8	no	—
279	C₂×C₄	—	151	2	no	—
280	C₂×C₄	—	151	2	no	—
281	C₂×C₄	—	151	4	no	—
282	C₂×C₄	—	151	2	no	—
283	C₂×C₄	—	151	8	no	—
284	C₂×C₄	#14559005	1	4	yes	complemented
285	C₂×C₄	C₂³×D₁₅₁	1	2	yes	—
286	C₂×C₄	C₂³×D₁₅₁	1	8	yes	—
287	C₂×C₄	C₂³×D₁₅₁	1	2	yes	—
288	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
290	C₂²	—	2	2	no	—
291	C₂²	—	302	8	no	—
292	C₂²	—	302	2	no	—
293	C₂²	—	302	2	no	—
294	C₂²	—	302	8	no	—
295	C₂²	—	302	4	no	—
296	C₂²	—	151	4	no	—
297	C₂²	C₂×#14559005	1	2	yes	—
298	C₂²	C₂⁴×D₁₅₁	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	2	8	no	—
301	C₄	—	302	2	no	—
302	C₄	—	302	2	no	—
303	C₄	—	302	4	no	—
304	C₄	—	151	4	no	—
305	C₄	#41×D₁₅₁	1	4	yes	—
306	C₄	C₂×#14559005	1	2	yes	—
307	C₄	C₂×#14559005	1	2	yes	—
308	C₂	—	2	2	no	—
309	C₂	—	302	4	no	—
310	C₂	—	604	4	no	—
311	C₂	C₂×#41×D₁₅₁	1	2	yes	—
312	C₂	C₂²×#14559005	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₅₁
commutator, socle, cosocle	C₂²×C₁₅₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₅₁
Fitting	#533×C₁₅₁	C₂
hypercenter	#533	D₁₅₁
hypocenter	C₁₅₁	#3444
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #23752767	C₂	semidirect
2. #23752763	C₂	semidirect
3. #23752762	C₂	semidirect
4. #23752749	C₂	semidirect
5. #23752748	C₂	semidirect
6. #23752747	C₂	semidirect
7. #23752725	C₂	semidirect
8. #23752724	C₂	semidirect
9. C₄×#14559005	C₂	semidirect
10. #23752680	C₂	semidirect
11. C₂×C₄²×D₁₅₁	C₂	semidirect
12. #533×C₁₅₁	C₂	semidirect
17. #18012739	C₂²	semidirect
18. #18012736	C₂²	semidirect
19. #18012734	C₂²	semidirect
20. #18012730	C₂²	semidirect
21. #18012728	C₂²	semidirect
22. #18012727	C₂²	semidirect
31. Q₈×Dic₁₅₁	C₂²	semidirect
32. #18012704	C₂²	semidirect
34. C₄×D₆₀₄	C₂²	semidirect
36. #18012701	C₂²	semidirect
37. #18012701	C₂²	semidirect
40. #18012697	C₂²	semidirect
41. #18012696	C₂²	semidirect
48. C₄²×D₁₅₁	C₂²	semidirect
49. C₄⋊Q₈×C₁₅₁	C₂²	semidirect
50. #125×C₁₅₁	C₂²	semidirect
52. C₄×Q₈×C₁₅₁	C₂²	semidirect
123. C₄×D₁₅₁	#41	semidirect
127. C₂×C₄×C₁₅₁	#41	semidirect
156. C₁₅₁	#3444	semidirect
159. #533	D₁₅₁	semidirect
175. C₄⋊Q₈	C₂×D₁₅₁	semidirect
179. #125	C₂×D₁₅₁	semidirect
193. C₄×Q₈	C₂×D₁₅₁	semidirect
284. C₂×C₄	#14559005	semidirect

Maximal subgroups

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

subgroup	index
#23752767	2
#23752763	2
#23752762	2
#23752749	2
#23752748	2
#23752747	2
#23752725	2
#23752724	2
C₄×#14559005	2
#23752680	2
C₂×C₄²×D₁₅₁	2
#533×C₁₅₁	2
#3444	151