Subgroups — C₂×#2155 #33403

Order 256.

329 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×#108	C₂	1	1	yes	maximal, characteristic
3	C₂×#427	C₂	1	1	yes	maximal, characteristic
4	#2155	C₂	1	8	yes	maximal, direct product factor
5	C₂×#365	C₂	1	4	yes	maximal, complemented
6	C₂²×M₁₆	—	2	1	no	—
7	C₂³×C₈	C₂²	1	1	yes	characteristic
8	C₂×#117	—	2	1	no	—
9	C₂×M₃₂	C₄	1	2	yes	complemented
10	C₂×M₃₂	C₄	1	4	yes	complemented
11	C₂×M₃₂	C₂²	1	2	yes	—
12	C₂×M₃₂	C₂²	1	4	yes	—
13	C₂²×C₁₆	C₄	1	2	yes	complemented
14	C₂×#108	—	2	1	no	—
15	C₂×#108	—	2	2	no	—
16	C₂×#108	C₂²	1	2	yes	—
17	C₂×#108	C₂²	1	4	yes	—
18	#427	—	2	2	no	—
19	#427	C₂²	1	4	yes	—
20	C₂×#107	—	2	1	no	—
21	C₂×#106	C₂²	1	2	yes	—
22	#365	C₂²	1	16	yes	complemented
23	C₂³×C₄	D₄	1	1	yes	characteristic
24	C₂×M₁₆	—	2	2	no	—
25	C₂×M₁₆	—	2	4	no	—
26	C₂×M₁₆	—	4	1	no	—
27	C₂×M₁₆	—	4	2	no	—
28	C₂²×C₈	—	4	1	no	—
29	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₈	D₄	1	1	yes	characteristic
33	C₂²×C₈	D₄	1	2	yes	—
34	C₂²×C₈	Q₈	1	1	yes	characteristic
35	C₂²×C₈	Q₈	1	2	yes	—
36	C₂²×C₈	Q₈	1	1	yes	characteristic
37	C₂²×C₈	C₂³	1	1	yes	characteristic
38	C₂²×C₈	C₂³	1	2	yes	—
39	#117	—	2	4	no	—
40	#117	—	4	2	no	—
41	C₂×C₄⋊C₄	—	2	2	no	—
42	C₂×C₂²⋊C₄	—	4	1	no	—
43	C₂×C₄²	—	4	1	no	—
44	M₃₂	C₂×C₄	1	4	yes	—
45	M₃₂	C₂×C₄	1	8	yes	complemented
46	M₃₂	C₂³	1	4	yes	—
47	C₂×C₁₆	C₂×C₄	1	4	yes	—
48	C₂×C₁₆	C₂×C₄	1	8	yes	complemented
49	#108	—	2	2	no	—
50	#108	—	2	4	no	—
51	#108	C₂³	1	4	yes	—
52	#107	—	2	4	no	—
53	#106	—	2	2	no	—
54	#106	C₂³	1	4	yes	—
55	C₂⁴	SD₁₆	1	1	yes	characteristic
56	C₂²×C₄	—	4	2	no	—
57	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄	SD₁₆	1	2	yes	—
61	C₂²×C₄	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄	SD₁₆	1	2	yes	—
64	C₂²×C₄	SD₁₆	1	1	yes	characteristic
65	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄	C₂×D₄	1	2	yes	—
67	M₁₆	—	2	4	no	—
68	M₁₆	—	4	2	no	—
69	M₁₆	—	4	4	no	—
70	C₂×C₈	—	4	2	no	—
71	C₂×C₈	—	4	4	no	—
72	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₈	C₄⋊C₄	1	2	yes	—
74	C₂×C₈	C₄⋊C₄	1	2	yes	—
75	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₈	C₄⋊C₄	1	2	yes	—
77	C₂×C₈	C₄⋊C₄	1	2	yes	—
78	C₂×C₈	C₄⋊C₄	1	2	yes	—
79	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈	C₂²×C₄	1	2	yes	—
81	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈	C₂²×C₄	1	2	yes	—
83	C₂×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈	C₂×D₄	1	2	yes	—
85	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₈	C₂×Q₈	1	2	yes	—
88	C₂×C₈	C₂×Q₈	1	2	yes	—
89	C₂×C₈	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄	—	2	4	no	—
91	C₄⋊C₄	—	4	2	no	—
92	C₂²⋊C₄	—	4	4	no	—
93	C₄²	—	4	4	no	—
94	C₁₆	C₂²×C₄	1	8	yes	—
95	C₂³	—	4	2	no	—
96	C₂³	#106	1	1	yes	characteristic
97	C₂³	#106	1	1	yes	characteristic
98	C₂³	#106	1	2	yes	—
99	C₂³	C₂×SD₁₆	1	1	yes	characteristic
100	C₂³	C₂×SD₁₆	1	2	yes	—
101	C₂×C₄	—	4	4	no	—
102	C₂×C₄	—	8	4	no	—
103	C₂×C₄	#106	1	2	yes	—
104	C₂×C₄	#106	1	2	yes	—
105	C₂×C₄	#106	1	1	yes	characteristic
106	C₂×C₄	#106	1	2	yes	—
107	C₂×C₄	#106	1	2	yes	—
108	C₂×C₄	#106	1	2	yes	—
109	C₂×C₄	#106	1	1	yes	characteristic
110	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄	C₂×SD₁₆	1	2	yes	—
116	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄	C₂×SD₁₆	1	2	yes	—
118	C₂×C₄	C₂×SD₁₆	1	2	yes	—
119	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
121	C₈	—	4	4	no	—
122	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₈	C₂×C₄⋊C₄	1	2	yes	—
124	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₈	C₂×C₄⋊C₄	1	2	yes	—
126	C₈	C₂³×C₄	1	1	yes	characteristic
127	C₈	C₂²×Q₈	1	1	yes	characteristic
128	C₂²	—	2	2	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	4	2	no	—
132	C₂²	—	4	2	no	—
133	C₂²	C₂×#106	1	2	yes	—
134	C₂²	C₂×#106	1	2	yes	—
135	C₂²	C₂×#106	1	1	yes	characteristic
136	C₂²	C₂×#106	1	1	yes	characteristic
137	C₂²	C₂²×SD₁₆	1	1	yes	characteristic
138	C₄	—	8	4	no	—
139	C₄	C₂×#106	1	1	yes	characteristic
140	C₄	C₂×#106	1	2	yes	—
141	C₄	C₂×#106	1	1	yes	characteristic
142	C₄	C₂×#106	1	2	yes	—
143	C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₄	C₂²×SD₁₆	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	2	1	no	—
147	C₂	—	2	2	no	—
148	C₂	—	2	2	no	—
149	C₂	#2155	1	2	yes	direct product factor
150	C₂	C₂²×#106	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#106
commutator	C₈	C₂³×C₄
Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂²	C₂×#106
prime radical	C₂⁴	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #2155	C₂	direct (N × H)
5. C₂×#365	C₂	semidirect
9. C₂×M₃₂	C₄	semidirect
10. C₂×M₃₂	C₄	semidirect
13. C₂²×C₁₆	C₄	semidirect
22. #365	C₂²	semidirect
45. M₃₂	C₂×C₄	semidirect
48. C₂×C₁₆	C₂×C₄	semidirect
149. C₂	#2155	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×M₃₂	2
C₂²×#108	2
C₂×#427	2
#2155	2
C₂×#365	2

Subgroups — C2×#2155 #33403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2155 #33403