Subgroups — #37575949

Order 24448.

604 conjugacy classes of subgroups.

The subgroup lattice has 326 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, complemented
2	C₂×#20259301	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#20259281	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132591	C₂	1	2	yes	maximal, complemented
5	#27132577	C₂	1	2	yes	maximal
6	#27132569	C₂	1	1	yes	maximal, characteristic, complemented
7	#27132561	C₂	1	1	yes	maximal, characteristic, complemented
8	#27132558	C₂	1	2	yes	maximal, complemented
9	#27132557	C₂	1	2	yes	maximal, complemented
10	#27132523	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×D₄×C₁₉₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₃₈₂	C₂²	1	1	yes	characteristic
13	#20259301	—	2	4	no	—
14	#20259301	—	2	4	no	—
15	#20259281	—	2	4	no	—
16	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
20	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
21	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₁₉₁	C₂²	1	2	yes	—
23	C₂×C₂²⋊Dic₁₉₁	C₂²	1	2	yes	—
24	C₂×C₂²⋊Dic₁₉₁	C₂²	1	2	yes	—
25	C₂×C₂²⋊Dic₁₉₁	C₂²	1	2	yes	—
26	C₂×C₄×Dic₁₉₁	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	C₁₉₁⋊#95	C₂²	1	2	yes	—
31	C₂²×D₄×C₁₉₁	C₂²	1	1	yes	characteristic
32	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
33	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
34	C₄×D₄×C₁₉₁	—	2	4	no	—
35	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
38	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
39	C₂×Dic₃₈₂	—	2	2	no	—
40	C₂×Dic₃₈₂	—	2	2	no	—
41	C₂×Dic₃₈₂	—	2	2	no	—
42	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
43	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
44	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
45	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
46	C₄⋊Dic₁₉₁	—	2	2	no	—
47	C₄⋊Dic₁₉₁	—	2	2	no	—
48	C₄⋊Dic₁₉₁	D₄	1	4	yes	complemented
49	Dic₁₉₁⋊C₄	—	2	4	no	—
50	Dic₁₉₁⋊C₄	—	2	4	no	—
51	Dic₁₉₁⋊C₄	—	2	4	no	—
52	C₂²⋊Dic₁₉₁	—	2	4	no	—
53	C₂²⋊Dic₁₉₁	—	2	4	no	—
54	C₂²⋊Dic₁₉₁	—	2	4	no	—
55	C₂²⋊Dic₁₉₁	—	2	4	no	—
56	C₄×Dic₁₉₁	—	2	4	no	—
57	C₂⁴×C₁₉₁	C₂³	1	1	yes	characteristic
58	C₂⁴×C₁₉₁	C₂³	1	1	yes	characteristic
59	C₂×D₄×C₁₉₁	—	2	2	no	—
60	C₂×D₄×C₁₉₁	—	2	2	no	—
61	C₂×D₄×C₁₉₁	—	2	2	no	—
62	C₂²×C₄×C₁₉₁	—	2	2	no	—
63	C₂²×C₄×C₁₉₁	—	2	2	no	—
64	C₂²×C₄×C₁₉₁	—	2	2	no	—
65	C₂²×C₄×C₁₉₁	—	2	2	no	—
66	C₂²×C₄×C₁₉₁	—	2	2	no	—
67	C₂²×C₄×C₁₉₁	D₄	1	4	yes	—
68	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₉₁	—	2	2	no	—
74	C₂²⋊C₄×C₁₉₁	—	2	2	no	—
75	C₂²⋊C₄×C₁₉₁	—	2	2	no	—
76	C₄²×C₁₉₁	—	2	2	no	—
77	Dic₃₈₂	—	4	4	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	—	2	2	no	—
83	C₂×Dic₁₉₁	—	2	2	no	—
84	C₂×Dic₁₉₁	—	2	2	no	—
85	C₂×Dic₁₉₁	—	2	2	no	—
86	C₂×Dic₁₉₁	—	2	2	no	—
87	C₂×Dic₁₉₁	—	2	2	no	—
88	C₂×Dic₁₉₁	—	2	2	no	—
89	C₂×Dic₁₉₁	C₂×D₄	1	4	yes	—
90	C₂³×C₁₉₁	—	2	1	no	—
91	C₂³×C₁₉₁	—	2	1	no	—
92	C₂³×C₁₉₁	—	2	1	no	—
93	C₂³×C₁₉₁	—	2	1	no	—
94	C₂³×C₁₉₁	—	2	1	no	—
95	C₂³×C₁₉₁	—	2	1	no	—
96	C₂³×C₁₉₁	—	2	1	no	—
97	C₂³×C₁₉₁	—	2	1	no	—
98	C₂³×C₁₉₁	—	2	1	no	—
99	C₂³×C₁₉₁	—	2	1	no	—
100	C₂³×C₁₉₁	—	2	1	no	—
101	C₂³×C₁₉₁	—	2	1	no	—
102	C₂³×C₁₉₁	C₂×D₄	1	2	yes	—
103	C₂³×C₁₉₁	#41	1	2	yes	—
104	C₂³×C₁₉₁	C₂⁴	1	1	yes	characteristic
105	D₄×C₁₉₁	—	2	8	no	—
106	C₂×C₄×C₁₉₁	—	2	1	no	—
107	C₂×C₄×C₁₉₁	—	2	1	no	—
108	C₂×C₄×C₁₉₁	—	2	1	no	—
109	C₂×C₄×C₁₉₁	—	2	1	no	—
110	C₂×C₄×C₁₉₁	—	2	8	no	—
111	C₂×C₄×C₁₉₁	—	2	1	no	—
112	C₂×C₄×C₁₉₁	—	2	1	no	—
113	C₂×C₄×C₁₉₁	—	2	1	no	—
114	C₂×C₄×C₁₉₁	—	2	1	no	—
115	C₂×C₄×C₁₉₁	—	2	1	no	—
116	C₂×C₄×C₁₉₁	—	2	1	no	—
117	C₂×C₄×C₁₉₁	—	4	4	no	—
118	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
119	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
120	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
121	C₂×C₄×C₁₉₁	#41	1	2	yes	—
122	C₂×C₄×C₁₉₁	#41	1	2	yes	—
123	C₂²×C₁₉₁	—	2	2	no	—
124	C₂²×C₁₉₁	—	2	2	no	—
125	C₂²×C₁₉₁	—	2	2	no	—
126	C₂²×C₁₉₁	—	2	2	no	—
127	C₂²×C₁₉₁	—	2	2	no	—
128	C₂²×C₁₉₁	—	2	2	no	—
129	C₂²×C₁₉₁	—	4	1	no	—
130	C₂²×C₁₉₁	—	4	1	no	—
131	C₂²×C₁₉₁	—	4	1	no	—
132	C₂²×C₁₉₁	—	4	1	no	—
133	C₂²×C₁₉₁	—	4	1	no	—
134	C₂²×C₁₉₁	—	4	1	no	—
135	C₂²×C₁₉₁	—	4	1	no	—
136	C₂²×C₁₉₁	—	4	1	no	—
137	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
138	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
139	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
140	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
141	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
142	C₂²×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂²×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
144	C₄×C₁₉₁	—	2	2	no	—
145	C₄×C₁₉₁	—	2	2	no	—
146	C₄×C₁₉₁	—	4	1	no	—
147	C₄×C₁₉₁	—	4	1	no	—
148	C₄×C₁₉₁	#121	1	4	yes	—
149	Dic₁₉₁	—	2	4	no	—
150	Dic₁₉₁	—	4	2	no	—
151	Dic₁₉₁	—	4	2	no	—
152	Dic₁₉₁	—	4	2	no	—
153	C₂×C₁₉₁	—	4	2	no	—
154	C₂×C₁₉₁	—	4	2	no	—
155	C₂×C₁₉₁	C₂×#121	1	1	yes	characteristic
156	C₂×C₁₉₁	#525	1	1	yes	characteristic
157	C₂×C₁₉₁	#536	1	1	yes	characteristic
158	C₂×C₁₉₁	#547	1	1	yes	characteristic
159	C₂×C₁₉₁	#548	1	1	yes	characteristic
160	C₂×C₁₉₁	#552	1	1	yes	characteristic
161	C₂×C₁₉₁	#556	1	1	yes	characteristic
162	C₁₉₁	#2415	1	1	yes	characteristic, complemented, Sylow
163	#2415	—	191	1	no	maximal, Sylow
164	C₂×C₂²⋊Q₈	—	191	1	no	—
165	C₂×C₂²⋊Q₈	—	191	1	no	—
166	C₂×C₄×D₄	D₁₉₁	1	1	yes	characteristic
167	C₂×#117	—	191	1	no	—
168	#397	—	191	2	no	—
169	#394	—	191	2	no	—
170	#391	—	191	1	no	—
171	#389	—	191	2	no	—
172	#389	—	191	1	no	—
173	#388	—	191	2	no	—
174	#380	—	191	1	no	—
175	C₂²×Q₈	—	191	1	no	—
176	C₂²×D₄	C₂×D₁₉₁	1	1	yes	characteristic
177	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
178	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
179	C₂²⋊Q₈	—	382	4	no	—
180	C₂²⋊Q₈	—	382	4	no	—
181	C₄×D₄	—	2	4	no	—
182	#117	—	382	4	no	—
183	C₂×C₄⋊C₄	—	191	1	no	—
184	C₂×C₄⋊C₄	—	191	1	no	—
185	C₂×C₄⋊C₄	—	191	2	no	—
186	C₂×C₄⋊C₄	—	191	2	no	—
187	C₂×C₄⋊C₄	—	191	2	no	—
188	C₂×C₄⋊C₄	—	191	1	no	—
189	C₂×C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
190	C₂×C₂²⋊C₄	—	191	2	no	—
191	C₂×C₂²⋊C₄	—	191	2	no	—
192	C₂×C₂²⋊C₄	—	191	2	no	—
193	C₂×C₂²⋊C₄	—	191	2	no	—
194	C₂×C₂²⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
195	C₂×C₂²⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
196	C₂×C₄²	—	191	2	no	—
197	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
198	#95	—	191	2	no	—
199	#95	—	191	2	no	—
200	#95	—	191	2	no	—
201	#95	—	191	2	no	—
202	C₂⁴	C₂²×D₁₉₁	1	1	yes	characteristic
203	C₂⁴	C₂²×D₁₉₁	1	1	yes	characteristic
204	C₂×Q₈	—	382	2	no	—
205	C₂×Q₈	—	382	2	no	—
206	C₂×Q₈	—	382	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	191	2	no	—
216	C₂²×C₄	—	191	2	no	—
217	C₂²×C₄	—	191	2	no	—
218	C₂²×C₄	—	191	2	no	—
219	C₂²×C₄	C₂²⋊D₁₉₁	1	4	yes	—
220	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
221	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
222	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
223	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
224	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	382	4	no	—
227	C₄⋊C₄	—	382	2	no	—
228	C₄⋊C₄	—	382	4	no	—
229	C₄⋊C₄	—	382	2	no	—
230	C₄⋊C₄	—	382	4	no	—
231	C₄⋊C₄	—	191	4	no	—
232	C₂²⋊C₄	—	2	2	no	—
233	C₂²⋊C₄	—	2	2	no	—
234	C₂²⋊C₄	—	382	4	no	—
235	C₂²⋊C₄	—	382	4	no	—
236	C₂²⋊C₄	—	382	4	no	—
237	C₂²⋊C₄	—	382	4	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	382	4	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	2	1	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	C₂×C₂²⋊D₁₉₁	1	2	yes	—
253	C₂³	#15690287	1	2	yes	—
254	C₂³	C₂³×D₁₉₁	1	1	yes	characteristic
255	Q₈	—	764	4	no	—
256	D₄	—	2	8	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	382	2	no	—
270	C₂×C₄	—	382	2	no	—
271	C₂×C₄	—	382	2	no	—
272	C₂×C₄	—	382	2	no	—
273	C₂×C₄	—	382	2	no	—
274	C₂×C₄	—	382	2	no	—
275	C₂×C₄	—	382	2	no	—
276	C₂×C₄	—	382	2	no	—
277	C₂×C₄	—	382	2	no	—
278	C₂×C₄	—	382	2	no	—
279	C₂×C₄	—	382	2	no	—
280	C₂×C₄	—	191	4	no	—
281	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
282	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
283	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
284	C₂×C₄	#15690286	1	2	yes	—
285	C₂×C₄	#15690287	1	2	yes	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
301	C₂²	C₂²×C₂²⋊D₁₉₁	1	1	yes	characteristic
302	C₂²	C₂×#15690286	1	1	yes	characteristic
303	C₂²	C₂×#15690287	1	1	yes	characteristic
304	C₂²	C₂×#15690287	1	1	yes	characteristic
305	C₂²	#20259369	1	1	yes	characteristic
306	C₂²	#20259369	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	382	4	no	—
312	C₄	—	764	2	no	—
313	C₄	—	764	2	no	—
314	C₄	—	764	2	no	—
315	C₄	#20259294	1	4	yes	—
316	C₂	—	4	2	no	—
317	C₂	—	4	2	no	—
318	C₂	C₂×#20259294	1	1	yes	characteristic
319	C₂	#27133174	1	1	yes	characteristic
320	C₂	#27133234	1	1	yes	characteristic
321	C₂	#27133318	1	1	yes	characteristic
322	C₂	#27133322	1	1	yes	characteristic
323	C₂	#27133353	1	1	yes	characteristic
324	C₂	#27133383	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉₁
commutator, socle, cosocle	C₂³×C₁₉₁	C₂⁴
Fitting	C₂×C₄×D₄×C₁₉₁	C₂
prime radical	C₂²×D₄×C₁₉₁	C₂²
hypercenter	C₂×C₄×D₄	D₁₉₁
hypocenter	C₁₉₁	#2415
LCS	G ▷ C₂³×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#20259301	C₂	semidirect
2. C₂×#20259301	C₂	semidirect
3. C₂×#20259281	C₂	semidirect
4. #27132591	C₂	semidirect
6. #27132569	C₂	semidirect
7. #27132561	C₂	semidirect
8. #27132558	C₂	semidirect
9. #27132557	C₂	semidirect
10. #27132523	C₂	semidirect
48. C₄⋊Dic₁₉₁	D₄	semidirect
162. C₁₉₁	#2415	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×#20259281	2
#27132591	2
#27132577	2
#27132569	2
#27132561	2
#27132558	2
#27132557	2
#27132523	2
C₂×C₄×D₄×C₁₉₁	2
#2415	191