Subgroups — #34058582

Order 19072.

1040 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23750358	C₂	1	4	yes	maximal, complemented
2	#23750345	C₂	1	4	yes	maximal, complemented
3	#23750301	C₂	1	8	yes	maximal, complemented
4	#23750240	C₂	1	4	yes	maximal, complemented
5	#23750201	C₂	1	1	yes	maximal, characteristic, complemented
6	#23750192	C₂	1	1	yes	maximal, characteristic
7	#23750178	C₂	1	1	yes	maximal, characteristic, complemented
8	#23750157	C₂	1	4	yes	maximal, complemented
9	C₂×#18011898	C₂	1	1	yes	maximal, characteristic, complemented
10	#23750102	C₂	1	1	yes	maximal, characteristic, complemented
11	#23750093	C₂	1	1	yes	maximal, characteristic, complemented
12	#518×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#14558736	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₉₈	C₂²	1	1	yes	characteristic
15	#18011911	C₂²	1	2	yes	complemented
16	#18011910	C₂²	1	2	yes	complemented
17	#18011907	C₂²	1	2	yes	—
18	#18011904	C₂²	1	8	yes	complemented
19	#18011904	C₂²	1	8	yes	complemented
20	#18011902	C₂²	1	2	yes	complemented
21	#18011899	C₂²	1	2	yes	complemented
22	#18011898	C₂²	1	2	yes	—
23	#18011898	C₂²	1	2	yes	—
24	#18011898	C₂²	1	2	yes	—
25	#18011898	C₂²	1	4	yes	—
26	#18011897	C₂²	1	8	yes	complemented
27	#18011895	C₂²	1	4	yes	—
28	#18011894	C₂²	1	8	yes	complemented
29	#18011893	C₂²	1	8	yes	complemented
30	#18011891	C₂²	1	4	yes	complemented
31	#18011890	C₂²	1	8	yes	complemented
32	#18011888	C₂²	1	4	yes	—
33	#18011882	C₂²	1	4	yes	—
34	#18011880	C₂²	1	4	yes	—
35	C₄×Dic₂₉₈	C₂²	1	4	yes	complemented
36	C₄×Dic₂₉₈	C₂²	1	4	yes	complemented
37	#18011876	C₂²	1	2	yes	complemented
38	#18011876	C₂²	1	4	yes	complemented
39	#18011873	C₂²	1	2	yes	complemented
40	C₄×C₂²⋊D₁₄₉	C₂²	1	8	yes	complemented
41	#18011871	C₂²	1	4	yes	—
42	#18011870	C₂²	1	1	yes	characteristic
43	#18011870	C₂²	1	8	yes	complemented
44	#18011868	C₂²	1	2	yes	complemented
45	#18011867	C₂²	1	4	yes	—
46	C₂×C₂²⋊Dic₁₄₉	C₂²	1	4	yes	—
47	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
48	C₄²×D₁₄₉	C₂²	1	2	yes	complemented
49	C₂²×D₄×C₁₄₉	C₂²	1	1	yes	characteristic
50	C₄×D₄×C₁₄₉	C₂²	1	4	yes	—
51	C₄×D₄×C₁₄₉	C₂²	1	4	yes	—
52	#117×C₁₄₉	C₂²	1	1	yes	characteristic
53	#117×C₁₄₉	C₂²	1	1	yes	characteristic
54	C₂×C₂²⋊C₄×C₁₄₉	C₂²	1	4	yes	—
55	#14558736	—	2	4	no	—
56	C₂×Dic₂₉₈	—	2	4	no	—
57	C₂×Dic₂₉₈	C₂³	1	1	yes	characteristic
58	C₂×Dic₂₉₈	C₂³	1	2	yes	—
59	C₂×Dic₂₉₈	C₂³	1	2	yes	—
60	C₂×Dic₂₉₈	C₂³	1	4	yes	—
61	C₂×D₅₉₆	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₄₉	C₂³	1	2	yes	—
63	C₂×C₂²⋊D₁₄₉	C₂³	1	8	yes	—
64	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
65	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
66	C₂×C₄×D₁₄₉	C₂³	1	2	yes	—
67	C₂×C₄×D₁₄₉	C₂³	1	4	yes	—
68	C₄⋊Dic₁₄₉	C₂³	1	8	yes	—
69	C₄⋊Dic₁₄₉	C₂³	1	2	yes	—
70	C₄⋊Dic₁₄₉	C₂³	1	4	yes	—
71	#14558715	C₂³	1	4	yes	—
72	#14558715	C₂³	1	4	yes	—
73	#14558715	C₂³	1	4	yes	—
74	#14558715	C₂³	1	8	yes	—
75	C₂²⋊Dic₁₄₉	C₂³	1	8	yes	—
76	C₂²⋊Dic₁₄₉	C₂³	1	8	yes	—
77	C₂²⋊Dic₁₄₉	C₂³	1	8	yes	—
78	C₂²⋊Dic₁₄₉	C₂³	1	2	yes	—
79	#14558713	C₂³	1	4	yes	—
80	#14558713	C₂³	1	8	yes	—
81	C₄×Dic₁₄₉	C₂³	1	4	yes	—
82	C₄×Dic₁₄₉	C₂³	1	8	yes	—
83	C₄×Dic₁₄₉	C₂³	1	2	yes	—
84	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
85	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
86	C₄×Dic₁₄₉	C₂³	1	2	yes	—
87	C₂⁴×C₁₄₉	C₂³	1	2	yes	—
88	C₂×D₄×C₁₄₉	—	2	4	no	—
89	C₂×D₄×C₁₄₉	C₂³	1	2	yes	—
90	C₂×D₄×C₁₄₉	C₂³	1	2	yes	—
91	C₂²×C₄×C₁₄₉	C₂³	1	8	yes	—
92	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₁₄₉	C₂³	1	2	yes	—
94	C₄⋊C₄×C₁₄₉	C₂³	1	2	yes	—
95	C₂²⋊C₄×C₁₄₉	C₂³	1	8	yes	—
96	C₂²⋊C₄×C₁₄₉	C₂³	1	2	yes	—
97	C₂²⋊C₄×C₁₄₉	C₂³	1	2	yes	—
98	C₄²×C₁₄₉	C₂³	1	2	yes	—
99	C₄²×C₁₄₉	C₂³	1	2	yes	—
100	C₂²×D₁₄₉	C₂⁴	1	2	yes	—
101	Dic₂₉₈	—	2	4	no	—
102	Dic₂₉₈	—	2	4	no	—
103	Dic₂₉₈	—	2	2	no	—
104	Dic₂₉₈	—	4	4	no	—
105	D₅₉₆	—	2	2	no	—
106	C₂²⋊D₁₄₉	—	2	4	no	—
107	C₂²⋊D₁₄₉	—	4	8	no	—
108	C₂×Dic₁₄₉	—	2	4	no	—
109	C₂×Dic₁₄₉	—	4	2	no	—
110	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
111	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
112	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
113	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
114	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
115	C₄×D₁₄₉	—	2	4	no	—
116	C₄×D₁₄₉	—	4	4	no	—
117	C₂³×C₁₄₉	—	2	4	no	—
118	C₂³×C₁₄₉	—	4	4	no	—
119	C₂³×C₁₄₉	C₂⁴	1	4	yes	—
120	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
121	D₄×C₁₄₉	—	2	4	no	—
122	D₄×C₁₄₉	—	2	4	no	—
123	C₂×C₄×C₁₄₉	—	4	8	no	—
124	C₂×C₄×C₁₄₉	#41	1	4	yes	—
125	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
126	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
127	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
128	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
129	C₂²×C₁₄₉	—	2	2	no	—
130	C₂²×C₁₄₉	—	2	4	no	—
131	C₂²×C₁₄₉	—	2	8	no	—
132	C₂²×C₁₄₉	—	8	4	no	—
133	C₂²×C₁₄₉	C₂×#41	1	2	yes	—
134	C₂²×C₁₄₉	C₂⁵	1	1	yes	characteristic
135	C₂×D₁₄₉	—	2	2	no	—
136	C₂×D₁₄₉	—	2	4	no	—
137	C₄×C₁₄₉	—	2	4	no	—
138	C₄×C₁₄₉	—	2	4	no	—
139	C₄×C₁₄₉	C₂×#41	1	2	yes	—
140	C₄×C₁₄₉	C₂×#41	1	2	yes	—
141	Dic₁₄₉	—	2	2	no	—
142	Dic₁₄₉	—	2	4	no	—
143	Dic₁₄₉	—	2	2	no	—
144	Dic₁₄₉	—	2	2	no	—
145	Dic₁₄₉	—	2	4	no	—
146	C₂×C₁₄₉	—	2	2	no	—
147	C₂×C₁₄₉	—	4	4	no	—
148	C₂×C₁₄₉	C₂²×#41	1	1	yes	characteristic
149	C₂×C₁₄₉	#585	1	2	yes	—
150	D₁₄₉	—	4	2	no	—
151	C₁₄₉	#3510	1	1	yes	characteristic, complemented, Sylow
152	#3510	—	149	1	no	maximal, Sylow
153	#559	—	149	4	no	—
154	#556	—	149	4	no	—
155	#551	—	149	8	no	—
156	#542	—	149	4	no	—
157	#536	—	149	1	no	—
158	#535	—	149	1	no	—
159	#533	—	149	1	no	—
160	#529	—	149	4	no	—
161	C₂×#124	—	149	1	no	—
162	#519	—	149	1	no	—
163	#518	D₁₄₉	1	1	yes	characteristic, complemented
164	C₄×#41	—	149	1	no	—
165	C₂×#41	—	149	1	no	—
166	C₂²×Q₈	—	149	1	no	—
167	C₂²×D₄	C₂×D₁₄₉	1	1	yes	characteristic
168	C₄⋊Q₈	—	149	2	no	—
169	C₄⋊Q₈	—	149	2	no	—
170	#127	—	149	2	no	—
171	#126	—	149	8	no	—
172	#126	—	149	8	no	—
173	#125	—	149	2	no	—
174	#124	—	149	2	no	—
175	#124	—	149	2	no	—
176	#124	—	149	4	no	—
177	#124	—	149	2	no	—
178	#124	—	149	8	no	—
179	#124	—	149	2	no	—
180	#123	—	149	8	no	—
181	#123	—	149	8	no	—
182	#123	—	149	4	no	—
183	C₂²⋊Q₈	—	149	8	no	—
184	C₂²⋊Q₈	—	149	4	no	—
185	C₂²⋊Q₈	—	149	4	no	—
186	#121	—	149	4	no	—
187	#120	—	149	4	no	—
188	C₄×Q₈	—	149	2	no	—
189	C₄×Q₈	—	149	4	no	—
190	C₄×Q₈	—	149	4	no	—
191	C₄×Q₈	—	149	4	no	—
192	C₄×D₄	—	149	8	no	—
193	C₄×D₄	—	149	2	no	—
194	C₄×D₄	—	149	4	no	—
195	C₄×D₄	C₂×D₁₄₉	1	4	yes	—
196	C₄×D₄	C₂×D₁₄₉	1	4	yes	—
197	#117	—	149	1	no	—
198	#117	—	149	8	no	—
199	#117	—	149	4	no	—
200	#117	—	149	2	no	—
201	#117	C₂×D₁₄₉	1	1	yes	characteristic
202	#117	C₂×D₁₄₉	1	1	yes	characteristic
203	C₂×C₂²⋊C₄	—	149	4	no	—
204	C₂×C₂²⋊C₄	C₂×D₁₄₉	1	4	yes	—
205	C₂×C₄²	—	149	1	no	—
206	C₂×C₄²	—	149	2	no	—
207	C₂⁴	C₂²×D₁₄₉	1	2	yes	—
208	#41	—	298	4	no	—
209	C₂×Q₈	—	298	4	no	—
210	C₂×Q₈	—	149	2	no	—
211	C₂×Q₈	—	149	2	no	—
212	C₂×Q₈	—	149	1	no	—
213	C₂×Q₈	—	149	4	no	—
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	—	149	2	no	—
216	C₂×D₄	—	149	8	no	—
217	C₂×D₄	—	149	1	no	—
218	C₂×D₄	C₂²×D₁₄₉	1	2	yes	—
219	C₂×D₄	C₂²×D₁₄₉	1	2	yes	—
220	C₂²×C₄	—	149	2	no	—
221	C₂²×C₄	—	149	2	no	—
222	C₂²×C₄	—	149	2	no	—
223	C₂²×C₄	—	149	4	no	—
224	C₂²×C₄	C₂²×D₁₄₉	1	8	yes	—
225	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
226	C₄⋊C₄	—	149	4	no	—
227	C₄⋊C₄	—	149	4	no	—
228	C₄⋊C₄	—	149	2	no	—
229	C₄⋊C₄	—	149	8	no	—
230	C₄⋊C₄	—	149	8	no	—
231	C₄⋊C₄	—	149	4	no	—
232	C₄⋊C₄	—	149	4	no	—
233	C₄⋊C₄	C₂²×D₁₄₉	1	2	yes	—
234	C₄⋊C₄	C₂²×D₁₄₉	1	2	yes	—
235	C₂²⋊C₄	—	149	8	no	—
236	C₂²⋊C₄	—	149	8	no	—
237	C₂²⋊C₄	—	149	2	no	—
238	C₂²⋊C₄	—	149	8	no	—
239	C₂²⋊C₄	—	149	4	no	—
240	C₂²⋊C₄	—	149	8	no	—
241	C₂²⋊C₄	C₂²×D₁₄₉	1	2	yes	—
242	C₂²⋊C₄	C₂²×D₁₄₉	1	2	yes	—
243	C₂²⋊C₄	C₂²×D₁₄₉	1	8	yes	—
244	C₄²	—	149	1	no	—
245	C₄²	—	149	4	no	—
246	C₄²	—	149	1	no	—
247	C₄²	—	149	2	no	—
248	C₄²	—	149	8	no	—
249	C₄²	—	149	2	no	—
250	C₄²	C₂²×D₁₄₉	1	2	yes	—
251	C₄²	C₂²×D₁₄₉	1	2	yes	—
252	C₂³	—	2	4	no	—
253	C₂³	—	4	4	no	—
254	C₂³	—	149	2	no	—
255	C₂³	C₂³×D₁₄₉	1	4	yes	—
256	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
257	Q₈	—	298	4	no	—
258	Q₈	—	298	4	no	—
259	Q₈	—	298	2	no	—
260	Q₈	—	596	4	no	—
261	D₄	—	2	4	no	—
262	D₄	—	2	4	no	—
263	D₄	—	298	4	no	—
264	D₄	—	298	2	no	—
265	D₄	—	596	8	no	—
266	C₂×C₄	—	4	8	no	—
267	C₂×C₄	—	298	4	no	—
268	C₂×C₄	—	298	4	no	—
269	C₂×C₄	—	596	4	no	—
270	C₂×C₄	—	596	2	no	—
271	C₂×C₄	—	149	2	no	—
272	C₂×C₄	—	149	4	no	—
273	C₂×C₄	—	149	4	no	—
274	C₂×C₄	—	149	2	no	—
275	C₂×C₄	—	149	2	no	—
276	C₂×C₄	#14558736	1	4	yes	—
277	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
278	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
279	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
280	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	8	no	—
284	C₂²	—	8	4	no	—
285	C₂²	—	298	2	no	—
286	C₂²	—	298	4	no	—
287	C₂²	C₂×#14558736	1	2	yes	—
288	C₂²	C₂⁴×D₁₄₉	1	1	yes	characteristic
289	C₄	—	2	4	no	—
290	C₄	—	2	4	no	—
291	C₄	—	298	2	no	—
292	C₄	—	298	4	no	—
293	C₄	—	298	2	no	—
294	C₄	—	298	2	no	—
295	C₄	—	298	4	no	—
296	C₄	C₂×#14558736	1	2	yes	—
297	C₄	C₂×#14558736	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	4	4	no	—
300	C₂	—	596	2	no	—
301	C₂	C₂²×#14558736	1	1	yes	characteristic
302	C₂	#23750460	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₄₉
commutator, socle, cosocle	C₂²×C₁₄₉	C₂⁵
Fitting	#518×C₁₄₉	C₂
prime radical	#23750192	C₂
hypercenter	#518	D₁₄₉
hypocenter	C₁₄₉	#3510
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ #518

Complements

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

N	H ≅ G/N	type
1. #23750358	C₂	semidirect
2. #23750345	C₂	semidirect
3. #23750301	C₂	semidirect
4. #23750240	C₂	semidirect
5. #23750201	C₂	semidirect
7. #23750178	C₂	semidirect
8. #23750157	C₂	semidirect
9. C₂×#18011898	C₂	semidirect
10. #23750102	C₂	semidirect
11. #23750093	C₂	semidirect
12. #518×C₁₄₉	C₂	semidirect
15. #18011911	C₂²	semidirect
16. #18011910	C₂²	semidirect
18. #18011904	C₂²	semidirect
19. #18011904	C₂²	semidirect
20. #18011902	C₂²	semidirect
21. #18011899	C₂²	semidirect
26. #18011897	C₂²	semidirect
28. #18011894	C₂²	semidirect
29. #18011893	C₂²	semidirect
30. #18011891	C₂²	semidirect
31. #18011890	C₂²	semidirect
35. C₄×Dic₂₉₈	C₂²	semidirect
36. C₄×Dic₂₉₈	C₂²	semidirect
37. #18011876	C₂²	semidirect
38. #18011876	C₂²	semidirect
39. #18011873	C₂²	semidirect
40. C₄×C₂²⋊D₁₄₉	C₂²	semidirect
43. #18011870	C₂²	semidirect
44. #18011868	C₂²	semidirect
48. C₄²×D₁₄₉	C₂²	semidirect
151. C₁₄₉	#3510	semidirect
163. #518	D₁₄₉	semidirect

Maximal subgroups

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

subgroup	index
#23750358	2
#23750345	2
#23750301	2
#23750240	2
#23750201	2
#23750192	2
#23750178	2
#23750157	2
C₂×#18011898	2
#23750102	2
#23750093	2
#518×C₁₄₉	2
#3510	149