Subgroups — C₂×#20265028 #27173466

Order 12928.

928 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13108053	C₂	1	2	yes	maximal
2	C₂²×C₄×Dic₁₀₁	C₂	1	2	yes	maximal
3	C₂²×C₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
4	#20265028	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15692162	C₂	1	1	yes	maximal, characteristic
6	C₂×#95×C₁₀₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
8	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
9	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
10	C₂×#13108053	C₄	1	16	yes	complemented
11	C₂×#13108053	C₂²	1	8	yes	—
12	C₂×C₄×Dic₁₀₁	—	2	8	no	—
13	C₂×C₄×Dic₁₀₁	C₄	1	8	yes	complemented
14	C₂×C₄×Dic₁₀₁	C₂²	1	8	yes	—
15	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	—
16	#15692162	C₂²	1	4	yes	—
17	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
19	#95×C₁₀₁	C₂²	1	4	yes	—
20	C₂²×Dic₁₀₁	—	2	4	no	—
21	C₂²×Dic₁₀₁	—	2	4	no	—
22	C₂²×Dic₁₀₁	C₂×C₄	1	8	yes	—
23	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	—
24	C₂²×Dic₁₀₁	C₂×C₄	1	8	yes	—
25	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	—
26	C₂²×Dic₁₀₁	D₄	1	4	yes	—
27	C₂²×Dic₁₀₁	Q₈	1	4	yes	—
28	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
29	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
30	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
31	#13108053	C₂×C₄	1	32	yes	complemented
32	C₄×Dic₁₀₁	—	2	16	no	—
33	C₄×Dic₁₀₁	C₂×C₄	1	16	yes	complemented
34	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₁₀₁	—	2	4	no	—
36	C₂²×C₄×C₁₀₁	—	2	4	no	—
37	C₂²×C₄×C₁₀₁	—	2	4	no	—
38	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
39	C₂²×C₄×C₁₀₁	C₂×C₄	1	8	yes	—
40	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
42	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
43	C₂³×C₁₀₁	C₂²×C₄	1	4	yes	—
44	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
45	C₂³×C₁₀₁	C₂×Q₈	1	2	yes	—
46	C₂³×C₁₀₁	#41	1	4	yes	—
47	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
48	C₂×C₄×C₁₀₁	—	2	8	no	—
49	C₂×C₄×C₁₀₁	—	2	16	no	—
50	C₂×C₄×C₁₀₁	—	2	4	no	—
51	C₂×C₄×C₁₀₁	—	2	4	no	—
52	C₂×C₄×C₁₀₁	—	2	4	no	—
53	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
54	C₂×C₄×C₁₀₁	C₂²×C₄	1	8	yes	—
55	C₂×Dic₁₀₁	—	2	16	no	—
56	C₂×Dic₁₀₁	—	2	4	no	—
57	C₂×Dic₁₀₁	—	2	4	no	—
58	C₂×Dic₁₀₁	C₄²	1	16	yes	complemented
59	C₂×Dic₁₀₁	C₄⋊C₄	1	16	yes	complemented
60	C₂×Dic₁₀₁	C₂²×C₄	1	8	yes	—
61	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
62	C₂×Dic₁₀₁	C₂²×C₄	1	8	yes	—
63	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
64	C₂×Dic₁₀₁	C₂×D₄	1	4	yes	—
65	C₂×Dic₁₀₁	C₂×Q₈	1	4	yes	—
66	C₂²×C₁₀₁	C₂×C₄²	1	4	yes	—
67	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
68	C₂²×C₁₀₁	#117	1	4	yes	—
69	C₂²×C₁₀₁	C₄×D₄	1	8	yes	—
70	C₂²×C₁₀₁	C₄×Q₈	1	8	yes	—
71	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
72	C₂²×C₁₀₁	C₂³×C₄	1	2	yes	—
73	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
74	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
75	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
76	C₄×C₁₀₁	—	2	4	no	—
77	C₄×C₁₀₁	—	2	8	no	—
78	Dic₁₀₁	—	2	8	no	—
79	Dic₁₀₁	C₂×C₄²	1	8	yes	complemented
80	Dic₁₀₁	C₂×C₄⋊C₄	1	8	yes	complemented
81	C₂×C₁₀₁	C₄×C₄⋊C₄	1	8	yes	complemented
82	C₂×C₁₀₁	C₂²×C₄²	1	1	yes	characteristic
83	C₂×C₁₀₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
85	C₂×C₁₀₁	C₂×C₄×D₄	1	2	yes	—
86	C₂×C₁₀₁	C₂×C₄×Q₈	1	2	yes	—
87	C₂×C₄×C₄⋊C₄	—	101	1	no	maximal, Sylow
88	C₁₀₁	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₄⋊C₄	—	101	2	no	—
90	C₂²×C₄²	—	101	1	no	—
91	C₂²×C₄²	—	101	2	no	—
92	C₄×C₄⋊C₄	—	101	8	no	—
93	C₂×#95	—	101	1	no	—
94	C₂×#95	D₁₀₁	1	1	yes	characteristic
95	C₂³×C₄	—	101	1	no	—
96	C₂³×C₄	—	101	1	no	—
97	C₂³×C₄	—	101	2	no	—
98	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
99	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
100	C₂×C₄⋊C₄	—	101	16	no	—
101	C₂×C₄⋊C₄	—	101	8	no	—
102	C₂×C₄²	—	101	8	no	—
103	C₂×C₄²	—	101	4	no	—
104	C₂×C₄²	—	101	8	no	—
105	C₂×C₄²	—	202	8	no	—
106	#95	—	101	4	no	—
107	#95	C₂×D₁₀₁	1	4	yes	—
108	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
109	C₂²×C₄	—	2	4	no	—
110	C₂²×C₄	—	2	4	no	—
111	C₂²×C₄	—	2	4	no	—
112	C₂²×C₄	—	101	4	no	—
113	C₂²×C₄	—	101	4	no	—
114	C₂²×C₄	—	101	8	no	—
115	C₂²×C₄	—	101	4	no	—
116	C₂²×C₄	—	101	4	no	—
117	C₂²×C₄	—	101	2	no	—
118	C₂²×C₄	—	101	2	no	—
119	C₂²×C₄	—	101	8	no	—
120	C₂²×C₄	—	101	4	no	—
121	C₂²×C₄	—	202	4	no	—
122	C₂²×C₄	—	202	4	no	—
123	C₂²×C₄	C₄×D₁₀₁	1	4	yes	—
124	C₂²×C₄	C₄×D₁₀₁	1	8	yes	—
125	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
126	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
127	C₄⋊C₄	—	101	32	no	—
128	C₄²	—	101	16	no	—
129	C₄²	—	202	16	no	—
130	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
131	C₂³	C₂×C₄×D₁₀₁	1	4	yes	—
132	C₂³	D₄×D₁₀₁	1	2	yes	—
133	C₂³	Q₈×D₁₀₁	1	2	yes	—
134	C₂³	#13108073	1	4	yes	—
135	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	2	16	no	—
141	C₂×C₄	—	101	4	no	—
142	C₂×C₄	—	101	4	no	—
143	C₂×C₄	—	101	8	no	—
144	C₂×C₄	—	101	4	no	—
145	C₂×C₄	—	101	4	no	—
146	C₂×C₄	—	101	16	no	—
147	C₂×C₄	—	101	16	no	—
148	C₂×C₄	—	101	8	no	—
149	C₂×C₄	—	202	4	no	—
150	C₂×C₄	—	202	4	no	—
151	C₂×C₄	—	202	16	no	—
152	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
153	C₂×C₄	C₂×C₄×D₁₀₁	1	8	yes	—
154	C₂²	C₄²×D₁₀₁	1	4	yes	—
155	C₂²	C₄⋊C₄×D₁₀₁	1	4	yes	—
156	C₂²	#15692211	1	4	yes	—
157	C₂²	#15692215	1	8	yes	—
158	C₂²	#15692220	1	8	yes	—
159	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
160	C₂²	C₂²×C₄×D₁₀₁	1	2	yes	—
161	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
162	C₂²	C₂×Q₈×D₁₀₁	1	1	yes	characteristic
163	C₂²	C₂×#13108073	1	2	yes	—
164	C₄	—	2	4	no	—
165	C₄	—	2	8	no	—
166	C₄	—	101	8	no	—
167	C₄	—	101	8	no	—
168	C₄	—	202	8	no	—
169	C₂	#20265028	1	8	yes	direct product factor
170	C₂	C₂×C₄²×D₁₀₁	1	1	yes	characteristic
171	C₂	C₂×C₄⋊C₄×D₁₀₁	1	1	yes	characteristic
172	C₂	C₂×#15692211	1	1	yes	characteristic
173	C₂	C₂×#15692215	1	2	yes	—
174	C₂	C₂×#15692220	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₁₀₁
commutator	C₂×C₁₀₁	C₂²×C₄²
Frattini	C₂³	C₂³×D₁₀₁
Fitting	C₂×#95×C₁₀₁	C₂
socle, prime radical	C₂⁴×C₁₀₁	C₂³
cosocle	C₂³×C₁₀₁	C₂⁴
hypercenter	C₂×#95	D₁₀₁
hypocenter	C₁₀₁	C₂×C₄×C₄⋊C₄
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#95

Complements

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

N	H ≅ G/N	type
4. #20265028	C₂	direct (N × H)
10. C₂×#13108053	C₄	semidirect
13. C₂×C₄×Dic₁₀₁	C₄	semidirect
31. #13108053	C₂×C₄	semidirect
33. C₄×Dic₁₀₁	C₂×C₄	semidirect
58. C₂×Dic₁₀₁	C₄²	semidirect
59. C₂×Dic₁₀₁	C₄⋊C₄	semidirect
79. Dic₁₀₁	C₂×C₄²	semidirect
80. Dic₁₀₁	C₂×C₄⋊C₄	semidirect
81. C₂×C₁₀₁	C₄×C₄⋊C₄	semidirect
88. C₁₀₁	C₂×C₄×C₄⋊C₄	semidirect
169. C₂	#20265028	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×#13108053	2
C₂²×C₄×Dic₁₀₁	2
#20265028	2
C₂×#15692162	2
C₂×#95×C₁₀₁	2
C₂×C₄×C₄⋊C₄	101

Subgroups — C2×#20265028 #27173466

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265028 #27173466