Subgroups — C₂×#20265751 #27184449

Order 12928.

1512 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13108074	C₂	1	1	yes	maximal, characteristic
2	#20265751	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15692253	C₂	1	2	yes	maximal, complemented
4	C₂×#15692244	C₂	1	2	yes	maximal, complemented
5	C₂×#15692238	C₂	1	4	yes	maximal, complemented
6	C₂×#15692231	C₂	1	4	yes	maximal, complemented
7	C₂×#15692214	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13108074	—	2	8	no	—
10	C₂×#13108074	C₂²	1	8	yes	—
11	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
12	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
13	C₂²×C₂²⋊D₁₀₁	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
15	#15692253	C₂²	1	16	yes	complemented
16	#15692244	C₂²	1	16	yes	complemented
17	#15692238	C₂²	1	32	yes	complemented
18	#15692231	C₂²	1	32	yes	complemented
19	#15692214	C₂²	1	8	yes	complemented
20	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
21	C₂×#13108053	C₂²	1	8	yes	—
22	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
23	C₂×#13108051	C₂²	1	8	yes	—
24	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
25	C₂²×Q₈×C₁₀₁	—	2	8	no	—
26	C₂²×Q₈×C₁₀₁	C₂²	1	2	yes	—
27	C₂²×Q₈×C₁₀₁	C₂²	1	2	yes	—
28	C₂²×Q₈×C₁₀₁	C₂²	1	8	yes	complemented
29	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
30	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
31	C₂³×D₁₀₁	C₂³	1	2	yes	—
32	#13108074	—	2	32	no	—
33	C₂×Dic₂₀₂	—	2	4	no	—
34	C₂×Dic₂₀₂	C₂³	1	4	yes	—
35	C₂×D₄₀₄	—	2	4	no	—
36	C₂×D₄₀₄	C₂³	1	4	yes	—
37	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
38	C₂×C₂²⋊D₁₀₁	C₂³	1	8	yes	—
39	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
40	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
41	C₂×C₄×D₁₀₁	—	2	8	no	—
42	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
43	C₄⋊Dic₁₀₁	C₂³	1	8	yes	—
44	#13108053	C₂³	1	32	yes	—
45	C₂²⋊Dic₁₀₁	C₂³	1	8	yes	—
46	#13108051	C₂³	1	32	yes	—
47	C₄×Dic₁₀₁	C₂³	1	8	yes	—
48	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₁₀₁	—	2	8	no	—
50	C₂×Q₈×C₁₀₁	—	2	8	no	—
51	C₂×Q₈×C₁₀₁	—	2	32	no	—
52	C₂×Q₈×C₁₀₁	C₂³	1	8	yes	—
53	C₂×Q₈×C₁₀₁	C₂³	1	8	yes	—
54	C₂²×C₄×C₁₀₁	—	2	8	no	—
55	C₂²×C₄×C₁₀₁	D₄	1	8	yes	—
56	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
57	C₂²×C₄×C₁₀₁	C₂³	1	8	yes	—
58	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
61	C₂³×C₁₀₁	—	2	4	no	—
62	C₂³×C₁₀₁	C₂×D₄	1	4	yes	—
63	C₂³×C₁₀₁	C₂⁴	1	2	yes	—
64	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
65	C₂²×D₁₀₁	—	2	4	no	—
66	C₂²×D₁₀₁	—	2	8	no	—
67	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
68	Q₈×C₁₀₁	—	2	16	no	—
69	Q₈×C₁₀₁	—	2	16	no	—
70	C₂×C₄×C₁₀₁	—	2	16	no	—
71	C₂×C₄×C₁₀₁	—	2	8	no	—
72	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
73	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
74	C₂×C₄×C₁₀₁	C₂×D₄	1	16	yes	—
75	C₂×C₄×C₁₀₁	C₂⁴	1	8	yes	—
76	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
77	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
78	C₂²⋊D₁₀₁	—	2	16	no	—
79	C₂×Dic₁₀₁	—	2	4	no	—
80	C₂×Dic₁₀₁	—	2	8	no	—
81	C₂×Dic₁₀₁	C₂⁴	1	8	yes	—
82	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
83	D₄₀₄	—	2	8	no	—
84	C₄×D₁₀₁	—	2	16	no	—
85	Dic₂₀₂	—	2	8	no	—
86	C₂²×C₁₀₁	—	2	8	no	—
87	C₂²×C₁₀₁	—	2	4	no	—
88	C₂²×C₁₀₁	C₂²×D₄	1	4	yes	—
89	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
90	C₂²×C₁₀₁	2₋¹⁺⁴	1	4	yes	—
91	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
92	C₂×D₁₀₁	—	2	4	no	—
93	C₂×D₁₀₁	—	2	8	no	—
94	C₂×D₁₀₁	—	4	8	no	—
95	C₄×C₁₀₁	—	2	8	no	—
96	C₄×C₁₀₁	C₂²×D₄	1	4	yes	—
97	C₄×C₁₀₁	C₂²×D₄	1	4	yes	—
98	Dic₁₀₁	—	2	4	no	—
99	Dic₁₀₁	—	2	8	no	—
100	C₂×C₁₀₁	—	2	4	no	—
101	C₂×C₁₀₁	#536	1	4	yes	complemented
102	C₂×C₁₀₁	C₂³×D₄	1	1	yes	characteristic
103	C₂×C₁₀₁	C₂×2₋¹⁺⁴	1	2	yes	—
104	D₁₀₁	—	4	4	no	—
105	C₂×#536	—	101	1	no	maximal, Sylow
106	C₁₀₁	C₂×#536	1	1	yes	characteristic, complemented, Sylow
107	C₂²×#41	—	101	1	no	—
108	C₂³×Q₈	D₁₀₁	1	1	yes	characteristic, complemented
109	#536	—	101	16	no	—
110	C₂×C₄⋊Q₈	—	101	2	no	—
111	C₂×#124	—	101	2	no	—
112	C₂×#123	—	101	4	no	—
113	C₂×C₂²⋊Q₈	—	101	4	no	—
114	C₂×#117	—	101	1	no	—
115	C₂×#41	—	101	8	no	—
116	C₂×#41	—	202	8	no	—
117	C₂²×Q₈	—	2	8	no	—
118	C₂²×Q₈	—	101	1	no	—
119	C₂²×Q₈	C₂×D₁₀₁	1	2	yes	—
120	C₂²×Q₈	C₂×D₁₀₁	1	2	yes	—
121	C₂²×Q₈	C₂×D₁₀₁	1	8	yes	complemented
122	C₂²×D₄	—	101	2	no	—
123	C₂²×D₄	—	101	1	no	—
124	C₂³×C₄	—	101	2	no	—
125	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
126	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
127	C₄⋊Q₈	—	101	16	no	—
128	#124	—	101	16	no	—
129	#123	—	101	32	no	—
130	C₂²⋊Q₈	—	101	32	no	—
131	#117	—	101	8	no	—
132	C₂×C₄⋊C₄	—	101	2	no	—
133	C₂×C₄⋊C₄	—	101	8	no	—
134	C₂×C₂²⋊C₄	—	101	2	no	—
135	C₂×C₂²⋊C₄	—	101	8	no	—
136	C₂×C₄²	—	101	2	no	—
137	C₂⁴	—	101	2	no	—
138	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
139	#41	—	202	32	no	—
140	C₂×Q₈	—	2	8	no	—
141	C₂×Q₈	—	2	8	no	—
142	C₂×Q₈	—	2	32	no	—
143	C₂×Q₈	—	101	4	no	—
144	C₂×Q₈	—	202	4	no	—
145	C₂×Q₈	C₂²×D₁₀₁	1	8	yes	—
146	C₂×Q₈	C₂²×D₁₀₁	1	8	yes	—
147	C₂×D₄	—	101	8	no	—
148	C₂×D₄	—	101	4	no	—
149	C₂×D₄	—	202	8	no	—
150	C₂×D₄	—	202	4	no	—
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	—	101	4	no	—
153	C₂²×C₄	—	101	2	no	—
154	C₂²×C₄	—	101	8	no	—
155	C₂²×C₄	—	202	8	no	—
156	C₂²×C₄	C₂²⋊D₁₀₁	1	8	yes	—
157	C₂²×C₄	C₂²×D₁₀₁	1	8	yes	—
158	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
159	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
160	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
161	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
162	C₄⋊C₄	—	101	32	no	—
163	C₄⋊C₄	—	101	8	no	—
164	C₂²⋊C₄	—	101	8	no	—
165	C₂²⋊C₄	—	101	32	no	—
166	C₄²	—	101	8	no	—
167	C₂³	—	2	4	no	—
168	C₂³	—	101	4	no	—
169	C₂³	—	202	4	no	—
170	C₂³	—	202	8	no	—
171	C₂³	C₂×C₂²⋊D₁₀₁	1	4	yes	—
172	C₂³	C₂³×D₁₀₁	1	2	yes	—
173	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
174	Q₈	—	2	16	no	—
175	Q₈	—	2	16	no	—
176	Q₈	—	202	8	no	—
177	D₄	—	202	16	no	—
178	D₄	—	202	8	no	—
179	C₂×C₄	—	2	16	no	—
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	101	8	no	—
182	C₂×C₄	—	101	4	no	—
183	C₂×C₄	—	202	4	no	—
184	C₂×C₄	—	202	8	no	—
185	C₂×C₄	—	202	16	no	—
186	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
187	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
188	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	16	yes	—
189	C₂×C₄	C₂³×D₁₀₁	1	8	yes	—
190	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
191	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
192	C₂²	—	2	8	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	404	8	no	—
195	C₂²	—	202	4	no	—
196	C₂²	—	202	8	no	—
197	C₂²	C₂²×C₂²⋊D₁₀₁	1	4	yes	—
198	C₂²	C₂²×C₂²⋊D₁₀₁	1	2	yes	—
199	C₂²	#15692303	1	4	yes	—
200	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
201	C₄	—	2	8	no	—
202	C₄	—	202	4	no	—
203	C₄	—	202	8	no	—
204	C₄	C₂²×C₂²⋊D₁₀₁	1	4	yes	—
205	C₄	C₂²×C₂²⋊D₁₀₁	1	4	yes	—
206	C₂	—	2	4	no	—
207	C₂	—	404	4	no	—
208	C₂	#20265751	1	4	yes	direct product factor
209	C₂	C₂³×C₂²⋊D₁₀₁	1	1	yes	characteristic
210	C₂	C₂×#15692303	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₁₀₁
commutator, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	C₂³×Q₈×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂²×#13108074	C₂
hypercenter	C₂³×Q₈	D₁₀₁
hypocenter	C₁₀₁	C₂×#536
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
2. #20265751	C₂	direct (N × H)
3. C₂×#15692253	C₂	semidirect
4. C₂×#15692244	C₂	semidirect
5. C₂×#15692238	C₂	semidirect
6. C₂×#15692231	C₂	semidirect
7. C₂×#15692214	C₂	semidirect
8. C₂³×Q₈×C₁₀₁	C₂	semidirect
15. #15692253	C₂²	semidirect
16. #15692244	C₂²	semidirect
17. #15692238	C₂²	semidirect
18. #15692231	C₂²	semidirect
19. #15692214	C₂²	semidirect
28. C₂²×Q₈×C₁₀₁	C₂²	semidirect
101. C₂×C₁₀₁	#536	semidirect
106. C₁₀₁	C₂×#536	semidirect
108. C₂³×Q₈	D₁₀₁	semidirect
121. C₂²×Q₈	C₂×D₁₀₁	semidirect
208. C₂	#20265751	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×#13108074	2
#20265751	2
C₂×#15692253	2
C₂×#15692244	2
C₂×#15692238	2
C₂×#15692231	2
C₂×#15692214	2
C₂³×Q₈×C₁₀₁	2
C₂×#536	101

Subgroups — C2×#20265751 #27184449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265751 #27184449