Subgroups — #34058586

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	#23750361	C₂	1	4	yes	maximal, complemented
2	#23750344	C₂	1	4	yes	maximal, complemented
3	#23750306	C₂	1	8	yes	maximal, complemented
4	#23750242	C₂	1	4	yes	maximal, complemented
5	#23750196	C₂	1	1	yes	maximal, characteristic, complemented
6	#23750178	C₂	1	1	yes	maximal, characteristic, complemented
7	#23750156	C₂	1	4	yes	maximal, complemented
8	C₂×#18011898	C₂	1	1	yes	maximal, characteristic, complemented
9	#23750102	C₂	1	1	yes	maximal, characteristic, complemented
10	#23750100	C₂	1	1	yes	maximal, characteristic, complemented
11	#41×Dic₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#535×C₁₄₉	C₂	1	1	yes	maximal, characteristic
13	C₂²×Dic₂₉₈	C₂²	1	1	yes	characteristic
14	#18011910	C₂²	1	2	yes	complemented
15	#18011909	C₂²	1	2	yes	complemented
16	#18011904	C₂²	1	8	yes	complemented
17	#18011904	C₂²	1	8	yes	complemented
18	#18011902	C₂²	1	2	yes	complemented
19	#18011898	C₂²	1	2	yes	—
20	#18011898	C₂²	1	2	yes	—
21	#18011898	C₂²	1	4	yes	—
22	#18011898	C₂²	1	2	yes	complemented
23	#18011898	C₂²	1	8	yes	complemented
24	#18011892	C₂²	1	4	yes	—
25	#18011892	C₂²	1	8	yes	complemented
26	#18011892	C₂²	1	8	yes	complemented
27	#18011890	C₂²	1	4	yes	complemented
28	#18011888	C₂²	1	4	yes	—
29	#18011888	C₂²	1	8	yes	complemented
30	Q₈×Dic₁₄₉	C₂²	1	2	yes	complemented
31	C₄×Dic₂₉₈	C₂²	1	4	yes	complemented
32	C₄×Dic₂₉₈	C₂²	1	4	yes	complemented
33	#18011876	C₂²	1	4	yes	complemented
34	D₄×Dic₁₄₉	C₂²	1	4	yes	—
35	D₄×Dic₁₄₉	C₂²	1	4	yes	—
36	D₄×Dic₁₄₉	C₂²	1	4	yes	—
37	D₄×Dic₁₄₉	C₂²	1	8	yes	complemented
38	D₄×Dic₁₄₉	C₂²	1	2	yes	complemented
39	#18011870	C₂²	1	1	yes	characteristic
40	#18011870	C₂²	1	1	yes	characteristic
41	#18011870	C₂²	1	2	yes	complemented
42	#18011867	C₂²	1	4	yes	—
43	#18011867	C₂²	1	8	yes	complemented
44	C₂×C₂²⋊Dic₁₄₉	C₂²	1	4	yes	—
45	C₂×C₂²⋊Dic₁₄₉	C₂²	1	4	yes	—
46	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
47	C₂×C₄×Dic₁₄₉	C₂²	1	2	yes	complemented
48	C₂×#41×C₁₄₉	C₂²	1	1	yes	characteristic
49	C₂²×D₄×C₁₄₉	C₂²	1	1	yes	characteristic
50	#127×C₁₄₉	C₂²	1	2	yes	—
51	#124×C₁₄₉	C₂²	1	2	yes	—
52	#121×C₁₄₉	C₂²	1	4	yes	—
53	#120×C₁₄₉	C₂²	1	4	yes	—
54	#117×C₁₄₉	C₂²	1	1	yes	characteristic
55	C₂×Dic₂₉₈	—	2	4	no	—
56	C₂×Dic₂₉₈	C₂³	1	2	yes	—
57	C₂×Dic₂₉₈	C₂³	1	2	yes	—
58	C₂×Dic₂₉₈	C₂³	1	4	yes	—
59	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
60	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
61	C₂²×Dic₁₄₉	C₂³	1	4	yes	—
62	C₂²×Dic₁₄₉	C₂³	1	8	yes	—
63	C₄⋊Dic₁₄₉	C₂³	1	4	yes	—
64	C₄⋊Dic₁₄₉	C₂³	1	4	yes	—
65	C₄⋊Dic₁₄₉	C₂³	1	2	yes	—
66	C₄⋊Dic₁₄₉	C₂³	1	2	yes	—
67	#14558715	C₂³	1	8	yes	—
68	#14558715	C₂³	1	4	yes	—
69	#14558715	C₂³	1	8	yes	—
70	#14558715	C₂³	1	4	yes	—
71	C₂²⋊Dic₁₄₉	C₂³	1	8	yes	—
72	C₂²⋊Dic₁₄₉	C₂³	1	8	yes	—
73	C₂²⋊Dic₁₄₉	C₂³	1	8	yes	—
74	C₂²⋊Dic₁₄₉	C₂³	1	2	yes	—
75	C₂²⋊Dic₁₄₉	C₂³	1	2	yes	—
76	C₂²⋊Dic₁₄₉	C₂³	1	4	yes	—
77	C₂²⋊Dic₁₄₉	C₂³	1	8	yes	—
78	C₄×Dic₁₄₉	C₂³	1	4	yes	—
79	C₄×Dic₁₄₉	C₂³	1	8	yes	—
80	C₄×Dic₁₄₉	C₂³	1	2	yes	—
81	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
82	C₄×Dic₁₄₉	C₂³	1	1	yes	characteristic
83	C₄×Dic₁₄₉	C₂³	1	2	yes	—
84	C₄×Dic₁₄₉	C₂³	1	2	yes	—
85	C₂⁴×C₁₄₉	C₂³	1	2	yes	—
86	#41×C₁₄₉	—	2	4	no	—
87	C₂×Q₈×C₁₄₉	C₂³	1	1	yes	characteristic
88	C₂×D₄×C₁₄₉	—	2	4	no	—
89	C₂×D₄×C₁₄₉	C₂³	1	2	yes	—
90	C₂×D₄×C₁₄₉	C₂³	1	2	yes	—
91	C₂×D₄×C₁₄₉	C₂³	1	2	yes	—
92	C₂×D₄×C₁₄₉	C₂³	1	1	yes	characteristic
93	C₂×D₄×C₁₄₉	C₂³	1	8	yes	—
94	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
95	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₁₄₉	C₂³	1	2	yes	—
97	C₂²⋊C₄×C₁₄₉	C₂³	1	8	yes	—
98	C₂²⋊C₄×C₁₄₉	C₂³	1	2	yes	—
99	C₄²×C₁₄₉	C₂³	1	2	yes	—
100	Dic₂₉₈	—	2	4	no	—
101	Dic₂₉₈	—	2	4	no	—
102	Dic₂₉₈	—	4	4	no	—
103	C₂×Dic₁₄₉	—	2	4	no	—
104	C₂×Dic₁₄₉	—	4	4	no	—
105	C₂×Dic₁₄₉	—	4	8	no	—
106	C₂×Dic₁₄₉	—	4	2	no	—
107	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
108	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
109	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
110	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
111	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
112	C₂³×C₁₄₉	—	2	4	no	—
113	C₂³×C₁₄₉	—	4	4	no	—
114	C₂³×C₁₄₉	C₂⁴	1	4	yes	—
115	C₂³×C₁₄₉	C₂⁴	1	2	yes	—
116	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
117	Q₈×C₁₄₉	—	2	2	no	—
118	D₄×C₁₄₉	—	2	4	no	—
119	D₄×C₁₄₉	—	2	4	no	—
120	D₄×C₁₄₉	—	2	4	no	—
121	D₄×C₁₄₉	—	2	2	no	—
122	D₄×C₁₄₉	—	4	8	no	—
123	C₂×C₄×C₁₄₉	—	2	4	no	—
124	C₂×C₄×C₁₄₉	#41	1	4	yes	—
125	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
126	C₂×C₄×C₁₄₉	C₂⁴	1	2	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	2	no	—
131	C₂²×C₁₄₉	—	2	4	no	—
132	C₂²×C₁₄₉	—	2	4	no	—
133	C₂²×C₁₄₉	—	2	8	no	—
134	C₂²×C₁₄₉	—	8	4	no	—
135	C₂²×C₁₄₉	C₂×#41	1	2	yes	—
136	C₂²×C₁₄₉	C₂⁵	1	1	yes	characteristic
137	C₄×C₁₄₉	—	2	2	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	4	no	—
142	Dic₁₄₉	—	2	2	no	—
143	Dic₁₄₉	—	2	4	no	—
144	Dic₁₄₉	—	2	4	no	—
145	Dic₁₄₉	—	2	2	no	—
146	C₂×C₁₄₉	—	2	2	no	—
147	C₂×C₁₄₉	—	4	2	no	—
148	C₂×C₁₄₉	—	4	4	no	—
149	C₂×C₁₄₉	C₂²×#41	1	1	yes	characteristic
150	C₂×C₁₄₉	#585	1	2	yes	—
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	D₁₄₉	1	1	yes	characteristic
159	#533	—	149	1	no	—
160	#529	—	149	4	no	—
161	C₂×#124	—	149	1	no	—
162	#519	—	149	1	no	—
163	#518	—	149	1	no	—
164	C₄×#41	—	149	1	no	—
165	C₂×#41	C₂×D₁₄₉	1	1	yes	characteristic
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	C₂×D₁₄₉	1	2	yes	—
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	C₂×D₁₄₉	1	2	yes	—
180	#123	—	149	8	no	—
181	#123	—	149	4	no	—
182	#123	—	149	8	no	—
183	C₂²⋊Q₈	—	149	4	no	—
184	C₂²⋊Q₈	—	149	4	no	—
185	C₂²⋊Q₈	—	149	8	no	—
186	#121	C₂×D₁₄₉	1	4	yes	—
187	#120	C₂×D₁₄₉	1	4	yes	—
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	4	no	—
193	C₄×D₄	—	149	2	no	—
194	C₄×D₄	—	149	4	no	—
195	C₄×D₄	—	149	4	no	—
196	C₄×D₄	—	149	8	no	—
197	#117	—	149	1	no	—
198	#117	—	149	2	no	—
199	#117	—	149	1	no	—
200	#117	—	149	4	no	—
201	#117	—	149	8	no	—
202	#117	C₂×D₁₄₉	1	1	yes	characteristic
203	C₂×C₂²⋊C₄	—	149	4	no	—
204	C₂×C₂²⋊C₄	—	149	4	no	—
205	C₂×C₄²	—	149	1	no	—
206	C₂×C₄²	—	149	2	no	—
207	C₂⁴	C₂²×D₁₄₉	1	2	yes	—
208	#41	—	2	4	no	—
209	C₂×Q₈	—	298	4	no	—
210	C₂×Q₈	—	149	2	no	—
211	C₂×Q₈	—	149	2	no	—
212	C₂×Q₈	—	149	4	no	—
213	C₂×Q₈	C₂²×D₁₄₉	1	1	yes	characteristic
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	C₂²×D₁₄₉	1	2	yes	—
216	C₂×D₄	C₂²×D₁₄₉	1	2	yes	—
217	C₂×D₄	C₂²×D₁₄₉	1	2	yes	—
218	C₂×D₄	C₂²×D₁₄₉	1	1	yes	characteristic
219	C₂×D₄	C₂²×D₁₄₉	1	8	yes	—
220	C₂²×C₄	—	149	2	no	—
221	C₂²×C₄	—	149	2	no	—
222	C₂²×C₄	—	149	4	no	—
223	C₂²×C₄	—	149	8	no	—
224	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
225	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
226	C₄⋊C₄	—	149	4	no	—
227	C₄⋊C₄	—	149	2	no	—
228	C₄⋊C₄	—	149	2	no	—
229	C₄⋊C₄	—	149	4	no	—
230	C₄⋊C₄	—	149	8	no	—
231	C₄⋊C₄	—	149	8	no	—
232	C₄⋊C₄	—	149	4	no	—
233	C₄⋊C₄	—	149	4	no	—
234	C₄⋊C₄	C₂²×D₁₄₉	1	2	yes	—
235	C₂²⋊C₄	—	149	2	no	—
236	C₂²⋊C₄	—	149	2	no	—
237	C₂²⋊C₄	—	149	8	no	—
238	C₂²⋊C₄	—	149	8	no	—
239	C₂²⋊C₄	—	149	8	no	—
240	C₂²⋊C₄	—	149	4	no	—
241	C₂²⋊C₄	—	149	8	no	—
242	C₂²⋊C₄	C₂²×D₁₄₉	1	2	yes	—
243	C₂²⋊C₄	C₂²×D₁₄₉	1	8	yes	—
244	C₄²	—	149	1	no	—
245	C₄²	—	149	2	no	—
246	C₄²	—	149	4	no	—
247	C₄²	—	149	1	no	—
248	C₄²	—	149	2	no	—
249	C₄²	—	149	8	no	—
250	C₄²	—	149	2	no	—
251	C₄²	C₂²×D₁₄₉	1	2	yes	—
252	C₂³	—	2	4	no	—
253	C₂³	—	4	4	no	—
254	C₂³	C₂³×D₁₄₉	1	4	yes	—
255	C₂³	C₂³×D₁₄₉	1	2	yes	—
256	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
257	Q₈	—	2	2	no	—
258	Q₈	—	298	4	no	—
259	Q₈	—	298	4	no	—
260	Q₈	—	596	4	no	—
261	D₄	—	2	4	no	—
262	D₄	—	2	4	no	—
263	D₄	—	2	4	no	—
264	D₄	—	2	2	no	—
265	D₄	—	4	8	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	298	4	no	—
268	C₂×C₄	—	596	4	no	—
269	C₂×C₄	—	596	8	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	4	no	—
276	C₂×C₄	#14558735	1	4	yes	—
277	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
278	C₂×C₄	C₂³×D₁₄₉	1	2	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	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	8	no	—
286	C₂²	—	8	4	no	—
287	C₂²	C₂×#14558735	1	2	yes	—
288	C₂²	C₂⁴×D₁₄₉	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	4	no	—
291	C₄	—	298	4	no	—
292	C₄	—	298	2	no	—
293	C₄	—	298	4	no	—
294	C₄	—	298	4	no	—
295	C₄	—	298	2	no	—
296	C₄	C₂×#14558735	1	2	yes	—
297	C₄	C₂×#14558735	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	4	4	no	—
300	C₂	—	4	2	no	—
301	C₂	C₂²×#14558735	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, prime radical	#535×C₁₄₉	C₂
hypercenter	#535	D₁₄₉
hypocenter	C₁₄₉	#3510
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ #535

Complements

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

N	H ≅ G/N	type
1. #23750361	C₂	semidirect
2. #23750344	C₂	semidirect
3. #23750306	C₂	semidirect
4. #23750242	C₂	semidirect
5. #23750196	C₂	semidirect
6. #23750178	C₂	semidirect
7. #23750156	C₂	semidirect
8. C₂×#18011898	C₂	semidirect
9. #23750102	C₂	semidirect
10. #23750100	C₂	semidirect
11. #41×Dic₁₄₉	C₂	semidirect
14. #18011910	C₂²	semidirect
15. #18011909	C₂²	semidirect
16. #18011904	C₂²	semidirect
17. #18011904	C₂²	semidirect
18. #18011902	C₂²	semidirect
22. #18011898	C₂²	semidirect
23. #18011898	C₂²	semidirect
25. #18011892	C₂²	semidirect
26. #18011892	C₂²	semidirect
27. #18011890	C₂²	semidirect
29. #18011888	C₂²	semidirect
30. Q₈×Dic₁₄₉	C₂²	semidirect
31. C₄×Dic₂₉₈	C₂²	semidirect
32. C₄×Dic₂₉₈	C₂²	semidirect
33. #18011876	C₂²	semidirect
37. D₄×Dic₁₄₉	C₂²	semidirect
38. D₄×Dic₁₄₉	C₂²	semidirect
41. #18011870	C₂²	semidirect
43. #18011867	C₂²	semidirect
47. C₂×C₄×Dic₁₄₉	C₂²	semidirect
151. C₁₄₉	#3510	semidirect

Maximal subgroups

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

subgroup	index
#23750361	2
#23750344	2
#23750306	2
#23750242	2
#23750196	2
#23750178	2
#23750156	2
C₂×#18011898	2
#23750102	2
#23750100	2
#41×Dic₁₄₉	2
#535×C₁₄₉	2
#3510	149