Subgroups — C₂×#20265731 #27184420

Order 12928.

1168 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265731	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15692254	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15692253	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15692245	C₂	1	2	yes	maximal, complemented
5	C₂×#15692235	C₂	1	1	yes	maximal, characteristic
6	C₂×#15692233	C₂	1	2	yes	maximal
7	C₂×#15692231	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₂₀₂	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×Dic₂₀₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15692212	C₂	1	2	yes	maximal
11	C₂×C₄²×D₁₀₁	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₂₀₂	C₂²	1	2	yes	—
14	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
18	#15692254	C₂²	1	8	yes	complemented
19	#15692253	C₂²	1	8	yes	complemented
20	#15692245	C₂²	1	16	yes	complemented
21	#15692235	C₂²	1	8	yes	—
22	#15692233	C₂²	1	16	yes	—
23	#15692231	C₂²	1	8	yes	—
24	C₄×Dic₂₀₂	C₂²	1	8	yes	complemented
25	C₄×Dic₂₀₂	C₂²	1	16	yes	complemented
26	#15692212	C₂²	1	16	yes	—
27	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
29	C₂×#13108053	C₂²	1	2	yes	—
30	C₂×#13108053	C₂²	1	2	yes	—
31	C₂×#13108053	C₂²	1	2	yes	—
32	C₂×#13108053	C₂²	1	2	yes	—
33	C₂×#13108053	C₂²	1	2	yes	—
34	C₂×#13108051	C₂²	1	2	yes	—
35	C₂×#13108051	C₂²	1	2	yes	—
36	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
37	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
38	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
40	C₄²×D₁₀₁	C₂²	1	8	yes	—
41	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
42	C₄×Q₈×C₁₀₁	C₂²	1	8	yes	complemented
43	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
44	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
45	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
46	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	—
47	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
48	C₂×Dic₂₀₂	—	2	4	no	—
49	C₂×Dic₂₀₂	—	2	8	no	—
50	C₂×Dic₂₀₂	C₂³	1	8	yes	—
51	C₂×Dic₂₀₂	C₂³	1	4	yes	—
52	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
53	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
54	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
55	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
56	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₁₀₁	—	2	4	no	—
58	C₂×C₄×D₁₀₁	—	2	4	no	—
59	C₂×C₄×D₁₀₁	Q₈	1	8	yes	complemented
60	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
61	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
62	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
63	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
64	C₄⋊Dic₁₀₁	C₂³	1	8	yes	—
65	#13108053	C₂³	1	8	yes	—
66	#13108053	C₂³	1	8	yes	—
67	#13108053	C₂³	1	8	yes	—
68	#13108053	C₂³	1	8	yes	—
69	#13108053	C₂³	1	8	yes	—
70	#13108051	C₂³	1	8	yes	—
71	#13108051	C₂³	1	8	yes	—
72	C₄×Dic₁₀₁	C₂³	1	4	yes	—
73	C₄×Dic₁₀₁	C₂³	1	4	yes	—
74	C₄×Dic₁₀₁	C₂³	1	4	yes	—
75	C₄×Dic₁₀₁	C₂³	1	8	yes	—
76	C₂×Q₈×C₁₀₁	—	2	4	no	—
77	C₂×Q₈×C₁₀₁	C₂³	1	4	yes	—
78	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
80	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
81	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₁₀₁	C₂³	1	8	yes	—
84	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
85	C₄²×C₁₀₁	C₂³	1	4	yes	—
86	C₄²×C₁₀₁	C₂³	1	8	yes	—
87	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
88	C₂²×D₁₀₁	—	2	2	no	—
89	C₂²×D₁₀₁	—	2	2	no	—
90	C₂²×D₁₀₁	C₂×Q₈	1	4	yes	—
91	C₂²×D₁₀₁	C₂⁴	1	2	yes	—
92	Q₈×C₁₀₁	—	2	8	no	—
93	C₂×C₄×C₁₀₁	—	2	4	no	—
94	C₂×C₄×C₁₀₁	—	2	2	no	—
95	C₂×C₄×C₁₀₁	—	2	4	no	—
96	C₂×C₄×C₁₀₁	C₂×Q₈	1	4	yes	—
97	C₂×C₄×C₁₀₁	#41	1	4	yes	—
98	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
99	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
100	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
101	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
102	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
103	C₂×Dic₁₀₁	—	2	2	no	—
104	C₂×Dic₁₀₁	—	2	4	no	—
105	C₂×Dic₁₀₁	—	2	4	no	—
106	C₂×Dic₁₀₁	C₂×Q₈	1	4	yes	—
107	C₂×Dic₁₀₁	#41	1	4	yes	—
108	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
109	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
110	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
111	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
112	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
113	C₄×D₁₀₁	—	2	8	no	—
114	C₄×D₁₀₁	—	2	8	no	—
115	C₄×D₁₀₁	C₂×Q₈	1	16	yes	complemented
116	Dic₂₀₂	—	2	8	no	—
117	Dic₂₀₂	—	2	16	no	—
118	C₂²×C₁₀₁	C₂²×Q₈	1	2	yes	—
119	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
120	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
121	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
122	C₂×D₁₀₁	—	2	8	no	—
123	C₂×D₁₀₁	—	2	2	no	—
124	C₂×D₁₀₁	—	2	2	no	—
125	C₂×D₁₀₁	C₂²×Q₈	1	4	yes	—
126	C₄×C₁₀₁	—	2	4	no	—
127	C₄×C₁₀₁	—	2	2	no	—
128	C₄×C₁₀₁	—	2	4	no	—
129	C₄×C₁₀₁	C₂²×Q₈	1	4	yes	—
130	C₄×C₁₀₁	C₂×#41	1	4	yes	—
131	Dic₁₀₁	—	2	4	no	—
132	Dic₁₀₁	—	2	4	no	—
133	Dic₁₀₁	—	2	2	no	—
134	Dic₁₀₁	C₂²×Q₈	1	4	yes	—
135	Dic₁₀₁	C₂×#41	1	4	yes	—
136	C₂×C₁₀₁	#533	1	4	yes	complemented
137	C₂×C₁₀₁	C₂³×Q₈	1	1	yes	characteristic
138	C₂×C₁₀₁	C₂²×#41	1	1	yes	characteristic
139	C₂×C₁₀₁	C₂²×#41	1	1	yes	characteristic
140	D₁₀₁	—	2	4	no	—
141	C₂×#533	—	101	1	no	maximal, Sylow
142	C₁₀₁	C₂×#533	1	1	yes	characteristic, complemented, Sylow
143	#533	—	101	16	no	—
144	C₂×C₄⋊Q₈	—	101	1	no	—
145	C₂×C₄⋊Q₈	—	101	1	no	—
146	C₂×#125	—	101	2	no	—
147	C₂×C₂²⋊Q₈	—	101	1	no	—
148	C₂×C₂²⋊Q₈	—	101	2	no	—
149	C₂×C₂²⋊Q₈	—	101	1	no	—
150	C₂×C₄×Q₈	—	101	1	no	—
151	C₂×C₄×Q₈	—	101	2	no	—
152	C₂×C₄×Q₈	D₁₀₁	1	1	yes	characteristic, complemented
153	C₂×#117	—	101	2	no	—
154	C₂²×C₄²	—	101	1	no	—
155	C₂²×Q₈	—	101	2	no	—
156	C₂²×Q₈	—	101	1	no	—
157	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
158	C₂³×C₄	—	101	1	no	—
159	C₂³×C₄	—	101	1	no	—
160	C₂³×C₄	—	101	1	no	—
161	C₄⋊Q₈	—	101	8	no	—
162	C₄⋊Q₈	—	101	8	no	—
163	#125	—	101	16	no	—
164	C₂²⋊Q₈	—	101	8	no	—
165	C₂²⋊Q₈	—	101	16	no	—
166	C₂²⋊Q₈	—	101	8	no	—
167	C₄×Q₈	—	101	16	no	—
168	C₄×Q₈	—	101	8	no	—
169	C₄×Q₈	C₂×D₁₀₁	1	8	yes	complemented
170	#117	—	101	16	no	—
171	C₂×C₄⋊C₄	—	101	2	no	—
172	C₂×C₄⋊C₄	—	101	2	no	—
173	C₂×C₄⋊C₄	—	101	2	no	—
174	C₂×C₄⋊C₄	—	101	2	no	—
175	C₂×C₄⋊C₄	—	101	2	no	—
176	C₂×C₄⋊C₄	—	101	2	no	—
177	C₂×C₄⋊C₄	—	101	1	no	—
178	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
179	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
180	C₂×C₂²⋊C₄	—	101	2	no	—
181	C₂×C₂²⋊C₄	—	101	2	no	—
182	C₂×C₄²	—	101	1	no	—
183	C₂×C₄²	—	101	1	no	—
184	C₂×C₄²	—	101	2	no	—
185	C₂×C₄²	—	101	1	no	—
186	C₂×C₄²	—	101	8	no	—
187	C₂×C₄²	C₂×D₁₀₁	1	2	yes	—
188	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
189	C₂⁴	—	101	1	no	—
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	—	101	8	no	—
192	C₂×Q₈	—	101	4	no	—
193	C₂×Q₈	—	202	8	no	—
194	C₂×Q₈	—	202	4	no	—
195	C₂×Q₈	C₂²×D₁₀₁	1	4	yes	—
196	C₂²×C₄	—	101	1	no	—
197	C₂²×C₄	—	101	1	no	—
198	C₂²×C₄	—	101	8	no	—
199	C₂²×C₄	—	101	4	no	—
200	C₂²×C₄	—	101	1	no	—
201	C₂²×C₄	—	101	2	no	—
202	C₂²×C₄	—	101	2	no	—
203	C₂²×C₄	—	101	4	no	—
204	C₂²×C₄	—	101	4	no	—
205	C₂²×C₄	—	202	4	no	—
206	C₂²×C₄	—	202	4	no	—
207	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
208	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
209	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
210	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
211	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
212	C₄⋊C₄	—	101	8	no	—
213	C₄⋊C₄	—	101	8	no	—
214	C₄⋊C₄	—	101	8	no	—
215	C₄⋊C₄	—	101	8	no	—
216	C₄⋊C₄	—	101	8	no	—
217	C₄⋊C₄	—	101	8	no	—
218	C₄⋊C₄	—	101	4	no	—
219	C₄⋊C₄	C₂²×D₁₀₁	1	8	yes	—
220	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
221	C₂²⋊C₄	—	101	8	no	—
222	C₂²⋊C₄	—	101	8	no	—
223	C₄²	—	101	4	no	—
224	C₄²	—	101	4	no	—
225	C₄²	—	101	4	no	—
226	C₄²	—	101	8	no	—
227	C₄²	C₂²×D₁₀₁	1	4	yes	—
228	C₄²	C₂²×D₁₀₁	1	8	yes	—
229	C₂³	—	101	4	no	—
230	C₂³	—	101	2	no	—
231	C₂³	—	202	2	no	—
232	C₂³	—	202	2	no	—
233	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
234	Q₈	—	2	8	no	—
235	Q₈	—	202	16	no	—
236	Q₈	—	202	8	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	101	4	no	—
241	C₂×C₄	—	101	4	no	—
242	C₂×C₄	—	101	2	no	—
243	C₂×C₄	—	101	2	no	—
244	C₂×C₄	—	101	16	no	—
245	C₂×C₄	—	101	2	no	—
246	C₂×C₄	—	101	4	no	—
247	C₂×C₄	—	101	4	no	—
248	C₂×C₄	—	202	8	no	—
249	C₂×C₄	—	202	4	no	—
250	C₂×C₄	—	202	2	no	—
251	C₂×C₄	—	202	4	no	—
252	C₂×C₄	—	202	8	no	—
253	C₂×C₄	Q₈×D₁₀₁	1	4	yes	—
254	C₂×C₄	#13108074	1	4	yes	—
255	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
256	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
257	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
258	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
259	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
260	C₂²	—	101	4	no	—
261	C₂²	—	202	8	no	—
262	C₂²	—	202	2	no	—
263	C₂²	—	202	2	no	—
264	C₂²	C₂×Q₈×D₁₀₁	1	2	yes	—
265	C₂²	#41×D₁₀₁	1	2	yes	—
266	C₂²	C₂×#13108074	1	2	yes	—
267	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	4	no	—
270	C₄	—	2	4	no	—
271	C₄	—	101	4	no	—
272	C₄	—	101	4	no	—
273	C₄	—	202	4	no	—
274	C₄	—	202	2	no	—
275	C₄	—	202	4	no	—
276	C₄	C₂×Q₈×D₁₀₁	1	4	yes	—
277	C₄	C₂×#13108074	1	4	yes	—
278	C₂	—	202	4	no	—
279	C₂	#20265731	1	4	yes	direct product factor
280	C₂	C₂²×Q₈×D₁₀₁	1	1	yes	characteristic
281	C₂	C₂×#41×D₁₀₁	1	1	yes	characteristic
282	C₂	C₂²×#13108074	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₁
commutator, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂×C₄×Q₈×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂³×D₁₀₁	C₂³
hypercenter	C₂×C₄×Q₈	D₁₀₁
hypocenter	C₁₀₁	C₂×#533
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #20265731	C₂	direct (N × H)
2. C₂×#15692254	C₂	semidirect
3. C₂×#15692253	C₂	semidirect
4. C₂×#15692245	C₂	semidirect
8. C₂×C₄×Dic₂₀₂	C₂	semidirect
9. C₂×C₄×Dic₂₀₂	C₂	semidirect
12. C₂×C₄×Q₈×C₁₀₁	C₂	semidirect
18. #15692254	C₂²	semidirect
19. #15692253	C₂²	semidirect
20. #15692245	C₂²	semidirect
24. C₄×Dic₂₀₂	C₂²	semidirect
25. C₄×Dic₂₀₂	C₂²	semidirect
42. C₄×Q₈×C₁₀₁	C₂²	semidirect
59. C₂×C₄×D₁₀₁	Q₈	semidirect
115. C₄×D₁₀₁	C₂×Q₈	semidirect
136. C₂×C₁₀₁	#533	semidirect
142. C₁₀₁	C₂×#533	semidirect
152. C₂×C₄×Q₈	D₁₀₁	semidirect
169. C₄×Q₈	C₂×D₁₀₁	semidirect
279. C₂	#20265731	direct (N × H)

Maximal subgroups

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

subgroup	index
#20265731	2
C₂×#15692254	2
C₂×#15692253	2
C₂×#15692245	2
C₂×#15692235	2
C₂×#15692233	2
C₂×#15692231	2
C₂×C₄×Dic₂₀₂	2
C₂×#15692212	2
C₂×C₄²×D₁₀₁	2
C₂×C₄×Q₈×C₁₀₁	2
C₂×#533	101

Subgroups — C2×#20265731 #27184420

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265731 #27184420