Subgroups — #37575270

Order 24448.

484 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20259299	C₂	1	1	yes	maximal, characteristic
2	#27132569	C₂	1	1	yes	maximal, characteristic, complemented
3	#27132567	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132556	C₂	1	1	yes	maximal, characteristic
5	#27132551	C₂	1	1	yes	maximal, characteristic
6	#27132551	C₂	1	1	yes	maximal, characteristic
7	#27132550	C₂	1	1	yes	maximal, characteristic, complemented
8	#27132548	C₂	1	1	yes	maximal, characteristic, complemented
9	#27132545	C₂	1	1	yes	maximal, characteristic, complemented
10	#27132544	C₂	1	1	yes	maximal, characteristic, complemented
11	#27132524	C₂	1	1	yes	maximal, characteristic
12	C₄⋊C₄×Dic₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×Dic₁₉₁⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²⋊C₄×Dic₁₉₁	C₂	1	1	yes	maximal, characteristic
15	#388×C₁₉₁	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₃₈₂	C₂²	1	1	yes	characteristic
17	C₂³×Dic₁₉₁	C₂²	1	1	yes	characteristic
18	#20259299	C₄	1	8	yes	complemented
19	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₉₁⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×Dic₁₉₁⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₉₁⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₉₁⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
32	C₂×C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
33	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
34	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
35	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
36	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
37	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
38	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
39	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
40	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
41	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
42	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
43	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
44	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
49	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
50	#95×C₁₉₁	C₂²	1	1	yes	characteristic
51	#95×C₁₉₁	C₂²	1	1	yes	characteristic
52	C₂×Dic₃₈₂	—	4	1	no	—
53	C₂×Dic₃₈₂	—	4	1	no	—
54	C₂×Dic₃₈₂	C₂×C₄	1	4	yes	—
55	C₂²×Dic₁₉₁	—	4	1	no	—
56	C₂²×Dic₁₉₁	—	4	1	no	—
57	C₂²×Dic₁₉₁	C₂×C₄	1	4	yes	—
58	C₂²×Dic₁₉₁	C₂³	1	1	yes	characteristic
59	C₂²×Dic₁₉₁	C₂³	1	1	yes	characteristic
60	C₂²×Dic₁₉₁	C₂³	1	1	yes	characteristic
61	C₂²×Dic₁₉₁	C₂³	1	1	yes	characteristic
62	C₂²×Dic₁₉₁	C₂³	1	1	yes	characteristic
63	C₂²×Dic₁₉₁	C₂³	1	1	yes	characteristic
64	C₂²×Dic₁₉₁	C₂³	1	1	yes	characteristic
65	C₂²×Dic₁₉₁	C₂³	1	1	yes	characteristic
66	C₄⋊Dic₁₉₁	—	4	1	no	—
67	C₄⋊Dic₁₉₁	C₂×C₄	1	4	yes	—
68	Dic₁₉₁⋊C₄	—	2	2	no	—
69	Dic₁₉₁⋊C₄	—	4	1	no	—
70	Dic₁₉₁⋊C₄	C₂×C₄	1	4	yes	—
71	Dic₁₉₁⋊C₄	C₂×C₄	1	4	yes	—
72	C₂²⋊Dic₁₉₁	—	2	2	no	—
73	C₂²⋊Dic₁₉₁	—	4	1	no	—
74	C₂²⋊Dic₁₉₁	C₂×C₄	1	4	yes	—
75	C₄×Dic₁₉₁	—	2	2	no	—
76	C₄×Dic₁₉₁	—	2	2	no	—
77	C₄×Dic₁₉₁	—	4	1	no	—
78	C₄×Dic₁₉₁	—	4	1	no	—
79	C₄×Dic₁₉₁	—	4	1	no	—
80	C₂⁴×C₁₉₁	C₂³	1	1	yes	characteristic
81	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
84	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
85	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
87	C₄⋊C₄×C₁₉₁	—	2	2	no	—
88	C₂²⋊C₄×C₁₉₁	—	2	2	no	—
89	C₂²⋊C₄×C₁₉₁	—	4	1	no	—
90	C₂²⋊C₄×C₁₉₁	C₂×C₄	1	4	yes	—
91	C₄²×C₁₉₁	—	4	1	no	—
92	Dic₃₈₂	—	4	4	no	—
93	C₂×Dic₁₉₁	—	2	1	no	—
94	C₂×Dic₁₉₁	—	2	1	no	—
95	C₂×Dic₁₉₁	—	2	1	no	—
96	C₂×Dic₁₉₁	—	2	1	no	—
97	C₂×Dic₁₉₁	—	2	1	no	—
98	C₂×Dic₁₉₁	—	2	1	no	—
99	C₂×Dic₁₉₁	—	2	1	no	—
100	C₂×Dic₁₉₁	—	2	1	no	—
101	C₂×Dic₁₉₁	—	2	1	no	—
102	C₂×Dic₁₉₁	—	2	1	no	—
103	C₂×Dic₁₉₁	—	2	1	no	—
104	C₂×Dic₁₉₁	—	2	1	no	—
105	C₂×Dic₁₉₁	—	2	1	no	—
106	C₂×Dic₁₉₁	—	2	1	no	—
107	C₂×Dic₁₉₁	—	2	1	no	—
108	C₂×Dic₁₉₁	—	2	1	no	—
109	C₂×Dic₁₉₁	—	2	1	no	—
110	C₂×Dic₁₉₁	—	2	1	no	—
111	C₂×Dic₁₉₁	—	2	1	no	—
112	C₂×Dic₁₉₁	—	4	4	no	—
113	C₂×Dic₁₉₁	C₂²×C₄	1	2	yes	—
114	C₂×Dic₁₉₁	C₂²×C₄	1	2	yes	—
115	C₂×Dic₁₉₁	C₂²×C₄	1	2	yes	—
116	C₂×Dic₁₉₁	C₂²×C₄	1	2	yes	—
117	C₂×Dic₁₉₁	#41	1	2	yes	—
118	C₂³×C₁₉₁	—	2	1	no	—
119	C₂³×C₁₉₁	—	2	1	no	—
120	C₂³×C₁₉₁	—	2	1	no	—
121	C₂³×C₁₉₁	—	2	1	no	—
122	C₂³×C₁₉₁	—	2	1	no	—
123	C₂³×C₁₉₁	—	2	1	no	—
124	C₂³×C₁₉₁	C₂²×C₄	1	2	yes	—
125	C₂³×C₁₉₁	C₂⁴	1	1	yes	characteristic
126	C₂×C₄×C₁₉₁	—	2	1	no	—
127	C₂×C₄×C₁₉₁	—	2	1	no	—
128	C₂×C₄×C₁₉₁	—	2	1	no	—
129	C₂×C₄×C₁₉₁	—	2	1	no	—
130	C₂×C₄×C₁₉₁	—	2	1	no	—
131	C₂×C₄×C₁₉₁	—	2	1	no	—
132	C₂×C₄×C₁₉₁	—	2	1	no	—
133	C₂×C₄×C₁₉₁	—	2	1	no	—
134	C₂×C₄×C₁₉₁	—	2	1	no	—
135	C₂×C₄×C₁₉₁	—	2	1	no	—
136	C₂×C₄×C₁₉₁	—	2	1	no	—
137	C₂×C₄×C₁₉₁	—	2	1	no	—
138	C₂×C₄×C₁₉₁	—	2	1	no	—
139	C₂×C₄×C₁₉₁	—	2	1	no	—
140	C₂×C₄×C₁₉₁	—	2	1	no	—
141	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
142	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
143	C₂×C₄×C₁₉₁	#41	1	2	yes	—
144	C₂²×C₁₉₁	—	2	2	no	—
145	C₂²×C₁₉₁	—	2	2	no	—
146	C₂²×C₁₉₁	—	2	2	no	—
147	C₂²×C₁₉₁	—	4	1	no	—
148	C₂²×C₁₉₁	—	4	1	no	—
149	C₂²×C₁₉₁	—	4	1	no	—
150	C₂²×C₁₉₁	—	4	1	no	—
151	C₂²×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
152	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
153	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
154	C₂²×C₁₉₁	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂²×C₁₉₁	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
157	C₂²×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
158	C₄×C₁₉₁	—	2	2	no	—
159	C₄×C₁₉₁	—	2	2	no	—
160	C₄×C₁₉₁	—	2	2	no	—
161	C₄×C₁₉₁	—	4	1	no	—
162	C₄×C₁₉₁	—	4	1	no	—
163	C₄×C₁₉₁	—	4	1	no	—
164	Dic₁₉₁	—	2	2	no	—
165	Dic₁₉₁	—	2	2	no	—
166	Dic₁₉₁	—	2	2	no	—
167	Dic₁₉₁	—	2	2	no	—
168	Dic₁₉₁	—	2	2	no	—
169	Dic₁₉₁	—	4	1	no	—
170	Dic₁₉₁	—	4	1	no	—
171	Dic₁₉₁	—	4	1	no	—
172	C₂×C₁₉₁	—	4	2	no	—
173	C₂×C₁₉₁	C₄×#41	1	1	yes	characteristic
174	C₂×C₁₉₁	#518	1	1	yes	characteristic
175	C₂×C₁₉₁	#519	1	1	yes	characteristic
176	C₂×C₁₉₁	#539	1	1	yes	characteristic
177	C₂×C₁₉₁	#539	1	1	yes	characteristic
178	C₂×C₁₉₁	#542	1	1	yes	characteristic
179	C₂×C₁₉₁	#542	1	1	yes	characteristic
180	C₁₉₁	#2332	1	1	yes	characteristic, complemented, Sylow
181	#2332	—	191	1	no	maximal, Sylow
182	C₂×C₂²⋊Q₈	—	191	1	no	—
183	#391	—	191	1	no	—
184	#391	—	191	1	no	—
185	#388	—	191	1	no	—
186	#388	—	191	1	no	—
187	#388	—	191	1	no	—
188	#388	D₁₉₁	1	1	yes	characteristic
189	#387	—	191	1	no	—
190	#387	—	191	1	no	—
191	#387	—	191	1	no	—
192	#387	—	191	1	no	—
193	#381	—	191	1	no	—
194	C₄×C₄⋊C₄	—	191	1	no	—
195	C₄×C₄⋊C₄	—	191	1	no	—
196	C₄×C₂²⋊C₄	—	191	1	no	—
197	C₂²×Q₈	—	191	1	no	—
198	C₂³×C₄	—	191	1	no	—
199	C₂²⋊Q₈	—	191	8	no	—
200	C₂×C₄⋊C₄	—	191	1	no	—
201	C₂×C₄⋊C₄	—	191	1	no	—
202	C₂×C₄⋊C₄	—	191	1	no	—
203	C₂×C₄⋊C₄	—	191	1	no	—
204	C₂×C₄⋊C₄	—	191	1	no	—
205	C₂×C₄⋊C₄	—	191	1	no	—
206	C₂×C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
207	C₂×C₂²⋊C₄	—	191	1	no	—
208	C₂×C₂²⋊C₄	—	191	1	no	—
209	C₂×C₂²⋊C₄	—	191	1	no	—
210	C₂×C₂²⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
211	C₂×C₂²⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
212	C₂×C₂²⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
213	C₂×C₄²	—	191	1	no	—
214	C₂×C₄²	—	191	1	no	—
215	C₂×C₄²	—	191	1	no	—
216	C₂×C₄²	—	191	1	no	—
217	C₂×C₄²	—	191	1	no	—
218	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
219	#95	—	191	1	no	—
220	#95	—	191	1	no	—
221	#95	—	191	1	no	—
222	#95	—	191	1	no	—
223	#95	—	191	1	no	—
224	#95	—	191	1	no	—
225	#95	—	191	1	no	—
226	#95	—	191	1	no	—
227	#95	—	191	1	no	—
228	#95	—	191	1	no	—
229	#95	—	191	1	no	—
230	#95	—	191	1	no	—
231	#95	C₂×D₁₉₁	1	1	yes	characteristic
232	#95	C₂×D₁₉₁	1	1	yes	characteristic
233	C₂⁴	C₂²×D₁₉₁	1	1	yes	characteristic
234	C₂×Q₈	—	191	4	no	—
235	C₂×Q₈	—	764	1	no	—
236	C₂×Q₈	—	764	1	no	—
237	C₂²×C₄	—	191	1	no	—
238	C₂²×C₄	—	191	1	no	—
239	C₂²×C₄	—	191	1	no	—
240	C₂²×C₄	—	191	1	no	—
241	C₂²×C₄	—	191	1	no	—
242	C₂²×C₄	—	191	1	no	—
243	C₂²×C₄	—	191	1	no	—
244	C₂²×C₄	—	191	1	no	—
245	C₂²×C₄	—	191	4	no	—
246	C₂²×C₄	—	764	1	no	—
247	C₂²×C₄	—	764	1	no	—
248	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
249	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
250	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
251	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
252	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
253	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	382	2	no	—
256	C₄⋊C₄	—	191	4	no	—
257	C₄⋊C₄	—	191	4	no	—
258	C₄⋊C₄	—	191	4	no	—
259	C₄⋊C₄	—	764	1	no	—
260	C₄⋊C₄	—	764	1	no	—
261	C₂²⋊C₄	—	2	2	no	—
262	C₂²⋊C₄	—	4	1	no	—
263	C₂²⋊C₄	—	382	2	no	—
264	C₂²⋊C₄	—	191	4	no	—
265	C₂²⋊C₄	—	764	1	no	—
266	C₂²⋊C₄	C₄×D₁₉₁	1	4	yes	—
267	C₄²	—	4	1	no	—
268	C₄²	—	382	2	no	—
269	C₄²	—	382	2	no	—
270	C₄²	—	764	1	no	—
271	C₄²	—	764	1	no	—
272	C₄²	—	764	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	2	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	—	2	1	no	—
279	C₂³	C₂×C₄×D₁₉₁	1	2	yes	—
280	C₂³	C₂³×D₁₉₁	1	1	yes	characteristic
281	Q₈	—	764	4	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	382	1	no	—
298	C₂×C₄	—	382	1	no	—
299	C₂×C₄	—	382	1	no	—
300	C₂×C₄	—	382	1	no	—
301	C₂×C₄	—	382	1	no	—
302	C₂×C₄	—	382	1	no	—
303	C₂×C₄	—	382	1	no	—
304	C₂×C₄	—	382	1	no	—
305	C₂×C₄	—	382	1	no	—
306	C₂×C₄	—	382	1	no	—
307	C₂×C₄	—	382	1	no	—
308	C₂×C₄	—	382	1	no	—
309	C₂×C₄	—	382	1	no	—
310	C₂×C₄	—	382	1	no	—
311	C₂×C₄	—	382	1	no	—
312	C₂×C₄	—	382	1	no	—
313	C₂×C₄	—	382	1	no	—
314	C₂×C₄	—	382	1	no	—
315	C₂×C₄	—	382	1	no	—
316	C₂×C₄	—	191	2	no	—
317	C₂×C₄	—	191	2	no	—
318	C₂×C₄	—	191	2	no	—
319	C₂×C₄	—	191	2	no	—
320	C₂×C₄	—	191	2	no	—
321	C₂×C₄	—	764	4	no	—
322	C₂×C₄	C₂×C₄×D₁₉₁	1	2	yes	—
323	C₂×C₄	C₂×C₄×D₁₉₁	1	2	yes	—
324	C₂×C₄	#15690286	1	2	yes	—
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	C₂²×C₄×D₁₉₁	1	1	yes	characteristic
333	C₂²	#41×D₁₉₁	1	1	yes	characteristic
334	C₂²	C₂×#15690286	1	1	yes	characteristic
335	C₂²	#20259367	1	1	yes	characteristic
336	C₂²	#20259368	1	1	yes	characteristic
337	C₂²	#20259369	1	1	yes	characteristic
338	C₂²	#20259369	1	1	yes	characteristic
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	4	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	382	2	no	—
346	C₄	—	382	2	no	—
347	C₄	—	382	2	no	—
348	C₄	—	382	2	no	—
349	C₄	—	382	2	no	—
350	C₄	—	764	1	no	—
351	C₄	—	764	1	no	—
352	C₄	—	764	1	no	—
353	C₂	—	4	2	no	—
354	C₂	C₄×#15690286	1	1	yes	characteristic
355	C₂	#27133135	1	1	yes	characteristic
356	C₂	#27133140	1	1	yes	characteristic
357	C₂	#27133256	1	1	yes	characteristic
358	C₂	#27133258	1	1	yes	characteristic
359	C₂	#27133279	1	1	yes	characteristic
360	C₂	#27133282	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉₁
commutator	C₂²×C₁₉₁	C₂³×C₄
Fitting	#388×C₁₉₁	C₂
socle, cosocle	C₂³×C₁₉₁	C₂⁴
prime radical	C₂⁴×C₁₉₁	C₂³
hypercenter	#388	D₁₉₁
hypocenter	C₁₉₁	#2332
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ #388

Complements

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

N	H ≅ G/N	type
2. #27132569	C₂	semidirect
3. #27132567	C₂	semidirect
7. #27132550	C₂	semidirect
8. #27132548	C₂	semidirect
9. #27132545	C₂	semidirect
10. #27132544	C₂	semidirect
12. C₄⋊C₄×Dic₁₉₁	C₂	semidirect
13. C₄×Dic₁₉₁⋊C₄	C₂	semidirect
18. #20259299	C₄	semidirect
180. C₁₉₁	#2332	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20259299	2
#27132569	2
#27132567	2
#27132556	2
#27132551	2
#27132550	2
#27132548	2
#27132545	2
#27132544	2
#27132524	2
C₄⋊C₄×Dic₁₉₁	2
C₄×Dic₁₉₁⋊C₄	2
C₂²⋊C₄×Dic₁₉₁	2
#388×C₁₉₁	2
#2332	191