Subgroups — C₂×#27132681 #37581063

Order 24448.

772 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15690287	C₂	1	1	yes	maximal, characteristic
2	C₂²×#15690271	C₂	1	1	yes	maximal, characteristic, complemented
3	#27132681	C₂	1	8	yes	maximal, direct product factor
4	C₂×#20259250	C₂	1	4	yes	maximal, complemented
5	C₂×#117×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15690287	—	2	1	no	—
7	C₂×#15690287	—	2	2	no	—
8	C₂×#15690287	—	2	1	no	—
9	C₂×#15690287	—	2	2	no	—
10	C₂×#15690287	C₄	1	2	yes	complemented
11	C₂×#15690287	C₄	1	4	yes	complemented
12	C₂×#15690287	C₂²	1	2	yes	—
13	C₂×#15690287	C₂²	1	4	yes	—
14	C₂²×Dic₃₈₂	C₄	1	1	yes	characteristic, complemented
15	C₂²×D₇₆₄	C₄	1	1	yes	characteristic, complemented
16	C₂²×C₂²⋊D₁₉₁	—	2	1	no	—
17	C₂²×C₄×D₁₉₁	—	2	1	no	—
18	C₂×#15690271	—	2	1	no	—
19	C₂×#15690271	—	2	2	no	—
20	C₂×#15690271	C₂²	1	2	yes	complemented
21	C₂×#15690271	C₂²	1	4	yes	complemented
22	C₂²×C₁₉₁⋊C₈	—	2	1	no	—
23	#20259250	C₂²	1	16	yes	complemented
24	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
25	#117×C₁₉₁	—	2	2	no	—
26	#117×C₁₉₁	C₂²	1	4	yes	complemented
27	C₂×C₄⋊C₄×C₁₉₁	—	2	1	no	—
28	C₂×C₂²⋊C₄×C₁₉₁	—	2	1	no	—
29	C₂×C₄²×C₁₉₁	C₂²	1	2	yes	complemented
30	C₂³×D₁₉₁	—	2	1	no	—
31	#15690287	—	2	4	no	—
32	#15690287	—	2	4	no	—
33	#15690287	—	2	4	no	—
34	#15690287	—	2	2	no	—
35	#15690287	—	2	4	no	—
36	#15690287	—	2	4	no	—
37	#15690287	—	2	2	no	—
38	#15690287	C₂×C₄	1	4	yes	—
39	#15690287	C₂×C₄	1	8	yes	complemented
40	#15690287	C₂³	1	4	yes	—
41	C₂×Dic₃₈₂	—	2	1	no	—
42	C₂×Dic₃₈₂	—	2	2	no	—
43	C₂×Dic₃₈₂	C₂×C₄	1	2	yes	—
44	C₂×Dic₃₈₂	C₂×C₄	1	4	yes	complemented
45	C₂×D₇₆₄	—	2	2	no	—
46	C₂×D₇₆₄	—	2	1	no	—
47	C₂×D₇₆₄	C₂×C₄	1	2	yes	—
48	C₂×D₇₆₄	C₂×C₄	1	4	yes	complemented
49	C₂×C₂²⋊D₁₉₁	—	2	2	no	—
50	C₂×C₂²⋊D₁₉₁	—	2	4	no	—
51	C₂×C₂²⋊D₁₉₁	—	2	2	no	—
52	C₂×C₂²⋊D₁₉₁	—	2	4	no	—
53	C₂²×Dic₁₉₁	—	2	1	no	—
54	C₂×C₄×D₁₉₁	—	2	4	no	—
55	C₂×C₄×D₁₉₁	—	2	2	no	—
56	C₂×C₄×D₁₉₁	—	2	4	no	—
57	C₂×C₄×D₁₉₁	—	2	2	no	—
58	#15690271	—	2	4	no	—
59	#15690271	—	2	2	no	—
60	#15690271	C₂³	1	4	yes	complemented
61	C₂×C₁₉₁⋊C₈	—	2	4	no	—
62	C₂×C₁₉₁⋊C₈	—	2	2	no	—
63	C₂⁴×C₁₉₁	D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉₁	—	2	2	no	—
65	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉₁	C₂×C₄	1	2	yes	—
68	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
70	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
73	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
75	C₄⋊C₄×C₁₉₁	—	2	4	no	—
76	C₂²⋊C₄×C₁₉₁	—	2	4	no	—
77	C₄²×C₁₉₁	—	2	2	no	—
78	C₄²×C₁₉₁	C₂³	1	4	yes	complemented
79	C₂²×D₁₉₁	—	2	2	no	—
80	C₂²×D₁₉₁	—	2	4	no	—
81	C₂²×D₁₉₁	—	4	4	no	—
82	Dic₃₈₂	—	2	2	no	—
83	Dic₃₈₂	—	2	4	no	—
84	Dic₃₈₂	C₂²×C₄	1	4	yes	—
85	D₇₆₄	—	2	2	no	—
86	D₇₆₄	—	2	4	no	—
87	D₇₆₄	C₂²×C₄	1	4	yes	—
88	C₂²⋊D₁₉₁	—	2	4	no	—
89	C₂²⋊D₁₉₁	—	2	8	no	—
90	C₂²⋊D₁₉₁	—	2	4	no	—
91	C₂×Dic₁₉₁	—	2	2	no	—
92	C₂×Dic₁₉₁	—	2	4	no	—
93	C₄×D₁₉₁	—	2	4	no	—
94	C₄×D₁₉₁	—	2	8	no	—
95	C₄×D₁₉₁	—	2	4	no	—
96	C₁₉₁⋊C₈	—	2	4	no	—
97	C₂³×C₁₉₁	—	4	2	no	—
98	C₂³×C₁₉₁	C₂²⋊C₄	1	1	yes	characteristic
99	C₂³×C₁₉₁	C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₁₉₁	C₂²⋊C₄	1	2	yes	—
101	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
102	C₂³×C₁₉₁	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₁₉₁	—	2	4	no	—
104	C₂×C₄×C₁₉₁	—	4	4	no	—
105	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
106	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
108	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
110	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
111	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
112	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
114	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
116	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
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	1	yes	characteristic
121	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
123	C₂²×C₁₉₁	—	2	2	no	—
124	C₂²×C₁₉₁	—	2	2	no	—
125	C₂²×C₁₉₁	—	2	2	no	—
126	C₂²×C₁₉₁	—	4	2	no	—
127	C₂²×C₁₉₁	—	4	2	no	—
128	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	2	yes	—
129	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	2	yes	—
131	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
133	C₂×D₁₉₁	—	2	4	no	—
134	C₂×D₁₉₁	—	4	8	no	—
135	C₂×D₁₉₁	—	4	4	no	—
136	C₄×C₁₉₁	—	4	4	no	—
137	C₄×C₁₉₁	C₂×C₂²⋊C₄	1	2	yes	—
138	C₄×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₄×C₁₉₁	C₂×C₂²⋊C₄	1	2	yes	—
141	C₄×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
142	C₄×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
143	Dic₁₉₁	—	2	4	no	—
144	C₂×C₁₉₁	—	2	1	no	—
145	C₂×C₁₉₁	—	2	2	no	—
146	C₂×C₁₉₁	—	2	1	no	—
147	C₂×C₁₉₁	—	2	2	no	—
148	C₂×C₁₉₁	#421	1	2	yes	complemented
149	C₂×C₁₉₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
150	D₁₉₁	—	4	4	no	—
151	C₁₉₁	C₂×#421	1	1	yes	characteristic, complemented, Sylow
152	C₂×#421	—	191	1	no	maximal, Sylow
153	C₂²×#41	—	191	1	no	—
154	C₂²×M₁₆	—	191	1	no	—
155	C₂×#117	D₁₉₁	1	1	yes	characteristic, complemented
156	#421	—	191	8	no	—
157	C₂×#104	—	191	4	no	—
158	C₂×#41	—	382	1	no	—
159	C₂×#41	—	382	2	no	—
160	C₂×#41	—	382	2	no	—
161	C₂×#41	—	382	1	no	—
162	C₂×#41	—	191	2	no	—
163	C₂×#41	—	191	4	no	—
164	C₂×#41	—	191	2	no	—
165	C₂×#41	—	191	4	no	—
166	C₂²×Q₈	—	191	1	no	—
167	C₂²×D₄	—	382	1	no	—
168	C₂²×D₄	—	191	1	no	—
169	C₂³×C₄	—	382	1	no	—
170	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
171	C₂×M₁₆	—	382	2	no	—
172	C₂×M₁₆	—	382	1	no	—
173	C₂×M₁₆	—	191	2	no	—
174	C₂×M₁₆	—	191	4	no	—
175	C₂²×C₈	—	382	1	no	—
176	#117	—	2	2	no	—
177	#117	C₂×D₁₉₁	1	4	yes	complemented
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₂²⋊C₄	—	2	1	no	—
180	C₂×C₄²	C₂×D₁₉₁	1	2	yes	complemented
181	#104	—	191	16	no	—
182	C₂⁴	—	382	1	no	—
183	C₂⁴	C₂²⋊D₁₉₁	1	1	yes	characteristic
184	#41	—	382	4	no	—
185	#41	—	382	4	no	—
186	#41	—	382	4	no	—
187	#41	—	382	2	no	—
188	#41	—	382	4	no	—
189	#41	—	382	4	no	—
190	#41	—	382	2	no	—
191	#41	—	191	4	no	—
192	#41	—	191	8	no	—
193	#41	—	191	4	no	—
194	C₂×Q₈	—	382	2	no	—
195	C₂×Q₈	—	382	1	no	—
196	C₂×Q₈	—	191	2	no	—
197	C₂×Q₈	—	191	4	no	—
198	C₂×D₄	—	382	2	no	—
199	C₂×D₄	—	382	4	no	—
200	C₂×D₄	—	382	2	no	—
201	C₂×D₄	—	382	4	no	—
202	C₂×D₄	—	382	2	no	—
203	C₂×D₄	—	382	1	no	—
204	C₂×D₄	—	191	2	no	—
205	C₂×D₄	—	191	4	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	382	1	no	—
208	C₂²×C₄	—	382	4	no	—
209	C₂²×C₄	—	382	2	no	—
210	C₂²×C₄	—	382	4	no	—
211	C₂²×C₄	—	382	2	no	—
212	C₂²×C₄	C₄×D₁₉₁	1	1	yes	characteristic
213	C₂²×C₄	C₄×D₁₉₁	1	1	yes	characteristic
214	C₂²×C₄	C₄×D₁₉₁	1	2	yes	—
215	C₂²×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
216	C₂²×C₄	C₂²⋊D₁₉₁	1	2	yes	—
217	C₂²×C₄	D₇₆₄	1	2	yes	—
218	C₂²×C₄	D₇₆₄	1	1	yes	characteristic
219	C₂²×C₄	D₇₆₄	1	1	yes	characteristic
220	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
221	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
222	M₁₆	—	382	4	no	—
223	M₁₆	—	382	2	no	—
224	M₁₆	—	191	4	no	—
225	C₂×C₈	—	382	4	no	—
226	C₂×C₈	—	382	2	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₂²⋊C₄	—	2	4	no	—
229	C₄²	—	2	2	no	—
230	C₄²	C₂²×D₁₉₁	1	4	yes	complemented
231	C₂³	—	4	2	no	—
232	C₂³	—	382	4	no	—
233	C₂³	—	382	2	no	—
234	C₂³	—	764	4	no	—
235	C₂³	(C₂×D₁₉₁)⋊C₄	1	1	yes	characteristic
236	C₂³	(C₂×D₁₉₁)⋊C₄	1	1	yes	characteristic
237	C₂³	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
238	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
239	C₂³	C₂×C₂²⋊D₁₉₁	1	2	yes	—
240	Q₈	—	382	2	no	—
241	Q₈	—	382	4	no	—
242	Q₈	—	191	4	no	—
243	D₄	—	382	4	no	—
244	D₄	—	382	4	no	—
245	D₄	—	382	8	no	—
246	D₄	—	382	2	no	—
247	D₄	—	382	4	no	—
248	D₄	—	191	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	382	4	no	—
252	C₂×C₄	—	382	2	no	—
253	C₂×C₄	—	382	4	no	—
254	C₂×C₄	—	382	8	no	—
255	C₂×C₄	—	382	4	no	—
256	C₂×C₄	(C₂×D₁₉₁)⋊C₄	1	1	yes	characteristic
257	C₂×C₄	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
258	C₂×C₄	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
259	C₂×C₄	(C₂×D₁₉₁)⋊C₄	1	1	yes	characteristic
260	C₂×C₄	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
261	C₂×C₄	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
262	C₂×C₄	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
263	C₂×C₄	C₂×C₄×D₁₉₁	1	2	yes	—
264	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄×D₁₉₁	1	2	yes	—
266	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
267	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
268	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
269	C₂×C₄	C₂×D₇₆₄	1	2	yes	—
270	C₂×C₄	C₂×D₇₆₄	1	2	yes	—
271	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
272	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
273	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
274	C₈	—	382	4	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	382	4	no	—
281	C₂²	—	764	8	no	—
282	C₂²	—	764	4	no	—
283	C₂²	C₂×(C₂×D₁₉₁)⋊C₄	1	2	yes	—
284	C₂²	C₂×(C₂×D₁₉₁)⋊C₄	1	2	yes	—
285	C₂²	C₂×(C₂×D₁₉₁)⋊C₄	1	1	yes	characteristic
286	C₂²	C₂×(C₂×D₁₉₁)⋊C₄	1	1	yes	characteristic
287	C₂²	C₂²×C₂²⋊D₁₉₁	1	1	yes	characteristic
288	C₄	—	4	4	no	—
289	C₄	—	382	4	no	—
290	C₄	C₂×(C₂×D₁₉₁)⋊C₄	1	2	yes	—
291	C₄	C₂×(C₂×D₁₉₁)⋊C₄	1	1	yes	characteristic
292	C₄	C₂×(C₂×D₁₉₁)⋊C₄	1	1	yes	characteristic
293	C₄	C₂×(C₂×D₁₉₁)⋊C₄	1	2	yes	—
294	C₄	C₂²×C₄×D₁₉₁	1	1	yes	characteristic
295	C₄	C₂²×D₇₆₄	1	1	yes	characteristic
296	C₂	—	2	1	no	—
297	C₂	—	2	2	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	—	764	4	no	—
301	C₂	#27132681	1	2	yes	direct product factor
302	C₂	C₂²×(C₂×D₁₉₁)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₇₆₄
commutator	C₄×C₁₉₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	C₂×#117×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×C₂²⋊C₄
prime radical	C₂²×#15690287	C₂
cosocle	C₂×C₄×C₁₉₁	C₂⁴
hypercenter	C₂×#117	D₁₉₁
hypocenter	C₁₉₁	C₂×#421
LCS	G ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. C₂²×#15690271	C₂	semidirect
3. #27132681	C₂	direct (N × H)
4. C₂×#20259250	C₂	semidirect
5. C₂×#117×C₁₉₁	C₂	semidirect
10. C₂×#15690287	C₄	semidirect
11. C₂×#15690287	C₄	semidirect
14. C₂²×Dic₃₈₂	C₄	semidirect
15. C₂²×D₇₆₄	C₄	semidirect
20. C₂×#15690271	C₂²	semidirect
21. C₂×#15690271	C₂²	semidirect
23. #20259250	C₂²	semidirect
26. #117×C₁₉₁	C₂²	semidirect
29. C₂×C₄²×C₁₉₁	C₂²	semidirect
39. #15690287	C₂×C₄	semidirect
44. C₂×Dic₃₈₂	C₂×C₄	semidirect
48. C₂×D₇₆₄	C₂×C₄	semidirect
60. #15690271	C₂³	semidirect
78. C₄²×C₁₉₁	C₂³	semidirect
148. C₂×C₁₉₁	#421	semidirect
151. C₁₉₁	C₂×#421	semidirect
155. C₂×#117	D₁₉₁	semidirect
177. #117	C₂×D₁₉₁	semidirect
180. C₂×C₄²	C₂×D₁₉₁	semidirect
230. C₄²	C₂²×D₁₉₁	semidirect
301. C₂	#27132681	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#15690287	2
C₂²×#15690271	2
#27132681	2
C₂×#20259250	2
C₂×#117×C₁₉₁	2
C₂×#421	191

Subgroups — C2×#27132681 #37581063

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27132681 #37581063