Subgroups — C₂×#20265802 #27184526

Order 12928.

1296 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	#20265802	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15692235	C₂	1	2	yes	maximal, complemented
3	C₂×#15692234	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15692233	C₂	1	2	yes	maximal, complemented
5	C₂×#15692233	C₂	1	2	yes	maximal, complemented
6	C₂×#15692232	C₂	1	2	yes	maximal, complemented
7	C₂×#15692231	C₂	1	2	yes	maximal, complemented
8	C₂²×#13108051	C₂	1	1	yes	maximal, characteristic
9	C₂×C₂²⋊C₄×D₁₀₁	C₂	1	2	yes	maximal
10	C₂×C₂²⋊Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×D₁₀₁	C₂²	1	1	yes	characteristic
12	C₂²×Dic₂₀₂	C₂²	1	2	yes	complemented
13	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic, complemented
14	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
16	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
17	#15692235	C₂²	1	16	yes	complemented
18	#15692234	C₂²	1	8	yes	complemented
19	#15692233	C₂²	1	16	yes	complemented
20	#15692233	C₂²	1	16	yes	complemented
21	#15692232	C₂²	1	16	yes	complemented
22	#15692231	C₂²	1	16	yes	complemented
23	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	complemented
24	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic, complemented
25	C₂×#13108053	C₂²	1	2	yes	complemented
26	C₂×#13108053	C₂²	1	2	yes	complemented
27	C₂×#13108053	C₂²	1	2	yes	complemented
28	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
29	C₂×#13108051	C₂²	1	8	yes	—
30	C₂×#13108051	C₂²	1	2	yes	—
31	C₂×#13108051	C₂²	1	2	yes	—
32	C₂×#13108051	C₂²	1	2	yes	—
33	C₂×#13108051	C₂²	1	2	yes	—
34	C₂²⋊C₄×D₁₀₁	C₂²	1	16	yes	—
35	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
36	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
37	C₂²⋊Q₈×C₁₀₁	C₂²	1	8	yes	complemented
38	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
39	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	complemented
40	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
41	C₂³×D₁₀₁	—	4	2	no	—
42	C₂³×D₁₀₁	—	4	2	no	—
43	C₂³×D₁₀₁	Q₈	1	8	yes	complemented
44	C₂³×D₁₀₁	C₂³	1	2	yes	—
45	C₂³×D₁₀₁	C₂³	1	4	yes	—
46	C₂×Dic₂₀₂	—	4	2	no	—
47	C₂×Dic₂₀₂	—	4	4	no	—
48	C₂×Dic₂₀₂	C₂³	1	8	yes	complemented
49	C₂×Dic₂₀₂	C₂³	1	4	yes	complemented
50	C₂²×Dic₁₀₁	—	4	2	no	—
51	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
52	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
53	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
54	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
55	C₂×C₄×D₁₀₁	—	4	4	no	—
56	C₂×C₄×D₁₀₁	—	4	4	no	—
57	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
58	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
59	C₄⋊Dic₁₀₁	C₂³	1	8	yes	complemented
60	C₄⋊Dic₁₀₁	C₂³	1	4	yes	complemented
61	#13108053	C₂³	1	8	yes	complemented
62	#13108053	C₂³	1	8	yes	complemented
63	#13108053	C₂³	1	8	yes	complemented
64	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
65	#13108051	C₂³	1	8	yes	—
66	#13108051	C₂³	1	8	yes	—
67	#13108051	C₂³	1	8	yes	—
68	#13108051	C₂³	1	8	yes	—
69	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
70	C₂×Q₈×C₁₀₁	—	4	2	no	—
71	C₂×Q₈×C₁₀₁	C₂³	1	4	yes	complemented
72	C₂²×C₄×C₁₀₁	—	4	2	no	—
73	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
74	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
75	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
78	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	complemented
79	C₄⋊C₄×C₁₀₁	C₂³	1	8	yes	complemented
80	C₂²⋊C₄×C₁₀₁	C₂³	1	8	yes	—
81	C₂³×C₁₀₁	—	2	2	no	—
82	C₂³×C₁₀₁	—	2	2	no	—
83	C₂³×C₁₀₁	C₂×Q₈	1	4	yes	—
84	C₂³×C₁₀₁	C₂⁴	1	2	yes	—
85	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
86	C₂²×D₁₀₁	—	2	4	no	—
87	C₂²×D₁₀₁	—	2	4	no	—
88	C₂²×D₁₀₁	—	4	16	no	—
89	C₂²×D₁₀₁	—	4	4	no	—
90	C₂²×D₁₀₁	—	4	4	no	—
91	C₂²×D₁₀₁	C₂×Q₈	1	8	yes	—
92	C₂²×D₁₀₁	C₂×Q₈	1	16	yes	complemented
93	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
94	Q₈×C₁₀₁	—	4	4	no	—
95	C₂×C₄×C₁₀₁	—	2	2	no	—
96	C₂×C₄×C₁₀₁	—	2	2	no	—
97	C₂×C₄×C₁₀₁	—	2	4	no	—
98	C₂×C₄×C₁₀₁	—	2	4	no	—
99	C₂×C₄×C₁₀₁	—	4	4	no	—
100	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
101	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
102	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
103	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
104	C₂×Dic₁₀₁	—	2	4	no	—
105	C₂×Dic₁₀₁	—	2	4	no	—
106	C₂×Dic₁₀₁	—	2	4	no	—
107	C₂×Dic₁₀₁	—	4	4	no	—
108	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
109	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
110	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
111	C₄×D₁₀₁	—	4	8	no	—
112	C₄×D₁₀₁	—	4	8	no	—
113	Dic₂₀₂	—	4	4	no	—
114	Dic₂₀₂	—	4	8	no	—
115	C₂²×C₁₀₁	—	2	8	no	—
116	C₂²×C₁₀₁	—	2	2	no	—
117	C₂²×C₁₀₁	—	2	2	no	—
118	C₂²×C₁₀₁	C₂²×Q₈	1	4	yes	—
119	C₂²×C₁₀₁	C₂²×Q₈	1	2	yes	—
120	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
121	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
122	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
123	C₂×D₁₀₁	—	2	16	no	—
124	C₂×D₁₀₁	—	2	4	no	—
125	C₂×D₁₀₁	—	2	4	no	—
126	C₂×D₁₀₁	—	4	16	no	—
127	C₂×D₁₀₁	C₂²×Q₈	1	8	yes	—
128	C₄×C₁₀₁	—	2	4	no	—
129	C₄×C₁₀₁	—	2	2	no	—
130	C₄×C₁₀₁	—	2	4	no	—
131	C₄×C₁₀₁	—	2	2	no	—
132	Dic₁₀₁	—	2	4	no	—
133	Dic₁₀₁	—	2	4	no	—
134	Dic₁₀₁	—	2	4	no	—
135	C₂×C₁₀₁	—	2	4	no	—
136	C₂×C₁₀₁	#543	1	4	yes	complemented
137	C₂×C₁₀₁	C₂³×Q₈	1	1	yes	characteristic
138	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
139	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
140	D₁₀₁	—	2	8	no	—
141	C₂×#543	—	101	1	no	maximal, Sylow
142	C₁₀₁	C₂×#543	1	1	yes	characteristic, complemented, Sylow
143	#543	—	101	16	no	—
144	C₂×C₂²⋊Q₈	—	101	2	no	—
145	C₂×C₂²⋊Q₈	—	101	1	no	—
146	C₂×C₂²⋊Q₈	—	101	2	no	—
147	C₂×C₂²⋊Q₈	—	101	2	no	—
148	C₂×C₂²⋊Q₈	—	101	2	no	—
149	C₂×C₂²⋊Q₈	—	101	2	no	—
150	C₂×C₂²⋊Q₈	D₁₀₁	1	1	yes	characteristic, complemented
151	C₂²×C₂²⋊C₄	—	101	1	no	—
152	C₂²×C₂²⋊C₄	—	101	2	no	—
153	C₂⁵	—	101	1	no	—
154	C₂²×Q₈	—	101	1	no	—
155	C₂²×Q₈	—	101	2	no	—
156	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic, complemented
157	C₂³×C₄	—	101	1	no	—
158	C₂³×C₄	—	101	2	no	—
159	C₂³×C₄	—	101	2	no	—
160	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
161	C₂²⋊Q₈	—	101	16	no	—
162	C₂²⋊Q₈	—	101	8	no	—
163	C₂²⋊Q₈	—	101	16	no	—
164	C₂²⋊Q₈	—	101	16	no	—
165	C₂²⋊Q₈	—	101	16	no	—
166	C₂²⋊Q₈	—	101	16	no	—
167	C₂²⋊Q₈	C₂×D₁₀₁	1	8	yes	complemented
168	C₂×C₄⋊C₄	—	101	2	no	—
169	C₂×C₄⋊C₄	—	101	2	no	—
170	C₂×C₄⋊C₄	—	101	2	no	—
171	C₂×C₄⋊C₄	—	101	1	no	—
172	C₂×C₄⋊C₄	—	101	2	no	—
173	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic, complemented
174	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	complemented
175	C₂×C₂²⋊C₄	—	101	2	no	—
176	C₂×C₂²⋊C₄	—	101	2	no	—
177	C₂×C₂²⋊C₄	—	101	2	no	—
178	C₂×C₂²⋊C₄	—	101	2	no	—
179	C₂×C₂²⋊C₄	—	101	2	no	—
180	C₂×C₂²⋊C₄	—	101	8	no	—
181	C₂×C₂²⋊C₄	—	101	16	no	—
182	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
183	C₂⁴	—	101	8	no	—
184	C₂⁴	—	101	4	no	—
185	C₂⁴	—	101	2	no	—
186	C₂⁴	—	404	2	no	—
187	C₂⁴	—	404	2	no	—
188	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	—	101	4	no	—
191	C₂×Q₈	—	101	8	no	—
192	C₂×Q₈	—	404	2	no	—
193	C₂×Q₈	—	404	4	no	—
194	C₂×Q₈	C₂²×D₁₀₁	1	4	yes	complemented
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	101	4	no	—
197	C₂²×C₄	—	101	2	no	—
198	C₂²×C₄	—	101	2	no	—
199	C₂²×C₄	—	101	2	no	—
200	C₂²×C₄	—	101	8	no	—
201	C₂²×C₄	—	101	8	no	—
202	C₂²×C₄	—	404	4	no	—
203	C₂²×C₄	—	404	4	no	—
204	C₂²×C₄	—	404	2	no	—
205	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
206	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
208	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
209	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
210	C₄⋊C₄	—	101	8	no	—
211	C₄⋊C₄	—	101	8	no	—
212	C₄⋊C₄	—	101	8	no	—
213	C₄⋊C₄	—	101	4	no	—
214	C₄⋊C₄	—	101	8	no	—
215	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	complemented
216	C₄⋊C₄	C₂²×D₁₀₁	1	8	yes	complemented
217	C₂²⋊C₄	—	101	8	no	—
218	C₂²⋊C₄	—	101	8	no	—
219	C₂²⋊C₄	—	101	8	no	—
220	C₂²⋊C₄	—	101	8	no	—
221	C₂²⋊C₄	—	101	8	no	—
222	C₂²⋊C₄	C₂²×D₁₀₁	1	8	yes	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	101	16	no	—
226	C₂³	—	101	8	no	—
227	C₂³	—	101	4	no	—
228	C₂³	—	404	4	no	—
229	C₂³	—	404	4	no	—
230	C₂³	—	404	16	no	—
231	C₂³	—	202	4	no	—
232	C₂³	—	202	4	no	—
233	C₂³	Q₈×D₁₀₁	1	4	yes	—
234	C₂³	C₂³×D₁₀₁	1	2	yes	—
235	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
236	Q₈	—	4	4	no	—
237	Q₈	—	404	4	no	—
238	Q₈	—	404	8	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	101	4	no	—
245	C₂×C₄	—	101	4	no	—
246	C₂×C₄	—	101	4	no	—
247	C₂×C₄	—	404	8	no	—
248	C₂×C₄	—	404	8	no	—
249	C₂×C₄	—	404	4	no	—
250	C₂×C₄	—	202	4	no	—
251	C₂×C₄	—	202	4	no	—
252	C₂×C₄	—	202	4	no	—
253	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
254	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
255	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
256	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	8	no	—
260	C₂²	—	101	8	no	—
261	C₂²	—	404	16	no	—
262	C₂²	—	202	4	no	—
263	C₂²	—	202	4	no	—
264	C₂²	—	202	16	no	—
265	C₂²	C₂×Q₈×D₁₀₁	1	4	yes	—
266	C₂²	C₂×Q₈×D₁₀₁	1	2	yes	—
267	C₂²	#15692300	1	2	yes	—
268	C₂²	#15692301	1	2	yes	—
269	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	2	4	no	—
273	C₄	—	2	4	no	—
274	C₄	—	202	4	no	—
275	C₄	—	202	4	no	—
276	C₄	—	202	4	no	—
277	C₂	—	2	4	no	—
278	C₂	—	202	8	no	—
279	C₂	#20265802	1	4	yes	direct product factor
280	C₂	C₂²×Q₈×D₁₀₁	1	1	yes	characteristic
281	C₂	C₂×#15692300	1	1	yes	characteristic
282	C₂	C₂×#15692301	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₁
commutator, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂×C₂²⋊Q₈×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂⁴×D₁₀₁	C₂²
hypercenter	C₂×C₂²⋊Q₈	D₁₀₁
hypocenter	C₁₀₁	C₂×#543
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #20265802	C₂	direct (N × H)
2. C₂×#15692235	C₂	semidirect
3. C₂×#15692234	C₂	semidirect
4. C₂×#15692233	C₂	semidirect
5. C₂×#15692233	C₂	semidirect
6. C₂×#15692232	C₂	semidirect
7. C₂×#15692231	C₂	semidirect
10. C₂×C₂²⋊Q₈×C₁₀₁	C₂	semidirect
12. C₂²×Dic₂₀₂	C₂²	semidirect
13. C₂²×Dic₂₀₂	C₂²	semidirect
17. #15692235	C₂²	semidirect
18. #15692234	C₂²	semidirect
19. #15692233	C₂²	semidirect
20. #15692233	C₂²	semidirect
21. #15692232	C₂²	semidirect
22. #15692231	C₂²	semidirect
23. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
24. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
25. C₂×#13108053	C₂²	semidirect
26. C₂×#13108053	C₂²	semidirect
27. C₂×#13108053	C₂²	semidirect
35. C₂²×Q₈×C₁₀₁	C₂²	semidirect
37. C₂²⋊Q₈×C₁₀₁	C₂²	semidirect
38. C₂×C₄⋊C₄×C₁₀₁	C₂²	semidirect
39. C₂×C₄⋊C₄×C₁₀₁	C₂²	semidirect
43. C₂³×D₁₀₁	Q₈	semidirect
48. C₂×Dic₂₀₂	C₂³	semidirect
49. C₂×Dic₂₀₂	C₂³	semidirect
59. C₄⋊Dic₁₀₁	C₂³	semidirect
60. C₄⋊Dic₁₀₁	C₂³	semidirect
61. #13108053	C₂³	semidirect
62. #13108053	C₂³	semidirect
63. #13108053	C₂³	semidirect
71. C₂×Q₈×C₁₀₁	C₂³	semidirect
78. C₄⋊C₄×C₁₀₁	C₂³	semidirect
79. C₄⋊C₄×C₁₀₁	C₂³	semidirect
92. C₂²×D₁₀₁	C₂×Q₈	semidirect
136. C₂×C₁₀₁	#543	semidirect
142. C₁₀₁	C₂×#543	semidirect
150. C₂×C₂²⋊Q₈	D₁₀₁	semidirect
156. C₂²×Q₈	C₂×D₁₀₁	semidirect
167. C₂²⋊Q₈	C₂×D₁₀₁	semidirect
173. C₂×C₄⋊C₄	C₂×D₁₀₁	semidirect
174. C₂×C₄⋊C₄	C₂×D₁₀₁	semidirect
194. C₂×Q₈	C₂²×D₁₀₁	semidirect
215. C₄⋊C₄	C₂²×D₁₀₁	semidirect
216. C₄⋊C₄	C₂²×D₁₀₁	semidirect
279. C₂	#20265802	direct (N × H)

Maximal subgroups

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

subgroup	index
#20265802	2
C₂×#15692235	2
C₂×#15692234	2
C₂×#15692233	2
C₂×#15692232	2
C₂×#15692231	2
C₂²×#13108051	2
C₂×C₂²⋊C₄×D₁₀₁	2
C₂×C₂²⋊Q₈×C₁₀₁	2
C₂×#543	101

Subgroups — C2×#20265802 #27184526

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265802 #27184526