Subgroups — #37581449

Order 24448.

364 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27133217	C₂	1	1	yes	maximal, characteristic
2	#27132794	C₂	1	1	yes	maximal, characteristic, complemented
3	#27132792	C₂	1	2	yes	maximal, complemented
4	#27132790	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×Dic₃₈₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₃₈₂	C₂	1	2	yes	maximal, complemented
7	#27132688	C₂	1	2	yes	maximal
8	#27132685	C₂	1	1	yes	maximal, characteristic
9	#27132617	C₂	1	2	yes	maximal
10	C₄×C₈×D₁₉₁	C₂	1	1	yes	maximal, characteristic
11	C₈×Q₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15690270	C₂²	1	2	yes	—
13	C₂×C₈×D₁₉₁	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₁₉₁	C₂²	1	1	yes	characteristic
15	#20259321	C₄	1	1	yes	characteristic
16	#20259320	C₄	1	1	yes	characteristic
17	#20259312	C₄	1	2	yes	—
18	#20259302	C₄	1	1	yes	characteristic
19	#20259300	C₄	1	2	yes	—
20	#20259298	C₄	1	1	yes	characteristic
21	C₄×Dic₃₈₂	C₂²	1	1	yes	characteristic
22	C₄×Dic₃₈₂	C₂²	1	2	yes	—
23	#20259279	C₂²	1	2	yes	—
24	C₄²×D₁₉₁	C₂²	1	1	yes	characteristic
25	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	2	yes	—
27	Dic₁₉₁⋊C₈	C₂²	1	2	yes	—
28	Dic₁₉₁⋊C₈	C₂²	1	2	yes	—
29	Dic₁₉₁⋊C₈	C₂²	1	2	yes	—
30	#20259233	C₂²	1	2	yes	—
31	#20259232	C₂²	1	2	yes	—
32	C₄×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
33	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
34	C₈×Dic₁₉₁	C₂²	1	2	yes	—
35	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₁₉₁	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₉₁	C₂²	1	2	yes	—
39	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₉₁	C₂²	1	2	yes	—
41	C₂×Dic₃₈₂	C₂×C₄	1	2	yes	—
42	C₂×Dic₃₈₂	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₉₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₉₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₉₁	C₂³	1	1	yes	characteristic
46	#15690270	—	2	4	no	—
47	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
48	C₂×C₁₉₁⋊C₈	C₂³	1	2	yes	—
49	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
50	C₈×D₁₉₁	—	2	2	no	—
51	C₈×D₁₉₁	Q₈	1	4	yes	complemented
52	C₄⋊Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
53	C₄⋊Dic₁₉₁	C₂×C₄	1	2	yes	—
54	Dic₁₉₁⋊C₄	C₂×C₄	1	2	yes	—
55	Dic₁₉₁⋊C₄	C₂×C₄	1	2	yes	—
56	Dic₁₉₁⋊C₄	C₂×C₄	1	2	yes	—
57	Dic₁₉₁⋊C₄	C₂×C₄	1	2	yes	—
58	Dic₁₉₁⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₁₉₁)⋊C₄	C₂×C₄	1	2	yes	—
60	(C₂×D₁₉₁)⋊C₄	C₂×C₄	1	2	yes	—
61	C₄×Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₁₉₁	C₂³	1	1	yes	characteristic
64	C₄×Dic₁₉₁	C₂³	1	2	yes	—
65	C₂×Q₈×C₁₉₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₉₁	C₂³	1	2	yes	—
68	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₁₉₁	C₂×C₄	1	2	yes	—
70	C₄⋊C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
72	C₄²×C₁₉₁	C₂³	1	2	yes	—
73	C₂²×D₁₉₁	C₂²×C₄	1	1	yes	characteristic
74	Dic₃₈₂	—	2	2	no	—
75	Dic₃₈₂	—	2	4	no	—
76	C₂×Dic₁₉₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₁₉₁	C₂²×C₄	1	2	yes	—
78	C₂×Dic₁₉₁	C₂²×C₄	1	2	yes	—
79	C₂×Dic₁₉₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₁₉₁	C₂²×C₄	1	1	yes	characteristic
81	C₄×D₁₉₁	—	2	2	no	—
82	C₄×D₁₉₁	—	2	2	no	—
83	C₄×D₁₉₁	C₂×Q₈	1	2	yes	—
84	C₄×D₁₉₁	#41	1	2	yes	—
85	C₁₉₁⋊C₈	—	2	1	no	—
86	C₁₉₁⋊C₈	—	2	2	no	—
87	C₁₉₁⋊C₈	C₂×Q₈	1	2	yes	—
88	Q₈×C₁₉₁	—	2	2	no	—
89	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
94	C₈×C₁₉₁	—	2	1	no	—
95	C₈×C₁₉₁	—	2	2	no	—
96	C₈×C₁₉₁	C₂×Q₈	1	2	yes	—
97	C₂²×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₁₉₁	—	2	1	no	—
99	C₂×D₁₉₁	—	2	1	no	—
100	C₂×D₁₉₁	C₄×Q₈	1	2	yes	—
101	C₄×C₁₉₁	—	2	2	no	—
102	C₄×C₁₉₁	—	2	1	no	—
103	C₄×C₁₉₁	—	2	2	no	—
104	C₄×C₁₉₁	#131	1	2	yes	—
105	C₄×C₁₉₁	C₂²×Q₈	1	1	yes	characteristic
106	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
107	Dic₁₉₁	—	2	1	no	—
108	Dic₁₉₁	—	2	2	no	—
109	Dic₁₉₁	—	2	2	no	—
110	Dic₁₉₁	C₄×Q₈	1	2	yes	—
111	Dic₁₉₁	#131	1	2	yes	—
112	C₂×C₁₉₁	C₂×C₄×Q₈	1	1	yes	characteristic
113	C₂×C₁₉₁	C₂×#131	1	1	yes	characteristic
114	C₂×C₁₉₁	C₂×#131	1	1	yes	characteristic
115	D₁₉₁	—	2	2	no	—
116	C₁₉₁	#2960	1	1	yes	characteristic, complemented, Sylow
117	#2960	—	191	1	no	maximal, Sylow
118	#533	—	191	1	no	—
119	#446	—	191	1	no	—
120	#446	—	191	1	no	—
121	#446	—	191	2	no	—
122	C₈×Q₈	—	191	2	no	—
123	C₈×Q₈	—	191	1	no	—
124	C₈×Q₈	D₁₉₁	1	1	yes	characteristic, complemented
125	#424	—	191	2	no	—
126	#423	—	191	1	no	—
127	#405	—	191	2	no	—
128	C₂×C₄×C₈	—	191	1	no	—
129	C₂×M₁₆	—	191	2	no	—
130	C₂²×C₈	—	191	1	no	—
131	C₂²×C₈	—	191	1	no	—
132	C₄⋊Q₈	—	191	1	no	—
133	C₄⋊Q₈	—	191	1	no	—
134	#125	—	191	2	no	—
135	C₂²⋊Q₈	—	191	1	no	—
136	C₂²⋊Q₈	—	191	2	no	—
137	C₂²⋊Q₈	—	191	1	no	—
138	C₄×Q₈	—	191	2	no	—
139	C₄×Q₈	—	191	1	no	—
140	C₄×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
141	#117	—	191	2	no	—
142	C₂×C₄²	—	191	1	no	—
143	C₄⋊C₈	—	191	2	no	—
144	C₄⋊C₈	—	191	2	no	—
145	C₄⋊C₈	—	191	2	no	—
146	C₄⋊C₈	—	191	1	no	—
147	C₄⋊C₈	—	191	2	no	—
148	C₄⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
149	C₄⋊C₈	C₂×D₁₉₁	1	2	yes	—
150	#97	—	191	2	no	—
151	#97	—	191	2	no	—
152	C₄×C₈	—	191	1	no	—
153	C₄×C₈	—	191	1	no	—
154	C₄×C₈	—	191	2	no	—
155	C₄×C₈	—	191	1	no	—
156	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
157	C₄×C₈	C₂×D₁₉₁	1	2	yes	—
158	C₂×Q₈	—	191	1	no	—
159	C₂×Q₈	—	191	2	no	—
160	C₂×Q₈	C₄×D₁₉₁	1	1	yes	characteristic
161	C₂²×C₄	—	191	1	no	—
162	C₂²×C₄	—	191	1	no	—
163	C₂²×C₄	—	191	1	no	—
164	M₁₆	—	382	4	no	—
165	C₂×C₈	—	382	2	no	—
166	C₂×C₈	—	191	1	no	—
167	C₂×C₈	—	191	2	no	—
168	C₂×C₈	—	191	1	no	—
169	C₂×C₈	—	191	4	no	—
170	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₁₉₁	1	2	yes	—
172	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
173	C₄⋊C₄	—	191	2	no	—
174	C₄⋊C₄	—	191	2	no	—
175	C₄⋊C₄	—	191	2	no	—
176	C₄⋊C₄	—	191	2	no	—
177	C₄⋊C₄	—	191	2	no	—
178	C₄⋊C₄	—	191	2	no	—
179	C₄⋊C₄	—	191	1	no	—
180	C₄⋊C₄	C₄×D₁₉₁	1	2	yes	—
181	C₄⋊C₄	C₄×D₁₉₁	1	1	yes	characteristic
182	C₂²⋊C₄	—	191	2	no	—
183	C₂²⋊C₄	—	191	2	no	—
184	C₄²	—	191	1	no	—
185	C₄²	—	191	1	no	—
186	C₄²	—	191	1	no	—
187	C₄²	—	191	2	no	—
188	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
189	C₄²	C₂²×D₁₉₁	1	2	yes	—
190	C₂³	—	191	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	382	4	no	—
193	Q₈	—	382	2	no	—
194	C₂×C₄	—	382	2	no	—
195	C₂×C₄	—	382	2	no	—
196	C₂×C₄	—	191	1	no	—
197	C₂×C₄	—	191	2	no	—
198	C₂×C₄	—	191	2	no	—
199	C₂×C₄	—	191	1	no	—
200	C₂×C₄	—	191	1	no	—
201	C₂×C₄	—	191	2	no	—
202	C₂×C₄	—	191	2	no	—
203	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₁₉₁	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₁₉₁	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
207	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	2	no	—
210	C₈	—	382	1	no	—
211	C₈	—	382	2	no	—
212	C₈	—	191	2	no	—
213	C₈	Q₈×D₁₉₁	1	2	yes	—
214	C₂²	—	382	1	no	—
215	C₂²	—	382	1	no	—
216	C₂²	—	191	2	no	—
217	C₂²	C₂²×C₄×D₁₉₁	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	382	2	no	—
222	C₄	—	382	1	no	—
223	C₄	—	382	2	no	—
224	C₄	—	191	2	no	—
225	C₄	—	191	2	no	—
226	C₄	#20259328	1	2	yes	—
227	C₄	C₂×Q₈×D₁₉₁	1	1	yes	characteristic
228	C₄	#41×D₁₉₁	1	1	yes	characteristic
229	C₂	—	382	2	no	—
230	C₂	C₄×Q₈×D₁₉₁	1	1	yes	characteristic
231	C₂	#131×D₁₉₁	1	1	yes	characteristic
232	C₂	C₂×#20259328	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₉₁
commutator, socle	C₂²×C₁₉₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	C₈×Q₈×C₁₉₁	C₂
prime radical	C₂²×D₁₉₁	C₂²×C₄
cosocle	C₂×C₄×C₁₉₁	C₂⁴
hypercenter	C₈×Q₈	D₁₉₁
hypocenter	C₁₉₁	#2960
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. #27132794	C₂	semidirect
3. #27132792	C₂	semidirect
4. #27132790	C₂	semidirect
5. C₈×Dic₃₈₂	C₂	semidirect
6. C₈×Dic₃₈₂	C₂	semidirect
11. C₈×Q₈×C₁₉₁	C₂	semidirect
51. C₈×D₁₉₁	Q₈	semidirect
116. C₁₉₁	#2960	semidirect
124. C₈×Q₈	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
#27133217	2
#27132794	2
#27132792	2
#27132790	2
C₈×Dic₃₈₂	2
#27132688	2
#27132685	2
#27132617	2
C₄×C₈×D₁₉₁	2
C₈×Q₈×C₁₉₁	2
#2960	191