Finite Group Databasebeta

← #135×D₂₃ #13093195

Subgroups — #135×D₂₃ #13093195

Order 1472.

266 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 101 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	#41×D₂₃	C₂	1	2	yes	maximal, complemented
2	C₂₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	#10625121	C₂	1	2	yes	maximal, complemented
4	#10625090	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₁₆×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#10625087	C₂	1	2	yes	maximal, complemented
7	SD₁₆×D₂₃	C₂	1	2	yes	maximal, complemented
8	#10625083	C₂	1	1	yes	maximal, characteristic, complemented
9	D₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#10625075	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×C₂₃	C₂²	1	2	yes	complemented
13	#67292	C₂²	1	2	yes	complemented
14	Q₈×D₂₃	C₂²	1	2	yes	—
15	#67290	—	2	2	no	—
16	#67290	C₂²	1	2	yes	complemented
17	D₄×D₂₃	—	2	2	no	—
18	D₄×D₂₃	C₂²	1	2	yes	—
19	#67288	—	2	2	no	—
20	#67288	C₂²	1	2	yes	complemented
21	C₂×C₄×D₂₃	—	2	2	no	—
22	C₂×C₄×D₂₃	C₂²	1	1	yes	characteristic
23	Q₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
24	SD₁₆×C₂₃	C₂²	1	2	yes	complemented
25	D₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
27	#67275	C₂²	1	2	yes	complemented
28	#67274	C₂²	1	2	yes	complemented
29	#67273	C₂²	1	2	yes	complemented
30	#67272	C₂²	1	2	yes	complemented
31	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
32	Dic₉₂	C₂²	1	1	yes	characteristic, complemented
33	D₁₈₄	C₂²	1	1	yes	characteristic, complemented
34	#67263	C₂²	1	2	yes	complemented
35	C₈×D₂₃	C₂²	1	1	yes	characteristic
36	C₈×D₂₃	C₂²	1	1	yes	characteristic
37	C₈×D₂₃	C₂²	1	1	yes	characteristic, complemented
38	C₈×D₂₃	C₂²	1	1	yes	characteristic, complemented
39	C₂²×D₂₃	—	2	2	no	—
40	C₂²×D₂₃	D₄	1	1	yes	characteristic
41	Q₈×C₂₃	C₂³	1	2	yes	—
42	D₄×C₂₃	—	2	2	no	—
43	D₄×C₂₃	C₂³	1	2	yes	—
44	C₂×C₄×C₂₃	—	2	2	no	—
45	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂²⋊D₂₃	—	2	2	no	—
47	C₂²⋊D₂₃	—	2	2	no	—
48	C₂²⋊D₂₃	—	2	2	no	—
49	C₂×Dic₂₃	—	2	2	no	—
50	C₂×Dic₂₃	D₄	1	1	yes	characteristic
51	D₉₂	—	2	2	no	—
52	D₉₂	C₂³	1	2	yes	—
53	C₄×D₂₃	—	2	2	no	—
54	C₄×D₂₃	—	2	2	no	—
55	C₄×D₂₃	—	2	2	no	—
56	C₄×D₂₃	D₄	1	1	yes	characteristic
57	C₄×D₂₃	D₄	1	1	yes	characteristic
58	C₄×D₂₃	C₂³	1	1	yes	characteristic
59	C₄×D₂₃	C₂³	1	1	yes	characteristic
60	Dic₄₆	—	2	2	no	—
61	Dic₄₆	C₂³	1	2	yes	—
62	C₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₂₃⋊C₈	C₂³	1	1	yes	characteristic
65	C₂₃⋊C₈	C₂³	1	1	yes	characteristic
66	C₂²×C₂₃	—	2	2	no	—
67	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
68	C₂×D₂₃	—	2	2	no	—
69	C₂×D₂₃	—	2	2	no	—
70	C₂×D₂₃	—	4	4	no	—
71	C₂×D₂₃	C₂×D₄	1	1	yes	characteristic
72	C₂×D₂₃	C₂×D₄	1	1	yes	characteristic
73	C₄×C₂₃	—	2	2	no	—
74	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
75	C₄×C₂₃	C₂⁴	1	1	yes	characteristic
76	Dic₂₃	—	2	2	no	—
77	Dic₂₃	C₂×D₄	1	1	yes	characteristic
78	Dic₂₃	C₂×D₄	1	1	yes	characteristic
79	C₂×#135	—	23	1	no	maximal, Sylow
80	C₂×C₂₃	—	2	1	no	—
81	C₂×C₂₃	—	4	2	no	—
82	C₂×C₂₃	C₂²×D₄	1	1	yes	characteristic
83	D₂₃	—	2	1	no	—
84	D₂₃	—	4	2	no	—
85	D₂₃	#135	1	2	yes	direct product factor
86	C₂×#41	—	23	2	no	—
87	#135	—	23	2	no	—
88	#135	—	23	1	no	—
89	#135	—	23	2	no	—
90	#135	—	23	1	no	—
91	#135	—	23	1	no	—
92	#135	D₂₃	1	1	yes	characteristic, direct product factor
93	C₂×Q₁₆	—	23	1	no	—
94	C₂×SD₁₆	—	23	2	no	—
95	C₂×D₈	—	23	1	no	—
96	C₂²×C₈	—	23	1	no	—
97	C₂₃	C₂×#135	1	1	yes	characteristic, complemented, Sylow
98	#41	—	23	2	no	—
99	#41	—	23	2	no	—
100	#41	—	23	2	no	—
101	#41	—	46	2	no	—
102	#41	—	46	2	no	—
103	#41	C₂×D₂₃	1	2	yes	complemented
104	C₂×Q₈	—	23	2	no	—
105	C₂×D₄	—	23	2	no	—
106	C₂×D₄	—	46	2	no	—
107	C₂²×C₄	—	23	1	no	—
108	C₂²×C₄	—	46	2	no	—
109	Q₁₆	—	23	2	no	—
110	Q₁₆	—	23	1	no	—
111	Q₁₆	C₂×D₂₃	1	1	yes	characteristic, complemented
112	SD₁₆	—	23	2	no	—
113	SD₁₆	—	23	2	no	—
114	SD₁₆	—	23	2	no	—
115	SD₁₆	C₂×D₂₃	1	2	yes	complemented
116	D₈	—	23	2	no	—
117	D₈	—	23	1	no	—
118	D₈	C₂×D₂₃	1	1	yes	characteristic, complemented
119	C₂×C₈	—	23	1	no	—
120	C₂×C₈	—	23	1	no	—
121	C₂×C₈	—	23	1	no	—
122	C₂×C₈	—	23	1	no	—
123	C₂×C₈	—	23	1	no	—
124	C₂×C₈	C₂×D₂₃	1	1	yes	characteristic, complemented
125	C₂³	—	23	1	no	—
126	C₂³	—	46	2	no	—
127	Q₈	—	23	2	no	—
128	Q₈	—	46	2	no	—
129	Q₈	C₂²×D₂₃	1	2	yes	—
130	D₄	—	2	2	no	—
131	D₄	—	23	2	no	—
132	D₄	—	46	2	no	—
133	D₄	—	46	2	no	—
134	D₄	—	46	2	no	—
135	D₄	—	46	2	no	—
136	D₄	C₂²×D₂₃	1	2	yes	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	23	1	no	—
139	C₂×C₄	—	23	1	no	—
140	C₂×C₄	—	23	1	no	—
141	C₂×C₄	—	23	1	no	—
142	C₂×C₄	—	23	1	no	—
143	C₂×C₄	—	46	2	no	—
144	C₂×C₄	—	46	2	no	—
145	C₂×C₄	—	46	2	no	—
146	C₂×C₄	—	46	2	no	—
147	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
148	C₈	—	23	1	no	—
149	C₈	—	23	1	no	—
150	C₈	C₂²×D₂₃	1	1	yes	characteristic
151	C₈	C₂²×D₂₃	1	1	yes	characteristic
152	C₂²	—	2	2	no	—
153	C₂²	—	23	1	no	—
154	C₂²	—	23	1	no	—
155	C₂²	—	46	2	no	—
156	C₂²	—	46	2	no	—
157	C₂²	—	92	4	no	—
158	C₂²	D₄×D₂₃	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	23	1	no	—
161	C₄	—	23	1	no	—
162	C₄	—	46	2	no	—
163	C₄	D₄×D₂₃	1	1	yes	characteristic
164	C₄	C₂³×D₂₃	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	4	2	no	—
167	C₂	—	23	2	no	—
168	C₂	—	46	1	no	—
169	C₂	—	92	2	no	—
170	C₂	C₂×D₄×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₂₃
commutator, cosocle	C₄×C₂₃	C₂⁴
Frattini	C₄	C₂³×D₂₃
Fitting	C₂₃×#135	C₂
socle	C₂×C₂₃	C₂²×D₄
hypercenter	#135	D₂₃
hypocenter	C₂₃	C₂×#135
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #41×D₂₃	C₂	semidirect
2. C₂₃×#135	C₂	semidirect
3. #10625121	C₂	semidirect
4. #10625090	C₂	semidirect
5. Q₁₆×D₂₃	C₂	semidirect
6. #10625087	C₂	semidirect
7. SD₁₆×D₂₃	C₂	semidirect
8. #10625083	C₂	semidirect
9. D₈×D₂₃	C₂	semidirect
10. #10625075	C₂	semidirect
11. C₂×C₈×D₂₃	C₂	semidirect
12. #41×C₂₃	C₂²	semidirect
13. #67292	C₂²	semidirect
16. #67290	C₂²	semidirect
20. #67288	C₂²	semidirect
23. Q₁₆×C₂₃	C₂²	semidirect
24. SD₁₆×C₂₃	C₂²	semidirect
25. D₈×C₂₃	C₂²	semidirect
26. C₂×C₈×C₂₃	C₂²	semidirect
27. #67275	C₂²	semidirect
28. #67274	C₂²	semidirect
29. #67273	C₂²	semidirect
30. #67272	C₂²	semidirect
31. C₂×C₂₃⋊C₈	C₂²	semidirect
32. Dic₉₂	C₂²	semidirect
33. D₁₈₄	C₂²	semidirect
34. #67263	C₂²	semidirect
37. C₈×D₂₃	C₂²	semidirect
38. C₈×D₂₃	C₂²	semidirect
85. D₂₃	#135	direct (N × H)
92. #135	D₂₃	direct (N × H)
97. C₂₃	C₂×#135	semidirect
103. #41	C₂×D₂₃	semidirect
111. Q₁₆	C₂×D₂₃	semidirect
115. SD₁₆	C₂×D₂₃	semidirect
118. D₈	C₂×D₂₃	semidirect
124. C₂×C₈	C₂×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
#41×D₂₃	2
C₂₃×#135	2
#10625121	2
#10625090	2
Q₁₆×D₂₃	2
#10625087	2
SD₁₆×D₂₃	2
#10625083	2
D₈×D₂₃	2
#10625075	2
C₂×C₈×D₂₃	2
C₂×#135	23