Subgroups — C₂×#1772 #20442

Order 256.

551 conjugacy classes of subgroups.

The subgroup lattice has 163 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×#427	C₂	1	1	yes	maximal, characteristic
3	C₂×#405	C₂	1	2	yes	maximal
4	C₂²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
5	#1772	C₂	1	8	yes	maximal, direct product factor
6	C₂×#339	C₂	1	2	yes	maximal
7	C₂²×M₁₆	C₂²	1	2	yes	—
8	C₂³×C₈	C₂²	1	1	yes	characteristic
9	C₂³×C₈	C₂²	1	1	yes	characteristic
10	C₂×#117	C₂²	1	2	yes	—
11	C₂²×C₄²	C₂²	1	1	yes	characteristic
12	C₂×#108	C₄	1	8	yes	complemented
13	C₂×#108	C₂²	1	4	yes	—
14	#427	C₄	1	4	yes	complemented
15	#427	C₂²	1	4	yes	—
16	C₂×#107	C₄	1	2	yes	complemented
17	C₂×#106	C₄	1	2	yes	complemented
18	#405	—	2	4	no	—
19	#405	C₂²	1	8	yes	—
20	C₂×#97	—	2	2	no	—
21	C₂×C₄×C₈	—	2	2	no	—
22	C₂×C₄×C₈	C₄	1	8	yes	complemented
23	C₂×C₄×C₈	C₂²	1	4	yes	—
24	#339	C₂²	1	8	yes	—
25	C₂³×C₄	—	2	1	no	—
26	C₂³×C₄	C₂³	1	1	yes	characteristic
27	C₂×M₁₆	—	2	2	no	—
28	C₂×M₁₆	—	2	4	no	—
29	C₂×M₁₆	C₂×C₄	1	8	yes	—
30	C₂×M₁₆	C₂³	1	4	yes	—
31	C₂²×C₈	—	2	2	no	—
32	C₂²×C₈	C₂×C₄	1	4	yes	—
33	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈	C₂×C₄	1	2	yes	—
36	C₂²×C₈	C₂×C₄	1	4	yes	—
37	C₂²×C₈	C₂×C₄	1	4	yes	—
38	C₂²×C₈	D₄	1	1	yes	characteristic
39	C₂²×C₈	D₄	1	2	yes	—
40	C₂²×C₈	D₄	1	1	yes	characteristic
41	C₂²×C₈	Q₈	1	4	yes	—
42	C₂²×C₈	C₂³	1	2	yes	—
43	C₂²×C₈	C₂³	1	2	yes	—
44	#117	—	2	2	no	—
45	#117	—	2	2	no	—
46	#117	C₂×C₄	1	4	yes	—
47	#117	C₂³	1	4	yes	—
48	C₂×C₄⋊C₄	C₂×C₄	1	4	yes	—
49	C₂×C₂²⋊C₄	—	2	2	no	—
50	C₂×C₄²	—	2	2	no	—
51	C₂×C₄²	—	2	1	no	—
52	C₂×C₄²	—	2	2	no	—
53	C₂×C₄²	C₂×C₄	1	4	yes	—
54	C₂×C₄²	C₂³	1	2	yes	—
55	#108	C₂×C₄	1	16	yes	complemented
56	#107	C₂×C₄	1	8	yes	complemented
57	#106	C₂×C₄	1	8	yes	complemented
58	#97	—	2	8	no	—
59	C₄×C₈	—	2	8	no	—
60	C₄×C₈	C₂×C₄	1	16	yes	complemented
61	C₂⁴	#41	1	1	yes	characteristic
62	C₂²×C₄	—	2	8	no	—
63	C₂²×C₄	—	2	4	no	—
64	C₂²×C₄	—	2	2	no	—
65	C₂²×C₄	—	2	4	no	—
66	C₂²×C₄	C₂²×C₄	1	2	yes	—
67	C₂²×C₄	C₂²×C₄	1	2	yes	—
68	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄	C₂×Q₈	1	2	yes	—
73	C₂²×C₄	#41	1	1	yes	characteristic
74	C₂²×C₄	#41	1	2	yes	—
75	C₂²×C₄	C₂⁴	1	1	yes	characteristic
76	M₁₆	—	2	8	no	—
77	M₁₆	—	2	4	no	—
78	M₁₆	C₂²×C₄	1	8	yes	—
79	C₂×C₈	—	2	8	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	C₄²	1	4	yes	—
82	C₂×C₈	C₄²	1	4	yes	—
83	C₂×C₈	C₄²	1	8	yes	—
84	C₂×C₈	C₄⋊C₄	1	4	yes	—
85	C₂×C₈	C₄⋊C₄	1	8	yes	—
86	C₂×C₈	C₄⋊C₄	1	4	yes	—
87	C₂×C₈	C₂²×C₄	1	4	yes	—
88	C₂×C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₈	C₂²×C₄	1	2	yes	—
90	C₂×C₈	C₂²×C₄	1	4	yes	—
91	C₂×C₈	C₂²×C₄	1	4	yes	—
92	C₂×C₈	C₂×D₄	1	2	yes	—
93	C₂×C₈	C₂×D₄	1	2	yes	—
94	C₂×C₈	C₂×Q₈	1	4	yes	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	C₂²×C₄	1	8	yes	—
97	C₂²⋊C₄	—	2	4	no	—
98	C₂²⋊C₄	—	2	4	no	—
99	C₄²	—	2	4	no	—
100	C₄²	—	2	4	no	—
101	C₄²	—	2	4	no	—
102	C₄²	—	2	2	no	—
103	C₄²	C₂²×C₄	1	4	yes	—
104	C₂³	—	2	4	no	—
105	C₂³	#117	1	2	yes	—
106	C₂³	C₄×D₄	1	1	yes	characteristic
107	C₂³	C₄×D₄	1	1	yes	characteristic
108	C₂³	C₄×Q₈	1	2	yes	—
109	C₂³	C₂×#41	1	1	yes	characteristic
110	C₂×C₄	—	2	8	no	—
111	C₂×C₄	—	2	16	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	2	8	no	—
114	C₂×C₄	—	4	8	no	—
115	C₂×C₄	C₂×C₄²	1	2	yes	—
116	C₂×C₄	C₂×C₄²	1	2	yes	—
117	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄	#117	1	2	yes	—
120	C₂×C₄	C₄×D₄	1	2	yes	—
121	C₂×C₄	C₄×D₄	1	2	yes	—
122	C₂×C₄	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄	C₄×Q₈	1	2	yes	—
125	C₂×C₄	C₄×Q₈	1	2	yes	—
126	C₂×C₄	C₄×Q₈	1	2	yes	—
127	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄	C₂×#41	1	1	yes	characteristic
133	C₈	—	2	8	no	—
134	C₈	C₂×C₄²	1	4	yes	—
135	C₈	C₂×C₄²	1	4	yes	—
136	C₈	C₂×C₄⋊C₄	1	4	yes	—
137	C₈	C₂×C₄⋊C₄	1	4	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	2	4	no	—
141	C₂²	—	2	4	no	—
142	C₂²	C₄×C₄⋊C₄	1	2	yes	—
143	C₂²	C₄×C₄⋊C₄	1	2	yes	—
144	C₂²	#443	1	4	yes	—
145	C₂²	C₂×#117	1	1	yes	characteristic
146	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
147	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
148	C₄	—	2	8	no	—
149	C₄	—	4	8	no	—
150	C₄	C₄×C₄⋊C₄	1	2	yes	—
151	C₄	C₄×C₄⋊C₄	1	2	yes	—
152	C₄	C₂²×C₄²	1	1	yes	characteristic
153	C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₄	C₂×C₄×D₄	1	1	yes	characteristic
155	C₄	C₂×C₄×Q₈	1	1	yes	characteristic
156	C₂	—	2	1	no	—
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	#1772	1	4	yes	direct product factor
160	C₂	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
161	C₂	C₂×#443	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	D₄
commutator	C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂³	C₄×D₄
prime radical	C₂⁴	#41
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
5. #1772	C₂	direct (N × H)
12. C₂×#108	C₄	semidirect
14. #427	C₄	semidirect
16. C₂×#107	C₄	semidirect
17. C₂×#106	C₄	semidirect
22. C₂×C₄×C₈	C₄	semidirect
55. #108	C₂×C₄	semidirect
56. #107	C₂×C₄	semidirect
57. #106	C₂×C₄	semidirect
60. C₄×C₈	C₂×C₄	semidirect
159. C₂	#1772	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×#108	2
C₂×#427	2
C₂×#405	2
C₂²×C₄×C₈	2
#1772	2
C₂×#339	2

Subgroups — C2×#1772 #20442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1772 #20442