Subgroups — #37575202

Order 24448.

556 conjugacy classes of subgroups.

The subgroup lattice has 288 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	C₂×C₄×Dic₃₈₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#27132569	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132561	C₂	1	1	yes	maximal, characteristic, complemented
5	#27132558	C₂	1	2	yes	maximal, complemented
6	#27132557	C₂	1	2	yes	maximal
7	#27132551	C₂	1	2	yes	maximal
8	#27132544	C₂	1	2	yes	maximal, complemented
9	#27132523	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊Dic₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#117×C₁₉₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₃₈₂	C₂²	1	1	yes	characteristic
13	#20259301	C₄	1	8	yes	complemented
14	C₄×Dic₃₈₂	—	2	4	no	—
15	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
19	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
20	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
21	C₂×C₂²⋊Dic₁₉₁	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₁₉₁	C₂²	1	2	yes	—
23	C₂×C₄×Dic₁₉₁	C₂²	1	2	yes	—
24	C₂×C₄×Dic₁₉₁	C₂²	1	2	yes	—
25	C₁₉₁⋊#95	C₂²	1	2	yes	—
26	C₁₉₁⋊#95	C₂²	1	2	yes	—
27	C₁₉₁⋊#95	C₂²	1	2	yes	—
28	C₁₉₁⋊#95	C₂²	1	2	yes	—
29	C₁₉₁⋊#95	C₂²	1	2	yes	—
30	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
31	#117×C₁₉₁	—	2	4	no	—
32	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
37	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
38	C₂×Dic₃₈₂	—	2	2	no	—
39	C₂×Dic₃₈₂	—	2	2	no	—
40	C₂×Dic₃₈₂	C₂×C₄	1	4	yes	—
41	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
42	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
43	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
44	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
45	C₄⋊Dic₁₉₁	—	2	2	no	—
46	C₄⋊Dic₁₉₁	—	2	2	no	—
47	C₄⋊Dic₁₉₁	C₂×C₄	1	4	yes	—
48	Dic₁₉₁⋊C₄	—	2	4	no	—
49	Dic₁₉₁⋊C₄	—	4	2	no	—
50	Dic₁₉₁⋊C₄	C₂×C₄	1	8	yes	—
51	C₂²⋊Dic₁₉₁	—	4	2	no	—
52	C₂²⋊Dic₁₉₁	C₂×C₄	1	8	yes	—
53	C₄×Dic₁₉₁	—	2	4	no	—
54	C₄×Dic₁₉₁	—	4	2	no	—
55	C₂⁴×C₁₉₁	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₉₁	—	2	2	no	—
57	C₂²×C₄×C₁₉₁	—	2	2	no	—
58	C₂²×C₄×C₁₉₁	C₂×C₄	1	4	yes	—
59	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₁₉₁	—	2	2	no	—
66	C₄⋊C₄×C₁₉₁	—	2	2	no	—
67	C₂²⋊C₄×C₁₉₁	—	2	2	no	—
68	C₂²⋊C₄×C₁₉₁	—	2	2	no	—
69	C₄²×C₁₉₁	—	2	2	no	—
70	C₄²×C₁₉₁	D₄	1	4	yes	—
71	Dic₃₈₂	—	2	8	no	—
72	C₂×Dic₁₉₁	—	2	2	no	—
73	C₂×Dic₁₉₁	—	2	2	no	—
74	C₂×Dic₁₉₁	—	2	2	no	—
75	C₂×Dic₁₉₁	—	2	2	no	—
76	C₂×Dic₁₉₁	—	2	2	no	—
77	C₂×Dic₁₉₁	—	2	2	no	—
78	C₂×Dic₁₉₁	—	2	2	no	—
79	C₂×Dic₁₉₁	—	2	2	no	—
80	C₂×Dic₁₉₁	—	2	2	no	—
81	C₂×Dic₁₉₁	—	2	2	no	—
82	C₂×Dic₁₉₁	C₂²×C₄	1	4	yes	—
83	C₂×Dic₁₉₁	C₂²×C₄	1	4	yes	—
84	C₂³×C₁₉₁	—	2	1	no	—
85	C₂³×C₁₉₁	—	2	1	no	—
86	C₂³×C₁₉₁	—	2	1	no	—
87	C₂³×C₁₉₁	—	2	1	no	—
88	C₂³×C₁₉₁	—	2	1	no	—
89	C₂³×C₁₉₁	—	2	1	no	—
90	C₂³×C₁₉₁	C₂²×C₄	1	2	yes	—
91	C₂³×C₁₉₁	C₂⁴	1	1	yes	characteristic
92	C₂×C₄×C₁₉₁	—	2	1	no	—
93	C₂×C₄×C₁₉₁	—	2	1	no	—
94	C₂×C₄×C₁₉₁	—	2	1	no	—
95	C₂×C₄×C₁₉₁	—	2	1	no	—
96	C₂×C₄×C₁₉₁	—	2	1	no	—
97	C₂×C₄×C₁₉₁	—	2	1	no	—
98	C₂×C₄×C₁₉₁	—	2	8	no	—
99	C₂×C₄×C₁₉₁	—	2	1	no	—
100	C₂×C₄×C₁₉₁	—	2	1	no	—
101	C₂×C₄×C₁₉₁	—	2	1	no	—
102	C₂×C₄×C₁₉₁	—	2	1	no	—
103	C₂×C₄×C₁₉₁	—	2	1	no	—
104	C₂×C₄×C₁₉₁	—	2	1	no	—
105	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
106	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₉₁	#41	1	2	yes	—
111	C₂²×C₁₉₁	—	2	2	no	—
112	C₂²×C₁₉₁	—	2	2	no	—
113	C₂²×C₁₉₁	—	2	2	no	—
114	C₂²×C₁₉₁	—	4	1	no	—
115	C₂²×C₁₉₁	—	4	1	no	—
116	C₂²×C₁₉₁	—	4	1	no	—
117	C₂²×C₁₉₁	—	4	1	no	—
118	C₂²×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
119	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
120	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
121	C₂²×C₁₉₁	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂²×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
123	C₂²×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
124	C₂²×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
125	C₄×C₁₉₁	—	2	2	no	—
126	C₄×C₁₉₁	—	2	2	no	—
127	C₄×C₁₉₁	—	2	2	no	—
128	C₄×C₁₉₁	—	4	1	no	—
129	C₄×C₁₉₁	—	4	1	no	—
130	C₄×C₁₉₁	C₄×D₄	1	4	yes	—
131	Dic₁₉₁	—	2	4	no	—
132	Dic₁₉₁	—	2	4	no	—
133	Dic₁₉₁	—	4	2	no	—
134	Dic₁₉₁	—	4	2	no	—
135	C₂×C₁₉₁	—	4	2	no	—
136	C₂×C₁₉₁	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂×C₁₉₁	#519	1	1	yes	characteristic
138	C₂×C₁₉₁	#520	1	1	yes	characteristic
139	C₂×C₁₉₁	#536	1	1	yes	characteristic
140	C₂×C₁₉₁	#537	1	1	yes	characteristic
141	C₂×C₁₉₁	#541	1	1	yes	characteristic
142	C₂×C₁₉₁	#542	1	1	yes	characteristic
143	C₁₉₁	#2323	1	1	yes	characteristic, complemented, Sylow
144	#2323	—	191	1	no	maximal, Sylow
145	C₂×C₂²⋊Q₈	—	191	1	no	—
146	C₂×C₄×Q₈	—	191	1	no	—
147	C₂×#117	D₁₉₁	1	1	yes	characteristic
148	#391	—	191	1	no	—
149	#389	—	191	2	no	—
150	#389	—	191	1	no	—
151	#388	—	191	2	no	—
152	#388	—	191	2	no	—
153	#387	—	191	2	no	—
154	#380	—	191	1	no	—
155	C₄×C₄⋊C₄	—	191	1	no	—
156	C₂²×Q₈	—	191	1	no	—
157	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
158	C₂²⋊Q₈	—	191	8	no	—
159	C₄×Q₈	—	382	4	no	—
160	#117	—	2	4	no	—
161	C₂×C₄⋊C₄	—	191	1	no	—
162	C₂×C₄⋊C₄	—	191	2	no	—
163	C₂×C₄⋊C₄	—	191	1	no	—
164	C₂×C₄⋊C₄	—	191	1	no	—
165	C₂×C₄⋊C₄	—	191	2	no	—
166	C₂×C₄⋊C₄	—	191	2	no	—
167	C₂×C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
168	C₂×C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	—	191	2	no	—
170	C₂×C₂²⋊C₄	—	191	2	no	—
171	C₂×C₂²⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
173	C₂×C₄²	—	191	2	no	—
174	C₂×C₄²	—	191	2	no	—
175	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
176	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
177	#95	—	191	2	no	—
178	#95	—	191	2	no	—
179	#95	—	191	2	no	—
180	#95	—	191	2	no	—
181	#95	—	191	2	no	—
182	C₂⁴	C₂²×D₁₉₁	1	1	yes	characteristic
183	C₂×Q₈	—	382	2	no	—
184	C₂×Q₈	—	382	2	no	—
185	C₂×Q₈	—	191	4	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	191	2	no	—
189	C₂²×C₄	—	191	2	no	—
190	C₂²×C₄	—	191	2	no	—
191	C₂²×C₄	—	191	2	no	—
192	C₂²×C₄	C₄×D₁₉₁	1	4	yes	—
193	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
194	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
195	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
196	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
197	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	382	4	no	—
202	C₄⋊C₄	—	382	2	no	—
203	C₄⋊C₄	—	382	2	no	—
204	C₄⋊C₄	—	191	8	no	—
205	C₄⋊C₄	—	191	4	no	—
206	C₄⋊C₄	—	764	2	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₂²⋊C₄	—	191	8	no	—
210	C₂²⋊C₄	—	764	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	382	4	no	—
213	C₄²	—	764	2	no	—
214	C₄²	C₂²⋊D₁₉₁	1	4	yes	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	1	no	—
221	C₂³	C₂×C₄×D₁₉₁	1	2	yes	—
222	C₂³	C₂³×D₁₉₁	1	1	yes	characteristic
223	Q₈	—	382	8	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	382	2	no	—
238	C₂×C₄	—	382	2	no	—
239	C₂×C₄	—	382	2	no	—
240	C₂×C₄	—	382	2	no	—
241	C₂×C₄	—	382	2	no	—
242	C₂×C₄	—	382	2	no	—
243	C₂×C₄	—	382	2	no	—
244	C₂×C₄	—	382	2	no	—
245	C₂×C₄	—	382	2	no	—
246	C₂×C₄	—	382	2	no	—
247	C₂×C₄	—	191	4	no	—
248	C₂×C₄	—	191	4	no	—
249	C₂×C₄	C₂×C₄×D₁₉₁	1	2	yes	—
250	C₂×C₄	C₂×C₄×D₁₉₁	1	2	yes	—
251	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
252	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
253	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
254	C₂×C₄	#15690287	1	2	yes	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	C₂²×C₄×D₁₉₁	1	1	yes	characteristic
263	C₂²	C₂²×C₂²⋊D₁₉₁	1	1	yes	characteristic
264	C₂²	C₂×#15690287	1	1	yes	characteristic
265	C₂²	#20259368	1	1	yes	characteristic
266	C₂²	#20259369	1	1	yes	characteristic
267	C₂²	#20259369	1	1	yes	characteristic
268	C₂²	#20259369	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	382	4	no	—
275	C₄	—	382	4	no	—
276	C₄	—	764	2	no	—
277	C₄	—	764	2	no	—
278	C₄	C₄×C₂²⋊D₁₉₁	1	4	yes	—
279	C₂	—	4	2	no	—
280	C₂	C₂×C₄×C₂²⋊D₁₉₁	1	1	yes	characteristic
281	C₂	#27133140	1	1	yes	characteristic
282	C₂	#27133143	1	1	yes	characteristic
283	C₂	#27133234	1	1	yes	characteristic
284	C₂	#27133241	1	1	yes	characteristic
285	C₂	#27133275	1	1	yes	characteristic
286	C₂	#27133284	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉₁
commutator	C₂²×C₁₉₁	C₂³×C₄
Fitting	C₂×#117×C₁₉₁	C₂
socle, cosocle	C₂³×C₁₉₁	C₂⁴
prime radical	C₂⁴×C₁₉₁	C₂³
hypercenter	C₂×#117	D₁₉₁
hypocenter	C₁₉₁	#2323
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₃₈₂	C₂	semidirect
3. #27132569	C₂	semidirect
4. #27132561	C₂	semidirect
5. #27132558	C₂	semidirect
8. #27132544	C₂	semidirect
10. C₄×C₄⋊Dic₁₉₁	C₂	semidirect
13. #20259301	C₄	semidirect
143. C₁₉₁	#2323	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
C₂×C₄×Dic₃₈₂	2
#27132569	2
#27132561	2
#27132558	2
#27132557	2
#27132551	2
#27132544	2
#27132523	2
C₄×C₄⋊Dic₁₉₁	2
C₂×#117×C₁₉₁	2
#2323	191