Subgroups — #34058380

Order 19072.

1328 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23750301	C₂	1	8	yes	maximal, complemented
2	#23750265	C₂	1	8	yes	maximal, complemented
3	#23750251	C₂	1	1	yes	maximal, characteristic, complemented
4	#23750213	C₂	1	4	yes	maximal, complemented
5	#23750187	C₂	1	1	yes	maximal, characteristic, complemented
6	#23750136	C₂	1	4	yes	maximal, complemented
7	C₂×#18011880	C₂	1	2	yes	maximal, complemented
8	#23750098	C₂	1	2	yes	maximal, complemented
9	C₂²×C₂²⋊C₄×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14558736	C₂²	1	4	yes	complemented
11	C₂²×C₂²⋊D₁₄₉	C₂²	1	2	yes	—
12	C₂²×C₂²⋊D₁₄₉	C₂²	1	4	yes	complemented
13	#18011904	C₂²	1	8	yes	complemented
14	#18011897	C₂²	1	8	yes	complemented
15	#18011895	C₂²	1	4	yes	complemented
16	#18011894	C₂²	1	8	yes	complemented
17	#18011893	C₂²	1	8	yes	complemented
18	#18011890	C₂²	1	4	yes	complemented
19	#18011890	C₂²	1	4	yes	complemented
20	#18011888	C₂²	1	8	yes	complemented
21	#18011885	C₂²	1	4	yes	complemented
22	#18011885	C₂²	1	4	yes	complemented
23	#18011884	C₂²	1	8	yes	complemented
24	#18011881	C₂²	1	4	yes	complemented
25	#18011880	C₂²	1	8	yes	—
26	#18011880	C₂²	1	4	yes	complemented
27	#18011880	C₂²	1	4	yes	complemented
28	C₄×C₂²⋊D₁₄₉	C₂²	1	8	yes	complemented
29	C₄×C₂²⋊D₁₄₉	C₂²	1	8	yes	complemented
30	#18011871	C₂²	1	8	yes	complemented
31	#18011870	C₂²	1	4	yes	complemented
32	#18011867	C₂²	1	4	yes	complemented
33	C₂×C₂²⋊Dic₁₄₉	C₂²	1	2	yes	—
34	C₂×C₂²⋊Dic₁₄₉	C₂²	1	4	yes	complemented
35	C₂²⋊C₄×D₁₄₉	C₂²	1	4	yes	complemented
36	C₂⁵×C₁₄₉	C₂²	1	1	yes	characteristic
37	C₂³×C₄×C₁₄₉	C₂²	1	2	yes	complemented
38	C₂×C₂²⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₁₄₉	C₂²	1	2	yes	complemented
41	C₂×C₂²⋊C₄×C₁₄₉	C₂²	1	8	yes	complemented
42	C₂³×D₁₄₉	C₂³	1	2	yes	—
43	#14558736	—	4	8	no	—
44	C₂×Dic₂₉₈	C₂³	1	4	yes	complemented
45	C₂×D₅₉₆	C₂³	1	4	yes	complemented
46	C₂×C₂²⋊D₁₄₉	—	2	8	no	—
47	C₂×C₂²⋊D₁₄₉	—	4	8	no	—
48	C₂×C₂²⋊D₁₄₉	C₂³	1	4	yes	—
49	C₂×C₂²⋊D₁₄₉	C₂³	1	4	yes	—
50	C₂×C₂²⋊D₁₄₉	C₂³	1	8	yes	—
51	C₂×C₂²⋊D₁₄₉	C₂³	1	8	yes	—
52	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
53	C₂²×Dic₁₄₉	C₂³	1	4	yes	—
54	C₂×C₄×D₁₄₉	C₂³	1	8	yes	complemented
55	C₄⋊Dic₁₄₉	C₂³	1	4	yes	complemented
56	C₄⋊Dic₁₄₉	C₂³	1	4	yes	complemented
57	#14558715	C₂³	1	8	yes	complemented
58	#14558715	C₂³	1	8	yes	complemented
59	C₂²⋊Dic₁₄₉	C₂³	1	8	yes	—
60	C₂²⋊Dic₁₄₉	C₂³	1	4	yes	—
61	C₂²⋊Dic₁₄₉	C₂³	1	4	yes	—
62	C₂²⋊Dic₁₄₉	C₂³	1	8	yes	—
63	#14558713	C₂³	1	8	yes	complemented
64	#14558713	C₂³	1	8	yes	complemented
65	C₄×Dic₁₄₉	C₂³	1	8	yes	complemented
66	C₂⁴×C₁₄₉	—	2	4	no	—
67	C₂⁴×C₁₄₉	—	4	4	no	—
68	C₂⁴×C₁₄₉	C₂³	1	1	yes	characteristic
69	C₂⁴×C₁₄₉	C₂³	1	4	yes	—
70	C₂⁴×C₁₄₉	C₂³	1	1	yes	characteristic
71	C₂⁴×C₁₄₉	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₁₄₉	—	4	4	no	—
73	C₂²×C₄×C₁₄₉	C₂³	1	4	yes	—
74	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
75	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
76	C₂²×C₄×C₁₄₉	C₂³	1	4	yes	—
77	C₂²⋊C₄×C₁₄₉	C₂³	1	8	yes	—
78	C₂²⋊C₄×C₁₄₉	C₂³	1	4	yes	—
79	C₂²⋊C₄×C₁₄₉	C₂³	1	4	yes	—
80	C₂²×D₁₄₉	—	2	4	no	—
81	C₂²×D₁₄₉	—	4	4	no	—
82	C₂²×D₁₄₉	C₂⁴	1	4	yes	—
83	Dic₂₉₈	—	4	4	no	—
84	D₅₉₆	—	4	4	no	—
85	C₂²⋊D₁₄₉	—	2	8	no	—
86	C₂²⋊D₁₄₉	—	2	8	no	—
87	C₂²⋊D₁₄₉	—	2	16	no	—
88	C₂²⋊D₁₄₉	—	4	8	no	—
89	C₂×Dic₁₄₉	—	2	4	no	—
90	C₂×Dic₁₄₉	—	4	4	no	—
91	C₂×Dic₁₄₉	C₂⁴	1	8	yes	—
92	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
93	C₄×D₁₄₉	—	4	8	no	—
94	C₂³×C₁₄₉	—	2	2	no	—
95	C₂³×C₁₄₉	—	2	4	no	—
96	C₂³×C₁₄₉	—	2	8	no	—
97	C₂³×C₁₄₉	—	2	2	no	—
98	C₂³×C₁₄₉	—	2	8	no	—
99	C₂³×C₁₄₉	—	4	2	no	—
100	C₂³×C₁₄₉	—	4	4	no	—
101	C₂³×C₁₄₉	—	4	2	no	—
102	C₂³×C₁₄₉	—	4	16	no	—
103	C₂³×C₁₄₉	#41	1	4	yes	complemented
104	C₂³×C₁₄₉	C₂⁴	1	2	yes	—
105	C₂³×C₁₄₉	C₂⁴	1	2	yes	—
106	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
107	C₂³×C₁₄₉	C₂⁴	1	2	yes	—
108	C₂×C₄×C₁₄₉	—	2	8	no	—
109	C₂×C₄×C₁₄₉	—	4	2	no	—
110	C₂×C₄×C₁₄₉	—	4	4	no	—
111	C₂×C₄×C₁₄₉	—	4	2	no	—
112	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
113	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
114	C₂²×C₁₄₉	—	2	8	no	—
115	C₂²×C₁₄₉	—	2	2	no	—
116	C₂²×C₁₄₉	—	2	2	no	—
117	C₂²×C₁₄₉	—	2	2	no	—
118	C₂²×C₁₄₉	—	2	4	no	—
119	C₂²×C₁₄₉	—	2	2	no	—
120	C₂²×C₁₄₉	—	2	4	no	—
121	C₂²×C₁₄₉	—	4	4	no	—
122	C₂²×C₁₄₉	—	4	16	no	—
123	C₂²×C₁₄₉	—	4	4	no	—
124	C₂²×C₁₄₉	C₂×#41	1	4	yes	—
125	C₂²×C₁₄₉	C₂×#41	1	2	yes	—
126	C₂²×C₁₄₉	H(5,𝔽₂)	1	4	yes	—
127	C₂²×C₁₄₉	C₂⁵	1	1	yes	characteristic
128	C₂×D₁₄₉	—	2	4	no	—
129	C₂×D₁₄₉	—	2	8	no	—
130	C₂×D₁₄₉	—	8	4	no	—
131	C₄×C₁₄₉	—	2	4	no	—
132	C₄×C₁₄₉	—	2	4	no	—
133	Dic₁₄₉	—	2	4	no	—
134	Dic₁₄₉	—	2	8	no	—
135	C₂×C₁₄₉	—	2	4	no	—
136	C₂×C₁₄₉	—	2	2	no	—
137	C₂×C₁₄₉	—	2	4	no	—
138	C₂×C₁₄₉	—	4	2	no	—
139	C₂×C₁₄₉	C₂²×#41	1	1	yes	characteristic
140	C₂×C₁₄₉	C₂×H(5,𝔽₂)	1	2	yes	—
141	D₁₄₉	—	4	4	no	—
142	C₁₄₉	#3490	1	1	yes	characteristic, complemented, Sylow
143	#3490	—	149	1	no	maximal, Sylow
144	#551	—	149	8	no	—
145	#546	—	149	8	no	—
146	#543	—	149	1	no	—
147	#538	—	149	4	no	—
148	#534	—	149	1	no	—
149	#525	—	149	4	no	—
150	C₂×#120	—	149	2	no	—
151	#518	—	149	2	no	—
152	C₂²×C₂²⋊C₄	D₁₄₉	1	1	yes	characteristic, complemented
153	C₂⁵	C₂×D₁₄₉	1	1	yes	characteristic
154	C₂×#41	—	149	4	no	—
155	C₂²×D₄	—	149	2	no	—
156	C₂²×D₄	—	149	4	no	—
157	C₂³×C₄	C₂×D₁₄₉	1	2	yes	complemented
158	#126	—	149	8	no	—
159	#124	—	149	8	no	—
160	#123	—	149	8	no	—
161	#123	—	149	4	no	—
162	#123	—	149	8	no	—
163	C₂²⋊Q₈	—	149	4	no	—
164	C₂²⋊Q₈	—	149	4	no	—
165	C₂²⋊Q₈	—	149	8	no	—
166	#121	—	149	4	no	—
167	#121	—	149	4	no	—
168	#121	—	149	8	no	—
169	#120	—	149	4	no	—
170	#120	—	149	4	no	—
171	#120	—	149	8	no	—
172	#120	—	149	4	no	—
173	C₄×D₄	—	149	8	no	—
174	C₄×D₄	—	149	8	no	—
175	C₄×D₄	—	149	8	no	—
176	#117	—	149	4	no	—
177	#117	—	149	4	no	—
178	C₂×C₂²⋊C₄	—	149	2	no	—
179	C₂×C₂²⋊C₄	—	149	4	no	—
180	C₂×C₂²⋊C₄	—	149	4	no	—
181	C₂×C₂²⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
182	C₂×C₂²⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
183	C₂×C₂²⋊C₄	C₂×D₁₄₉	1	2	yes	complemented
184	C₂×C₂²⋊C₄	C₂×D₁₄₉	1	8	yes	complemented
185	C₂⁴	—	2	4	no	—
186	C₂⁴	—	4	4	no	—
187	C₂⁴	—	149	2	no	—
188	C₂⁴	C₂²×D₁₄₉	1	1	yes	characteristic
189	C₂⁴	C₂²×D₁₄₉	1	4	yes	—
190	C₂⁴	C₂²×D₁₄₉	1	1	yes	characteristic
191	C₂⁴	C₂²×D₁₄₉	1	1	yes	characteristic
192	#41	—	596	8	no	—
193	C₂×Q₈	—	149	4	no	—
194	C₂×D₄	—	298	8	no	—
195	C₂×D₄	—	596	8	no	—
196	C₂×D₄	—	149	4	no	—
197	C₂×D₄	—	149	4	no	—
198	C₂×D₄	—	149	8	no	—
199	C₂×D₄	—	149	8	no	—
200	C₂×D₄	—	149	4	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	—	149	2	no	—
203	C₂²×C₄	—	149	4	no	—
204	C₂²×C₄	—	149	8	no	—
205	C₂²×C₄	C₂²×D₁₄₉	1	4	yes	—
206	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
207	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
208	C₂²×C₄	C₂²×D₁₄₉	1	4	yes	—
209	C₄⋊C₄	—	149	8	no	—
210	C₄⋊C₄	—	149	4	no	—
211	C₄⋊C₄	—	149	4	no	—
212	C₄⋊C₄	—	149	8	no	—
213	C₂²⋊C₄	—	149	4	no	—
214	C₂²⋊C₄	—	149	8	no	—
215	C₂²⋊C₄	—	149	4	no	—
216	C₂²⋊C₄	—	149	8	no	—
217	C₂²⋊C₄	—	149	8	no	—
218	C₂²⋊C₄	—	149	8	no	—
219	C₂²⋊C₄	C₂²×D₁₄₉	1	4	yes	—
220	C₂²⋊C₄	C₂²×D₁₄₉	1	4	yes	—
221	C₂²⋊C₄	C₂²×D₁₄₉	1	8	yes	—
222	C₄²	—	149	8	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	4	no	—
225	C₂³	—	2	8	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	8	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	4	no	—
230	C₂³	—	4	2	no	—
231	C₂³	—	4	16	no	—
232	C₂³	—	298	4	no	—
233	C₂³	—	596	4	no	—
234	C₂³	—	149	4	no	—
235	C₂³	#14558736	1	4	yes	complemented
236	C₂³	C₂³×D₁₄₉	1	2	yes	—
237	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
238	C₂³	C₂³×D₁₄₉	1	2	yes	—
239	C₂³	C₂³×D₁₄₉	1	2	yes	—
240	Q₈	—	596	4	no	—
241	D₄	—	298	8	no	—
242	D₄	—	298	8	no	—
243	D₄	—	298	16	no	—
244	D₄	—	596	4	no	—
245	D₄	—	596	8	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	298	4	no	—
251	C₂×C₄	—	596	8	no	—
252	C₂×C₄	—	596	4	no	—
253	C₂×C₄	—	149	4	no	—
254	C₂×C₄	—	149	8	no	—
255	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
256	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
257	C₂²	—	2	8	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	4	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	4	4	no	—
265	C₂²	—	4	16	no	—
266	C₂²	—	4	4	no	—
267	C₂²	—	298	4	no	—
268	C₂²	—	298	8	no	—
269	C₂²	—	1192	4	no	—
270	C₂²	C₂×#14558736	1	4	yes	—
271	C₂²	C₂×#14558736	1	2	yes	—
272	C₂²	#18011956	1	4	yes	—
273	C₂²	C₂⁴×D₁₄₉	1	1	yes	characteristic
274	C₄	—	2	4	no	—
275	C₄	—	2	4	no	—
276	C₄	—	298	4	no	—
277	C₄	—	298	8	no	—
278	C₂	—	2	2	no	—
279	C₂	—	2	4	no	—
280	C₂	—	2	4	no	—
281	C₂	—	4	2	no	—
282	C₂	—	596	4	no	—
283	C₂	C₂²×#14558736	1	1	yes	characteristic
284	C₂	C₂×#18011956	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₄₉
commutator, socle, cosocle	C₂²×C₁₄₉	C₂⁵
Fitting	C₂²×C₂²⋊C₄×C₁₄₉	C₂
hypercenter	C₂²×C₂²⋊C₄	D₁₄₉
hypocenter	C₁₄₉	#3490
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. #23750301	C₂	semidirect
2. #23750265	C₂	semidirect
3. #23750251	C₂	semidirect
4. #23750213	C₂	semidirect
5. #23750187	C₂	semidirect
6. #23750136	C₂	semidirect
7. C₂×#18011880	C₂	semidirect
8. #23750098	C₂	semidirect
9. C₂²×C₂²⋊C₄×C₁₄₉	C₂	semidirect
10. C₂×#14558736	C₂²	semidirect
12. C₂²×C₂²⋊D₁₄₉	C₂²	semidirect
13. #18011904	C₂²	semidirect
14. #18011897	C₂²	semidirect
15. #18011895	C₂²	semidirect
16. #18011894	C₂²	semidirect
17. #18011893	C₂²	semidirect
18. #18011890	C₂²	semidirect
19. #18011890	C₂²	semidirect
20. #18011888	C₂²	semidirect
21. #18011885	C₂²	semidirect
22. #18011885	C₂²	semidirect
23. #18011884	C₂²	semidirect
24. #18011881	C₂²	semidirect
26. #18011880	C₂²	semidirect
27. #18011880	C₂²	semidirect
28. C₄×C₂²⋊D₁₄₉	C₂²	semidirect
29. C₄×C₂²⋊D₁₄₉	C₂²	semidirect
30. #18011871	C₂²	semidirect
31. #18011870	C₂²	semidirect
32. #18011867	C₂²	semidirect
34. C₂×C₂²⋊Dic₁₄₉	C₂²	semidirect
35. C₂²⋊C₄×D₁₄₉	C₂²	semidirect
37. C₂³×C₄×C₁₄₉	C₂²	semidirect
40. C₂×C₂²⋊C₄×C₁₄₉	C₂²	semidirect
41. C₂×C₂²⋊C₄×C₁₄₉	C₂²	semidirect
44. C₂×Dic₂₉₈	C₂³	semidirect
45. C₂×D₅₉₆	C₂³	semidirect
54. C₂×C₄×D₁₄₉	C₂³	semidirect
55. C₄⋊Dic₁₄₉	C₂³	semidirect
56. C₄⋊Dic₁₄₉	C₂³	semidirect
57. #14558715	C₂³	semidirect
58. #14558715	C₂³	semidirect
63. #14558713	C₂³	semidirect
64. #14558713	C₂³	semidirect
65. C₄×Dic₁₄₉	C₂³	semidirect
103. C₂³×C₁₄₉	#41	semidirect
142. C₁₄₉	#3490	semidirect
152. C₂²×C₂²⋊C₄	D₁₄₉	semidirect
157. C₂³×C₄	C₂×D₁₄₉	semidirect
183. C₂×C₂²⋊C₄	C₂×D₁₄₉	semidirect
184. C₂×C₂²⋊C₄	C₂×D₁₄₉	semidirect
235. C₂³	#14558736	semidirect

Maximal subgroups

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

subgroup	index
#23750301	2
#23750265	2
#23750251	2
#23750213	2
#23750187	2
#23750136	2
C₂×#18011880	2
#23750098	2
C₂²×C₂²⋊C₄×C₁₄₉	2
#3490	149