Subgroups — C₂×#20265912 #27184720

Order 12928.

1240 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265912	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15692252	C₂	1	1	yes	maximal, characteristic
3	C₂×#15692244	C₂	1	2	yes	maximal, complemented
4	C₂×#15692243	C₂	1	2	yes	maximal, complemented
5	C₂×#15692240	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×Dic₂₀₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×D₄₀₄	C₂	1	2	yes	maximal, complemented
8	C₂×#15692217	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
11	C₂²×D₄₀₄	C₂²	1	2	yes	—
12	C₂²×D₄₀₄	C₂²	1	2	yes	—
13	C₂²×D₄₀₄	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₀₁	C₂²	1	4	yes	—
15	#15692252	C₂²	1	8	yes	—
16	#15692244	C₂²	1	16	yes	complemented
17	#15692243	C₂²	1	16	yes	complemented
18	#15692240	C₂²	1	32	yes	complemented
19	C₄×Dic₂₀₂	C₂²	1	8	yes	complemented
20	C₄×D₄₀₄	C₂²	1	16	yes	complemented
21	#15692217	C₂²	1	16	yes	complemented
22	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
23	C₂×#13108053	C₂²	1	2	yes	—
24	C₂×#13108051	C₂²	1	4	yes	—
25	C₂×#13108051	C₂²	1	4	yes	—
26	C₂×#13108051	C₂²	1	4	yes	—
27	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
28	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
29	C₄×Q₈×C₁₀₁	C₂²	1	8	yes	complemented
30	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
32	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	—
33	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
34	C₂³×D₁₀₁	C₂³	1	4	yes	—
35	C₂×Dic₂₀₂	—	2	4	no	—
36	C₂×Dic₂₀₂	C₂³	1	4	yes	—
37	C₂×D₄₀₄	—	2	8	no	—
38	C₂×D₄₀₄	—	2	8	no	—
39	C₂×D₄₀₄	—	4	4	no	—
40	C₂×D₄₀₄	C₂³	1	8	yes	—
41	C₂×D₄₀₄	C₂³	1	8	yes	—
42	C₂×D₄₀₄	C₂³	1	8	yes	—
43	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
44	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
45	C₂×C₄×D₁₀₁	—	4	8	no	—
46	C₂×C₄×D₁₀₁	C₂³	1	16	yes	—
47	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
48	#13108053	C₂³	1	8	yes	—
49	#13108051	C₂³	1	16	yes	—
50	#13108051	C₂³	1	16	yes	—
51	#13108051	C₂³	1	16	yes	—
52	C₄×Dic₁₀₁	C₂³	1	8	yes	—
53	C₂×Q₈×C₁₀₁	—	2	4	no	—
54	C₂×Q₈×C₁₀₁	C₂³	1	4	yes	—
55	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
56	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
61	C₄⋊C₄×C₁₀₁	C₂³	1	8	yes	—
62	C₄²×C₁₀₁	C₂³	1	8	yes	—
63	C₄²×C₁₀₁	C₂³	1	4	yes	—
64	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
65	C₂²×D₁₀₁	—	2	8	no	—
66	C₂²×D₁₀₁	—	2	8	no	—
67	C₂²×D₁₀₁	—	2	8	no	—
68	C₂²×D₁₀₁	C₂⁴	1	8	yes	—
69	Q₈×C₁₀₁	—	2	8	no	—
70	C₂×C₄×C₁₀₁	—	2	4	no	—
71	C₂×C₄×C₁₀₁	—	2	4	no	—
72	C₂×C₄×C₁₀₁	—	2	2	no	—
73	C₂×C₄×C₁₀₁	#41	1	4	yes	—
74	C₂×C₄×C₁₀₁	#41	1	4	yes	—
75	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
76	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
77	C₂×C₄×C₁₀₁	C₂⁴	1	4	yes	—
78	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
79	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
80	C₂×Dic₁₀₁	—	2	4	no	—
81	C₂×Dic₁₀₁	—	2	4	no	—
82	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
83	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
84	D₄₀₄	—	2	16	no	—
85	D₄₀₄	—	2	16	no	—
86	D₄₀₄	—	4	8	no	—
87	C₄×D₁₀₁	—	4	16	no	—
88	Dic₂₀₂	—	2	8	no	—
89	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
90	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
91	C₂²×C₁₀₁	H(5,𝔽₂)	1	2	yes	—
92	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
93	C₂×D₁₀₁	—	2	8	no	—
94	C₂×D₁₀₁	—	2	8	no	—
95	C₂×D₁₀₁	—	2	8	no	—
96	C₂×D₁₀₁	—	4	16	no	—
97	C₄×C₁₀₁	—	2	4	no	—
98	C₄×C₁₀₁	—	2	4	no	—
99	C₄×C₁₀₁	—	2	2	no	—
100	C₄×C₁₀₁	C₂×#41	1	4	yes	—
101	C₄×C₁₀₁	C₂×#41	1	4	yes	—
102	Dic₁₀₁	—	2	4	no	—
103	Dic₁₀₁	—	2	4	no	—
104	C₂×C₁₀₁	#559	1	4	yes	complemented
105	C₂×C₁₀₁	C₂²×#41	1	1	yes	characteristic
106	C₂×C₁₀₁	C₂²×#41	1	1	yes	characteristic
107	C₂×C₁₀₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
108	D₁₀₁	—	4	8	no	—
109	C₂×#559	—	101	1	no	maximal, Sylow
110	C₁₀₁	C₂×#559	1	1	yes	characteristic, complemented, Sylow
111	#559	—	101	16	no	—
112	C₂×#127	—	101	1	no	—
113	C₂×#124	—	101	2	no	—
114	C₂×#124	—	101	2	no	—
115	C₂×#123	—	101	4	no	—
116	C₂×C₄×Q₈	—	101	1	no	—
117	C₂×C₄×Q₈	D₁₀₁	1	1	yes	characteristic, complemented
118	C₂×C₄×D₄	—	101	2	no	—
119	C₂×C₄×D₄	—	101	2	no	—
120	C₂²×Q₈	—	101	1	no	—
121	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
122	C₂²×D₄	—	101	2	no	—
123	C₂²×D₄	—	101	2	no	—
124	C₂²×D₄	—	101	2	no	—
125	C₂³×C₄	—	101	4	no	—
126	#127	—	101	8	no	—
127	#124	—	101	16	no	—
128	#124	—	101	16	no	—
129	#123	—	101	32	no	—
130	C₄×Q₈	—	101	8	no	—
131	C₄×Q₈	C₂×D₁₀₁	1	8	yes	complemented
132	C₄×D₄	—	101	16	no	—
133	C₄×D₄	—	101	16	no	—
134	C₂×C₄⋊C₄	—	101	1	no	—
135	C₂×C₄⋊C₄	—	101	2	no	—
136	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
137	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
138	C₂×C₂²⋊C₄	—	101	4	no	—
139	C₂×C₂²⋊C₄	—	101	4	no	—
140	C₂×C₂²⋊C₄	—	101	4	no	—
141	C₂×C₄²	—	101	2	no	—
142	C₂×C₄²	C₂×D₁₀₁	1	2	yes	—
143	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
144	C₂⁴	—	101	4	no	—
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	—	101	4	no	—
147	C₂×Q₈	—	202	4	no	—
148	C₂×Q₈	C₂²×D₁₀₁	1	4	yes	—
149	C₂×D₄	—	101	8	no	—
150	C₂×D₄	—	101	8	no	—
151	C₂×D₄	—	101	8	no	—
152	C₂×D₄	—	404	4	no	—
153	C₂×D₄	—	202	8	no	—
154	C₂×D₄	—	202	8	no	—
155	C₂²×C₄	—	101	2	no	—
156	C₂²×C₄	—	101	2	no	—
157	C₂²×C₄	—	101	16	no	—
158	C₂²×C₄	—	404	8	no	—
159	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
160	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
161	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
163	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
164	C₄⋊C₄	—	101	8	no	—
165	C₄⋊C₄	—	101	4	no	—
166	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
167	C₄⋊C₄	C₂²×D₁₀₁	1	8	yes	—
168	C₂²⋊C₄	—	101	16	no	—
169	C₂²⋊C₄	—	101	16	no	—
170	C₂²⋊C₄	—	101	16	no	—
171	C₄²	—	101	8	no	—
172	C₄²	C₂²×D₁₀₁	1	8	yes	—
173	C₄²	C₂²×D₁₀₁	1	4	yes	—
174	C₂³	—	101	8	no	—
175	C₂³	—	202	8	no	—
176	C₂³	—	202	8	no	—
177	C₂³	—	202	8	no	—
178	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
179	Q₈	—	2	8	no	—
180	Q₈	—	202	8	no	—
181	D₄	—	404	8	no	—
182	D₄	—	202	16	no	—
183	D₄	—	202	16	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	101	4	no	—
188	C₂×C₄	—	101	4	no	—
189	C₂×C₄	—	404	16	no	—
190	C₂×C₄	—	202	4	no	—
191	C₂×C₄	—	202	4	no	—
192	C₂×C₄	#13108074	1	4	yes	—
193	C₂×C₄	#13108075	1	4	yes	—
194	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
195	C₂×C₄	C₂³×D₁₀₁	1	4	yes	—
196	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
197	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
198	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
199	C₂²	—	404	16	no	—
200	C₂²	—	202	8	no	—
201	C₂²	—	202	8	no	—
202	C₂²	—	202	8	no	—
203	C₂²	C₂×#13108074	1	2	yes	—
204	C₂²	C₂×#13108075	1	2	yes	—
205	C₂²	#15692301	1	2	yes	—
206	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
207	C₄	—	2	4	no	—
208	C₄	—	2	4	no	—
209	C₄	—	2	2	no	—
210	C₄	—	202	4	no	—
211	C₄	—	202	4	no	—
212	C₄	C₂×#13108074	1	4	yes	—
213	C₄	C₂×#13108075	1	4	yes	—
214	C₂	—	404	8	no	—
215	C₂	#20265912	1	4	yes	direct product factor
216	C₂	C₂²×#13108074	1	1	yes	characteristic
217	C₂	C₂²×#13108075	1	1	yes	characteristic
218	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×C₄×Q₈×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂×#15692252	C₂
hypercenter	C₂×C₄×Q₈	D₁₀₁
hypocenter	C₁₀₁	C₂×#559
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #20265912	C₂	direct (N × H)
3. C₂×#15692244	C₂	semidirect
4. C₂×#15692243	C₂	semidirect
5. C₂×#15692240	C₂	semidirect
6. C₂×C₄×Dic₂₀₂	C₂	semidirect
7. C₂×C₄×D₄₀₄	C₂	semidirect
8. C₂×#15692217	C₂	semidirect
9. C₂×C₄×Q₈×C₁₀₁	C₂	semidirect
16. #15692244	C₂²	semidirect
17. #15692243	C₂²	semidirect
18. #15692240	C₂²	semidirect
19. C₄×Dic₂₀₂	C₂²	semidirect
20. C₄×D₄₀₄	C₂²	semidirect
21. #15692217	C₂²	semidirect
29. C₄×Q₈×C₁₀₁	C₂²	semidirect
104. C₂×C₁₀₁	#559	semidirect
110. C₁₀₁	C₂×#559	semidirect
117. C₂×C₄×Q₈	D₁₀₁	semidirect
131. C₄×Q₈	C₂×D₁₀₁	semidirect
215. C₂	#20265912	direct (N × H)

Maximal subgroups

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

subgroup	index
#20265912	2
C₂×#15692252	2
C₂×#15692244	2
C₂×#15692243	2
C₂×#15692240	2
C₂×C₄×Dic₂₀₂	2
C₂×C₄×D₄₀₄	2
C₂×#15692217	2
C₂×C₄×Q₈×C₁₀₁	2
C₂×#559	101

Subgroups — C2×#20265912 #27184720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265912 #27184720