Subgroups — #34058562

Order 19072.

1232 conjugacy classes of subgroups.

The subgroup lattice has 324 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	#23750296	C₂	1	4	yes	maximal, complemented
3	#23750265	C₂	1	8	yes	maximal, complemented
4	#23750226	C₂	1	4	yes	maximal, complemented
5	#23750194	C₂	1	1	yes	maximal, characteristic, complemented
6	#23750192	C₂	1	1	yes	maximal, characteristic, complemented
7	#23750178	C₂	1	1	yes	maximal, characteristic, complemented
8	#23750172	C₂	1	4	yes	maximal, complemented
9	C₂×#18011907	C₂	1	1	yes	maximal, characteristic, complemented
10	#23750099	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	C₂×#14558736	C₂²	1	8	yes	complemented
15	C₂²×D₅₉₆	C₂²	1	1	yes	characteristic
16	C₂²×C₂²⋊D₁₄₉	C₂²	1	4	yes	—
17	#18011910	C₂²	1	2	yes	complemented
18	#18011908	C₂²	1	2	yes	complemented
19	#18011907	C₂²	1	2	yes	—
20	#18011907	C₂²	1	2	yes	—
21	#18011907	C₂²	1	4	yes	—
22	#18011907	C₂²	1	2	yes	complemented
23	#18011902	C₂²	1	2	yes	complemented
24	#18011899	C₂²	1	2	yes	complemented
25	#18011898	C₂²	1	2	yes	complemented
26	#18011897	C₂²	1	8	yes	complemented
27	#18011894	C₂²	1	8	yes	complemented
28	#18011893	C₂²	1	8	yes	complemented
29	#18011888	C₂²	1	4	yes	—
30	#18011886	C₂²	1	4	yes	complemented
31	#18011885	C₂²	1	8	yes	complemented
32	#18011884	C₂²	1	8	yes	complemented
33	#18011884	C₂²	1	8	yes	complemented
34	#18011882	C₂²	1	4	yes	complemented
35	#18011881	C₂²	1	4	yes	—
36	#18011880	C₂²	1	4	yes	—
37	C₄×Dic₂₉₈	C₂²	1	4	yes	complemented
38	#18011876	C₂²	1	2	yes	complemented
39	C₄×D₅₉₆	C₂²	1	4	yes	complemented
40	#18011873	C₂²	1	2	yes	complemented
41	#18011873	C₂²	1	4	yes	complemented
42	C₄×C₂²⋊D₁₄₉	C₂²	1	8	yes	complemented
43	#18011871	C₂²	1	4	yes	—
44	#18011870	C₂²	1	1	yes	characteristic
45	#18011868	C₂²	1	2	yes	complemented
46	C₂²⋊C₄×D₁₄₉	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	complemented
51	C₄×D₄×C₁₄₉	C₂²	1	4	yes	complemented
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	C₂³×D₁₄₉	C₂³	1	2	yes	—
56	#14558736	—	2	4	no	—
57	#14558736	—	4	16	no	—
58	C₂×Dic₂₉₈	C₂³	1	4	yes	—
59	C₂×Dic₂₉₈	C₂³	1	1	yes	characteristic
60	C₂×D₅₉₆	—	2	4	no	—
61	C₂×D₅₉₆	C₂³	1	2	yes	—
62	C₂×D₅₉₆	C₂³	1	2	yes	—
63	C₂×D₅₉₆	C₂³	1	1	yes	characteristic
64	C₂×D₅₉₆	C₂³	1	8	yes	—
65	C₂×C₂²⋊D₁₄₉	—	4	8	no	—
66	C₂×C₂²⋊D₁₄₉	C₂³	1	8	yes	—
67	C₂×C₂²⋊D₁₄₉	C₂³	1	2	yes	—
68	C₂×C₂²⋊D₁₄₉	C₂³	1	8	yes	—
69	C₂×C₂²⋊D₁₄₉	C₂³	1	8	yes	—
70	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
71	C₂×C₄×D₁₄₉	C₂³	1	2	yes	—
72	C₂×C₄×D₁₄₉	C₂³	1	4	yes	—
73	C₂×C₄×D₁₄₉	C₂³	1	8	yes	—
74	C₄⋊Dic₁₄₉	C₂³	1	2	yes	—
75	C₄⋊Dic₁₄₉	C₂³	1	4	yes	—
76	#14558715	C₂³	1	4	yes	—
77	#14558715	C₂³	1	4	yes	—
78	#14558715	C₂³	1	4	yes	—
79	C₂²⋊Dic₁₄₉	C₂³	1	2	yes	—
80	C₂²⋊Dic₁₄₉	C₂³	1	8	yes	—
81	#14558713	C₂³	1	8	yes	—
82	#14558713	C₂³	1	4	yes	—
83	#14558713	C₂³	1	8	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	C₄×Dic₁₄₉	C₂³	1	2	yes	—
88	C₄×Dic₁₄₉	C₂³	1	4	yes	—
89	C₂⁴×C₁₄₉	C₂³	1	2	yes	—
90	C₂×D₄×C₁₄₉	—	2	4	no	—
91	C₂×D₄×C₁₄₉	C₂³	1	2	yes	—
92	C₂×D₄×C₁₄₉	C₂³	1	2	yes	—
93	C₂²×C₄×C₁₄₉	C₂³	1	8	yes	—
94	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
95	C₄⋊C₄×C₁₄₉	C₂³	1	2	yes	—
96	C₄⋊C₄×C₁₄₉	C₂³	1	2	yes	—
97	C₂²⋊C₄×C₁₄₉	C₂³	1	2	yes	—
98	C₂²⋊C₄×C₁₄₉	C₂³	1	2	yes	—
99	C₂²⋊C₄×C₁₄₉	C₂³	1	8	yes	—
100	C₄²×C₁₄₉	C₂³	1	2	yes	—
101	C₄²×C₁₄₉	C₂³	1	2	yes	—
102	C₂²×D₁₄₉	—	2	4	no	—
103	C₂²×D₁₄₉	—	4	4	no	—
104	C₂²×D₁₄₉	C₂⁴	1	2	yes	—
105	C₂²×D₁₄₉	C₂⁴	1	4	yes	—
106	Dic₂₉₈	—	2	2	no	—
107	Dic₂₉₈	—	2	8	no	—
108	D₅₉₆	—	2	4	no	—
109	D₅₉₆	—	2	4	no	—
110	D₅₉₆	—	2	2	no	—
111	D₅₉₆	—	4	8	no	—
112	C₂²⋊D₁₄₉	—	2	4	no	—
113	C₂²⋊D₁₄₉	—	2	16	no	—
114	C₂²⋊D₁₄₉	—	4	8	no	—
115	C₂²⋊D₁₄₉	—	4	8	no	—
116	C₂×Dic₁₄₉	—	2	4	no	—
117	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
118	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
119	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
120	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
121	C₄×D₁₄₉	—	2	4	no	—
122	C₄×D₁₄₉	—	2	8	no	—
123	C₄×D₁₄₉	—	4	8	no	—
124	C₂³×C₁₄₉	—	2	4	no	—
125	C₂³×C₁₄₉	—	4	4	no	—
126	C₂³×C₁₄₉	C₂⁴	1	4	yes	—
127	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
128	D₄×C₁₄₉	—	2	4	no	—
129	D₄×C₁₄₉	—	2	4	no	—
130	C₂×C₄×C₁₄₉	—	4	8	no	—
131	C₂×C₄×C₁₄₉	#41	1	4	yes	complemented
132	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
133	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
134	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
135	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
136	C₂²×C₁₄₉	—	2	2	no	—
137	C₂²×C₁₄₉	—	2	4	no	—
138	C₂²×C₁₄₉	—	2	8	no	—
139	C₂²×C₁₄₉	—	8	4	no	—
140	C₂²×C₁₄₉	C₂×#41	1	2	yes	—
141	C₂²×C₁₄₉	C₂⁵	1	1	yes	characteristic
142	C₂×D₁₄₉	—	2	2	no	—
143	C₂×D₁₄₉	—	2	4	no	—
144	C₂×D₁₄₉	—	2	4	no	—
145	C₂×D₁₄₉	—	2	8	no	—
146	C₂×D₁₄₉	—	8	4	no	—
147	C₄×C₁₄₉	—	2	4	no	—
148	C₄×C₁₄₉	—	2	4	no	—
149	C₄×C₁₄₉	C₂×#41	1	2	yes	—
150	C₄×C₁₄₉	C₂×#41	1	2	yes	—
151	Dic₁₄₉	—	2	2	no	—
152	Dic₁₄₉	—	2	4	no	—
153	Dic₁₄₉	—	2	2	no	—
154	Dic₁₄₉	H(5,𝔽₂)	1	4	yes	—
155	C₂×C₁₄₉	—	2	2	no	—
156	C₂×C₁₄₉	—	4	4	no	—
157	C₂×C₁₄₉	C₂²×#41	1	1	yes	characteristic
158	C₂×C₁₄₉	C₂×H(5,𝔽₂)	1	2	yes	—
159	D₁₄₉	—	4	2	no	—
160	D₁₄₉	—	4	4	no	—
161	C₁₄₉	#3508	1	1	yes	characteristic, complemented, Sylow
162	#3508	—	149	1	no	maximal, Sylow
163	#559	—	149	4	no	—
164	#550	—	149	4	no	—
165	#546	—	149	8	no	—
166	#540	—	149	4	no	—
167	#535	—	149	1	no	—
168	#535	—	149	1	no	—
169	#533	—	149	1	no	—
170	#532	—	149	4	no	—
171	C₂×#127	—	149	1	no	—
172	#518	—	149	1	no	—
173	#518	D₁₄₉	1	1	yes	characteristic, complemented
174	C₄×#41	—	149	1	no	—
175	C₂×#41	—	149	1	no	—
176	C₂×#41	—	149	8	no	—
177	C₂²×D₄	—	149	4	no	—
178	C₂²×D₄	—	149	1	no	—
179	C₂²×D₄	C₂×D₁₄₉	1	1	yes	characteristic
180	C₄⋊Q₈	—	149	2	no	—
181	#127	—	149	2	no	—
182	#127	—	149	2	no	—
183	#127	—	149	4	no	—
184	#127	—	149	2	no	—
185	#127	—	149	2	no	—
186	#125	—	149	2	no	—
187	#124	—	149	2	no	—
188	#124	—	149	8	no	—
189	#124	—	149	2	no	—
190	#123	—	149	8	no	—
191	#123	—	149	8	no	—
192	C₂²⋊Q₈	—	149	4	no	—
193	#121	—	149	4	no	—
194	#121	—	149	8	no	—
195	#121	—	149	4	no	—
196	#121	—	149	8	no	—
197	#121	—	149	8	no	—
198	#120	—	149	4	no	—
199	#120	—	149	4	no	—
200	C₄×Q₈	—	149	2	no	—
201	C₄×Q₈	—	149	4	no	—
202	C₄×D₄	—	149	8	no	—
203	C₄×D₄	—	149	2	no	—
204	C₄×D₄	—	149	4	no	—
205	C₄×D₄	—	149	4	no	—
206	C₄×D₄	—	149	4	no	—
207	C₄×D₄	C₂×D₁₄₉	1	4	yes	complemented
208	C₄×D₄	C₂×D₁₄₉	1	4	yes	complemented
209	#117	—	149	1	no	—
210	#117	—	149	2	no	—
211	#117	C₂×D₁₄₉	1	1	yes	characteristic
212	#117	C₂×D₁₄₉	1	1	yes	characteristic
213	C₂×C₂²⋊C₄	—	149	4	no	—
214	C₂×C₂²⋊C₄	C₂×D₁₄₉	1	4	yes	—
215	C₂×C₄²	—	149	1	no	—
216	C₂×C₄²	—	149	2	no	—
217	C₂⁴	—	149	2	no	—
218	C₂⁴	C₂²×D₁₄₉	1	2	yes	—
219	#41	—	298	4	no	—
220	#41	—	596	16	no	—
221	C₂×Q₈	—	149	1	no	—
222	C₂×Q₈	—	149	4	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	298	4	no	—
225	C₂×D₄	—	596	8	no	—
226	C₂×D₄	—	149	2	no	—
227	C₂×D₄	—	149	8	no	—
228	C₂×D₄	—	149	8	no	—
229	C₂×D₄	—	149	8	no	—
230	C₂×D₄	—	149	2	no	—
231	C₂×D₄	—	149	2	no	—
232	C₂×D₄	—	149	1	no	—
233	C₂×D₄	—	149	8	no	—
234	C₂×D₄	C₂²×D₁₄₉	1	2	yes	—
235	C₂×D₄	C₂²×D₁₄₉	1	2	yes	—
236	C₂²×C₄	—	149	2	no	—
237	C₂²×C₄	—	149	2	no	—
238	C₂²×C₄	—	149	4	no	—
239	C₂²×C₄	—	149	8	no	—
240	C₂²×C₄	C₂²×D₁₄₉	1	8	yes	—
241	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
242	C₄⋊C₄	—	149	2	no	—
243	C₄⋊C₄	—	149	4	no	—
244	C₄⋊C₄	—	149	4	no	—
245	C₄⋊C₄	—	149	4	no	—
246	C₄⋊C₄	—	149	4	no	—
247	C₄⋊C₄	C₂²×D₁₄₉	1	2	yes	—
248	C₄⋊C₄	C₂²×D₁₄₉	1	2	yes	—
249	C₂²⋊C₄	—	149	2	no	—
250	C₂²⋊C₄	—	149	8	no	—
251	C₂²⋊C₄	—	149	8	no	—
252	C₂²⋊C₄	—	149	4	no	—
253	C₂²⋊C₄	—	149	8	no	—
254	C₂²⋊C₄	C₂²×D₁₄₉	1	2	yes	—
255	C₂²⋊C₄	C₂²×D₁₄₉	1	2	yes	—
256	C₂²⋊C₄	C₂²×D₁₄₉	1	8	yes	—
257	C₄²	—	149	1	no	—
258	C₄²	—	149	1	no	—
259	C₄²	—	149	4	no	—
260	C₄²	—	149	2	no	—
261	C₄²	—	149	2	no	—
262	C₄²	C₂²×D₁₄₉	1	2	yes	—
263	C₄²	C₂²×D₁₄₉	1	2	yes	—
264	C₂³	—	2	4	no	—
265	C₂³	—	4	4	no	—
266	C₂³	—	298	4	no	—
267	C₂³	—	596	4	no	—
268	C₂³	—	149	4	no	—
269	C₂³	—	149	2	no	—
270	C₂³	C₂³×D₁₄₉	1	4	yes	—
271	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
272	Q₈	—	298	8	no	—
273	Q₈	—	298	2	no	—
274	D₄	—	2	4	no	—
275	D₄	—	2	4	no	—
276	D₄	—	298	4	no	—
277	D₄	—	298	16	no	—
278	D₄	—	298	4	no	—
279	D₄	—	298	4	no	—
280	D₄	—	298	2	no	—
281	D₄	—	596	8	no	—
282	D₄	—	596	8	no	—
283	D₄	—	596	8	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	—	298	8	no	—
286	C₂×C₄	—	298	4	no	—
287	C₂×C₄	—	298	4	no	—
288	C₂×C₄	—	596	8	no	—
289	C₂×C₄	—	149	2	no	—
290	C₂×C₄	—	149	2	no	—
291	C₂×C₄	—	149	4	no	—
292	C₂×C₄	—	149	2	no	—
293	C₂×C₄	#14558736	1	4	yes	complemented
294	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
295	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
296	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
298	C₂²	—	2	2	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	8	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	298	2	no	—
303	C₂²	—	298	4	no	—
304	C₂²	—	298	4	no	—
305	C₂²	—	298	8	no	—
306	C₂²	—	1192	4	no	—
307	C₂²	C₂×#14558736	1	2	yes	—
308	C₂²	C₂⁴×D₁₄₉	1	1	yes	characteristic
309	C₄	—	2	4	no	—
310	C₄	—	2	4	no	—
311	C₄	—	298	2	no	—
312	C₄	—	298	2	no	—
313	C₄	—	298	4	no	—
314	C₄	—	149	4	no	—
315	C₄	C₂×#14558736	1	2	yes	—
316	C₄	C₂×#14558736	1	2	yes	—
317	C₂	—	2	2	no	—
318	C₂	—	4	4	no	—
319	C₂	—	596	4	no	—
320	C₂	—	596	2	no	—
321	C₂	C₂²×#14558736	1	1	yes	characteristic
322	C₂	H(5,𝔽₂)×D₁₄₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₄₉
commutator, socle, cosocle	C₂²×C₁₄₉	C₂⁵
Fitting	#518×C₁₄₉	C₂
hypercenter	#518	D₁₄₉
hypocenter	C₁₄₉	#3508
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. #23750296	C₂	semidirect
3. #23750265	C₂	semidirect
4. #23750226	C₂	semidirect
5. #23750194	C₂	semidirect
6. #23750192	C₂	semidirect
7. #23750178	C₂	semidirect
8. #23750172	C₂	semidirect
9. C₂×#18011907	C₂	semidirect
10. #23750099	C₂	semidirect
11. #23750093	C₂	semidirect
12. #518×C₁₄₉	C₂	semidirect
14. C₂×#14558736	C₂²	semidirect
17. #18011910	C₂²	semidirect
18. #18011908	C₂²	semidirect
22. #18011907	C₂²	semidirect
23. #18011902	C₂²	semidirect
24. #18011899	C₂²	semidirect
25. #18011898	C₂²	semidirect
26. #18011897	C₂²	semidirect
27. #18011894	C₂²	semidirect
28. #18011893	C₂²	semidirect
30. #18011886	C₂²	semidirect
31. #18011885	C₂²	semidirect
32. #18011884	C₂²	semidirect
33. #18011884	C₂²	semidirect
34. #18011882	C₂²	semidirect
37. C₄×Dic₂₉₈	C₂²	semidirect
38. #18011876	C₂²	semidirect
39. C₄×D₅₉₆	C₂²	semidirect
40. #18011873	C₂²	semidirect
41. #18011873	C₂²	semidirect
42. C₄×C₂²⋊D₁₄₉	C₂²	semidirect
45. #18011868	C₂²	semidirect
48. C₄²×D₁₄₉	C₂²	semidirect
50. C₄×D₄×C₁₄₉	C₂²	semidirect
51. C₄×D₄×C₁₄₉	C₂²	semidirect
131. C₂×C₄×C₁₄₉	#41	semidirect
161. C₁₄₉	#3508	semidirect
173. #518	D₁₄₉	semidirect
207. C₄×D₄	C₂×D₁₄₉	semidirect
208. C₄×D₄	C₂×D₁₄₉	semidirect
293. C₂×C₄	#14558736	semidirect

Maximal subgroups

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

subgroup	index
#23750358	2
#23750296	2
#23750265	2
#23750226	2
#23750194	2
#23750192	2
#23750178	2
#23750172	2
C₂×#18011907	2
#23750099	2
#23750093	2
#518×C₁₄₉	2
#3508	149