Finite Group Databasebeta

Subgroups — #13093227

Order 1472.

258 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10625161	C₂	1	1	yes	maximal, characteristic, complemented
2	#10625155	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
4	#10625121	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#67273	C₂	1	1	yes	maximal, characteristic, complemented
6	#10625086	C₂	1	2	yes	maximal, complemented
7	SD₁₆×D₂₃	C₂	1	2	yes	maximal, complemented
8	#10625083	C₂	1	2	yes	maximal, complemented
9	#10625082	C₂	1	2	yes	maximal, complemented
10	#10625080	C₂	1	1	yes	maximal, characteristic, complemented
11	#10625078	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×C₂₃	C₂²	1	1	yes	characteristic, complemented
13	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₂₃	—	2	2	no	—
15	Q₈×D₂₃	C₂²	1	2	yes	complemented
16	#67290	—	2	2	no	—
17	#67290	—	2	1	no	—
18	#67290	C₂²	1	2	yes	—
19	#67290	C₂²	1	2	yes	complemented
20	D₄×D₂₃	—	2	1	no	—
21	D₄×D₂₃	C₂²	1	2	yes	—
22	#67288	—	2	1	no	—
23	#67288	C₂²	1	1	yes	characteristic
24	#67288	C₂²	1	1	yes	characteristic
25	C₂×Dic₄₆	—	2	1	no	—
26	C₂×Dic₄₆	C₂²	1	1	yes	characteristic, complemented
27	SD₁₆×C₂₃	C₂²	1	2	yes	complemented
28	D₈×C₂₃	C₂²	1	2	yes	complemented
29	M₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
30	#67275	C₂²	1	1	yes	characteristic, complemented
31	#67274	C₂²	1	1	yes	characteristic, complemented
32	#67273	C₂²	1	2	yes	complemented
33	#67273	C₂²	1	1	yes	characteristic, complemented
34	#67273	C₂²	1	2	yes	complemented
35	#67272	C₂²	1	1	yes	characteristic, complemented
36	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
37	Dic₉₂	C₂²	1	2	yes	complemented
38	#67263	C₂²	1	2	yes	complemented
39	#67262	C₂²	1	2	yes	complemented
40	C₈×D₂₃	C₂²	1	2	yes	complemented
41	C₂³×C₂₃	—	2	1	no	—
42	C₂²×D₂₃	—	2	2	no	—
43	Q₈×C₂₃	C₂³	1	1	yes	characteristic
44	D₄×C₂₃	—	2	1	no	—
45	D₄×C₂₃	—	2	1	no	—
46	D₄×C₂₃	C₂³	1	1	yes	characteristic
47	D₄×C₂₃	C₂³	1	2	yes	—
48	C₂×C₄×C₂₃	—	2	1	no	—
49	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂²⋊D₂₃	—	2	1	no	—
51	C₂²⋊D₂₃	—	2	2	no	—
52	C₂²⋊D₂₃	—	2	2	no	—
53	C₂²⋊D₂₃	—	2	2	no	—
54	C₂²⋊D₂₃	D₄	1	2	yes	—
55	C₂×Dic₂₃	—	2	1	no	—
56	C₂×Dic₂₃	—	2	2	no	—
57	C₂×Dic₂₃	—	2	2	no	—
58	D₉₂	D₄	1	1	yes	characteristic
59	D₉₂	C₂³	1	1	yes	characteristic
60	C₄×D₂₃	—	2	2	no	—
61	C₄×D₂₃	—	2	1	no	—
62	C₄×D₂₃	C₂³	1	2	yes	—
63	Dic₄₆	—	2	1	no	—
64	Dic₄₆	—	2	2	no	—
65	Dic₄₆	D₄	1	1	yes	characteristic
66	Dic₄₆	C₂³	1	2	yes	—
67	Dic₄₆	C₂³	1	1	yes	characteristic
68	C₈×C₂₃	C₂³	1	2	yes	—
69	C₂₃⋊C₈	C₂³	1	1	yes	characteristic
70	C₂₃⋊C₈	C₂³	1	1	yes	characteristic
71	C₂²×C₂₃	—	2	1	no	—
72	C₂²×C₂₃	—	2	2	no	—
73	C₂²×C₂₃	—	8	1	no	—
74	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
75	C₂×D₂₃	—	2	1	no	—
76	C₂×D₂₃	—	8	2	no	—
77	C₂×D₂₃	C₂×D₄	1	2	yes	—
78	C₄×C₂₃	—	2	1	no	—
79	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₄×C₂₃	C₂⁴	1	1	yes	characteristic
81	Dic₂₃	—	2	2	no	—
82	Dic₂₃	—	2	1	no	—
83	Dic₂₃	C₂×D₄	1	2	yes	—
84	#577	—	23	1	no	maximal, Sylow
85	C₂×C₂₃	—	2	1	no	—
86	C₂×C₂₃	—	4	2	no	—
87	C₂×C₂₃	—	4	1	no	—
88	C₂×C₂₃	C₂²×D₄	1	1	yes	characteristic
89	D₂₃	—	2	2	no	—
90	D₂₃	—	4	1	no	—
91	2₋¹⁺⁴	—	23	1	no	—
92	H(5,𝔽₂)	—	23	1	no	—
93	#137	—	23	2	no	—
94	#137	—	23	1	no	—
95	AGL(1,ℤ/8)	—	23	2	no	—
96	AGL(1,ℤ/8)	D₂₃	1	1	yes	characteristic, complemented
97	#135	—	23	1	no	—
98	#135	—	23	2	no	—
99	C₂×SD₁₆	—	23	1	no	—
100	C₂×SD₁₆	—	23	2	no	—
101	#131	—	23	1	no	—
102	C₂₃	#577	1	1	yes	characteristic, complemented, Sylow
103	#41	—	23	1	no	—
104	#41	—	23	2	no	—
105	#41	—	23	2	no	—
106	#41	—	23	1	no	—
107	#41	—	46	1	no	—
108	#41	—	46	2	no	—
109	#41	—	46	1	no	—
110	#41	C₂×D₂₃	1	1	yes	characteristic, complemented
111	C₂×Q₈	—	23	2	no	—
112	C₂×Q₈	—	23	1	no	—
113	C₂×Q₈	—	46	1	no	—
114	C₂×D₄	—	23	2	no	—
115	C₂×D₄	—	46	1	no	—
116	C₂×D₄	—	46	2	no	—
117	C₂×D₄	C₂×D₂₃	1	1	yes	characteristic
118	Q₁₆	—	23	1	no	—
119	Q₁₆	—	23	2	no	—
120	SD₁₆	—	23	1	no	—
121	SD₁₆	—	23	2	no	—
122	SD₁₆	—	23	2	no	—
123	SD₁₆	—	23	1	no	—
124	SD₁₆	—	23	2	no	—
125	SD₁₆	C₂×D₂₃	1	2	yes	complemented
126	D₈	—	23	1	no	—
127	D₈	C₂×D₂₃	1	2	yes	complemented
128	M₁₆	—	23	2	no	—
129	M₁₆	C₂×D₂₃	1	1	yes	characteristic, complemented
130	C₂×C₈	—	23	1	no	—
131	C₂×C₈	—	23	2	no	—
132	C₂³	—	2	1	no	—
133	C₂³	—	46	2	no	—
134	Q₈	—	23	1	no	—
135	Q₈	—	23	2	no	—
136	Q₈	—	23	1	no	—
137	Q₈	—	46	2	no	—
138	Q₈	—	46	1	no	—
139	Q₈	C₂²×D₂₃	1	1	yes	characteristic
140	D₄	—	2	1	no	—
141	D₄	—	2	1	no	—
142	D₄	—	23	2	no	—
143	D₄	—	23	1	no	—
144	D₄	—	23	1	no	—
145	D₄	—	46	2	no	—
146	D₄	—	46	2	no	—
147	D₄	—	46	2	no	—
148	D₄	—	46	1	no	—
149	D₄	C₂²×D₂₃	1	1	yes	characteristic
150	D₄	C₂²×D₂₃	1	2	yes	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	23	2	no	—
153	C₂×C₄	—	46	2	no	—
154	C₂×C₄	—	46	2	no	—
155	C₂×C₄	—	46	2	no	—
156	C₂×C₄	—	46	1	no	—
157	C₂×C₄	—	46	1	no	—
158	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
159	C₈	—	23	1	no	—
160	C₈	—	23	1	no	—
161	C₈	C₂²×D₂₃	1	2	yes	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	8	1	no	—
165	C₂²	—	23	2	no	—
166	C₂²	—	46	1	no	—
167	C₂²	—	184	2	no	—
168	C₂²	D₄×D₂₃	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	23	2	no	—
171	C₄	—	46	1	no	—
172	C₄	—	46	2	no	—
173	C₄	D₄×D₂₃	1	1	yes	characteristic
174	C₄	C₂³×D₂₃	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	4	2	no	—
177	C₂	—	4	1	no	—
178	C₂	—	46	2	no	—
179	C₂	—	92	1	no	—
180	C₂	C₂×D₄×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×D₂₃
commutator, cosocle	C₄×C₂₃	C₂⁴
Frattini	C₄	C₂³×D₂₃
Fitting	C₂₃×AGL(1,ℤ/8)	C₂
socle	C₂×C₂₃	C₂²×D₄
hypercenter	AGL(1,ℤ/8)	D₂₃
hypocenter	C₂₃	#577
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. #10625161	C₂	semidirect
2. #10625155	C₂	semidirect
3. C₂₃×AGL(1,ℤ/8)	C₂	semidirect
4. #10625121	C₂	semidirect
5. C₂×#67273	C₂	semidirect
6. #10625086	C₂	semidirect
7. SD₁₆×D₂₃	C₂	semidirect
8. #10625083	C₂	semidirect
9. #10625082	C₂	semidirect
10. #10625080	C₂	semidirect
11. #10625078	C₂	semidirect
12. #41×C₂₃	C₂²	semidirect
15. Q₈×D₂₃	C₂²	semidirect
19. #67290	C₂²	semidirect
26. C₂×Dic₄₆	C₂²	semidirect
27. SD₁₆×C₂₃	C₂²	semidirect
28. D₈×C₂₃	C₂²	semidirect
29. M₁₆×C₂₃	C₂²	semidirect
30. #67275	C₂²	semidirect
31. #67274	C₂²	semidirect
32. #67273	C₂²	semidirect
33. #67273	C₂²	semidirect
34. #67273	C₂²	semidirect
35. #67272	C₂²	semidirect
36. C₂×C₂₃⋊C₈	C₂²	semidirect
37. Dic₉₂	C₂²	semidirect
38. #67263	C₂²	semidirect
39. #67262	C₂²	semidirect
40. C₈×D₂₃	C₂²	semidirect
96. AGL(1,ℤ/8)	D₂₃	semidirect
102. C₂₃	#577	semidirect
110. #41	C₂×D₂₃	semidirect
125. SD₁₆	C₂×D₂₃	semidirect
127. D₈	C₂×D₂₃	semidirect
129. M₁₆	C₂×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
#10625161	2
#10625155	2
C₂₃×AGL(1,ℤ/8)	2
#10625121	2
C₂×#67273	2
#10625086	2
SD₁₆×D₂₃	2
#10625083	2
#10625082	2
#10625080	2
#10625078	2
#577	23