Subgroups — C₂×#27132783 #37581414

Order 24448.

464 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20259342	C₂	1	1	yes	maximal, characteristic
2	C₂×#20259328	C₂	1	1	yes	maximal, characteristic, complemented
3	#27132783	C₂	1	8	yes	maximal, direct product factor
4	C₂×#20259257	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#20259249	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#20259248	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#20259232	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15690287	C₂²	1	1	yes	characteristic
10	#20259342	C₄	1	4	yes	complemented
11	#20259342	C₂²	1	4	yes	—
12	C₂×#15690277	C₄	1	1	yes	characteristic, complemented
13	C₂×#15690275	C₄	1	1	yes	characteristic, complemented
14	C₂×#15690274	C₄	1	1	yes	characteristic, complemented
15	C₂×#15690272	C₄	1	1	yes	characteristic, complemented
16	#20259328	—	2	2	no	—
17	#20259328	C₂²	1	4	yes	complemented
18	C₂×#15690271	C₂²	1	1	yes	characteristic
19	C₂×#15690270	—	2	1	no	—
20	C₂²×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₁₉₁	—	2	1	no	—
22	C₂×C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
23	#20259257	C₂²	1	4	yes	complemented
24	#20259249	C₂²	1	4	yes	complemented
25	#20259248	C₂²	1	4	yes	complemented
26	#20259232	C₂²	1	4	yes	complemented
27	C₂×#41×C₁₉₁	C₂²	1	1	yes	characteristic
28	#131×C₁₉₁	—	2	2	no	—
29	#131×C₁₉₁	C₂²	1	4	yes	complemented
30	C₂×M₁₆×C₁₉₁	—	2	1	no	—
31	C₂×M₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
32	C₂²×C₈×C₁₉₁	—	2	1	no	—
33	C₂²×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
34	#15690287	—	2	1	no	—
35	#15690287	—	2	1	no	—
36	#15690287	C₂×C₄	1	2	yes	—
37	#15690287	C₂³	1	2	yes	—
38	C₂×Dic₃₈₂	C₂×C₄	1	1	yes	characteristic
39	C₂×D₇₆₄	C₂×C₄	1	1	yes	characteristic
40	C₂×C₂²⋊D₁₉₁	—	2	1	no	—
41	C₂²×Dic₁₉₁	—	2	1	no	—
42	C₂×C₄×D₁₉₁	—	2	1	no	—
43	#15690277	C₂×C₄	1	4	yes	complemented
44	#15690275	C₂×C₄	1	4	yes	complemented
45	#15690274	C₂×C₄	1	4	yes	complemented
46	#15690272	C₂×C₄	1	4	yes	complemented
47	#15690271	—	2	1	no	—
48	#15690271	C₂³	1	2	yes	—
49	#15690270	—	2	4	no	—
50	C₂×C₁₉₁⋊C₈	—	2	1	no	—
51	C₂×C₁₉₁⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₁₉₁⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₂×C₁₉₁⋊C₈	C₂×C₄	1	2	yes	—
54	C₂×C₁₉₁⋊C₈	C₂³	1	2	yes	—
55	C₈×D₁₉₁	—	2	4	no	—
56	C₄×Dic₁₉₁	—	2	1	no	—
57	C₄×Dic₁₉₁	C₂³	1	2	yes	—
58	#41×C₁₉₁	—	2	1	no	—
59	#41×C₁₉₁	—	2	1	no	—
60	#41×C₁₉₁	C₂×C₄	1	2	yes	—
61	#41×C₁₉₁	C₂³	1	2	yes	—
62	C₂×Q₈×C₁₉₁	C₂×C₄	1	1	yes	characteristic
63	C₂×D₄×C₁₉₁	—	2	1	no	—
64	C₂×D₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉₁	—	2	1	no	—
66	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
67	M₁₆×C₁₉₁	—	2	1	no	—
68	M₁₆×C₁₉₁	—	2	4	no	—
69	M₁₆×C₁₉₁	C₂³	1	2	yes	—
70	C₂×C₈×C₁₉₁	—	2	1	no	—
71	C₂×C₈×C₁₉₁	—	2	4	no	—
72	C₂×C₈×C₁₉₁	D₄	1	2	yes	—
73	C₂×C₈×C₁₉₁	D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₉₁	D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₉₁	C₂³	1	2	yes	—
76	C₂²×D₁₉₁	—	2	1	no	—
77	Dic₃₈₂	—	2	1	no	—
78	Dic₃₈₂	C₂²×C₄	1	2	yes	—
79	D₇₆₄	—	2	1	no	—
80	D₇₆₄	C₂²×C₄	1	2	yes	—
81	C₂²⋊D₁₉₁	—	2	2	no	—
82	C₂²⋊D₁₉₁	—	2	2	no	—
83	C₂×Dic₁₉₁	—	2	1	no	—
84	C₂×Dic₁₉₁	—	2	2	no	—
85	C₂×Dic₁₉₁	—	4	2	no	—
86	C₄×D₁₉₁	—	2	2	no	—
87	C₄×D₁₉₁	—	2	2	no	—
88	C₁₉₁⋊C₈	—	2	2	no	—
89	C₁₉₁⋊C₈	C₂²×C₄	1	2	yes	—
90	C₁₉₁⋊C₈	C₂²×C₄	1	2	yes	—
91	C₂³×C₁₉₁	—	2	1	no	—
92	C₂³×C₁₉₁	#41	1	1	yes	characteristic
93	Q₈×C₁₉₁	—	2	1	no	—
94	Q₈×C₁₉₁	C₂²×C₄	1	2	yes	—
95	D₄×C₁₉₁	—	2	2	no	—
96	D₄×C₁₉₁	—	2	2	no	—
97	D₄×C₁₉₁	—	2	1	no	—
98	D₄×C₁₉₁	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₁₉₁	—	2	1	no	—
100	C₂×C₄×C₁₉₁	—	2	2	no	—
101	C₂×C₄×C₁₉₁	—	2	2	no	—
102	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
107	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
108	C₈×C₁₉₁	—	2	2	no	—
109	C₈×C₁₉₁	C₂×D₄	1	2	yes	—
110	C₈×C₁₉₁	C₂×D₄	1	2	yes	—
111	C₂²×C₁₉₁	—	2	2	no	—
112	C₂²×C₁₉₁	—	2	2	no	—
113	C₂²×C₁₉₁	—	4	2	no	—
114	C₂²×C₁₉₁	C₄×D₄	1	1	yes	characteristic
115	C₂²×C₁₉₁	C₄×D₄	1	1	yes	characteristic
116	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
117	C₂×D₁₉₁	—	2	2	no	—
118	C₂×D₁₉₁	—	4	2	no	—
119	C₄×C₁₉₁	—	2	2	no	—
120	C₄×C₁₉₁	C₄×D₄	1	1	yes	characteristic
121	C₄×C₁₉₁	C₄×D₄	1	1	yes	characteristic
122	C₄×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
123	C₄×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
124	Dic₁₉₁	—	2	2	no	—
125	Dic₁₉₁	—	4	2	no	—
126	C₂×C₁₉₁	—	2	1	no	—
127	C₂×C₁₉₁	—	2	1	no	—
128	C₂×C₁₉₁	—	4	2	no	—
129	C₂×C₁₉₁	#444	1	2	yes	complemented
130	C₂×C₁₉₁	C₂×C₄×D₄	1	1	yes	characteristic
131	D₁₉₁	—	4	2	no	—
132	C₁₉₁	C₂×#444	1	1	yes	characteristic, complemented, Sylow
133	C₂×#444	—	191	1	no	maximal, Sylow
134	C₂×#135	—	191	1	no	—
135	C₂×#131	—	191	1	no	—
136	C₂×#131	D₁₉₁	1	1	yes	characteristic, complemented
137	#444	—	191	8	no	—
138	C₂×#108	—	191	1	no	—
139	C₂×#104	—	191	1	no	—
140	C₂×#104	—	191	1	no	—
141	C₂×#97	—	191	1	no	—
142	C₂×#41	—	191	1	no	—
143	C₂×#41	C₂×D₁₉₁	1	1	yes	characteristic
144	#135	—	191	4	no	—
145	#135	—	191	4	no	—
146	C₂×Q₁₆	—	191	1	no	—
147	C₂×SD₁₆	—	191	1	no	—
148	C₂×SD₁₆	—	191	1	no	—
149	C₂×D₈	—	191	1	no	—
150	#131	—	2	2	no	—
151	#131	—	382	2	no	—
152	#131	—	191	4	no	—
153	#131	C₂×D₁₉₁	1	4	yes	complemented
154	C₂×M₁₆	—	2	1	no	—
155	C₂×M₁₆	—	382	1	no	—
156	C₂×M₁₆	—	191	1	no	—
157	C₂×M₁₆	C₂×D₁₉₁	1	1	yes	characteristic
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	—	382	1	no	—
160	C₂²×C₈	—	191	1	no	—
161	C₂²×C₈	C₂×D₁₉₁	1	1	yes	characteristic
162	C₂×C₄²	—	191	1	no	—
163	#108	—	191	4	no	—
164	#104	—	191	4	no	—
165	#104	—	191	4	no	—
166	#97	—	191	4	no	—
167	#41	—	2	1	no	—
168	#41	—	2	1	no	—
169	#41	—	382	1	no	—
170	#41	—	382	1	no	—
171	#41	—	191	2	no	—
172	#41	—	191	2	no	—
173	#41	C₄×D₁₉₁	1	2	yes	—
174	#41	C₂²×D₁₉₁	1	2	yes	—
175	C₂×Q₈	—	191	1	no	—
176	C₂×Q₈	C₄×D₁₉₁	1	1	yes	characteristic
177	C₂×D₄	—	2	1	no	—
178	C₂×D₄	—	382	1	no	—
179	C₂×D₄	—	191	1	no	—
180	C₂×D₄	C₄×D₁₉₁	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	382	1	no	—
183	C₂²×C₄	—	382	1	no	—
184	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
185	Q₁₆	—	191	4	no	—
186	SD₁₆	—	191	4	no	—
187	SD₁₆	—	191	4	no	—
188	D₈	—	191	4	no	—
189	M₁₆	—	2	1	no	—
190	M₁₆	—	2	4	no	—
191	M₁₆	—	382	1	no	—
192	M₁₆	—	382	4	no	—
193	M₁₆	—	191	2	no	—
194	M₁₆	C₂²×D₁₉₁	1	2	yes	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	382	1	no	—
198	C₂×C₈	—	382	4	no	—
199	C₂×C₈	—	191	1	no	—
200	C₂×C₈	—	191	1	no	—
201	C₂×C₈	—	191	2	no	—
202	C₂×C₈	—	191	2	no	—
203	C₂×C₈	C₂²⋊D₁₉₁	1	2	yes	—
204	C₂×C₈	C₂²⋊D₁₉₁	1	1	yes	characteristic
205	C₂×C₈	C₂²⋊D₁₉₁	1	1	yes	characteristic
206	C₂×C₈	C₂²×D₁₉₁	1	2	yes	—
207	C₄²	—	382	1	no	—
208	C₄²	—	191	2	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	382	1	no	—
211	C₂³	#15690287	1	1	yes	characteristic
212	Q₈	—	2	1	no	—
213	Q₈	—	382	1	no	—
214	Q₈	—	191	2	no	—
215	Q₈	C₂×C₄×D₁₉₁	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	—	382	2	no	—
220	D₄	—	382	2	no	—
221	D₄	—	382	1	no	—
222	D₄	—	191	2	no	—
223	D₄	C₂×C₄×D₁₉₁	1	2	yes	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	382	1	no	—
228	C₂×C₄	—	382	2	no	—
229	C₂×C₄	—	382	2	no	—
230	C₂×C₄	—	382	2	no	—
231	C₂×C₄	—	764	2	no	—
232	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
235	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
236	C₂×C₄	#15690287	1	1	yes	characteristic
237	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	382	2	no	—
240	C₈	—	191	2	no	—
241	C₈	—	191	2	no	—
242	C₈	C₂×C₂²⋊D₁₉₁	1	2	yes	—
243	C₈	C₂×C₂²⋊D₁₉₁	1	2	yes	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	382	2	no	—
248	C₂²	—	764	2	no	—
249	C₂²	C₄×C₂²⋊D₁₉₁	1	1	yes	characteristic
250	C₂²	C₄×C₂²⋊D₁₉₁	1	1	yes	characteristic
251	C₂²	C₂×#15690287	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	382	2	no	—
254	C₄	—	764	2	no	—
255	C₄	C₄×C₂²⋊D₁₉₁	1	1	yes	characteristic
256	C₄	C₄×C₂²⋊D₁₉₁	1	1	yes	characteristic
257	C₄	C₂²×C₄×D₁₉₁	1	1	yes	characteristic
258	C₄	C₂²×C₂²⋊D₁₉₁	1	1	yes	characteristic
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	4	2	no	—
262	C₂	—	764	2	no	—
263	C₂	#27132783	1	2	yes	direct product factor
264	C₂	C₂×C₄×C₂²⋊D₁₉₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₉₁
commutator	C₄×C₁₉₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	C₂×#131×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₄×D₄
prime radical	C₂×#20259342	C₂
cosocle	C₂×C₄×C₁₉₁	C₂⁴
hypercenter	C₂×#131	D₁₉₁
hypocenter	C₁₉₁	C₂×#444
LCS	G ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. C₂×#20259328	C₂	semidirect
3. #27132783	C₂	direct (N × H)
4. C₂×#20259257	C₂	semidirect
5. C₂×#20259249	C₂	semidirect
6. C₂×#20259248	C₂	semidirect
7. C₂×#20259232	C₂	semidirect
8. C₂×#131×C₁₉₁	C₂	semidirect
10. #20259342	C₄	semidirect
12. C₂×#15690277	C₄	semidirect
13. C₂×#15690275	C₄	semidirect
14. C₂×#15690274	C₄	semidirect
15. C₂×#15690272	C₄	semidirect
17. #20259328	C₂²	semidirect
23. #20259257	C₂²	semidirect
24. #20259249	C₂²	semidirect
25. #20259248	C₂²	semidirect
26. #20259232	C₂²	semidirect
29. #131×C₁₉₁	C₂²	semidirect
43. #15690277	C₂×C₄	semidirect
44. #15690275	C₂×C₄	semidirect
45. #15690274	C₂×C₄	semidirect
46. #15690272	C₂×C₄	semidirect
129. C₂×C₁₉₁	#444	semidirect
132. C₁₉₁	C₂×#444	semidirect
136. C₂×#131	D₁₉₁	semidirect
153. #131	C₂×D₁₉₁	semidirect
263. C₂	#27132783	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#20259342	2
C₂×#20259328	2
#27132783	2
C₂×#20259257	2
C₂×#20259249	2
C₂×#20259248	2
C₂×#20259232	2
C₂×#131×C₁₉₁	2
C₂×#444	191

Subgroups — C2×#27132783 #37581414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27132783 #37581414