Subgroups — C₂×#20265227 #27179745

Order 12928.

528 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
2	#20265227	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15692184	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊#11717579	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×Dic₁₀₁	C₂	1	2	yes	maximal, complemented
6	C₂×C₂²⋊C₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₀₁	C₄	1	1	yes	characteristic
8	C₂³×Dic₁₀₁	C₄	1	1	yes	characteristic
9	C₂²×#11717579	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₀₁	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₁₀₁	C₄	1	4	yes	—
12	C₂×C₄×Dic₁₀₁	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	—
14	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
16	#15692184	C₂²	1	8	yes	complemented
17	C₂²⋊#11717579	C₂²	1	4	yes	—
18	C₈×Dic₁₀₁	C₂²	1	8	yes	complemented
19	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₁₀₁	C₂²	1	2	yes	—
21	C₂²⋊C₈×C₁₀₁	C₂²	1	4	yes	—
22	C₂²×Dic₁₀₁	—	2	4	no	—
23	C₂²×Dic₁₀₁	C₈	1	8	yes	complemented
24	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	—
26	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	—
29	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
30	C₂×#11717579	—	2	4	no	—
31	C₂×#11717579	C₂³	1	4	yes	—
32	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₀₁	C₂×C₄	1	2	yes	—
34	C₄×Dic₁₀₁	C₂×C₄	1	4	yes	—
35	C₄×Dic₁₀₁	C₂×C₄	1	2	yes	—
36	C₄×Dic₁₀₁	C₂³	1	2	yes	—
37	C₄×Dic₁₀₁	C₂³	1	2	yes	—
38	C₂⁴×C₁₀₁	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₀₁	—	2	4	no	—
40	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₁₀₁	—	2	4	no	—
45	C₂×C₈×C₁₀₁	C₂³	1	4	yes	—
46	C₂³×C₁₀₁	—	2	2	no	—
47	C₂³×C₁₀₁	—	2	2	no	—
48	C₂³×C₁₀₁	C₂×C₈	1	4	yes	—
49	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
50	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₀₁	—	2	2	no	—
52	C₂×C₄×C₁₀₁	—	2	4	no	—
53	C₂×C₄×C₁₀₁	—	2	4	no	—
54	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₀₁	#41	1	4	yes	—
57	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₁₀₁	—	2	2	no	—
59	C₂×Dic₁₀₁	—	2	8	no	—
60	C₂×Dic₁₀₁	C₂×C₈	1	4	yes	—
61	C₂×Dic₁₀₁	C₂×C₈	1	4	yes	—
62	C₂×Dic₁₀₁	C₂×C₈	1	16	yes	complemented
63	C₂×Dic₁₀₁	M₁₆	1	4	yes	—
64	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
65	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
66	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
67	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
68	C₈×C₁₀₁	—	2	4	no	—
69	#11717579	—	2	4	no	—
70	C₂²×C₁₀₁	—	2	2	no	—
71	C₂²×C₁₀₁	—	2	2	no	—
72	C₂²×C₁₀₁	—	2	8	no	—
73	C₂²×C₁₀₁	#117	1	2	yes	—
74	C₂²×C₁₀₁	C₂²×C₈	1	4	yes	—
75	C₂²×C₁₀₁	C₂²×C₈	1	2	yes	—
76	C₂²×C₁₀₁	C₂×M₁₆	1	2	yes	—
77	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
78	C₄×C₁₀₁	—	2	2	no	—
79	C₄×C₁₀₁	#117	1	2	yes	—
80	C₄×C₁₀₁	C₂×#41	1	2	yes	—
81	Dic₁₀₁	—	2	2	no	—
82	Dic₁₀₁	C₂²×C₈	1	4	yes	—
83	Dic₁₀₁	C₂²×C₈	1	4	yes	—
84	Dic₁₀₁	C₂×M₁₆	1	4	yes	—
85	C₂×C₁₀₁	—	2	4	no	—
86	C₂×C₁₀₁	#431	1	4	yes	complemented
87	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
88	C₂×C₁₀₁	C₂³×C₈	1	1	yes	characteristic
89	C₂×C₁₀₁	C₂²×M₁₆	1	1	yes	characteristic
90	C₂×#431	—	101	1	no	maximal, Sylow
91	C₁₀₁	C₂×#431	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₄²	—	101	1	no	—
93	#431	—	101	8	no	—
94	C₂×C₄⋊C₈	—	101	2	no	—
95	C₂×C₂²⋊C₈	—	101	1	no	—
96	C₂×C₂²⋊C₈	D₁₀₁	1	1	yes	characteristic
97	C₂×C₄×C₈	—	101	2	no	—
98	C₂³×C₄	—	101	1	no	—
99	C₂³×C₄	—	101	1	no	—
100	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
101	C₂²×C₈	—	101	2	no	—
102	C₂²×C₈	C₂×D₁₀₁	1	2	yes	—
103	C₂×C₄²	—	101	1	no	—
104	C₂×C₄²	—	101	4	no	—
105	C₂×C₄²	—	101	1	no	—
106	C₂×C₄²	—	101	4	no	—
107	C₂×C₄²	—	101	1	no	—
108	C₂×C₄²	—	101	1	no	—
109	C₄⋊C₈	—	101	8	no	—
110	C₂²⋊C₈	—	101	4	no	—
111	C₂²⋊C₈	C₂×D₁₀₁	1	4	yes	—
112	C₄×C₈	—	101	8	no	—
113	C₂⁴	C₄×D₁₀₁	1	1	yes	characteristic
114	C₂²×C₄	—	2	4	no	—
115	C₂²×C₄	—	101	8	no	—
116	C₂²×C₄	—	101	1	no	—
117	C₂²×C₄	—	101	1	no	—
118	C₂²×C₄	—	101	4	no	—
119	C₂²×C₄	—	101	1	no	—
120	C₂²×C₄	—	101	4	no	—
121	C₂²×C₄	—	101	1	no	—
122	C₂²×C₄	—	202	4	no	—
123	C₂²×C₄	C₄×D₁₀₁	1	1	yes	characteristic
124	C₂²×C₄	C₄×D₁₀₁	1	2	yes	—
125	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
126	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
127	C₂×C₈	—	2	4	no	—
128	C₂×C₈	—	101	4	no	—
129	C₂×C₈	—	202	4	no	—
130	C₂×C₈	C₂²×D₁₀₁	1	4	yes	—
131	C₄²	—	101	4	no	—
132	C₄²	—	101	2	no	—
133	C₄²	—	101	4	no	—
134	C₄²	—	101	2	no	—
135	C₄²	—	101	2	no	—
136	C₄²	—	101	2	no	—
137	C₂³	—	2	2	no	—
138	C₂³	—	2	2	no	—
139	C₂³	C₈×D₁₀₁	1	4	yes	—
140	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
141	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	101	2	no	—
146	C₂×C₄	—	101	2	no	—
147	C₂×C₄	—	101	2	no	—
148	C₂×C₄	—	101	4	no	—
149	C₂×C₄	—	101	4	no	—
150	C₂×C₄	—	101	16	no	—
151	C₂×C₄	—	101	4	no	—
152	C₂×C₄	—	101	2	no	—
153	C₂×C₄	—	202	8	no	—
154	C₂×C₄	—	202	2	no	—
155	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
156	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
157	C₂×C₄	#13108073	1	4	yes	—
158	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
159	C₈	—	2	4	no	—
160	C₈	—	202	4	no	—
161	C₂²	—	2	8	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	2	no	—
164	C₂²	#15692211	1	2	yes	—
165	C₂²	C₂×C₈×D₁₀₁	1	4	yes	—
166	C₂²	C₂×C₈×D₁₀₁	1	2	yes	—
167	C₂²	M₁₆×D₁₀₁	1	2	yes	—
168	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	101	4	no	—
171	C₄	—	101	4	no	—
172	C₄	—	101	4	no	—
173	C₄	—	202	2	no	—
174	C₄	#15692211	1	2	yes	—
175	C₄	C₂×#13108073	1	2	yes	—
176	C₂	—	2	4	no	—
177	C₂	#20265227	1	4	yes	direct product factor
178	C₂	C₂×#15692211	1	1	yes	characteristic
179	C₂	C₂²×C₈×D₁₀₁	1	1	yes	characteristic
180	C₂	C₂×M₁₆×D₁₀₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₀₁
commutator	C₂×C₁₀₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	C₂×C₂²⋊C₈×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂²×C₄
prime radical	C₂⁴×C₁₀₁	C₂×C₄
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	C₂×C₂²⋊C₈	D₁₀₁
hypocenter	C₁₀₁	C₂×#431
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. #20265227	C₂	direct (N × H)
3. C₂×#15692184	C₂	semidirect
5. C₂×C₈×Dic₁₀₁	C₂	semidirect
16. #15692184	C₂²	semidirect
18. C₈×Dic₁₀₁	C₂²	semidirect
23. C₂²×Dic₁₀₁	C₈	semidirect
62. C₂×Dic₁₀₁	C₂×C₈	semidirect
86. C₂×C₁₀₁	#431	semidirect
91. C₁₀₁	C₂×#431	semidirect
177. C₂	#20265227	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×C₄×Dic₁₀₁	2
#20265227	2
C₂×#15692184	2
C₂×C₂²⋊#11717579	2
C₂×C₈×Dic₁₀₁	2
C₂×C₂²⋊C₈×C₁₀₁	2
C₂×#431	101

Subgroups — C2×#20265227 #27179745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265227 #27179745