Subgroups — C₂×C₄×#13108053 #27173468

Order 12928.

928 conjugacy classes of subgroups.

The subgroup lattice has 180 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	1	yes	maximal, characteristic
3	C₂²×C₄×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
4	C₄×#13108053	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15692162	C₂	1	2	yes	maximal
6	C₂²×C₄²×C₁₀₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
8	C₂³×Dic₁₀₁	C₂²	1	2	yes	—
9	C₂³×Dic₁₀₁	C₂²	1	1	yes	characteristic
10	C₂×#13108053	C₄	1	16	yes	direct product factor
11	C₂×#13108053	C₂²	1	8	yes	—
12	C₂×C₄×Dic₁₀₁	—	2	4	no	—
13	C₂×C₄×Dic₁₀₁	C₄	1	8	yes	complemented
14	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	—
15	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	—
16	#15692162	C₂²	1	8	yes	—
17	C₂³×C₄×C₁₀₁	C₂²	1	2	yes	—
18	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₁₀₁	—	2	4	no	—
20	C₂×C₄²×C₁₀₁	C₂²	1	4	yes	—
21	C₂²×Dic₁₀₁	—	2	4	no	—
22	C₂²×Dic₁₀₁	—	2	4	no	—
23	C₂²×Dic₁₀₁	—	2	4	no	—
24	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	—
25	C₂²×Dic₁₀₁	C₂×C₄	1	8	yes	—
26	C₂²×Dic₁₀₁	C₂×C₄	1	4	yes	—
27	C₂²×Dic₁₀₁	C₂×C₄	1	8	yes	—
28	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
29	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
30	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
31	#13108053	C₂×C₄	1	32	yes	direct product factor
32	C₄×Dic₁₀₁	—	2	8	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	C₂×C₄	1	8	yes	—
38	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
39	C₂²×C₄×C₁₀₁	D₄	1	4	yes	—
40	C₂²×C₄×C₁₀₁	Q₈	1	4	yes	—
41	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
42	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
43	C₄²×C₁₀₁	—	2	8	no	—
44	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
45	C₂³×C₁₀₁	C₂²×C₄	1	4	yes	—
46	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
47	C₂³×C₁₀₁	C₂×Q₈	1	2	yes	—
48	C₂³×C₁₀₁	#41	1	4	yes	—
49	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
50	C₂×C₄×C₁₀₁	—	2	16	no	—
51	C₂×C₄×C₁₀₁	—	2	4	no	—
52	C₂×C₄×C₁₀₁	—	2	4	no	—
53	C₂×C₄×C₁₀₁	C₄⋊C₄	1	16	yes	complemented
54	C₂×C₄×C₁₀₁	C₂²×C₄	1	8	yes	—
55	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
57	C₂×C₄×C₁₀₁	C₂×Q₈	1	4	yes	—
58	C₂×Dic₁₀₁	—	2	8	no	—
59	C₂×Dic₁₀₁	—	2	16	no	—
60	C₂×Dic₁₀₁	—	2	4	no	—
61	C₂×Dic₁₀₁	—	2	4	no	—
62	C₂×Dic₁₀₁	—	2	4	no	—
63	C₂×Dic₁₀₁	C₄²	1	16	yes	complemented
64	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
65	C₂×Dic₁₀₁	C₂²×C₄	1	8	yes	—
66	C₂×Dic₁₀₁	C₂²×C₄	1	4	yes	—
67	C₂×Dic₁₀₁	C₂²×C₄	1	8	yes	—
68	C₂²×C₁₀₁	C₂×C₄²	1	4	yes	—
69	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	4	yes	—
70	C₂²×C₁₀₁	#117	1	4	yes	—
71	C₂²×C₁₀₁	C₄×D₄	1	8	yes	—
72	C₂²×C₁₀₁	C₄×Q₈	1	8	yes	—
73	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
74	C₂²×C₁₀₁	C₂³×C₄	1	2	yes	—
75	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
76	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
77	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
78	C₄×C₁₀₁	—	2	8	no	—
79	C₄×C₁₀₁	C₂×C₄⋊C₄	1	8	yes	complemented
80	Dic₁₀₁	—	2	4	no	—
81	Dic₁₀₁	—	2	8	no	—
82	Dic₁₀₁	C₂×C₄²	1	8	yes	complemented
83	C₂×C₁₀₁	C₄×C₄⋊C₄	1	8	yes	complemented
84	C₂×C₁₀₁	C₂²×C₄²	1	1	yes	characteristic
85	C₂×C₁₀₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₁₀₁	C₂×#117	1	1	yes	characteristic
87	C₂×C₁₀₁	C₂×C₄×D₄	1	2	yes	—
88	C₂×C₁₀₁	C₂×C₄×Q₈	1	2	yes	—
89	C₂×C₄×C₄⋊C₄	—	101	1	no	maximal, Sylow
90	C₁₀₁	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₄⋊C₄	—	101	2	no	—
92	C₂²×C₄²	—	101	1	no	—
93	C₂²×C₄²	—	101	1	no	—
94	C₂²×C₄²	D₁₀₁	1	1	yes	characteristic
95	C₄×C₄⋊C₄	—	101	8	no	—
96	C₂×#95	—	101	2	no	—
97	C₂³×C₄	—	101	1	no	—
98	C₂³×C₄	—	101	2	no	—
99	C₂³×C₄	—	101	1	no	—
100	C₂³×C₄	C₂×D₁₀₁	1	2	yes	—
101	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
102	C₂×C₄⋊C₄	—	101	16	no	—
103	C₂×C₄⋊C₄	—	101	8	no	—
104	C₂×C₄²	—	2	4	no	—
105	C₂×C₄²	—	101	8	no	—
106	C₂×C₄²	—	101	4	no	—
107	C₂×C₄²	—	101	4	no	—
108	C₂×C₄²	—	202	4	no	—
109	C₂×C₄²	C₂×D₁₀₁	1	4	yes	—
110	#95	—	101	8	no	—
111	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
112	C₂²×C₄	—	2	4	no	—
113	C₂²×C₄	—	2	4	no	—
114	C₂²×C₄	—	101	4	no	—
115	C₂²×C₄	—	101	8	no	—
116	C₂²×C₄	—	101	2	no	—
117	C₂²×C₄	—	101	4	no	—
118	C₂²×C₄	—	101	8	no	—
119	C₂²×C₄	—	101	2	no	—
120	C₂²×C₄	—	101	4	no	—
121	C₂²×C₄	—	202	4	no	—
122	C₂²×C₄	—	202	4	no	—
123	C₂²×C₄	—	202	4	no	—
124	C₂²×C₄	Dic₂₀₂	1	4	yes	—
125	C₂²×C₄	C₄×D₁₀₁	1	8	yes	—
126	C₂²×C₄	C₄×D₁₀₁	1	4	yes	—
127	C₂²×C₄	C₂²⋊D₁₀₁	1	4	yes	—
128	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
129	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
130	C₄⋊C₄	—	101	32	no	—
131	C₄²	—	2	8	no	—
132	C₄²	—	101	16	no	—
133	C₄²	—	202	8	no	—
134	C₂³	C₂×C₄×D₁₀₁	1	2	yes	—
135	C₂³	C₂×C₄×D₁₀₁	1	4	yes	—
136	C₂³	C₂×C₂²⋊D₁₀₁	1	2	yes	—
137	C₂³	C₂×Dic₂₀₂	1	2	yes	—
138	C₂³	#13108074	1	4	yes	—
139	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	16	no	—
143	C₂×C₄	—	101	4	no	—
144	C₂×C₄	—	101	8	no	—
145	C₂×C₄	—	101	16	no	—
146	C₂×C₄	—	101	4	no	—
147	C₂×C₄	—	101	8	no	—
148	C₂×C₄	—	202	4	no	—
149	C₂×C₄	—	202	4	no	—
150	C₂×C₄	—	202	4	no	—
151	C₂×C₄	—	202	8	no	—
152	C₂×C₄	—	202	16	no	—
153	C₂×C₄	#13108053	1	16	yes	direct product factor
154	C₂×C₄	C₂×C₄×D₁₀₁	1	8	yes	—
155	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
156	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
157	C₂×C₄	C₂×Dic₂₀₂	1	4	yes	—
158	C₂²	C₄²×D₁₀₁	1	4	yes	—
159	C₂²	C₂×#13108053	1	4	yes	—
160	C₂²	#15692212	1	4	yes	—
161	C₂²	C₄×C₂²⋊D₁₀₁	1	8	yes	—
162	C₂²	C₄×Dic₂₀₂	1	8	yes	—
163	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
164	C₂²	C₂²×C₄×D₁₀₁	1	2	yes	—
165	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
166	C₂²	C₂²×Dic₂₀₂	1	1	yes	characteristic
167	C₂²	C₂×#13108074	1	2	yes	—
168	C₄	—	2	8	no	—
169	C₄	—	101	8	no	—
170	C₄	—	202	4	no	—
171	C₄	—	202	8	no	—
172	C₄	C₂×#13108053	1	8	yes	direct product factor
173	C₂	C₄×#13108053	1	8	yes	direct product factor
174	C₂	C₂×C₄²×D₁₀₁	1	1	yes	characteristic
175	C₂	C₂²×#13108053	1	1	yes	characteristic
176	C₂	C₂×#15692212	1	1	yes	characteristic
177	C₂	C₂×C₄×C₂²⋊D₁₀₁	1	2	yes	—
178	C₂	C₂×C₄×Dic₂₀₂	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₄×#13108053	C₂	direct (N × H)
10. C₂×#13108053	C₄	direct (N × H)
13. C₂×C₄×Dic₁₀₁	C₄	semidirect
31. #13108053	C₂×C₄	direct (N × H)
33. C₄×Dic₁₀₁	C₂×C₄	semidirect
53. C₂×C₄×C₁₀₁	C₄⋊C₄	semidirect
63. C₂×Dic₁₀₁	C₄²	semidirect
79. C₄×C₁₀₁	C₂×C₄⋊C₄	semidirect
82. Dic₁₀₁	C₂×C₄²	semidirect
83. C₂×C₁₀₁	C₄×C₄⋊C₄	semidirect
90. C₁₀₁	C₂×C₄×C₄⋊C₄	semidirect
153. C₂×C₄	#13108053	direct (N × H)
172. C₄	C₂×#13108053	direct (N × H)
173. C₂	C₄×#13108053	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
C₄×#13108053	2
C₂×#15692162	2
C₂²×C₄²×C₁₀₁	2
C₂×C₄×C₄⋊C₄	101

Subgroups — C2×C4×#13108053 #27173468

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#13108053 #27173468