Subgroups — #37575605

Order 24448.

548 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27132570	C₂	1	2	yes	maximal, complemented
2	#27132549	C₂	1	2	yes	maximal, complemented
3	#27132549	C₂	1	2	yes	maximal, complemented
4	#27132538	C₂	1	2	yes	maximal
5	C₄⋊C₄×Dic₁₉₁	C₂	1	2	yes	maximal, complemented
6	C₂²⋊C₄×Dic₁₉₁	C₂	1	2	yes	maximal
7	C₄×C₂²⋊Dic₁₉₁	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₉₁⋊#95	C₂	1	1	yes	maximal, characteristic
9	C₂×C₂²⋊Q₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic
10	C₂³×Dic₁₉₁	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
13	C₂×C₂²⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₁₉₁	C₂²	1	2	yes	—
16	C₂×C₄×Dic₁₉₁	C₂²	1	2	yes	—
17	C₂×C₄×Dic₁₉₁	C₂²	1	2	yes	—
18	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
19	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
20	C₁₉₁⋊#95	C₂²	1	2	yes	—
21	C₁₉₁⋊#95	C₂²	1	2	yes	—
22	C₁₉₁⋊#95	C₂²	1	2	yes	—
23	C₁₉₁⋊#95	C₂²	1	2	yes	—
24	C₁₉₁⋊#95	C₂²	1	2	yes	—
25	C₁₉₁⋊#95	C₂²	1	2	yes	—
26	C₂²×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
27	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
28	C₂²⋊Q₈×C₁₉₁	C₄	1	8	yes	complemented
29	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	2	yes	—
30	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊C₄×C₁₉₁	C₂²	1	2	yes	—
32	C₂²×Dic₁₉₁	—	2	4	no	—
33	C₂²×Dic₁₉₁	—	2	4	no	—
34	C₂²×Dic₁₉₁	—	4	2	no	—
35	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
36	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
37	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
38	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
39	C₄⋊Dic₁₉₁	—	2	4	no	—
40	C₄⋊Dic₁₉₁	—	4	2	no	—
41	C₂²⋊Dic₁₉₁	—	2	2	no	—
42	C₂²⋊Dic₁₉₁	Q₈	1	4	yes	complemented
43	C₄×Dic₁₉₁	—	2	4	no	—
44	C₄×Dic₁₉₁	—	4	2	no	—
45	C₄×Dic₁₉₁	—	4	2	no	—
46	C₂⁴×C₁₉₁	C₂³	1	1	yes	characteristic
47	C₂×Q₈×C₁₉₁	—	4	2	no	—
48	C₂×Q₈×C₁₉₁	C₂×C₄	1	4	yes	complemented
49	C₂²×C₄×C₁₉₁	—	4	2	no	—
50	C₂²×C₄×C₁₉₁	C₂×C₄	1	4	yes	—
51	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
53	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₁₉₁	C₂×C₄	1	4	yes	complemented
56	C₄⋊C₄×C₁₉₁	C₂×C₄	1	8	yes	complemented
57	C₂²⋊C₄×C₁₉₁	C₂×C₄	1	8	yes	—
58	C₂×Dic₁₉₁	—	2	2	no	—
59	C₂×Dic₁₉₁	—	2	2	no	—
60	C₂×Dic₁₉₁	—	2	2	no	—
61	C₂×Dic₁₉₁	—	2	2	no	—
62	C₂×Dic₁₉₁	—	2	2	no	—
63	C₂×Dic₁₉₁	—	2	2	no	—
64	C₂×Dic₁₉₁	—	2	2	no	—
65	C₂×Dic₁₉₁	—	2	2	no	—
66	C₂×Dic₁₉₁	—	2	2	no	—
67	C₂×Dic₁₉₁	—	2	2	no	—
68	C₂×Dic₁₉₁	—	4	8	no	—
69	C₂×Dic₁₉₁	C₂×Q₈	1	4	yes	—
70	C₂×Dic₁₉₁	#41	1	4	yes	—
71	C₂³×C₁₉₁	—	2	2	no	—
72	C₂³×C₁₉₁	—	2	2	no	—
73	C₂³×C₁₉₁	C₂²×C₄	1	2	yes	—
74	C₂³×C₁₉₁	C₂×Q₈	1	2	yes	—
75	C₂³×C₁₉₁	#41	1	2	yes	—
76	C₂³×C₁₉₁	C₂⁴	1	1	yes	characteristic
77	Q₈×C₁₉₁	—	4	4	no	—
78	C₂×C₄×C₁₉₁	—	2	2	no	—
79	C₂×C₄×C₁₉₁	—	2	2	no	—
80	C₂×C₄×C₁₉₁	—	2	2	no	—
81	C₂×C₄×C₁₉₁	—	2	2	no	—
82	C₂×C₄×C₁₉₁	—	2	2	no	—
83	C₂×C₄×C₁₉₁	—	2	2	no	—
84	C₂×C₄×C₁₉₁	—	4	4	no	—
85	C₂×C₄×C₁₉₁	C₂²×C₄	1	4	yes	—
86	C₂×C₄×C₁₉₁	C₂²×C₄	1	4	yes	—
87	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
89	C₂²×C₁₉₁	—	2	2	no	—
90	C₂²×C₁₉₁	—	2	2	no	—
91	C₂²×C₁₉₁	—	2	4	no	—
92	C₂²×C₁₉₁	—	2	4	no	—
93	C₂²×C₁₉₁	C₄×Q₈	1	4	yes	—
94	C₂²×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
95	C₂²×C₁₉₁	C₂²×Q₈	1	1	yes	characteristic
96	C₂²×C₁₉₁	C₂×#41	1	2	yes	—
97	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
98	C₂²×C₁₉₁	H(5,𝔽₂)	1	2	yes	—
99	C₄×C₁₉₁	—	2	4	no	—
100	C₄×C₁₉₁	—	2	2	no	—
101	C₄×C₁₉₁	—	2	2	no	—
102	C₄×C₁₉₁	—	2	4	no	—
103	Dic₁₉₁	—	2	4	no	—
104	Dic₁₉₁	—	2	4	no	—
105	Dic₁₉₁	—	4	2	no	—
106	Dic₁₉₁	—	4	2	no	—
107	C₂×C₁₉₁	—	2	4	no	—
108	C₂×C₁₉₁	C₂×C₄×Q₈	1	1	yes	characteristic
109	C₂×C₁₉₁	C₄×#41	1	1	yes	characteristic
110	C₂×C₁₉₁	#518	1	1	yes	characteristic
111	C₂×C₁₉₁	#551	1	2	yes	—
112	C₂×C₁₉₁	#554	1	2	yes	—
113	C₁₉₁	#2368	1	1	yes	characteristic, complemented, Sylow
114	#2368	—	191	1	no	maximal, Sylow
115	C₂×C₂²⋊Q₈	D₁₉₁	1	1	yes	characteristic
116	#391	—	191	2	no	—
117	#387	—	191	2	no	—
118	#387	—	191	2	no	—
119	#385	—	191	2	no	—
120	C₄×C₄⋊C₄	—	191	2	no	—
121	C₄×C₂²⋊C₄	—	191	1	no	—
122	C₄×C₂²⋊C₄	—	191	2	no	—
123	C₂×#95	—	191	1	no	—
124	C₂²×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
125	C₂³×C₄	—	191	2	no	—
126	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
127	C₂²⋊Q₈	Dic₁₉₁	1	8	yes	complemented
128	C₂×C₄⋊C₄	—	191	2	no	—
129	C₂×C₄⋊C₄	—	191	2	no	—
130	C₂×C₄⋊C₄	C₂×D₁₉₁	1	2	yes	—
131	C₂×C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
132	C₂×C₂²⋊C₄	—	191	1	no	—
133	C₂×C₂²⋊C₄	—	191	1	no	—
134	C₂×C₂²⋊C₄	C₂×D₁₉₁	1	2	yes	—
135	C₂×C₄²	—	191	2	no	—
136	C₂×C₄²	—	191	2	no	—
137	C₂×C₄²	—	191	2	no	—
138	#95	—	191	1	no	—
139	#95	—	191	1	no	—
140	#95	—	191	2	no	—
141	#95	—	191	2	no	—
142	#95	—	191	2	no	—
143	#95	—	191	2	no	—
144	#95	—	191	2	no	—
145	#95	—	191	2	no	—
146	C₂⁴	C₂²×D₁₉₁	1	1	yes	characteristic
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	C₂×Dic₁₉₁	1	4	yes	complemented
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	—	382	4	no	—
151	C₂²×C₄	—	382	4	no	—
152	C₂²×C₄	—	191	2	no	—
153	C₂²×C₄	—	191	2	no	—
154	C₂²×C₄	—	191	2	no	—
155	C₂²×C₄	—	191	2	no	—
156	C₂²×C₄	—	764	2	no	—
157	C₂²×C₄	C₂×Dic₁₉₁	1	4	yes	—
158	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
160	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
161	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
162	C₄⋊C₄	—	382	4	no	—
163	C₄⋊C₄	—	764	2	no	—
164	C₄⋊C₄	C₂×Dic₁₉₁	1	4	yes	complemented
165	C₄⋊C₄	C₂×Dic₁₉₁	1	8	yes	complemented
166	C₂²⋊C₄	—	382	2	no	—
167	C₂²⋊C₄	—	191	4	no	—
168	C₂²⋊C₄	C₂×Dic₁₉₁	1	8	yes	—
169	C₄²	—	382	4	no	—
170	C₄²	—	764	2	no	—
171	C₄²	—	764	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	—	2	2	no	—
174	C₂³	C₂²×Dic₁₉₁	1	2	yes	—
175	C₂³	Q₈×D₁₉₁	1	2	yes	—
176	C₂³	#15690288	1	2	yes	—
177	C₂³	C₂³×D₁₉₁	1	1	yes	characteristic
178	Q₈	—	4	4	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	382	2	no	—
187	C₂×C₄	—	382	2	no	—
188	C₂×C₄	—	382	2	no	—
189	C₂×C₄	—	382	2	no	—
190	C₂×C₄	—	382	2	no	—
191	C₂×C₄	—	382	2	no	—
192	C₂×C₄	—	382	2	no	—
193	C₂×C₄	—	382	2	no	—
194	C₂×C₄	—	382	2	no	—
195	C₂×C₄	—	382	2	no	—
196	C₂×C₄	—	191	4	no	—
197	C₂×C₄	—	191	4	no	—
198	C₂×C₄	—	764	8	no	—
199	C₂×C₄	C₂²×Dic₁₉₁	1	4	yes	—
200	C₂×C₄	C₂²×Dic₁₉₁	1	4	yes	—
201	C₂×C₄	C₂²×Dic₁₉₁	1	2	yes	—
202	C₂×C₄	C₂²×Dic₁₉₁	1	2	yes	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	Q₈×Dic₁₉₁	1	4	yes	—
208	C₂²	C₂³×Dic₁₉₁	1	1	yes	characteristic
209	C₂²	C₂×Q₈×D₁₉₁	1	1	yes	characteristic
210	C₂²	#41×D₁₉₁	1	2	yes	—
211	C₂²	C₂×#15690288	1	1	yes	characteristic
212	C₂²	#20259367	1	2	yes	—
213	C₄	—	2	4	no	—
214	C₄	—	2	2	no	—
215	C₄	—	2	2	no	—
216	C₄	—	2	4	no	—
217	C₄	—	382	4	no	—
218	C₄	—	382	4	no	—
219	C₄	—	764	2	no	—
220	C₄	—	764	2	no	—
221	C₂	—	2	4	no	—
222	C₂	C₂×Q₈×Dic₁₉₁	1	1	yes	characteristic
223	C₂	#41×Dic₁₉₁	1	1	yes	characteristic
224	C₂	#27133138	1	1	yes	characteristic
225	C₂	#27133340	1	2	yes	—
226	C₂	#27133367	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉₁
commutator	C₂²×C₁₉₁	C₂³×C₄
Fitting	C₂×C₂²⋊Q₈×C₁₉₁	C₂
socle, cosocle	C₂³×C₁₉₁	C₂⁴
prime radical	C₂⁴×C₁₉₁	C₂³
hypercenter	C₂×C₂²⋊Q₈	D₁₉₁
hypocenter	C₁₉₁	#2368
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #27132570	C₂	semidirect
2. #27132549	C₂	semidirect
3. #27132549	C₂	semidirect
5. C₄⋊C₄×Dic₁₉₁	C₂	semidirect
28. C₂²⋊Q₈×C₁₉₁	C₄	semidirect
42. C₂²⋊Dic₁₉₁	Q₈	semidirect
48. C₂×Q₈×C₁₉₁	C₂×C₄	semidirect
55. C₄⋊C₄×C₁₉₁	C₂×C₄	semidirect
56. C₄⋊C₄×C₁₉₁	C₂×C₄	semidirect
113. C₁₉₁	#2368	semidirect
127. C₂²⋊Q₈	Dic₁₉₁	semidirect
148. C₂×Q₈	C₂×Dic₁₉₁	semidirect
164. C₄⋊C₄	C₂×Dic₁₉₁	semidirect
165. C₄⋊C₄	C₂×Dic₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
#27132570	2
#27132549	2
#27132538	2
C₄⋊C₄×Dic₁₉₁	2
C₂²⋊C₄×Dic₁₉₁	2
C₄×C₂²⋊Dic₁₉₁	2
C₂×C₁₉₁⋊#95	2
C₂×C₂²⋊Q₈×C₁₉₁	2
#2368	191