Subgroups — #37575260

Order 24448.

444 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20259313	C₂	1	1	yes	maximal, characteristic
2	#27132561	C₂	1	1	yes	maximal, characteristic
3	#27132559	C₂	1	2	yes	maximal
4	#27132558	C₂	1	2	yes	maximal
5	#27132548	C₂	1	2	yes	maximal
6	#27132545	C₂	1	2	yes	maximal
7	#27132532	C₂	1	1	yes	maximal, characteristic
8	C₄⋊C₄×Dic₁₉₁	C₂	1	1	yes	maximal, characteristic
9	C₄⋊C₄×Dic₁₉₁	C₂	1	1	yes	maximal, characteristic
10	#27132509	C₂	1	1	yes	maximal, characteristic
11	#389×C₁₉₁	C₂	1	1	yes	maximal, characteristic
12	#20259313	C₄	1	8	yes	complemented
13	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
16	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
17	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
18	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
19	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
20	C₂×C₄×Dic₁₉₁	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₉₁	C₂²	1	2	yes	—
22	C₂×C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
24	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
25	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
26	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
27	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
28	C₁₉₁⋊#95	C₂²	1	2	yes	—
29	C₁₉₁⋊#95	C₂²	1	2	yes	—
30	C₁₉₁⋊#95	C₂²	1	2	yes	—
31	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	2	yes	—
33	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
35	#95×C₁₉₁	C₂²	1	2	yes	—
36	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
37	C₂²×Dic₁₉₁	C₂³	1	1	yes	characteristic
38	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
39	C₂²×Dic₁₉₁	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₉₁	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₉₁	C₂³	1	1	yes	characteristic
42	C₄⋊Dic₁₉₁	—	2	2	no	—
43	C₄⋊Dic₁₉₁	C₂×C₄	1	8	yes	—
44	C₄⋊Dic₁₉₁	C₂×C₄	1	4	yes	—
45	Dic₁₉₁⋊C₄	—	4	2	no	—
46	Dic₁₉₁⋊C₄	—	4	2	no	—
47	Dic₁₉₁⋊C₄	—	4	2	no	—
48	Dic₁₉₁⋊C₄	C₂×C₄	1	8	yes	—
49	C₄×Dic₁₉₁	—	4	2	no	—
50	C₄×Dic₁₉₁	—	4	2	no	—
51	C₄×Dic₁₉₁	C₂×C₄	1	4	yes	—
52	C₄×Dic₁₉₁	Q₈	1	4	yes	—
53	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
57	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₁₉₁	—	2	2	no	—
59	C₄⋊C₄×C₁₉₁	—	4	2	no	—
60	C₄⋊C₄×C₁₉₁	C₂×C₄	1	4	yes	—
61	C₄²×C₁₉₁	—	2	2	no	—
62	C₂×Dic₁₉₁	—	2	1	no	—
63	C₂×Dic₁₉₁	—	2	1	no	—
64	C₂×Dic₁₉₁	—	2	2	no	—
65	C₂×Dic₁₉₁	—	2	1	no	—
66	C₂×Dic₁₉₁	—	2	1	no	—
67	C₂×Dic₁₉₁	—	2	1	no	—
68	C₂×Dic₁₉₁	—	2	1	no	—
69	C₂×Dic₁₉₁	—	2	2	no	—
70	C₂×Dic₁₉₁	—	2	1	no	—
71	C₂×Dic₁₉₁	—	2	1	no	—
72	C₂×Dic₁₉₁	—	2	2	no	—
73	C₂×Dic₁₉₁	—	2	2	no	—
74	C₂×Dic₁₉₁	—	2	2	no	—
75	C₂×Dic₁₉₁	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₉₁	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₉₁	C₂²×C₄	1	4	yes	—
78	C₂×Dic₁₉₁	C₂×Q₈	1	2	yes	—
79	C₂×Dic₁₉₁	C₂×Q₈	1	2	yes	—
80	C₂³×C₁₉₁	C₂⁴	1	1	yes	characteristic
81	C₂×C₄×C₁₉₁	—	2	2	no	—
82	C₂×C₄×C₁₉₁	—	2	2	no	—
83	C₂×C₄×C₁₉₁	—	2	1	no	—
84	C₂×C₄×C₁₉₁	—	2	1	no	—
85	C₂×C₄×C₁₉₁	—	2	2	no	—
86	C₂×C₄×C₁₉₁	—	2	1	no	—
87	C₂×C₄×C₁₉₁	—	2	2	no	—
88	C₂×C₄×C₁₉₁	—	2	2	no	—
89	C₂×C₄×C₁₉₁	—	2	1	no	—
90	C₂×C₄×C₁₉₁	—	2	1	no	—
91	C₂×C₄×C₁₉₁	—	2	1	no	—
92	C₂×C₄×C₁₉₁	C₂²×C₄	1	4	yes	—
93	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₉₁	C₂×Q₈	1	2	yes	—
95	C₂×C₄×C₁₉₁	#41	1	2	yes	—
96	C₂²×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
97	C₂²×C₁₉₁	C₂²×Q₈	1	1	yes	characteristic
98	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
99	C₂²×C₁₉₁	H(5,𝔽₂)	1	1	yes	characteristic
100	C₂²×C₁₉₁	H(5,𝔽₂)	1	1	yes	characteristic
101	C₂²×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
102	C₂²×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
103	C₄×C₁₉₁	—	2	4	no	—
104	C₄×C₁₉₁	—	4	1	no	—
105	C₄×C₁₉₁	—	4	2	no	—
106	C₄×C₁₉₁	—	4	1	no	—
107	C₄×C₁₉₁	C₄×Q₈	1	4	yes	—
108	Dic₁₉₁	—	2	2	no	—
109	Dic₁₉₁	—	2	4	no	—
110	Dic₁₉₁	—	2	2	no	—
111	Dic₁₉₁	—	2	2	no	—
112	Dic₁₉₁	—	2	2	no	—
113	Dic₁₉₁	—	4	2	no	—
114	C₂×C₁₉₁	C₂×C₄×Q₈	1	1	yes	characteristic
115	C₂×C₁₉₁	#520	1	1	yes	characteristic
116	C₂×C₁₉₁	#520	1	1	yes	characteristic
117	C₂×C₁₉₁	#540	1	1	yes	characteristic
118	C₂×C₁₉₁	#541	1	1	yes	characteristic
119	C₂×C₁₉₁	#544	1	1	yes	characteristic
120	C₂×C₁₉₁	#544	1	1	yes	characteristic
121	C₁₉₁	#2330	1	1	yes	characteristic, complemented, Sylow
122	#2330	—	191	1	no	maximal, Sylow
123	C₂×#125	—	191	1	no	—
124	#389	—	191	2	no	—
125	#389	—	191	2	no	—
126	#389	—	191	1	no	—
127	#389	D₁₉₁	1	1	yes	characteristic
128	#387	—	191	2	no	—
129	#387	—	191	2	no	—
130	#384	—	191	1	no	—
131	C₄×C₄⋊C₄	—	191	1	no	—
132	C₄×C₄⋊C₄	—	191	1	no	—
133	#376	—	191	1	no	—
134	#125	—	191	8	no	—
135	C₂×C₄⋊C₄	—	191	1	no	—
136	C₂×C₄⋊C₄	—	191	1	no	—
137	C₂×C₄⋊C₄	—	191	2	no	—
138	C₂×C₄⋊C₄	—	191	2	no	—
139	C₂×C₄⋊C₄	—	191	2	no	—
140	C₂×C₄⋊C₄	—	191	2	no	—
141	C₂×C₄⋊C₄	—	191	2	no	—
142	C₂×C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
143	C₂×C₄⋊C₄	C₂×D₁₉₁	1	2	yes	—
144	C₂×C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
145	C₂×C₄²	—	191	1	no	—
146	C₂×C₄²	—	191	1	no	—
147	C₂×C₄²	—	191	2	no	—
148	C₂×C₄²	—	191	2	no	—
149	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
150	#95	—	191	1	no	—
151	#95	—	191	1	no	—
152	#95	—	191	1	no	—
153	#95	—	191	1	no	—
154	#95	—	191	2	no	—
155	#95	—	191	2	no	—
156	#95	—	191	2	no	—
157	#95	C₂×D₁₉₁	1	2	yes	—
158	C₂²×C₄	—	191	2	no	—
159	C₂²×C₄	—	191	1	no	—
160	C₂²×C₄	—	191	1	no	—
161	C₂²×C₄	—	191	1	no	—
162	C₂²×C₄	—	191	2	no	—
163	C₂²×C₄	—	191	1	no	—
164	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
165	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
166	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
167	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
168	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	382	2	no	—
172	C₄⋊C₄	—	191	8	no	—
173	C₄⋊C₄	—	191	8	no	—
174	C₄⋊C₄	—	191	4	no	—
175	C₄⋊C₄	—	764	2	no	—
176	C₄⋊C₄	—	764	2	no	—
177	C₄⋊C₄	—	764	2	no	—
178	C₄⋊C₄	C₄×D₁₉₁	1	4	yes	—
179	C₄²	—	2	2	no	—
180	C₄²	—	191	4	no	—
181	C₄²	—	191	4	no	—
182	C₄²	—	764	2	no	—
183	C₄²	—	764	2	no	—
184	C₂³	C₂³×D₁₉₁	1	1	yes	characteristic
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	382	1	no	—
197	C₂×C₄	—	382	1	no	—
198	C₂×C₄	—	382	1	no	—
199	C₂×C₄	—	382	2	no	—
200	C₂×C₄	—	382	1	no	—
201	C₂×C₄	—	382	1	no	—
202	C₂×C₄	—	382	1	no	—
203	C₂×C₄	—	382	1	no	—
204	C₂×C₄	—	382	2	no	—
205	C₂×C₄	—	382	1	no	—
206	C₂×C₄	—	382	2	no	—
207	C₂×C₄	—	382	2	no	—
208	C₂×C₄	—	382	2	no	—
209	C₂×C₄	—	191	2	no	—
210	C₂×C₄	—	191	2	no	—
211	C₂×C₄	—	191	4	no	—
212	C₂×C₄	—	191	2	no	—
213	C₂×C₄	—	191	2	no	—
214	C₂×C₄	C₂×C₄×D₁₉₁	1	4	yes	—
215	C₂×C₄	C₂×C₄×D₁₉₁	1	2	yes	—
216	C₂×C₄	Q₈×D₁₉₁	1	2	yes	—
217	C₂×C₄	#15690286	1	2	yes	—
218	C₂²	C₂²×C₄×D₁₉₁	1	1	yes	characteristic
219	C₂²	C₂×Q₈×D₁₉₁	1	1	yes	characteristic
220	C₂²	C₂×#15690286	1	1	yes	characteristic
221	C₂²	#20259367	1	1	yes	characteristic
222	C₂²	#20259368	1	1	yes	characteristic
223	C₂²	#20259369	1	1	yes	characteristic
224	C₂²	#20259370	1	1	yes	characteristic
225	C₄	—	2	4	no	—
226	C₄	—	4	1	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	1	no	—
229	C₄	—	382	2	no	—
230	C₄	—	382	2	no	—
231	C₄	—	382	2	no	—
232	C₄	—	382	4	no	—
233	C₄	—	382	2	no	—
234	C₄	—	764	2	no	—
235	C₄	#20259287	1	4	yes	—
236	C₂	C₂×#20259287	1	1	yes	characteristic
237	C₂	#27133142	1	1	yes	characteristic
238	C₂	#27133145	1	1	yes	characteristic
239	C₂	#27133266	1	1	yes	characteristic
240	C₂	#27133273	1	1	yes	characteristic
241	C₂	#27133290	1	1	yes	characteristic
242	C₂	#27133293	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉₁
commutator	C₂²×C₁₉₁	C₂³×C₄
Fitting	#389×C₁₉₁	C₂
socle, prime radical, cosocle	C₂³×C₁₉₁	C₂⁴
hypercenter	#389	D₁₉₁
hypocenter	C₁₉₁	#2330
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
12. #20259313	C₄	semidirect
121. C₁₉₁	#2330	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20259313	2
#27132561	2
#27132559	2
#27132558	2
#27132548	2
#27132545	2
#27132532	2
C₄⋊C₄×Dic₁₉₁	2
#27132509	2
#389×C₁₉₁	2
#2330	191