Subgroups — #34058467

Order 19072.

1748 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	C₂×#18011956	C₂	1	2	yes	maximal, complemented
2	#23750295	C₂	1	4	yes	maximal, complemented
3	#23750260	C₂	1	4	yes	maximal, complemented
4	D₄×C₂²⋊D₁₄₉	C₂	1	8	yes	maximal, complemented
5	#23750192	C₂	1	2	yes	maximal, complemented
6	#23750188	C₂	1	4	yes	maximal, complemented
7	#23750175	C₂	1	1	yes	maximal, characteristic, complemented
8	#23750173	C₂	1	2	yes	maximal, complemented
9	#127×D₁₄₉	C₂	1	2	yes	maximal, complemented
10	C₄×#14558736	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#127×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#18011956	—	2	16	no	—
13	C₂×#14558736	C₂²	1	1	yes	characteristic
14	C₂×#14558736	C₂²	1	2	yes	complemented
15	C₂×#14558735	C₂²	1	4	yes	complemented
16	C₂×#14558735	C₂²	1	4	yes	complemented
17	C₂²×C₂²⋊D₁₄₉	C₂²	1	8	yes	complemented
18	C₂×D₄×D₁₄₉	C₂²	1	4	yes	complemented
19	C₂×D₄×D₁₄₉	C₂²	1	4	yes	complemented
20	C₂×D₄×D₁₄₉	C₂²	1	8	yes	complemented
21	#18011911	C₂²	1	1	yes	characteristic, complemented
22	#18011908	C₂²	1	1	yes	characteristic, complemented
23	#18011907	C₂²	1	4	yes	complemented
24	#18011907	C₂²	1	8	yes	complemented
25	#18011899	C₂²	1	2	yes	complemented
26	#18011898	C₂²	1	4	yes	complemented
27	#18011885	C₂²	1	4	yes	complemented
28	#18011883	C₂²	1	16	yes	complemented
29	#18011882	C₂²	1	8	yes	complemented
30	#18011882	C₂²	1	8	yes	complemented
31	#18011880	C₂²	1	8	yes	complemented
32	#18011879	C₂²	1	16	yes	complemented
33	C₄×Dic₂₉₈	C₂²	1	2	yes	complemented
34	C₄×D₅₉₆	C₂²	1	4	yes	complemented
35	C₄×D₅₉₆	C₂²	1	2	yes	complemented
36	D₄×Dic₁₄₉	C₂²	1	8	yes	complemented
37	C₄×C₂²⋊D₁₄₉	C₂²	1	4	yes	complemented
38	#18011870	C₂²	1	1	yes	characteristic, complemented
39	#18011868	C₂²	1	2	yes	complemented
40	C₄²×D₁₄₉	C₂²	1	2	yes	complemented
41	C₂²×D₄×C₁₄₉	C₂²	1	2	yes	complemented
42	C₂²×D₄×C₁₄₉	C₂²	1	4	yes	complemented
43	#127×C₁₄₉	C₂²	1	2	yes	complemented
44	#127×C₁₄₉	C₂²	1	2	yes	complemented
45	#127×C₁₄₉	C₂²	1	4	yes	complemented
46	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic, complemented
47	C₂³×D₁₄₉	C₂³	1	8	yes	—
48	#14558736	—	2	8	no	—
49	#14558736	D₄	1	8	yes	complemented
50	#14558735	—	2	16	no	—
51	#14558735	—	2	16	no	—
52	C₂×Dic₂₉₈	C₂³	1	2	yes	—
53	C₂×Dic₂₉₈	C₂³	1	1	yes	characteristic
54	C₂×D₅₉₆	C₂³	1	2	yes	—
55	C₂×D₅₉₆	C₂³	1	4	yes	—
56	C₂×D₅₉₆	C₂³	1	1	yes	characteristic
57	C₂×C₂²⋊D₁₄₉	—	2	32	no	—
58	C₂×C₂²⋊D₁₄₉	C₂³	1	8	yes	—
59	C₂×C₂²⋊D₁₄₉	C₂³	1	8	yes	—
60	C₂×C₂²⋊D₁₄₉	C₂³	1	4	yes	—
61	C₂×C₂²⋊D₁₄₉	C₂³	1	16	yes	—
62	C₂×C₂²⋊D₁₄₉	C₂³	1	2	yes	—
63	D₄×D₁₄₉	—	2	16	no	—
64	D₄×D₁₄₉	—	2	16	no	—
65	D₄×D₁₄₉	—	4	16	no	—
66	C₂²×Dic₁₄₉	C₂³	1	8	yes	—
67	C₂×C₄×D₁₄₉	C₂³	1	4	yes	—
68	C₂×C₄×D₁₄₉	C₂³	1	4	yes	—
69	C₂×C₄×D₁₄₉	C₂³	1	2	yes	—
70	C₄⋊Dic₁₄₉	C₂³	1	4	yes	—
71	C₄⋊Dic₁₄₉	C₂³	1	2	yes	—
72	#14558715	C₂³	1	4	yes	—
73	C₂²⋊Dic₁₄₉	C₂³	1	2	yes	—
74	C₂²⋊Dic₁₄₉	C₂³	1	16	yes	—
75	#14558713	C₂³	1	4	yes	—
76	#14558713	C₂³	1	8	yes	—
77	C₄×Dic₁₄₉	C₂³	1	2	yes	—
78	C₄×Dic₁₄₉	C₂³	1	4	yes	—
79	C₂⁴×C₁₄₉	C₂³	1	4	yes	—
80	C₂×D₄×C₁₄₉	—	2	8	no	—
81	C₂×D₄×C₁₄₉	—	4	8	no	—
82	C₂×D₄×C₁₄₉	C₂³	1	4	yes	—
83	C₂×D₄×C₁₄₉	C₂³	1	8	yes	—
84	C₂×D₄×C₁₄₉	C₂³	1	4	yes	—
85	C₂×D₄×C₁₄₉	C₂³	1	8	yes	—
86	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
87	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
88	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
89	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
90	C₄²×C₁₄₉	C₂³	1	2	yes	—
91	C₂²×D₁₄₉	—	2	16	no	—
92	C₂²×D₁₄₉	—	4	16	no	—
93	C₂²×D₁₄₉	C₂⁴	1	4	yes	—
94	C₂²×D₁₄₉	C₂⁴	1	2	yes	—
95	Dic₂₉₈	—	2	4	no	—
96	Dic₂₉₈	C₂×D₄	1	4	yes	complemented
97	D₅₉₆	—	2	4	no	—
98	D₅₉₆	—	2	8	no	—
99	D₅₉₆	C₂×D₄	1	4	yes	complemented
100	C₂²⋊D₁₄₉	—	2	8	no	—
101	C₂²⋊D₁₄₉	—	2	16	no	—
102	C₂²⋊D₁₄₉	—	2	16	no	—
103	C₂²⋊D₁₄₉	—	4	16	no	—
104	C₂²⋊D₁₄₉	C₂×D₄	1	8	yes	complemented
105	C₂×Dic₁₄₉	—	2	16	no	—
106	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
107	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
108	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
109	C₄×D₁₄₉	—	2	8	no	—
110	C₄×D₁₄₉	—	2	8	no	—
111	C₄×D₁₄₉	C₂×D₄	1	8	yes	complemented
112	C₂³×C₁₄₉	—	2	8	no	—
113	C₂³×C₁₄₉	—	4	8	no	—
114	C₂³×C₁₄₉	C₂⁴	1	8	yes	—
115	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
116	D₄×C₁₄₉	—	2	8	no	—
117	D₄×C₁₄₉	—	2	8	no	—
118	D₄×C₁₄₉	—	2	16	no	—
119	D₄×C₁₄₉	—	4	8	no	—
120	C₂×C₄×C₁₄₉	—	2	4	no	—
121	C₂×C₄×C₁₄₉	C₂×D₄	1	4	yes	complemented
122	C₂×C₄×C₁₄₉	C₂⁴	1	2	yes	—
123	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
124	C₂×C₄×C₁₄₉	C₂⁴	1	1	yes	characteristic
125	C₂×C₄×C₁₄₉	C₂⁴	1	2	yes	—
126	C₂²×C₁₄₉	—	2	2	no	—
127	C₂²×C₁₄₉	—	2	8	no	—
128	C₂²×C₁₄₉	—	2	16	no	—
129	C₂²×C₁₄₉	—	8	8	no	—
130	C₂²×C₁₄₉	C₂²×D₄	1	2	yes	—
131	C₂²×C₁₄₉	C₂⁵	1	1	yes	characteristic
132	C₂×D₁₄₉	—	2	4	no	—
133	C₂×D₁₄₉	—	2	4	no	—
134	C₂×D₁₄₉	—	2	8	no	—
135	C₂×D₁₄₉	—	8	16	no	—
136	C₂×D₁₄₉	C₂²×D₄	1	4	yes	—
137	C₄×C₁₄₉	—	2	2	no	—
138	C₄×C₁₄₉	C₂²×D₄	1	2	yes	—
139	C₄×C₁₄₉	C₂²×D₄	1	2	yes	—
140	C₄×C₁₄₉	H(5,𝔽₂)	1	4	yes	complemented
141	Dic₁₄₉	—	2	4	no	—
142	Dic₁₄₉	—	2	4	no	—
143	Dic₁₄₉	C₂²×D₄	1	4	yes	—
144	C₂×C₁₄₉	—	2	2	no	—
145	C₂×C₁₄₉	—	4	8	no	—
146	C₂×C₁₄₉	C₂³×D₄	1	1	yes	characteristic
147	C₂×C₁₄₉	C₂×H(5,𝔽₂)	1	2	yes	—
148	D₁₄₉	—	2	4	no	—
149	D₁₄₉	—	4	4	no	—
150	C₁₄₉	#3498	1	1	yes	characteristic, complemented, Sylow
151	#3498	—	149	1	no	maximal, Sylow
152	C₂×H(5,𝔽₂)	—	149	2	no	—
153	#550	—	149	4	no	—
154	D₄²	—	149	8	no	—
155	D₄²	—	149	4	no	—
156	#535	—	149	2	no	—
157	#535	—	149	4	no	—
158	#532	—	149	2	no	—
159	#532	—	149	1	no	—
160	C₂×#127	—	149	2	no	—
161	C₂×#127	D₁₄₉	1	1	yes	characteristic, complemented
162	C₄×#41	—	149	1	no	—
163	H(5,𝔽₂)	—	298	16	no	—
164	C₂×#41	—	149	4	no	—
165	C₂×#41	—	149	4	no	—
166	C₂×#41	—	149	2	no	—
167	C₂×#41	—	149	1	no	—
168	C₂²×D₄	—	149	8	no	—
169	C₂²×D₄	—	149	4	no	—
170	C₂²×D₄	—	149	4	no	—
171	C₂²×D₄	—	149	8	no	—
172	C₂²×D₄	C₂×D₁₄₉	1	2	yes	complemented
173	C₂²×D₄	C₂×D₁₄₉	1	4	yes	complemented
174	C₄⋊Q₈	—	149	1	no	—
175	#127	—	149	4	no	—
176	#127	—	149	8	no	—
177	#127	—	149	1	no	—
178	#127	C₂×D₁₄₉	1	2	yes	complemented
179	#127	C₂×D₁₄₉	1	2	yes	complemented
180	#127	C₂×D₁₄₉	1	4	yes	complemented
181	#124	—	149	4	no	—
182	#124	—	149	2	no	—
183	#121	—	149	8	no	—
184	#121	—	149	8	no	—
185	#121	—	149	16	no	—
186	#121	—	149	4	no	—
187	#120	—	149	8	no	—
188	#120	—	149	16	no	—
189	C₄×Q₈	—	149	2	no	—
190	C₄×D₄	—	149	8	no	—
191	C₄×D₄	—	149	4	no	—
192	C₄×D₄	—	149	2	no	—
193	C₄×D₄	—	149	4	no	—
194	#117	—	149	1	no	—
195	#117	—	149	2	no	—
196	C₂×C₄²	—	149	2	no	—
197	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic, complemented
198	C₂⁴	—	149	8	no	—
199	C₂⁴	C₂²×D₁₄₉	1	4	yes	—
200	#41	—	298	16	no	—
201	#41	—	298	16	no	—
202	#41	—	298	8	no	—
203	#41	—	149	8	no	—
204	C₂×Q₈	—	149	2	no	—
205	C₂×Q₈	—	149	1	no	—
206	C₂×D₄	—	2	8	no	—
207	C₂×D₄	—	4	8	no	—
208	C₂×D₄	—	298	16	no	—
209	C₂×D₄	—	298	32	no	—
210	C₂×D₄	—	298	16	no	—
211	C₂×D₄	—	596	16	no	—
212	C₂×D₄	—	149	8	no	—
213	C₂×D₄	—	149	4	no	—
214	C₂×D₄	—	149	8	no	—
215	C₂×D₄	—	149	16	no	—
216	C₂×D₄	—	149	2	no	—
217	C₂×D₄	—	149	4	no	—
218	C₂×D₄	—	149	2	no	—
219	C₂×D₄	—	149	1	no	—
220	C₂×D₄	C₂²×D₁₄₉	1	8	yes	—
221	C₂×D₄	C₂²×D₁₄₉	1	4	yes	—
222	C₂×D₄	C₂²×D₁₄₉	1	4	yes	—
223	C₂×D₄	C₂²×D₁₄₉	1	8	yes	—
224	C₂²×C₄	—	149	8	no	—
225	C₂²×C₄	—	149	4	no	—
226	C₂²×C₄	—	149	4	no	—
227	C₂²×C₄	—	149	2	no	—
228	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
229	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
230	C₄⋊C₄	—	149	4	no	—
231	C₄⋊C₄	—	149	2	no	—
232	C₄⋊C₄	—	149	4	no	—
233	C₂²⋊C₄	—	149	2	no	—
234	C₂²⋊C₄	—	149	16	no	—
235	C₂²⋊C₄	—	149	4	no	—
236	C₂²⋊C₄	—	149	8	no	—
237	C₄²	—	149	2	no	—
238	C₄²	—	149	4	no	—
239	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
240	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
241	C₄²	C₂²×D₁₄₉	1	2	yes	—
242	C₂³	—	2	8	no	—
243	C₂³	—	4	8	no	—
244	C₂³	—	298	16	no	—
245	C₂³	—	596	16	no	—
246	C₂³	—	149	4	no	—
247	C₂³	—	149	2	no	—
248	C₂³	C₂³×D₁₄₉	1	8	yes	—
249	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
250	Q₈	—	298	4	no	—
251	Q₈	—	149	4	no	—
252	D₄	—	2	8	no	—
253	D₄	—	2	8	no	—
254	D₄	—	2	16	no	—
255	D₄	—	4	8	no	—
256	D₄	—	298	16	no	—
257	D₄	—	298	8	no	—
258	D₄	—	298	16	no	—
259	D₄	—	298	4	no	—
260	D₄	—	298	8	no	—
261	D₄	—	596	16	no	—
262	D₄	—	149	8	no	—
263	D₄	—	149	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	298	16	no	—
266	C₂×C₄	—	298	8	no	—
267	C₂×C₄	—	298	8	no	—
268	C₂×C₄	—	149	4	no	—
269	C₂×C₄	—	149	4	no	—
270	C₂×C₄	—	149	2	no	—
271	C₂×C₄	—	149	8	no	—
272	C₂×C₄	D₄×D₁₄₉	1	4	yes	complemented
273	C₂×C₄	C₂³×D₁₄₉	1	2	yes	—
274	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
275	C₂×C₄	C₂³×D₁₄₉	1	1	yes	characteristic
276	C₂×C₄	C₂³×D₁₄₉	1	2	yes	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	8	no	—
279	C₂²	—	2	16	no	—
280	C₂²	—	8	8	no	—
281	C₂²	—	298	4	no	—
282	C₂²	—	298	4	no	—
283	C₂²	—	298	8	no	—
284	C₂²	—	149	4	no	—
285	C₂²	—	1192	16	no	—
286	C₂²	C₂×D₄×D₁₄₉	1	2	yes	—
287	C₂²	C₂⁴×D₁₄₉	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	298	4	no	—
290	C₄	—	298	4	no	—
291	C₄	—	149	4	no	—
292	C₄	C₂×D₄×D₁₄₉	1	2	yes	—
293	C₄	C₂×D₄×D₁₄₉	1	2	yes	—
294	C₄	#18011956	1	4	yes	complemented
295	C₂	—	2	2	no	—
296	C₂	—	4	8	no	—
297	C₂	—	298	4	no	—
298	C₂	—	596	4	no	—
299	C₂	C₂²×D₄×D₁₄₉	1	1	yes	characteristic
300	C₂	C₂×#18011956	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₄₉
commutator, socle, cosocle	C₂²×C₁₄₉	C₂⁵
Fitting	C₂×#127×C₁₄₉	C₂
hypercenter	C₂×#127	D₁₄₉
hypocenter	C₁₄₉	#3498
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
1. C₂×#18011956	C₂	semidirect
2. #23750295	C₂	semidirect
3. #23750260	C₂	semidirect
4. D₄×C₂²⋊D₁₄₉	C₂	semidirect
5. #23750192	C₂	semidirect
6. #23750188	C₂	semidirect
7. #23750175	C₂	semidirect
8. #23750173	C₂	semidirect
9. #127×D₁₄₉	C₂	semidirect
10. C₄×#14558736	C₂	semidirect
11. C₂×#127×C₁₄₉	C₂	semidirect
14. C₂×#14558736	C₂²	semidirect
15. C₂×#14558735	C₂²	semidirect
16. C₂×#14558735	C₂²	semidirect
17. C₂²×C₂²⋊D₁₄₉	C₂²	semidirect
18. C₂×D₄×D₁₄₉	C₂²	semidirect
19. C₂×D₄×D₁₄₉	C₂²	semidirect
20. C₂×D₄×D₁₄₉	C₂²	semidirect
21. #18011911	C₂²	semidirect
22. #18011908	C₂²	semidirect
23. #18011907	C₂²	semidirect
24. #18011907	C₂²	semidirect
25. #18011899	C₂²	semidirect
26. #18011898	C₂²	semidirect
27. #18011885	C₂²	semidirect
28. #18011883	C₂²	semidirect
29. #18011882	C₂²	semidirect
30. #18011882	C₂²	semidirect
31. #18011880	C₂²	semidirect
32. #18011879	C₂²	semidirect
33. C₄×Dic₂₉₈	C₂²	semidirect
34. C₄×D₅₉₆	C₂²	semidirect
35. C₄×D₅₉₆	C₂²	semidirect
36. D₄×Dic₁₄₉	C₂²	semidirect
37. C₄×C₂²⋊D₁₄₉	C₂²	semidirect
38. #18011870	C₂²	semidirect
39. #18011868	C₂²	semidirect
40. C₄²×D₁₄₉	C₂²	semidirect
41. C₂²×D₄×C₁₄₉	C₂²	semidirect
42. C₂²×D₄×C₁₄₉	C₂²	semidirect
43. #127×C₁₄₉	C₂²	semidirect
44. #127×C₁₄₉	C₂²	semidirect
45. #127×C₁₄₉	C₂²	semidirect
46. C₂×C₄²×C₁₄₉	C₂²	semidirect
49. #14558736	D₄	semidirect
96. Dic₂₉₈	C₂×D₄	semidirect
99. D₅₉₆	C₂×D₄	semidirect
104. C₂²⋊D₁₄₉	C₂×D₄	semidirect
111. C₄×D₁₄₉	C₂×D₄	semidirect
121. C₂×C₄×C₁₄₉	C₂×D₄	semidirect
140. C₄×C₁₄₉	H(5,𝔽₂)	semidirect
150. C₁₄₉	#3498	semidirect
161. C₂×#127	D₁₄₉	semidirect
172. C₂²×D₄	C₂×D₁₄₉	semidirect
173. C₂²×D₄	C₂×D₁₄₉	semidirect
178. #127	C₂×D₁₄₉	semidirect
179. #127	C₂×D₁₄₉	semidirect
180. #127	C₂×D₁₄₉	semidirect
197. C₂×C₄²	C₂×D₁₄₉	semidirect
272. C₂×C₄	D₄×D₁₄₉	semidirect
294. C₄	#18011956	semidirect

Maximal subgroups

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

subgroup	index
C₂×#18011956	2
#23750295	2
#23750260	2
D₄×C₂²⋊D₁₄₉	2
#23750192	2
#23750188	2
#23750175	2
#23750173	2
#127×D₁₄₉	2
C₄×#14558736	2
C₂×#127×C₁₄₉	2
#3498	149