Subgroups — C₂×#23750160 #34057958

Order 19072.

1232 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23750160	C₂	1	16	yes	maximal, direct product factor
2	C₂×#18011906	C₂	1	2	yes	maximal, complemented
3	C₂×#18011899	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#18011896	C₂	1	2	yes	maximal, complemented
5	C₂×#18011891	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#18011886	C₂	1	1	yes	maximal, characteristic
7	C₂×#18011876	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#18011873	C₂	1	2	yes	maximal, complemented
9	C₂×#18011873	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#18011869	C₂	1	2	yes	maximal, complemented
11	C₂×C₄²×D₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#125×C₁₄₉	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₂₉₈	C₂²	1	1	yes	characteristic
14	C₂²×D₅₉₆	C₂²	1	2	yes	—
15	C₂²×D₅₉₆	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₄₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₄₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₁₄₉	C₂²	1	1	yes	characteristic
19	C₂²×C₄×D₁₄₉	C₂²	1	2	yes	—
20	#18011906	C₂²	1	16	yes	complemented
21	#18011899	C₂²	1	8	yes	complemented
22	#18011896	C₂²	1	16	yes	complemented
23	#18011891	C₂²	1	8	yes	complemented
24	#18011886	C₂²	1	8	yes	—
25	#18011876	C₂²	1	8	yes	complemented
26	#18011873	C₂²	1	16	yes	complemented
27	#18011873	C₂²	1	8	yes	complemented
28	#18011869	C₂²	1	16	yes	complemented
29	C₂×C₄⋊Dic₁₄₉	C₂²	1	2	yes	—
30	C₂×#14558715	C₂²	1	2	yes	—
31	C₂×#14558713	C₂²	1	2	yes	—
32	C₂×#14558713	C₂²	1	2	yes	—
33	C₂×#14558713	C₂²	1	2	yes	—
34	C₂×#14558713	C₂²	1	2	yes	—
35	C₂×#14558713	C₂²	1	2	yes	—
36	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
37	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
38	C₂×C₄×Dic₁₄₉	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₁₄₉	C₂²	1	2	yes	—
40	C₄²×D₁₄₉	C₂²	1	8	yes	complemented
41	#125×C₁₄₉	C₂²	1	8	yes	complemented
42	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	2	yes	—
43	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	2	yes	—
44	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₁₄₉	C₂²	1	1	yes	characteristic
46	C₂×C₄²×C₁₄₉	C₂²	1	1	yes	characteristic
47	C₂³×D₁₄₉	C₂³	1	2	yes	—
48	C₂³×D₁₄₉	C₂³	1	1	yes	characteristic
49	C₂×Dic₂₉₈	—	2	4	no	—
50	C₂×Dic₂₉₈	C₂³	1	4	yes	—
51	C₂×D₅₉₆	—	2	8	no	—
52	C₂×D₅₉₆	—	2	4	no	—
53	C₂×D₅₉₆	C₂³	1	8	yes	—
54	C₂×D₅₉₆	C₂³	1	4	yes	—
55	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
56	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
57	C₂²×Dic₁₄₉	C₂³	1	2	yes	—
58	C₂²×Dic₁₄₉	C₂³	1	1	yes	characteristic
59	C₂×C₄×D₁₄₉	—	2	4	no	—
60	C₂×C₄×D₁₄₉	—	2	4	no	—
61	C₂×C₄×D₁₄₉	—	2	8	no	—
62	C₂×C₄×D₁₄₉	—	2	4	no	—
63	C₂×C₄×D₁₄₉	C₂³	1	4	yes	—
64	C₂×C₄×D₁₄₉	C₂³	1	4	yes	—
65	C₂×C₄×D₁₄₉	C₂³	1	4	yes	—
66	C₂×C₄×D₁₄₉	C₂³	1	8	yes	—
67	C₄⋊Dic₁₄₉	C₂³	1	8	yes	—
68	#14558715	C₂³	1	8	yes	—
69	#14558713	C₂³	1	8	yes	—
70	#14558713	C₂³	1	8	yes	—
71	#14558713	C₂³	1	8	yes	—
72	#14558713	C₂³	1	8	yes	—
73	#14558713	C₂³	1	8	yes	—
74	C₄×Dic₁₄₉	C₂³	1	4	yes	—
75	C₄×Dic₁₄₉	C₂³	1	4	yes	—
76	C₄×Dic₁₄₉	C₂³	1	4	yes	—
77	C₄×Dic₁₄₉	C₂³	1	8	yes	—
78	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
80	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
81	C₂²×C₄×C₁₄₉	C₂³	1	2	yes	—
82	C₂²×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₁₄₉	C₂³	1	8	yes	—
84	C₄⋊C₄×C₁₄₉	C₂³	1	8	yes	—
85	C₄⋊C₄×C₁₄₉	C₂³	1	4	yes	—
86	C₄⋊C₄×C₁₄₉	C₂³	1	4	yes	—
87	C₄²×C₁₄₉	C₂³	1	4	yes	—
88	C₂²×D₁₄₉	—	2	2	no	—
89	C₂²×D₁₄₉	—	2	2	no	—
90	C₂²×D₁₄₉	—	2	4	no	—
91	C₂²×D₁₄₉	—	2	4	no	—
92	C₂²×D₁₄₉	—	2	4	no	—
93	C₂²×D₁₄₉	#41	1	4	yes	—
94	C₂²×D₁₄₉	C₂⁴	1	4	yes	—
95	C₂²×D₁₄₉	C₂⁴	1	2	yes	—
96	Dic₂₉₈	—	2	8	no	—
97	D₅₉₆	—	2	16	no	—
98	D₅₉₆	—	2	8	no	—
99	C₂×Dic₁₄₉	—	2	2	no	—
100	C₂×Dic₁₄₉	—	2	2	no	—
101	C₂×Dic₁₄₉	—	2	4	no	—
102	C₂×Dic₁₄₉	#41	1	4	yes	—
103	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
104	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
105	C₂×Dic₁₄₉	C₂⁴	1	4	yes	—
106	C₂×Dic₁₄₉	C₂⁴	1	2	yes	—
107	C₄×D₁₄₉	—	2	8	no	—
108	C₄×D₁₄₉	—	2	8	no	—
109	C₄×D₁₄₉	—	2	8	no	—
110	C₄×D₁₄₉	—	2	16	no	—
111	C₂³×C₁₄₉	C₂⁴	1	1	yes	characteristic
112	C₂×C₄×C₁₄₉	—	2	2	no	—
113	C₂×C₄×C₁₄₉	—	2	2	no	—
114	C₂×C₄×C₁₄₉	—	2	4	no	—
115	C₂×C₄×C₁₄₉	—	2	4	no	—
116	C₂×C₄×C₁₄₉	#41	1	4	yes	—
117	C₂×C₄×C₁₄₉	C₂⁴	1	2	yes	—
118	C₂×C₄×C₁₄₉	C₂⁴	1	2	yes	—
119	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
120	C₂×C₄×C₁₄₉	C₂⁴	1	4	yes	—
121	C₂×C₄×C₁₄₉	C₂⁴	1	2	yes	—
122	C₂²×C₁₄₉	C₂×#41	1	2	yes	—
123	C₂²×C₁₄₉	C₂×#41	1	2	yes	—
124	C₂²×C₁₄₉	C₂×#41	1	2	yes	—
125	C₂²×C₁₄₉	C₂⁵	1	1	yes	characteristic
126	C₂×D₁₄₉	—	2	8	no	—
127	C₂×D₁₄₉	—	2	2	no	—
128	C₂×D₁₄₉	—	2	2	no	—
129	C₂×D₁₄₉	—	2	4	no	—
130	C₂×D₁₄₉	—	2	4	no	—
131	C₂×D₁₄₉	—	2	4	no	—
132	C₂×D₁₄₉	—	4	8	no	—
133	C₂×D₁₄₉	C₂×#41	1	4	yes	—
134	C₄×C₁₄₉	—	2	2	no	—
135	C₄×C₁₄₉	—	2	2	no	—
136	C₄×C₁₄₉	—	2	4	no	—
137	C₄×C₁₄₉	—	2	4	no	—
138	C₄×C₁₄₉	C₂×#41	1	4	yes	—
139	Dic₁₄₉	—	2	2	no	—
140	Dic₁₄₉	—	2	2	no	—
141	Dic₁₄₉	—	2	4	no	—
142	Dic₁₄₉	C₂×#41	1	4	yes	—
143	C₂×C₁₄₉	#529	1	4	yes	complemented
144	C₂×C₁₄₉	C₂²×#41	1	1	yes	characteristic
145	C₂×C₁₄₉	C₂²×#41	1	1	yes	characteristic
146	C₂×C₁₄₉	C₂²×#41	1	1	yes	characteristic
147	D₁₄₉	—	2	4	no	—
148	D₁₄₉	—	4	4	no	—
149	C₁₄₉	C₂×#529	1	1	yes	characteristic, complemented, Sylow
150	C₂×#529	—	149	1	no	maximal, Sylow
151	#529	—	149	16	no	—
152	C₂×#126	—	149	2	no	—
153	C₂×#125	D₁₄₉	1	1	yes	characteristic, complemented
154	C₂×#124	—	149	1	no	—
155	C₂×#123	—	149	2	no	—
156	C₂×C₂²⋊Q₈	—	149	1	no	—
157	C₂×#121	—	149	1	no	—
158	C₂×C₄×Q₈	—	149	1	no	—
159	C₂×C₄×D₄	—	149	2	no	—
160	C₂×C₄×D₄	—	149	1	no	—
161	C₂×#117	—	149	2	no	—
162	C₂²×C₄²	—	149	1	no	—
163	C₂²×Q₈	—	149	1	no	—
164	C₂²×D₄	—	149	2	no	—
165	C₂²×D₄	—	149	1	no	—
166	C₂³×C₄	—	149	2	no	—
167	C₂³×C₄	—	149	1	no	—
168	C₂³×C₄	—	149	1	no	—
169	C₂³×C₄	—	149	1	no	—
170	#126	—	149	16	no	—
171	#125	C₂×D₁₄₉	1	8	yes	complemented
172	#124	—	149	8	no	—
173	#123	—	149	16	no	—
174	C₂²⋊Q₈	—	149	8	no	—
175	#121	—	149	8	no	—
176	C₄×Q₈	—	149	8	no	—
177	C₄×D₄	—	149	8	no	—
178	C₄×D₄	—	149	16	no	—
179	#117	—	149	16	no	—
180	C₂×C₄⋊C₄	—	149	2	no	—
181	C₂×C₄⋊C₄	—	149	2	no	—
182	C₂×C₄⋊C₄	C₂×D₁₄₉	1	2	yes	—
183	C₂×C₄⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂×D₁₄₉	1	2	yes	—
185	C₂×C₄⋊C₄	C₂×D₁₄₉	1	1	yes	characteristic
186	C₂×C₂²⋊C₄	—	149	2	no	—
187	C₂×C₂²⋊C₄	—	149	2	no	—
188	C₂×C₂²⋊C₄	—	149	2	no	—
189	C₂×C₂²⋊C₄	—	149	2	no	—
190	C₂×C₂²⋊C₄	—	149	2	no	—
191	C₂×C₄²	—	149	1	no	—
192	C₂×C₄²	—	149	2	no	—
193	C₂×C₄²	—	149	1	no	—
194	C₂×C₄²	—	149	1	no	—
195	C₂×C₄²	—	149	8	no	—
196	C₂×C₄²	C₂×D₁₄₉	1	1	yes	characteristic
197	C₂⁴	—	149	2	no	—
198	C₂⁴	—	149	1	no	—
199	C₂×Q₈	—	298	4	no	—
200	C₂×Q₈	—	149	4	no	—
201	C₂×D₄	—	298	8	no	—
202	C₂×D₄	—	298	4	no	—
203	C₂×D₄	—	149	8	no	—
204	C₂×D₄	—	149	4	no	—
205	C₂²×C₄	—	298	4	no	—
206	C₂²×C₄	—	298	4	no	—
207	C₂²×C₄	—	298	8	no	—
208	C₂²×C₄	—	298	4	no	—
209	C₂²×C₄	—	149	1	no	—
210	C₂²×C₄	—	149	1	no	—
211	C₂²×C₄	—	149	2	no	—
212	C₂²×C₄	—	149	4	no	—
213	C₂²×C₄	—	149	4	no	—
214	C₂²×C₄	—	149	4	no	—
215	C₂²×C₄	—	149	8	no	—
216	C₂²×C₄	—	149	1	no	—
217	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
218	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
219	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
220	C₂²×C₄	C₂²×D₁₄₉	1	2	yes	—
221	C₂²×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
222	C₄⋊C₄	—	149	8	no	—
223	C₄⋊C₄	—	149	8	no	—
224	C₄⋊C₄	C₂²×D₁₄₉	1	8	yes	—
225	C₄⋊C₄	C₂²×D₁₄₉	1	4	yes	—
226	C₄⋊C₄	C₂²×D₁₄₉	1	8	yes	—
227	C₄⋊C₄	C₂²×D₁₄₉	1	4	yes	—
228	C₂²⋊C₄	—	149	8	no	—
229	C₂²⋊C₄	—	149	8	no	—
230	C₂²⋊C₄	—	149	8	no	—
231	C₂²⋊C₄	—	149	8	no	—
232	C₂²⋊C₄	—	149	8	no	—
233	C₄²	—	149	4	no	—
234	C₄²	—	149	4	no	—
235	C₄²	—	149	4	no	—
236	C₄²	—	149	8	no	—
237	C₄²	C₂²×D₁₄₉	1	4	yes	—
238	C₂³	—	298	2	no	—
239	C₂³	—	298	2	no	—
240	C₂³	—	298	4	no	—
241	C₂³	—	298	4	no	—
242	C₂³	—	298	4	no	—
243	C₂³	—	149	4	no	—
244	C₂³	—	149	4	no	—
245	C₂³	—	149	2	no	—
246	C₂³	C₂³×D₁₄₉	1	1	yes	characteristic
247	Q₈	—	298	8	no	—
248	D₄	—	298	16	no	—
249	D₄	—	298	8	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	298	8	no	—
255	C₂×C₄	—	298	8	no	—
256	C₂×C₄	—	298	16	no	—
257	C₂×C₄	—	298	2	no	—
258	C₂×C₄	—	298	2	no	—
259	C₂×C₄	—	298	4	no	—
260	C₂×C₄	—	298	8	no	—
261	C₂×C₄	—	149	2	no	—
262	C₂×C₄	—	149	2	no	—
263	C₂×C₄	—	149	4	no	—
264	C₂×C₄	—	149	4	no	—
265	C₂×C₄	—	149	2	no	—
266	C₂×C₄	#14558737	1	4	yes	—
267	C₂×C₄	C₂³×D₁₄₉	1	2	yes	—
268	C₂×C₄	C₂³×D₁₄₉	1	2	yes	—
269	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
270	C₂×C₄	C₂³×D₁₄₉	1	4	yes	—
271	C₂×C₄	C₂³×D₁₄₉	1	2	yes	—
272	C₂²	—	298	8	no	—
273	C₂²	—	298	2	no	—
274	C₂²	—	298	2	no	—
275	C₂²	—	298	4	no	—
276	C₂²	—	298	4	no	—
277	C₂²	—	298	4	no	—
278	C₂²	—	596	8	no	—
279	C₂²	—	149	4	no	—
280	C₂²	#41×D₁₄₉	1	2	yes	—
281	C₂²	#41×D₁₄₉	1	2	yes	—
282	C₂²	C₂×#14558737	1	2	yes	—
283	C₂²	C₂⁴×D₁₄₉	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	4	no	—
287	C₄	—	2	4	no	—
288	C₄	—	298	2	no	—
289	C₄	—	298	2	no	—
290	C₄	—	298	4	no	—
291	C₄	—	149	4	no	—
292	C₄	C₂×#14558737	1	4	yes	—
293	C₂	—	298	4	no	—
294	C₂	—	596	4	no	—
295	C₂	#23750160	1	4	yes	direct product factor
296	C₂	C₂×#41×D₁₄₉	1	1	yes	characteristic
297	C₂	C₂×#41×D₁₄₉	1	1	yes	characteristic
298	C₂	C₂²×#14558737	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₄₉
commutator, cosocle	C₂²×C₁₄₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₄₉
Fitting	C₂×#125×C₁₄₉	C₂
socle	C₂³×C₁₄₉	C₂⁴
prime radical	C₂×#18011886	C₂
hypercenter	C₂×#125	D₁₄₉
hypocenter	C₁₄₉	C₂×#529
LCS	G ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂³ ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
1. #23750160	C₂	direct (N × H)
2. C₂×#18011906	C₂	semidirect
3. C₂×#18011899	C₂	semidirect
4. C₂×#18011896	C₂	semidirect
5. C₂×#18011891	C₂	semidirect
7. C₂×#18011876	C₂	semidirect
8. C₂×#18011873	C₂	semidirect
9. C₂×#18011873	C₂	semidirect
10. C₂×#18011869	C₂	semidirect
11. C₂×C₄²×D₁₄₉	C₂	semidirect
12. C₂×#125×C₁₄₉	C₂	semidirect
20. #18011906	C₂²	semidirect
21. #18011899	C₂²	semidirect
22. #18011896	C₂²	semidirect
23. #18011891	C₂²	semidirect
25. #18011876	C₂²	semidirect
26. #18011873	C₂²	semidirect
27. #18011873	C₂²	semidirect
28. #18011869	C₂²	semidirect
40. C₄²×D₁₄₉	C₂²	semidirect
41. #125×C₁₄₉	C₂²	semidirect
143. C₂×C₁₄₉	#529	semidirect
149. C₁₄₉	C₂×#529	semidirect
153. C₂×#125	D₁₄₉	semidirect
171. #125	C₂×D₁₄₉	semidirect
295. C₂	#23750160	direct (N × H)

Maximal subgroups

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

subgroup	index
#23750160	2
C₂×#18011906	2
C₂×#18011899	2
C₂×#18011896	2
C₂×#18011891	2
C₂×#18011886	2
C₂×#18011876	2
C₂×#18011873	2
C₂×#18011869	2
C₂×C₄²×D₁₄₉	2
C₂×#125×C₁₄₉	2
C₂×#529	149

Subgroups — C2×#23750160 #34057958

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23750160 #34057958