Subgroups — C₂×#13092967 #15686680

Order 2944.

1728 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#67288	C₂	1	2	yes	maximal, complemented
2	#13092967	C₂	1	16	yes	maximal, direct product factor
3	C₂×#10625040	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#10625045	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10625046	C₂	1	2	yes	maximal, complemented
6	C₂×#10625101	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×D₉₂	C₂	1	4	yes	maximal, complemented
8	C₂²×C₄²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
10	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×C₂²⋊D₂₃	C₂²	1	4	yes	—
12	C₂×#67288	—	2	16	no	—
13	C₂×#67288	C₂²	1	16	yes	complemented
14	C₂²×D₉₂	C₂²	1	4	yes	—
15	C₂²×D₉₂	C₂²	1	2	yes	—
16	C₂²×C₄×D₂₃	C₂²	1	4	yes	—
17	C₂²×Dic₄₆	C₂²	1	2	yes	—
18	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₂₃	C₂²	1	8	yes	complemented
22	#10625101	C₂²	1	32	yes	complemented
23	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	8	yes	—
24	C₂×C₄⋊Dic₂₃	C₂²	1	4	yes	—
25	#10625046	C₂²	1	16	yes	complemented
26	#10625045	C₂²	1	8	yes	complemented
27	C₄×D₉₂	C₂²	1	32	yes	complemented
28	#10625040	C₂²	1	8	yes	complemented
29	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
30	C₂³×D₂₃	C₂³	1	4	yes	—
31	C₂²×C₄×C₂₃	—	2	8	no	—
32	C₂²×C₄×C₂₃	D₄	1	8	yes	complemented
33	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
34	C₂²×C₄×C₂₃	C₂³	1	8	yes	—
35	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
37	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
39	C₂×C₂²⋊D₂₃	—	2	16	no	—
40	C₂×C₂²⋊D₂₃	C₂³	1	16	yes	—
41	C₂²×Dic₂₃	C₂³	1	4	yes	—
42	#67288	—	2	64	no	—
43	C₂×D₉₂	—	2	8	no	—
44	C₂×D₉₂	—	2	16	no	—
45	C₂×D₉₂	C₂³	1	16	yes	—
46	C₂×D₉₂	C₂³	1	8	yes	—
47	C₂×C₄×D₂₃	—	2	16	no	—
48	C₂×C₄×D₂₃	C₂³	1	16	yes	—
49	C₂×Dic₄₆	—	2	8	no	—
50	C₂×Dic₄₆	C₂³	1	8	yes	—
51	C₄²×C₂₃	C₂³	1	4	yes	—
52	C₄²×C₂₃	C₂³	1	8	yes	—
53	C₄²×C₂₃	C₂³	1	4	yes	—
54	(C₂×D₂₃)⋊C₄	C₂³	1	32	yes	—
55	C₄⋊Dic₂₃	C₂³	1	16	yes	—
56	C₂³×C₂₃	—	2	4	no	—
57	C₂³×C₂₃	C₂×D₄	1	4	yes	—
58	C₂³×C₂₃	C₂⁴	1	2	yes	—
59	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
60	C₂²×D₂₃	—	2	8	no	—
61	C₂²×D₂₃	—	2	16	no	—
62	C₂²×D₂₃	C₂⁴	1	8	yes	—
63	C₂×C₄×C₂₃	—	2	4	no	—
64	C₂×C₄×C₂₃	—	2	16	no	—
65	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
66	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
67	C₂×C₄×C₂₃	C₂×D₄	1	16	yes	complemented
68	C₂×C₄×C₂₃	#41	1	8	yes	complemented
69	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
70	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
71	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
72	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
73	C₂²⋊D₂₃	—	2	32	no	—
74	C₂×Dic₂₃	—	2	8	no	—
75	C₂×Dic₂₃	C₂⁴	1	8	yes	—
76	D₉₂	—	2	16	no	—
77	D₉₂	—	2	32	no	—
78	C₄×D₂₃	—	2	32	no	—
79	Dic₄₆	—	2	16	no	—
80	C₂×#532	—	23	1	no	maximal, Sylow
81	C₂²×C₂₃	—	2	8	no	—
82	C₂²×C₂₃	—	2	4	no	—
83	C₂²×C₂₃	C₂²×D₄	1	4	yes	—
84	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
85	C₂²×C₂₃	C₂×#41	1	4	yes	—
86	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
87	C₂×D₂₃	—	2	8	no	—
88	C₂×D₂₃	—	2	16	no	—
89	C₂×D₂₃	—	4	16	no	—
90	C₄×C₂₃	—	2	4	no	—
91	C₄×C₂₃	C₂²×D₄	1	4	yes	—
92	C₄×C₂₃	C₂²×D₄	1	4	yes	—
93	C₄×C₂₃	C₂×#41	1	8	yes	complemented
94	Dic₂₃	—	2	8	no	—
95	C₂²×#41	—	23	2	no	—
96	#532	—	23	16	no	—
97	C₂×C₄⋊Q₈	—	23	1	no	—
98	C₂×#127	—	23	1	no	—
99	C₂×#124	—	23	2	no	—
100	C₂×#121	—	23	4	no	—
101	C₂×C₄×D₄	—	23	4	no	—
102	C₂²×C₄²	D₂₃	1	1	yes	characteristic, complemented
103	C₂×C₂₃	—	2	4	no	—
104	C₂×C₂₃	#532	1	4	yes	complemented
105	C₂×C₂₃	C₂³×D₄	1	1	yes	characteristic
106	C₂×C₂₃	C₂²×#41	1	2	yes	—
107	D₂₃	—	4	8	no	—
108	C₂×#41	—	23	16	no	—
109	C₂×#41	—	46	16	no	—
110	C₂²×Q₈	—	23	2	no	—
111	C₂²×D₄	—	23	4	no	—
112	C₂²×D₄	—	23	4	no	—
113	C₂²×D₄	—	23	2	no	—
114	C₂³×C₄	—	23	4	no	—
115	C₂³×C₄	C₂×D₂₃	1	2	yes	—
116	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
117	C₄⋊Q₈	—	23	8	no	—
118	#127	—	23	8	no	—
119	#124	—	23	16	no	—
120	#121	—	23	32	no	—
121	C₄×D₄	—	23	32	no	—
122	C₂×C₄⋊C₄	—	23	4	no	—
123	C₂×C₂²⋊C₄	—	23	8	no	—
124	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
125	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
126	C₂×C₄²	C₂×D₂₃	1	2	yes	—
127	C₂×C₄²	C₂×D₂₃	1	8	yes	complemented
128	C₂₃	C₂×#532	1	1	yes	characteristic, complemented, Sylow
129	C₂⁴	—	23	4	no	—
130	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
131	#41	—	46	64	no	—
132	C₂×Q₈	—	23	8	no	—
133	C₂×Q₈	—	46	8	no	—
134	C₂×D₄	—	23	16	no	—
135	C₂×D₄	—	23	16	no	—
136	C₂×D₄	—	23	8	no	—
137	C₂×D₄	—	46	16	no	—
138	C₂×D₄	—	46	8	no	—
139	C₂×D₄	—	46	16	no	—
140	C₂²×C₄	—	2	8	no	—
141	C₂²×C₄	—	23	4	no	—
142	C₂²×C₄	—	23	16	no	—
143	C₂²×C₄	—	46	16	no	—
144	C₂²×C₄	D₉₂	1	8	yes	complemented
145	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
146	C₂²×C₄	C₂²×D₂₃	1	8	yes	—
147	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
148	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
149	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
150	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
151	C₄⋊C₄	—	23	16	no	—
152	C₂²⋊C₄	—	23	32	no	—
153	C₄²	C₂²×D₂₃	1	8	yes	—
154	C₄²	C₂²×D₂₃	1	4	yes	—
155	C₄²	C₂²×D₂₃	1	4	yes	—
156	C₂³	—	2	4	no	—
157	C₂³	—	23	8	no	—
158	C₂³	—	46	8	no	—
159	C₂³	—	46	16	no	—
160	C₂³	C₂×D₉₂	1	4	yes	—
161	C₂³	C₂³×D₂₃	1	2	yes	—
162	C₂³	C₂³×D₂₃	1	1	yes	characteristic
163	Q₈	—	46	16	no	—
164	D₄	—	46	32	no	—
165	D₄	—	46	16	no	—
166	D₄	—	46	32	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	16	no	—
169	C₂×C₄	—	23	8	no	—
170	C₂×C₄	—	46	8	no	—
171	C₂×C₄	—	46	32	no	—
172	C₂×C₄	C₂×D₉₂	1	4	yes	—
173	C₂×C₄	C₂×D₉₂	1	4	yes	—
174	C₂×C₄	C₂×D₉₂	1	16	yes	complemented
175	C₂×C₄	#67288	1	8	yes	complemented
176	C₂×C₄	C₂³×D₂₃	1	4	yes	—
177	C₂×C₄	C₂³×D₂₃	1	2	yes	—
178	C₂×C₄	C₂³×D₂₃	1	2	yes	—
179	C₂×C₄	C₂³×D₂₃	1	4	yes	—
180	C₂²	—	2	8	no	—
181	C₂²	—	2	4	no	—
182	C₂²	—	46	8	no	—
183	C₂²	—	46	16	no	—
184	C₂²	—	92	16	no	—
185	C₂²	C₂²×D₉₂	1	4	yes	—
186	C₂²	C₂²×D₉₂	1	2	yes	—
187	C₂²	C₂×#67288	1	4	yes	—
188	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
189	C₄	—	2	4	no	—
190	C₄	—	46	8	no	—
191	C₄	C₂²×D₉₂	1	4	yes	—
192	C₄	C₂²×D₉₂	1	4	yes	—
193	C₄	C₂×#67288	1	8	yes	complemented
194	C₂	—	2	4	no	—
195	C₂	—	92	8	no	—
196	C₂	#13092967	1	4	yes	direct product factor
197	C₂	C₂³×D₉₂	1	1	yes	characteristic
198	C₂	C₂²×#67288	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₃
commutator, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂²×C₄²×C₂₃	C₂
socle	C₂³×C₂₃	C₂⁴
hypercenter	C₂²×C₄²	D₂₃
hypocenter	C₂₃	C₂×#532
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂²×#67288	C₂	semidirect
2. #13092967	C₂	direct (N × H)
3. C₂×#10625040	C₂	semidirect
4. C₂×#10625045	C₂	semidirect
5. C₂×#10625046	C₂	semidirect
6. C₂×#10625101	C₂	semidirect
7. C₂×C₄×D₉₂	C₂	semidirect
8. C₂²×C₄²×C₂₃	C₂	semidirect
13. C₂×#67288	C₂²	semidirect
21. C₂×C₄²×C₂₃	C₂²	semidirect
22. #10625101	C₂²	semidirect
25. #10625046	C₂²	semidirect
26. #10625045	C₂²	semidirect
27. C₄×D₉₂	C₂²	semidirect
28. #10625040	C₂²	semidirect
32. C₂²×C₄×C₂₃	D₄	semidirect
67. C₂×C₄×C₂₃	C₂×D₄	semidirect
68. C₂×C₄×C₂₃	#41	semidirect
93. C₄×C₂₃	C₂×#41	semidirect
102. C₂²×C₄²	D₂₃	semidirect
104. C₂×C₂₃	#532	semidirect
127. C₂×C₄²	C₂×D₂₃	semidirect
128. C₂₃	C₂×#532	semidirect
144. C₂²×C₄	D₉₂	semidirect
174. C₂×C₄	C₂×D₉₂	semidirect
175. C₂×C₄	#67288	semidirect
193. C₄	C₂×#67288	semidirect
196. C₂	#13092967	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×#67288	2
#13092967	2
C₂×#10625040	2
C₂×#10625045	2
C₂×#10625046	2
C₂×#10625101	2
C₂×C₄×D₉₂	2
C₂²×C₄²×C₂₃	2
C₂×#532	23

Subgroups — C2×#13092967 #15686680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13092967 #15686680