Subgroups — #34083349

Order 19328.

1270 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752952	C₂	1	4	yes	maximal, complemented
2	#23752908	C₂	1	4	yes	maximal, complemented
3	D₄×D₆₀₄	C₂	1	4	yes	maximal, complemented
4	#23752844	C₂	1	4	yes	maximal, complemented
5	#23752817	C₂	1	2	yes	maximal, complemented
6	#23752802	C₂	1	4	yes	maximal, complemented
7	#23752776	C₂	1	2	yes	maximal, complemented
8	#23752770	C₂	1	4	yes	maximal, complemented
9	C₂×#18012713	C₂	1	2	yes	maximal, complemented
10	#525×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×D₆₀₄	C₂²	1	2	yes	complemented
12	C₂²×D₆₀₄	C₂²	1	2	yes	complemented
13	C₂²×C₂²⋊D₁₅₁	C₂²	1	4	yes	complemented
14	C₂×D₄×D₁₅₁	C₂²	1	8	yes	complemented
15	#18012736	C₂²	1	2	yes	complemented
16	#18012735	C₂²	1	4	yes	complemented
17	#18012734	C₂²	1	8	yes	complemented
18	#18012731	C₂²	1	2	yes	complemented
19	#18012728	C₂²	1	4	yes	complemented
20	#18012724	C₂²	1	8	yes	complemented
21	#18012721	C₂²	1	8	yes	complemented
22	#18012714	C₂²	1	8	yes	complemented
23	#18012713	C₂²	1	4	yes	complemented
24	#18012713	C₂²	1	4	yes	complemented
25	#18012713	C₂²	1	4	yes	complemented
26	#18012713	C₂²	1	4	yes	complemented
27	#18012713	C₂²	1	4	yes	complemented
28	#18012712	C₂²	1	8	yes	complemented
29	#18012711	C₂²	1	8	yes	complemented
30	#18012709	C₂²	1	8	yes	complemented
31	#18012709	C₂²	1	8	yes	complemented
32	#18012707	C₂²	1	4	yes	complemented
33	#18012707	C₂²	1	4	yes	complemented
34	C₄×D₆₀₄	C₂²	1	4	yes	complemented
35	(C₂²⋊D₁₅₁)⋊C₄	C₂²	1	8	yes	complemented
36	#18012698	C₂²	1	2	yes	complemented
37	C₂×C₄⋊Dic₁₅₁	C₂²	1	2	yes	complemented
38	C₂×(C₂×D₁₅₁)⋊C₄	C₂²	1	4	yes	complemented
39	C₂×#41×C₁₅₁	C₂²	1	1	yes	characteristic, complemented
40	C₂³×C₄×C₁₅₁	C₂²	1	1	yes	characteristic, complemented
41	#123×C₁₅₁	C₂²	1	2	yes	complemented
42	C₂²⋊Q₈×C₁₅₁	C₂²	1	2	yes	complemented
43	#121×C₁₅₁	C₂²	1	2	yes	complemented
44	#120×C₁₅₁	C₂²	1	2	yes	complemented
45	C₄×D₄×C₁₅₁	C₂²	1	4	yes	complemented
46	#117×C₁₅₁	C₂²	1	1	yes	characteristic, complemented
47	C₂³×D₁₅₁	C₂³	1	4	yes	—
48	C₂³×D₁₅₁	C₂³	1	4	yes	—
49	C₂×D₆₀₄	—	4	4	no	—
50	C₂×D₆₀₄	—	4	4	no	—
51	C₂×D₆₀₄	C₂³	1	4	yes	—
52	C₂×D₆₀₄	C₂³	1	8	yes	—
53	C₂×D₆₀₄	C₂³	1	4	yes	—
54	C₂×D₆₀₄	C₂³	1	4	yes	—
55	C₂×C₂²⋊D₁₅₁	—	4	8	no	—
56	C₂×C₂²⋊D₁₅₁	C₂³	1	8	yes	—
57	C₂×C₂²⋊D₁₅₁	C₂³	1	8	yes	—
58	C₂×C₂²⋊D₁₅₁	C₂³	1	8	yes	—
59	D₄×D₁₅₁	—	4	16	no	—
60	C₂²×Dic₁₅₁	C₂³	1	4	yes	—
61	C₂×C₄×D₁₅₁	C₂³	1	8	yes	—
62	C₄⋊Dic₁₅₁	C₂³	1	4	yes	—
63	C₄⋊Dic₁₅₁	C₂³	1	4	yes	—
64	Dic₁₅₁⋊C₄	C₂³	1	8	yes	—
65	C₂²⋊Dic₁₅₁	C₂³	1	4	yes	—
66	(C₂×D₁₅₁)⋊C₄	C₂³	1	8	yes	—
67	(C₂×D₁₅₁)⋊C₄	C₂³	1	8	yes	—
68	(C₂×D₁₅₁)⋊C₄	C₂³	1	8	yes	—
69	(C₂×D₁₅₁)⋊C₄	C₂³	1	8	yes	—
70	C₄×Dic₁₅₁	C₂³	1	4	yes	—
71	C₂⁴×C₁₅₁	C₂³	1	1	yes	characteristic
72	#41×C₁₅₁	—	4	2	no	—
73	C₂×Q₈×C₁₅₁	C₂³	1	1	yes	characteristic
74	C₂×D₄×C₁₅₁	C₂³	1	4	yes	—
75	C₂×D₄×C₁₅₁	C₂³	1	1	yes	characteristic
76	C₂×D₄×C₁₅₁	C₂³	1	2	yes	—
77	C₂²×C₄×C₁₅₁	—	4	2	no	—
78	C₂²×C₄×C₁₅₁	C₂³	1	2	yes	—
79	C₂²×C₄×C₁₅₁	C₂³	1	2	yes	—
80	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
81	C₂²×C₄×C₁₅₁	C₂³	1	2	yes	—
82	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₁₅₁	C₂³	1	4	yes	—
84	C₄⋊C₄×C₁₅₁	C₂³	1	2	yes	—
85	C₂²⋊C₄×C₁₅₁	C₂³	1	4	yes	—
86	C₂²⋊C₄×C₁₅₁	C₂³	1	4	yes	—
87	C₂²⋊C₄×C₁₅₁	C₂³	1	2	yes	—
88	C₄²×C₁₅₁	C₂³	1	2	yes	—
89	C₂²×D₁₅₁	—	2	8	no	—
90	C₂²×D₁₅₁	—	4	8	no	—
91	C₂²×D₁₅₁	—	4	4	no	—
92	C₂²×D₁₅₁	—	4	4	no	—
93	C₂²×D₁₅₁	—	4	4	no	—
94	C₂²×D₁₅₁	C₂⁴	1	8	yes	—
95	D₆₀₄	—	2	8	no	—
96	D₆₀₄	—	4	8	no	—
97	D₆₀₄	—	4	4	no	—
98	D₆₀₄	—	4	4	no	—
99	C₂²⋊D₁₅₁	—	2	16	no	—
100	C₂²⋊D₁₅₁	—	4	8	no	—
101	C₂²⋊D₁₅₁	—	4	8	no	—
102	C₂×Dic₁₅₁	—	4	4	no	—
103	C₂×Dic₁₅₁	C₂⁴	1	8	yes	—
104	C₄×D₁₅₁	—	4	8	no	—
105	C₂³×C₁₅₁	—	2	4	no	—
106	C₂³×C₁₅₁	—	4	1	no	—
107	C₂³×C₁₅₁	C₂⁴	1	1	yes	characteristic
108	C₂³×C₁₅₁	C₂⁴	1	2	yes	—
109	C₂³×C₁₅₁	C₂⁴	1	2	yes	—
110	Q₈×C₁₅₁	—	4	1	no	—
111	D₄×C₁₅₁	—	2	8	no	—
112	D₄×C₁₅₁	—	4	1	no	—
113	D₄×C₁₅₁	—	4	2	no	—
114	C₂×C₄×C₁₅₁	—	2	4	no	—
115	C₂×C₄×C₁₅₁	—	2	2	no	—
116	C₂×C₄×C₁₅₁	—	2	4	no	—
117	C₂×C₄×C₁₅₁	—	4	2	no	—
118	C₂×C₄×C₁₅₁	—	4	1	no	—
119	C₂×C₄×C₁₅₁	C₂⁴	1	2	yes	—
120	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
121	C₂×C₄×C₁₅₁	C₂⁴	1	2	yes	—
122	C₂×C₄×C₁₅₁	C₂⁴	1	4	yes	—
123	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
124	C₂²×C₁₅₁	—	2	2	no	—
125	C₂²×C₁₅₁	—	2	2	no	—
126	C₂²×C₁₅₁	—	2	1	no	—
127	C₂²×C₁₅₁	—	2	2	no	—
128	C₂²×C₁₅₁	—	2	4	no	—
129	C₂²×C₁₅₁	—	2	2	no	—
130	C₂²×C₁₅₁	—	4	4	no	—
131	C₂²×C₁₅₁	H(5,𝔽₂)	1	4	yes	—
132	C₂²×C₁₅₁	C₂⁵	1	1	yes	characteristic
133	C₂×D₁₅₁	—	2	8	no	—
134	C₂×D₁₅₁	—	2	8	no	—
135	C₂×D₁₅₁	—	2	8	no	—
136	C₂×D₁₅₁	—	8	8	no	—
137	C₂×D₁₅₁	—	8	8	no	—
138	C₄×C₁₅₁	—	2	1	no	—
139	C₄×C₁₅₁	—	2	2	no	—
140	C₄×C₁₅₁	—	2	2	no	—
141	C₄×C₁₅₁	—	2	4	no	—
142	C₄×C₁₅₁	H(5,𝔽₂)	1	2	yes	—
143	Dic₁₅₁	—	2	8	no	—
144	C₂×C₁₅₁	—	2	4	no	—
145	C₂×C₁₅₁	—	4	2	no	—
146	C₂×C₁₅₁	—	4	1	no	—
147	C₂×C₁₅₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×C₁₅₁	C₂×H(5,𝔽₂)	1	2	yes	—
149	D₁₅₁	—	4	8	no	—
150	C₁₅₁	#3537	1	1	yes	characteristic, complemented, Sylow
151	#3537	—	151	1	no	maximal, Sylow
152	#560	—	151	4	no	—
153	#553	—	151	4	no	—
154	D₄²	—	151	4	no	—
155	D₄²	—	151	4	no	—
156	#540	—	151	2	no	—
157	#538	—	151	4	no	—
158	#535	—	151	2	no	—
159	#534	—	151	4	no	—
160	#525	D₁₅₁	1	1	yes	characteristic, complemented
161	C₂×#121	—	151	2	no	—
162	C₂×#41	C₂×D₁₅₁	1	1	yes	characteristic, complemented
163	C₂²×D₄	—	151	4	no	—
164	C₂²×D₄	—	151	8	no	—
165	C₂²×D₄	—	151	2	no	—
166	C₂²×D₄	—	151	2	no	—
167	C₂³×C₄	C₂×D₁₅₁	1	1	yes	characteristic, complemented
168	#127	—	151	4	no	—
169	#127	—	151	2	no	—
170	#126	—	151	8	no	—
171	#125	—	151	2	no	—
172	#124	—	151	4	no	—
173	#123	—	151	8	no	—
174	#123	—	151	8	no	—
175	#123	C₂×D₁₅₁	1	2	yes	complemented
176	C₂²⋊Q₈	C₂×D₁₅₁	1	2	yes	complemented
177	#121	—	151	8	no	—
178	#121	—	151	4	no	—
179	#121	—	151	4	no	—
180	#121	—	151	4	no	—
181	#121	—	151	4	no	—
182	#121	—	151	4	no	—
183	#121	—	151	8	no	—
184	#121	—	151	8	no	—
185	#121	C₂×D₁₅₁	1	2	yes	complemented
186	#120	—	151	4	no	—
187	#120	—	151	4	no	—
188	#120	—	151	8	no	—
189	#120	—	151	8	no	—
190	#120	C₂×D₁₅₁	1	2	yes	complemented
191	C₄×D₄	—	151	8	no	—
192	C₄×D₄	—	151	4	no	—
193	C₄×D₄	C₂×D₁₅₁	1	4	yes	complemented
194	#117	—	151	2	no	—
195	#117	C₂×D₁₅₁	1	1	yes	characteristic, complemented
196	C₂×C₄⋊C₄	—	151	2	no	—
197	C₂×C₂²⋊C₄	—	151	4	no	—
198	C₂⁴	—	151	4	no	—
199	C₂⁴	—	151	4	no	—
200	C₂⁴	C₂²×D₁₅₁	1	1	yes	characteristic
201	#41	—	4	2	no	—
202	C₂×Q₈	C₂²×D₁₅₁	1	1	yes	characteristic
203	C₂×D₄	—	604	4	no	—
204	C₂×D₄	—	604	4	no	—
205	C₂×D₄	—	604	16	no	—
206	C₂×D₄	—	604	8	no	—
207	C₂×D₄	—	151	8	no	—
208	C₂×D₄	—	151	8	no	—
209	C₂×D₄	—	151	8	no	—
210	C₂×D₄	—	151	4	no	—
211	C₂×D₄	—	151	4	no	—
212	C₂×D₄	—	151	8	no	—
213	C₂×D₄	—	151	4	no	—
214	C₂×D₄	C₂²×D₁₅₁	1	4	yes	—
215	C₂×D₄	C₂²×D₁₅₁	1	1	yes	characteristic
216	C₂×D₄	C₂²×D₁₅₁	1	2	yes	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	—	151	4	no	—
219	C₂²×C₄	—	151	8	no	—
220	C₂²×C₄	C₂²×D₁₅₁	1	2	yes	—
221	C₂²×C₄	C₂²×D₁₅₁	1	2	yes	—
222	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
223	C₂²×C₄	C₂²×D₁₅₁	1	2	yes	—
224	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
225	C₄⋊C₄	—	151	8	no	—
226	C₄⋊C₄	—	151	4	no	—
227	C₄⋊C₄	—	151	4	no	—
228	C₄⋊C₄	C₂²×D₁₅₁	1	2	yes	—
229	C₄⋊C₄	C₂²×D₁₅₁	1	4	yes	—
230	C₂²⋊C₄	—	151	4	no	—
231	C₂²⋊C₄	—	151	8	no	—
232	C₂²⋊C₄	—	151	8	no	—
233	C₂²⋊C₄	—	151	8	no	—
234	C₂²⋊C₄	—	151	8	no	—
235	C₂²⋊C₄	C₂²×D₁₅₁	1	2	yes	—
236	C₂²⋊C₄	C₂²×D₁₅₁	1	4	yes	—
237	C₂²⋊C₄	C₂²×D₁₅₁	1	4	yes	—
238	C₄²	—	151	4	no	—
239	C₄²	C₂²×D₁₅₁	1	2	yes	—
240	C₂³	—	2	4	no	—
241	C₂³	—	4	1	no	—
242	C₂³	—	302	8	no	—
243	C₂³	—	604	4	no	—
244	C₂³	—	604	4	no	—
245	C₂³	—	604	4	no	—
246	C₂³	—	604	8	no	—
247	C₂³	—	151	8	no	—
248	C₂³	C₂³×D₁₅₁	1	2	yes	—
249	C₂³	C₂³×D₁₅₁	1	1	yes	characteristic
250	C₂³	C₂³×D₁₅₁	1	2	yes	—
251	Q₈	—	4	1	no	—
252	D₄	—	2	8	no	—
253	D₄	—	4	1	no	—
254	D₄	—	4	2	no	—
255	D₄	—	302	16	no	—
256	D₄	—	302	8	no	—
257	D₄	—	604	4	no	—
258	D₄	—	604	4	no	—
259	D₄	—	604	8	no	—
260	D₄	—	604	8	no	—
261	D₄	—	604	8	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	604	8	no	—
268	C₂×C₄	—	604	4	no	—
269	C₂×C₄	—	151	8	no	—
270	C₂×C₄	C₂³×D₁₅₁	1	2	yes	—
271	C₂×C₄	C₂³×D₁₅₁	1	4	yes	—
272	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
273	C₂×C₄	C₂³×D₁₅₁	1	2	yes	—
274	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	4	4	no	—
282	C₂²	—	302	8	no	—
283	C₂²	—	302	8	no	—
284	C₂²	—	302	8	no	—
285	C₂²	—	1208	8	no	—
286	C₂²	—	1208	8	no	—
287	C₂²	#18012785	1	4	yes	—
288	C₂²	C₂⁴×D₁₅₁	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	4	no	—
293	C₄	—	302	8	no	—
294	C₄	#18012784	1	2	yes	—
295	C₂	—	2	4	no	—
296	C₂	—	4	1	no	—
297	C₂	—	4	2	no	—
298	C₂	—	604	8	no	—
299	C₂	C₂×#18012784	1	1	yes	characteristic
300	C₂	C₂×#18012785	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₅₁
commutator, socle, cosocle	C₂²×C₁₅₁	C₂⁵
Fitting	#525×C₁₅₁	C₂
hypercenter	#525	D₁₅₁
hypocenter	C₁₅₁	#3537
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ #525

Complements

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

N	H ≅ G/N	type
1. #23752952	C₂	semidirect
2. #23752908	C₂	semidirect
3. D₄×D₆₀₄	C₂	semidirect
4. #23752844	C₂	semidirect
5. #23752817	C₂	semidirect
6. #23752802	C₂	semidirect
7. #23752776	C₂	semidirect
8. #23752770	C₂	semidirect
9. C₂×#18012713	C₂	semidirect
10. #525×C₁₅₁	C₂	semidirect
11. C₂²×D₆₀₄	C₂²	semidirect
12. C₂²×D₆₀₄	C₂²	semidirect
13. C₂²×C₂²⋊D₁₅₁	C₂²	semidirect
14. C₂×D₄×D₁₅₁	C₂²	semidirect
15. #18012736	C₂²	semidirect
16. #18012735	C₂²	semidirect
17. #18012734	C₂²	semidirect
18. #18012731	C₂²	semidirect
19. #18012728	C₂²	semidirect
20. #18012724	C₂²	semidirect
21. #18012721	C₂²	semidirect
22. #18012714	C₂²	semidirect
23. #18012713	C₂²	semidirect
24. #18012713	C₂²	semidirect
25. #18012713	C₂²	semidirect
26. #18012713	C₂²	semidirect
27. #18012713	C₂²	semidirect
28. #18012712	C₂²	semidirect
29. #18012711	C₂²	semidirect
30. #18012709	C₂²	semidirect
31. #18012709	C₂²	semidirect
32. #18012707	C₂²	semidirect
33. #18012707	C₂²	semidirect
34. C₄×D₆₀₄	C₂²	semidirect
35. (C₂²⋊D₁₅₁)⋊C₄	C₂²	semidirect
36. #18012698	C₂²	semidirect
37. C₂×C₄⋊Dic₁₅₁	C₂²	semidirect
38. C₂×(C₂×D₁₅₁)⋊C₄	C₂²	semidirect
39. C₂×#41×C₁₅₁	C₂²	semidirect
40. C₂³×C₄×C₁₅₁	C₂²	semidirect
41. #123×C₁₅₁	C₂²	semidirect
42. C₂²⋊Q₈×C₁₅₁	C₂²	semidirect
43. #121×C₁₅₁	C₂²	semidirect
44. #120×C₁₅₁	C₂²	semidirect
45. C₄×D₄×C₁₅₁	C₂²	semidirect
46. #117×C₁₅₁	C₂²	semidirect
150. C₁₅₁	#3537	semidirect
160. #525	D₁₅₁	semidirect
162. C₂×#41	C₂×D₁₅₁	semidirect
167. C₂³×C₄	C₂×D₁₅₁	semidirect
175. #123	C₂×D₁₅₁	semidirect
176. C₂²⋊Q₈	C₂×D₁₅₁	semidirect
185. #121	C₂×D₁₅₁	semidirect
190. #120	C₂×D₁₅₁	semidirect
193. C₄×D₄	C₂×D₁₅₁	semidirect
195. #117	C₂×D₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#23752952	2
#23752908	2
D₄×D₆₀₄	2
#23752844	2
#23752817	2
#23752802	2
#23752776	2
#23752770	2
C₂×#18012713	2
#525×C₁₅₁	2
#3537	151