Subgroups — #37575253

Order 24448.

528 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20259301	C₂	1	1	yes	maximal, characteristic
2	#27132567	C₂	1	2	yes	maximal, complemented
3	#27132560	C₂	1	2	yes	maximal, complemented
4	#27132550	C₂	1	2	yes	maximal, complemented
5	#27132546	C₂	1	2	yes	maximal, complemented
6	#27132536	C₂	1	2	yes	maximal
7	#27132523	C₂	1	1	yes	maximal, characteristic
8	C₄×C₂²⋊Dic₁₉₁	C₂	1	2	yes	maximal
9	C₂×#95×C₁₉₁	C₂	1	1	yes	maximal, characteristic
10	C₂²×Dic₃₈₂	C₂²	1	1	yes	characteristic
11	#20259301	C₄	1	8	yes	complemented
12	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
16	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
17	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
18	C₂×C₂²⋊Dic₁₉₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊Dic₁₉₁	C₂²	1	2	yes	—
20	C₂×C₄×Dic₁₉₁	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₉₁	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	C₁₉₁⋊#95	C₂²	1	2	yes	—
27	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
28	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
29	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
30	#95×C₁₉₁	C₂²	1	2	yes	—
31	#95×C₁₉₁	C₂²	1	2	yes	—
32	C₂×Dic₃₈₂	—	4	1	no	—
33	C₂×Dic₃₈₂	—	4	1	no	—
34	C₂×Dic₃₈₂	C₂×C₄	1	4	yes	complemented
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	—	4	1	no	—
40	C₄⋊Dic₁₉₁	—	4	1	no	—
41	C₄⋊Dic₁₉₁	C₂×C₄	1	4	yes	complemented
42	Dic₁₉₁⋊C₄	—	4	2	no	—
43	Dic₁₉₁⋊C₄	—	4	2	no	—
44	Dic₁₉₁⋊C₄	C₂×C₄	1	8	yes	complemented
45	C₂²⋊Dic₁₉₁	—	2	4	no	—
46	C₂²⋊Dic₁₉₁	C₂×C₄	1	8	yes	—
47	C₄×Dic₁₉₁	—	4	2	no	—
48	C₄×Dic₁₉₁	—	4	2	no	—
49	C₂⁴×C₁₉₁	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₉₁	—	2	2	no	—
51	C₂²×C₄×C₁₉₁	—	4	1	no	—
52	C₂²×C₄×C₁₉₁	—	4	1	no	—
53	C₂²×C₄×C₁₉₁	—	4	1	no	—
54	C₂²×C₄×C₁₉₁	—	4	1	no	—
55	C₂²×C₄×C₁₉₁	—	4	1	no	—
56	C₂²×C₄×C₁₉₁	—	4	1	no	—
57	C₂²×C₄×C₁₉₁	C₂×C₄	1	4	yes	—
58	C₂²×C₄×C₁₉₁	Q₈	1	4	yes	complemented
59	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
60	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
63	Dic₃₈₂	—	4	4	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	—	2	2	no	—
69	C₂×Dic₁₉₁	—	2	2	no	—
70	C₂×Dic₁₉₁	—	2	2	no	—
71	C₂×Dic₁₉₁	—	2	2	no	—
72	C₂×Dic₁₉₁	—	2	2	no	—
73	C₂×Dic₁₉₁	—	2	2	no	—
74	C₂×Dic₁₉₁	C₂²×C₄	1	4	yes	—
75	C₂×Dic₁₉₁	C₂²×C₄	1	4	yes	—
76	C₂³×C₁₉₁	—	2	1	no	—
77	C₂³×C₁₉₁	—	2	1	no	—
78	C₂³×C₁₉₁	—	2	1	no	—
79	C₂³×C₁₉₁	—	2	1	no	—
80	C₂³×C₁₉₁	C₂²×C₄	1	2	yes	—
81	C₂³×C₁₉₁	C₂×Q₈	1	2	yes	—
82	C₂³×C₁₉₁	#41	1	2	yes	—
83	C₂³×C₁₉₁	C₂⁴	1	1	yes	characteristic
84	C₂×C₄×C₁₉₁	—	2	2	no	—
85	C₂×C₄×C₁₉₁	—	2	2	no	—
86	C₂×C₄×C₁₉₁	—	2	1	no	—
87	C₂×C₄×C₁₉₁	—	2	1	no	—
88	C₂×C₄×C₁₉₁	—	2	1	no	—
89	C₂×C₄×C₁₉₁	—	2	1	no	—
90	C₂×C₄×C₁₉₁	—	2	2	no	—
91	C₂×C₄×C₁₉₁	—	2	2	no	—
92	C₂×C₄×C₁₉₁	—	2	2	no	—
93	C₂×C₄×C₁₉₁	—	4	4	no	—
94	C₂×C₄×C₁₉₁	—	4	4	no	—
95	C₂×C₄×C₁₉₁	—	4	4	no	—
96	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₁₉₁	C₂×Q₈	1	4	yes	—
99	C₂²×C₁₉₁	—	2	2	no	—
100	C₂²×C₁₉₁	—	2	2	no	—
101	C₂²×C₁₉₁	—	2	2	no	—
102	C₂²×C₁₉₁	—	2	2	no	—
103	C₂²×C₁₉₁	—	2	2	no	—
104	C₂²×C₁₉₁	—	2	2	no	—
105	C₂²×C₁₉₁	C₄×Q₈	1	4	yes	—
106	C₂²×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
107	C₂²×C₁₉₁	C₂²×Q₈	1	1	yes	characteristic
108	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
109	C₂²×C₁₉₁	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂²×C₁₉₁	H(5,𝔽₂)	1	1	yes	characteristic
111	C₂²×C₁₉₁	H(5,𝔽₂)	1	1	yes	characteristic
112	C₂²×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
113	C₄×C₁₉₁	—	2	2	no	—
114	C₄×C₁₉₁	—	2	2	no	—
115	C₄×C₁₉₁	—	2	4	no	—
116	C₄×C₁₉₁	—	4	2	no	—
117	Dic₁₉₁	—	2	4	no	—
118	Dic₁₉₁	—	2	4	no	—
119	Dic₁₉₁	—	4	2	no	—
120	Dic₁₉₁	—	4	2	no	—
121	C₂×C₁₉₁	—	2	4	no	—
122	C₂×C₁₉₁	C₂×C₄×Q₈	1	1	yes	characteristic
123	C₂×C₁₉₁	#518	1	1	yes	characteristic
124	C₂×C₁₉₁	#520	1	1	yes	characteristic
125	C₂×C₁₉₁	#538	1	1	yes	characteristic
126	C₂×C₁₉₁	#539	1	1	yes	characteristic
127	C₂×C₁₉₁	#543	1	1	yes	characteristic
128	C₂×C₁₉₁	#544	1	1	yes	characteristic
129	C₁₉₁	#2329	1	1	yes	characteristic, complemented, Sylow
130	#2329	—	191	1	no	maximal, Sylow
131	C₂×C₂²⋊Q₈	—	191	1	no	—
132	#391	—	191	2	no	—
133	#389	—	191	2	no	—
134	#387	—	191	2	no	—
135	#387	—	191	2	no	—
136	#385	—	191	2	no	—
137	#380	—	191	1	no	—
138	C₄×C₂²⋊C₄	—	191	2	no	—
139	C₂×#95	D₁₉₁	1	1	yes	characteristic
140	C₂²×Q₈	—	191	1	no	—
141	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
142	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
143	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
144	C₂²⋊Q₈	—	191	8	no	—
145	C₂×C₄⋊C₄	—	191	1	no	—
146	C₂×C₄⋊C₄	—	191	2	no	—
147	C₂×C₄⋊C₄	—	191	1	no	—
148	C₂×C₄⋊C₄	—	191	1	no	—
149	C₂×C₄⋊C₄	—	191	2	no	—
150	C₂×C₄⋊C₄	—	191	2	no	—
151	C₂×C₂²⋊C₄	—	191	2	no	—
152	C₂×C₂²⋊C₄	—	191	2	no	—
153	C₂×C₄²	—	191	2	no	—
154	C₂×C₄²	—	191	2	no	—
155	#95	—	191	2	no	—
156	#95	—	191	2	no	—
157	#95	—	191	2	no	—
158	#95	—	191	2	no	—
159	#95	—	191	2	no	—
160	#95	C₂×D₁₉₁	1	2	yes	—
161	#95	C₂×D₁₉₁	1	2	yes	—
162	C₂⁴	C₂²×D₁₉₁	1	1	yes	characteristic
163	C₂×Q₈	—	191	4	no	—
164	C₂×Q₈	—	764	1	no	—
165	C₂×Q₈	—	764	1	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	4	1	no	—
168	C₂²×C₄	—	4	1	no	—
169	C₂²×C₄	—	4	1	no	—
170	C₂²×C₄	—	4	1	no	—
171	C₂²×C₄	—	4	1	no	—
172	C₂²×C₄	—	4	1	no	—
173	C₂²×C₄	—	191	2	no	—
174	C₂²×C₄	—	191	2	no	—
175	C₂²×C₄	—	191	2	no	—
176	C₂²×C₄	—	191	2	no	—
177	C₂²×C₄	C₄×D₁₉₁	1	4	yes	—
178	C₂²×C₄	Dic₃₈₂	1	4	yes	complemented
179	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
180	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
181	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
182	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
183	C₄⋊C₄	—	191	8	no	—
184	C₄⋊C₄	—	191	4	no	—
185	C₄⋊C₄	—	764	2	no	—
186	C₄⋊C₄	—	764	2	no	—
187	C₄⋊C₄	—	764	1	no	—
188	C₄⋊C₄	—	764	1	no	—
189	C₂²⋊C₄	—	382	4	no	—
190	C₂²⋊C₄	—	191	8	no	—
191	C₄²	—	764	2	no	—
192	C₄²	—	764	2	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	1	no	—
195	C₂³	—	2	1	no	—
196	C₂³	—	2	1	no	—
197	C₂³	C₂×C₄×D₁₉₁	1	2	yes	—
198	C₂³	C₂×Dic₃₈₂	1	2	yes	—
199	C₂³	#15690287	1	2	yes	—
200	C₂³	C₂³×D₁₉₁	1	1	yes	characteristic
201	Q₈	—	764	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	382	2	no	—
215	C₂×C₄	—	382	2	no	—
216	C₂×C₄	—	382	2	no	—
217	C₂×C₄	—	382	2	no	—
218	C₂×C₄	—	382	2	no	—
219	C₂×C₄	—	382	2	no	—
220	C₂×C₄	—	382	2	no	—
221	C₂×C₄	—	382	2	no	—
222	C₂×C₄	—	382	2	no	—
223	C₂×C₄	—	382	2	no	—
224	C₂×C₄	—	191	4	no	—
225	C₂×C₄	—	191	4	no	—
226	C₂×C₄	C₂×C₄×D₁₉₁	1	2	yes	—
227	C₂×C₄	C₂×C₄×D₁₉₁	1	2	yes	—
228	C₂×C₄	C₂×Dic₃₈₂	1	4	yes	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	C₄×Dic₃₈₂	1	4	yes	—
236	C₂²	C₂²×C₄×D₁₉₁	1	1	yes	characteristic
237	C₂²	C₂²×Dic₃₈₂	1	1	yes	characteristic
238	C₂²	C₂×#15690287	1	1	yes	characteristic
239	C₂²	#20259367	1	1	yes	characteristic
240	C₂²	#20259367	1	1	yes	characteristic
241	C₂²	#20259367	1	1	yes	characteristic
242	C₂²	#20259370	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	4	2	no	—
247	C₄	—	382	4	no	—
248	C₄	—	382	4	no	—
249	C₄	—	764	2	no	—
250	C₄	—	764	2	no	—
251	C₂	—	2	4	no	—
252	C₂	C₂×C₄×Dic₃₈₂	1	1	yes	characteristic
253	C₂	#27133136	1	1	yes	characteristic
254	C₂	#27133144	1	1	yes	characteristic
255	C₂	#27133251	1	1	yes	characteristic
256	C₂	#27133260	1	1	yes	characteristic
257	C₂	#27133289	1	1	yes	characteristic
258	C₂	#27133294	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉₁
commutator	C₂²×C₁₉₁	C₂³×C₄
Fitting	C₂×#95×C₁₉₁	C₂
socle, cosocle	C₂³×C₁₉₁	C₂⁴
prime radical	C₂⁴×C₁₉₁	C₂³
hypercenter	C₂×#95	D₁₉₁
hypocenter	C₁₉₁	#2329
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂×#95

Complements

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

N	H ≅ G/N	type
2. #27132567	C₂	semidirect
3. #27132560	C₂	semidirect
4. #27132550	C₂	semidirect
5. #27132546	C₂	semidirect
11. #20259301	C₄	semidirect
34. C₂×Dic₃₈₂	C₂×C₄	semidirect
41. C₄⋊Dic₁₉₁	C₂×C₄	semidirect
44. Dic₁₉₁⋊C₄	C₂×C₄	semidirect
58. C₂²×C₄×C₁₉₁	Q₈	semidirect
129. C₁₉₁	#2329	semidirect
178. C₂²×C₄	Dic₃₈₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20259301	2
#27132567	2
#27132560	2
#27132550	2
#27132546	2
#27132536	2
#27132523	2
C₄×C₂²⋊Dic₁₉₁	2
C₂×#95×C₁₉₁	2
#2329	191