Subgroups — C₂×#1953 #21183

Order 256.

429 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#430	C₂	1	2	yes	maximal
2	C₂²×#108	C₂	1	1	yes	maximal, characteristic
3	C₂²×#108	C₂	1	1	yes	maximal, characteristic
4	C₂×#405	C₂	1	1	yes	maximal, characteristic
5	#1953	C₂	1	8	yes	maximal, direct product factor
6	C₂×#344	C₂	1	2	yes	maximal
7	C₂²×M₁₆	C₂²	1	2	yes	—
8	C₂²×M₁₆	C₂²	1	2	yes	—
9	C₂²×M₁₆	C₂²	1	1	yes	characteristic
10	C₂³×C₈	C₂²	1	1	yes	characteristic
11	C₂×#117	C₂²	1	1	yes	characteristic
12	#430	—	2	4	no	—
13	#430	C₂²	1	8	yes	—
14	C₂×#108	—	2	4	no	—
15	C₂×#108	—	2	1	no	—
16	C₂×#108	—	2	1	no	—
17	C₂×#108	—	2	2	no	—
18	C₂×#108	C₄	1	4	yes	complemented
19	C₂×#108	C₄	1	4	yes	complemented
20	C₂×#108	C₂²	1	4	yes	—
21	C₂×#108	C₂²	1	4	yes	—
22	#405	—	2	2	no	—
23	#405	C₂²	1	4	yes	—
24	C₂×#97	—	2	1	no	—
25	C₂×C₄×C₈	—	2	1	no	—
26	#344	C₂²	1	8	yes	—
27	C₂³×C₄	C₂³	1	1	yes	characteristic
28	C₂×M₁₆	—	2	1	no	—
29	C₂×M₁₆	—	2	2	no	—
30	C₂×M₁₆	—	2	2	no	—
31	C₂×M₁₆	—	2	2	no	—
32	C₂×M₁₆	—	2	1	no	—
33	C₂×M₁₆	—	2	1	no	—
34	C₂×M₁₆	—	2	2	no	—
35	C₂×M₁₆	—	2	2	no	—
36	C₂×M₁₆	—	2	2	no	—
37	C₂×M₁₆	—	2	2	no	—
38	C₂×M₁₆	—	2	2	no	—
39	C₂×M₁₆	C₂×C₄	1	4	yes	—
40	C₂×M₁₆	C₂×C₄	1	4	yes	—
41	C₂×M₁₆	D₄	1	2	yes	—
42	C₂×M₁₆	D₄	1	2	yes	—
43	C₂×M₁₆	C₂³	1	4	yes	—
44	C₂×M₁₆	C₂³	1	4	yes	—
45	C₂×M₁₆	C₂³	1	2	yes	—
46	C₂²×C₈	—	2	1	no	—
47	C₂²×C₈	—	2	2	no	—
48	C₂²×C₈	—	2	2	no	—
49	C₂²×C₈	C₂×C₄	1	2	yes	—
50	C₂²×C₈	C₂×C₄	1	2	yes	—
51	C₂²×C₈	D₄	1	1	yes	characteristic
52	C₂²×C₈	D₄	1	1	yes	characteristic
53	C₂²×C₈	D₄	1	2	yes	—
54	C₂²×C₈	Q₈	1	2	yes	—
55	C₂²×C₈	Q₈	1	2	yes	—
56	C₂²×C₈	C₂³	1	2	yes	—
57	#117	—	2	1	no	—
58	#117	—	2	1	no	—
59	#117	Q₈	1	2	yes	—
60	#117	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄	—	2	1	no	—
62	C₂×C₂²⋊C₄	Q₈	1	2	yes	—
63	C₂×C₄²	—	2	1	no	—
64	#108	—	2	16	no	—
65	#108	—	2	4	no	—
66	#108	—	2	4	no	—
67	#108	C₂×C₄	1	16	yes	complemented
68	#97	—	2	4	no	—
69	C₄×C₈	—	2	4	no	—
70	C₂⁴	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄	—	2	2	no	—
72	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
81	C₂²×C₄	#41	1	1	yes	characteristic
82	C₂²×C₄	#41	1	1	yes	characteristic
83	C₂²×C₄	#41	1	1	yes	characteristic
84	C₂²×C₄	#41	1	1	yes	characteristic
85	C₂²×C₄	C₂⁴	1	1	yes	characteristic
86	M₁₆	—	2	4	no	—
87	M₁₆	—	2	4	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	2	4	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	2	4	no	—
93	M₁₆	—	2	4	no	—
94	M₁₆	—	4	2	no	—
95	M₁₆	—	4	2	no	—
96	M₁₆	C₂²×C₄	1	8	yes	—
97	M₁₆	C₂×D₄	1	4	yes	—
98	C₂×C₈	—	2	4	no	—
99	C₂×C₈	—	2	4	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	2	4	no	—
104	C₂×C₈	—	2	4	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	C₄⋊C₄	1	2	yes	—
108	C₂×C₈	C₄⋊C₄	1	2	yes	—
109	C₂×C₈	C₄⋊C₄	1	2	yes	—
110	C₂×C₈	C₄⋊C₄	1	2	yes	—
111	C₂×C₈	C₄⋊C₄	1	4	yes	—
112	C₂×C₈	C₄⋊C₄	1	4	yes	—
113	C₂×C₈	C₂²×C₄	1	4	yes	—
114	C₂×C₈	C₂×D₄	1	2	yes	—
115	C₂×C₈	C₂×D₄	1	2	yes	—
116	C₂×C₈	C₂×Q₈	1	4	yes	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	C₂×Q₈	1	4	yes	—
121	C₄²	—	2	2	no	—
122	C₄²	—	2	2	no	—
123	C₂³	—	4	2	no	—
124	C₂³	C₄×Q₈	1	1	yes	characteristic
125	C₂³	C₄×Q₈	1	1	yes	characteristic
126	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂³	C₄⋊Q₈	1	1	yes	characteristic
129	C₂³	C₄⋊Q₈	1	1	yes	characteristic
130	C₂³	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	4	4	no	—
133	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
142	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
143	C₂×C₄	#121	1	1	yes	characteristic
144	C₂×C₄	#121	1	1	yes	characteristic
145	C₂×C₄	#121	1	1	yes	characteristic
146	C₂×C₄	#121	1	1	yes	characteristic
147	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
148	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
149	C₂×C₄	#125	1	1	yes	characteristic
150	C₂×C₄	#125	1	1	yes	characteristic
151	C₂×C₄	#125	1	1	yes	characteristic
152	C₂×C₄	#125	1	1	yes	characteristic
153	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
154	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
155	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
159	C₂×C₄	C₂×#41	1	1	yes	characteristic
160	C₂×C₄	C₂×#41	1	1	yes	characteristic
161	C₈	—	2	8	no	—
162	C₈	—	2	4	no	—
163	C₈	—	4	4	no	—
164	C₈	C₂×C₄⋊C₄	1	4	yes	—
165	C₈	C₂×C₄⋊C₄	1	4	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	—	2	2	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	4	1	no	—
170	C₂²	—	4	1	no	—
171	C₂²	—	4	1	no	—
172	C₂²	—	4	1	no	—
173	C₂²	#389	1	1	yes	characteristic
174	C₂²	#389	1	1	yes	characteristic
175	C₂²	#389	1	1	yes	characteristic
176	C₂²	#389	1	1	yes	characteristic
177	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
178	C₂²	C₂×C₂²⋊Q₈	1	1	yes	characteristic
179	C₂²	C₂×C₄⋊Q₈	1	1	yes	characteristic
180	C₄	—	4	4	no	—
181	C₄	#389	1	1	yes	characteristic
182	C₄	#389	1	1	yes	characteristic
183	C₄	#389	1	1	yes	characteristic
184	C₄	#389	1	1	yes	characteristic
185	C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
186	C₄	C₂×C₄×D₄	1	1	yes	characteristic
187	C₄	C₂×#121	1	1	yes	characteristic
188	C₄	C₂×#125	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#1953	1	2	yes	direct product factor
196	C₂	C₂×#389	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#121
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	#389
prime radical	C₂⁴	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
5. #1953	C₂	direct (N × H)
18. C₂×#108	C₄	semidirect
19. C₂×#108	C₄	semidirect
67. #108	C₂×C₄	semidirect
195. C₂	#1953	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#430	2
C₂²×#108	2
C₂×#405	2
#1953	2
C₂×#344	2

Subgroups — C2×#1953 #21183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1953 #21183