Subgroups — C₂×C₄⋊Q₈×D₂₃ #15686670

Order 2944.

1624 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×D₂₃	C₂	1	2	yes	maximal
2	C₂×#10625040	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10625059	C₂	1	4	yes	maximal, complemented
4	C₂×#10625115	C₂	1	2	yes	maximal, complemented
5	C₄⋊Q₈×D₂₃	C₂	1	16	yes	maximal, direct product factor
6	C₂×C₄⋊C₄×D₂₃	C₂	1	4	yes	maximal
7	C₂×C₄²×D₂₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
10	C₂×Q₈×D₂₃	—	2	16	no	—
11	C₂×Q₈×D₂₃	C₂²	1	16	yes	—
12	C₂²×C₄×D₂₃	C₂²	1	4	yes	—
13	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
15	C₂²×Dic₄₆	C₂²	1	2	yes	—
16	C₂²×Dic₄₆	C₂²	1	4	yes	—
17	C₂₃×C₄⋊Q₈	C₂²	1	8	yes	complemented
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
19	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
20	#10625115	C₂²	1	16	yes	complemented
21	C₂×C₄⋊Dic₂₃	C₂²	1	4	yes	—
22	C₂×Dic₂₃⋊C₄	C₂²	1	8	yes	—
23	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
25	C₄⋊C₄×D₂₃	C₂²	1	32	yes	—
26	#10625059	C₂²	1	32	yes	complemented
27	C₄²×D₂₃	C₂²	1	8	yes	—
28	#10625040	C₂²	1	8	yes	complemented
29	C₂³×D₂₃	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₂₃	—	2	8	no	—
31	C₂×Q₈×C₂₃	C₂³	1	8	yes	—
32	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
33	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
35	C₂²×Dic₂₃	C₂³	1	4	yes	—
36	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
37	C₂²×Dic₂₃	C₂³	1	2	yes	—
38	Q₈×D₂₃	—	2	64	no	—
39	C₂×C₄×D₂₃	—	2	16	no	—
40	C₂×C₄×D₂₃	D₄	1	8	yes	—
41	C₂×C₄×D₂₃	Q₈	1	16	yes	complemented
42	C₂×C₄×D₂₃	C₂³	1	16	yes	—
43	C₂×C₄×D₂₃	C₂³	1	4	yes	—
44	C₂×C₄×D₂₃	C₂³	1	8	yes	—
45	C₂×Dic₄₆	—	2	8	no	—
46	C₂×Dic₄₆	—	2	16	no	—
47	C₂×Dic₄₆	C₂³	1	8	yes	—
48	C₂×Dic₄₆	C₂³	1	16	yes	—
49	C₄⋊C₄×C₂₃	C₂³	1	16	yes	—
50	C₄²×C₂₃	C₂³	1	4	yes	—
51	C₄⋊Dic₂₃	C₂³	1	16	yes	—
52	Dic₂₃⋊C₄	C₂³	1	32	yes	—
53	C₄×Dic₂₃	C₂³	1	8	yes	—
54	C₄×Dic₂₃	C₂³	1	4	yes	—
55	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
56	C₂²×D₂₃	C₂×D₄	1	4	yes	—
57	C₂²×D₂₃	C₂×Q₈	1	8	yes	—
58	C₂²×D₂₃	C₂⁴	1	2	yes	—
59	Q₈×C₂₃	—	2	16	no	—
60	C₂×C₄×C₂₃	—	2	8	no	—
61	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
62	C₂×C₄×C₂₃	C₂×Q₈	1	8	yes	complemented
63	C₂×C₄×C₂₃	C₂⁴	1	8	yes	—
64	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
65	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
66	C₂×Dic₂₃	—	2	8	no	—
67	C₂×Dic₂₃	C₂×D₄	1	4	yes	—
68	C₂×Dic₂₃	C₂×Q₈	1	8	yes	complemented
69	C₂×Dic₂₃	C₂⁴	1	8	yes	—
70	C₂×Dic₂₃	C₂⁴	1	2	yes	—
71	C₂×Dic₂₃	C₂⁴	1	4	yes	—
72	C₄×D₂₃	—	2	32	no	—
73	C₄×D₂₃	C₂×D₄	1	16	yes	—
74	C₄×D₂₃	C₂×Q₈	1	32	yes	complemented
75	Dic₄₆	—	2	16	no	—
76	Dic₄₆	—	2	32	no	—
77	C₂²×C₄⋊Q₈	—	23	1	no	maximal, Sylow
78	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
79	C₂²×C₂₃	C₂²×Q₈	1	4	yes	—
80	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
81	C₂×D₂₃	C₄⋊Q₈	1	16	yes	direct product factor
82	C₂×D₂₃	C₂²×D₄	1	4	yes	—
83	C₂×D₂₃	C₂²×Q₈	1	8	yes	—
84	C₄×C₂₃	—	2	8	no	—
85	C₄×C₂₃	C₂²×D₄	1	4	yes	—
86	C₄×C₂₃	C₂²×Q₈	1	8	yes	complemented
87	Dic₂₃	—	2	8	no	—
88	Dic₂₃	C₂²×D₄	1	4	yes	—
89	Dic₂₃	C₂²×Q₈	1	8	yes	complemented
90	C₂³×Q₈	—	23	2	no	—
91	C₂×C₄⋊Q₈	—	23	2	no	—
92	C₂×C₄⋊Q₈	—	23	16	no	—
93	C₂×C₄⋊Q₈	—	23	4	no	—
94	C₂×C₄⋊Q₈	—	23	1	no	—
95	C₂×C₄⋊Q₈	D₂₃	1	1	yes	characteristic, direct product factor
96	C₂²×C₄⋊C₄	—	23	4	no	—
97	C₂²×C₄²	—	23	1	no	—
98	C₂×C₂₃	C₂×C₄⋊Q₈	1	4	yes	complemented
99	C₂×C₂₃	C₂³×D₄	1	1	yes	characteristic
100	C₂×C₂₃	C₂³×Q₈	1	2	yes	—
101	D₂₃	C₂×C₄⋊Q₈	1	8	yes	direct product factor
102	C₂²×Q₈	—	23	16	no	—
103	C₂²×Q₈	—	23	4	no	—
104	C₂²×Q₈	—	23	2	no	—
105	C₂²×Q₈	—	46	16	no	—
106	C₂²×Q₈	C₂×D₂₃	1	2	yes	—
107	C₂³×C₄	—	23	2	no	—
108	C₂³×C₄	—	23	1	no	—
109	C₂³×C₄	—	23	4	no	—
110	C₄⋊Q₈	—	23	16	no	—
111	C₄⋊Q₈	—	23	32	no	—
112	C₄⋊Q₈	—	23	8	no	—
113	C₄⋊Q₈	C₂×D₂₃	1	8	yes	direct product factor
114	C₂×C₄⋊C₄	—	23	8	no	—
115	C₂×C₄⋊C₄	—	23	4	no	—
116	C₂×C₄⋊C₄	—	23	32	no	—
117	C₂×C₄⋊C₄	C₂×D₂₃	1	4	yes	—
118	C₂×C₄²	—	23	1	no	—
119	C₂×C₄²	—	23	2	no	—
120	C₂×C₄²	—	23	8	no	—
121	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
122	C₂₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
123	C₂⁴	—	23	1	no	—
124	C₂×Q₈	—	2	8	no	—
125	C₂×Q₈	—	23	16	no	—
126	C₂×Q₈	—	23	8	no	—
127	C₂×Q₈	—	46	16	no	—
128	C₂×Q₈	—	46	64	no	—
129	C₂×Q₈	—	46	8	no	—
130	C₂×Q₈	C₂²×D₂₃	1	8	yes	—
131	C₂²×C₄	—	23	2	no	—
132	C₂²×C₄	—	23	1	no	—
133	C₂²×C₄	—	23	8	no	—
134	C₂²×C₄	—	23	16	no	—
135	C₂²×C₄	—	23	8	no	—
136	C₂²×C₄	—	23	4	no	—
137	C₂²×C₄	—	23	4	no	—
138	C₂²×C₄	—	23	16	no	—
139	C₂²×C₄	—	46	16	no	—
140	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
141	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
142	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
143	C₄⋊C₄	—	23	32	no	—
144	C₄⋊C₄	—	23	16	no	—
145	C₄⋊C₄	C₂²×D₂₃	1	16	yes	—
146	C₄²	—	23	4	no	—
147	C₄²	—	23	8	no	—
148	C₄²	C₂²×D₂₃	1	4	yes	—
149	C₂³	—	23	4	no	—
150	C₂³	—	23	8	no	—
151	C₂³	—	23	2	no	—
152	C₂³	C₂³×D₂₃	1	1	yes	characteristic
153	Q₈	—	2	16	no	—
154	Q₈	—	46	32	no	—
155	Q₈	—	46	16	no	—
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	23	4	no	—
158	C₂×C₄	—	23	8	no	—
159	C₂×C₄	—	23	4	no	—
160	C₂×C₄	—	23	2	no	—
161	C₂×C₄	—	23	16	no	—
162	C₂×C₄	—	23	32	no	—
163	C₂×C₄	—	23	8	no	—
164	C₂×C₄	—	46	32	no	—
165	C₂×C₄	—	46	8	no	—
166	C₂×C₄	D₄×D₂₃	1	4	yes	—
167	C₂×C₄	Q₈×D₂₃	1	8	yes	complemented
168	C₂×C₄	C₂³×D₂₃	1	8	yes	—
169	C₂×C₄	C₂³×D₂₃	1	2	yes	—
170	C₂×C₄	C₂³×D₂₃	1	4	yes	—
171	C₂²	—	23	16	no	—
172	C₂²	—	23	4	no	—
173	C₂²	—	23	8	no	—
174	C₂²	C₂×D₄×D₂₃	1	2	yes	—
175	C₂²	C₂×Q₈×D₂₃	1	4	yes	—
176	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
177	C₄	—	2	8	no	—
178	C₄	—	23	4	no	—
179	C₄	—	23	8	no	—
180	C₄	—	46	8	no	—
181	C₄	C₂×D₄×D₂₃	1	4	yes	—
182	C₄	C₂×Q₈×D₂₃	1	8	yes	complemented
183	C₂	—	23	8	no	—
184	C₂	C₄⋊Q₈×D₂₃	1	4	yes	direct product factor
185	C₂	C₂²×D₄×D₂₃	1	1	yes	characteristic
186	C₂	C₂²×Q₈×D₂₃	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₂₃
commutator, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂×C₂₃×C₄⋊Q₈	C₂
socle	C₂³×C₂₃	C₂⁴
prime radical	C₂³×D₂₃	C₂³
hypercenter	C₂×C₄⋊Q₈	D₂₃
hypocenter	C₂₃	C₂²×C₄⋊Q₈
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#10625040	C₂	semidirect
3. C₂×#10625059	C₂	semidirect
4. C₂×#10625115	C₂	semidirect
5. C₄⋊Q₈×D₂₃	C₂	direct (N × H)
8. C₂×C₂₃×C₄⋊Q₈	C₂	semidirect
17. C₂₃×C₄⋊Q₈	C₂²	semidirect
20. #10625115	C₂²	semidirect
26. #10625059	C₂²	semidirect
28. #10625040	C₂²	semidirect
41. C₂×C₄×D₂₃	Q₈	semidirect
62. C₂×C₄×C₂₃	C₂×Q₈	semidirect
68. C₂×Dic₂₃	C₂×Q₈	semidirect
74. C₄×D₂₃	C₂×Q₈	semidirect
81. C₂×D₂₃	C₄⋊Q₈	direct (N × H)
86. C₄×C₂₃	C₂²×Q₈	semidirect
89. Dic₂₃	C₂²×Q₈	semidirect
95. C₂×C₄⋊Q₈	D₂₃	direct (N × H)
98. C₂×C₂₃	C₂×C₄⋊Q₈	semidirect
101. D₂₃	C₂×C₄⋊Q₈	direct (N × H)
113. C₄⋊Q₈	C₂×D₂₃	direct (N × H)
122. C₂₃	C₂²×C₄⋊Q₈	semidirect
167. C₂×C₄	Q₈×D₂₃	semidirect
182. C₄	C₂×Q₈×D₂₃	semidirect
184. C₂	C₄⋊Q₈×D₂₃	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×Q₈×D₂₃	2
C₂×#10625040	2
C₂×#10625059	2
C₂×#10625115	2
C₄⋊Q₈×D₂₃	2
C₂×C₄⋊C₄×D₂₃	2
C₂×C₄²×D₂₃	2
C₂×C₂₃×C₄⋊Q₈	2
C₂²×C₄⋊Q₈	23

Subgroups — C2×C4⋊Q8×D23 #15686670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊Q₈×D₂₃ #15686670