Subgroups — C₂²×C₂₃×#395 #19182160

Order 5888.

6046 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊C₄×C₂₃	C₂	1	3	yes	maximal
3	C₂×C₂₃×#395	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₂₃×#95	C₂	1	3	yes	maximal
5	C₂³×Q₈×C₂₃	—	2	24	no	—
6	C₂³×Q₈×C₂₃	C₂²	1	12	yes	—
7	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₂₃	C₂²	1	3	yes	—
9	C₂⁴×C₄×C₂₃	C₂²	1	3	yes	—
10	C₂²×C₄⋊C₄×C₂₃	—	2	24	no	—
11	C₂²×C₄⋊C₄×C₂₃	C₂²	1	36	yes	—
12	C₂₃×#395	C₂²	1	64	yes	direct product factor
13	C₂×C₂₃×#95	C₂²	1	36	yes	—
14	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
15	C₂²×Q₈×C₂₃	—	2	128	no	—
16	C₂²×Q₈×C₂₃	—	2	144	no	—
17	C₂²×Q₈×C₂₃	C₂³	1	16	yes	—
18	C₂³×C₄×C₂₃	—	2	24	no	—
19	C₂³×C₄×C₂₃	—	2	12	no	—
20	C₂³×C₄×C₂₃	—	2	12	no	—
21	C₂³×C₄×C₂₃	—	2	12	no	—
22	C₂³×C₄×C₂₃	D₄	1	24	yes	—
23	C₂³×C₄×C₂₃	Q₈	1	24	yes	complemented
24	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
25	C₂³×C₄×C₂₃	C₂³	1	18	yes	—
26	C₂³×C₄×C₂₃	C₂³	1	18	yes	—
27	C₂×C₄⋊C₄×C₂₃	—	2	144	no	—
28	C₂×C₄⋊C₄×C₂₃	C₂³	1	48	yes	—
29	C₂₃×#95	C₂³	1	48	yes	—
30	C₂⁴×C₂₃	C₂×D₄	1	12	yes	—
31	C₂⁴×C₂₃	C₂×Q₈	1	12	yes	—
32	C₂⁴×C₂₃	#41	1	4	yes	—
33	C₂⁴×C₂₃	C₂⁴	1	3	yes	—
34	C₂×Q₈×C₂₃	—	2	384	no	—
35	C₂×Q₈×C₂₃	—	2	96	no	—
36	C₂²×C₄×C₂₃	—	2	96	no	—
37	C₂²×C₄×C₂₃	—	2	96	no	—
38	C₂²×C₄×C₂₃	—	2	36	no	—
39	C₂²×C₄×C₂₃	—	2	72	no	—
40	C₂²×C₄×C₂₃	—	2	36	no	—
41	C₂²×C₄×C₂₃	—	2	36	no	—
42	C₂²×C₄×C₂₃	—	4	16	no	—
43	C₂²×C₄×C₂₃	C₂×D₄	1	72	yes	—
44	C₂²×C₄×C₂₃	C₂×Q₈	1	72	yes	complemented
45	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
46	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
47	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
48	C₄⋊C₄×C₂₃	—	2	96	no	—
49	C₂²×#395	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂³×C₂₃	#120	1	16	yes	—
51	C₂³×C₂₃	C₂²⋊Q₈	1	48	yes	—
52	C₂³×C₂₃	C₄⋊Q₈	1	48	yes	—
53	C₂³×C₂₃	C₂²×D₄	1	18	yes	—
54	C₂³×C₂₃	C₂²×Q₈	1	18	yes	—
55	C₂³×C₂₃	C₂×#41	1	6	yes	—
56	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
57	Q₈×C₂₃	—	2	128	no	—
58	C₂×C₄×C₂₃	—	2	144	no	—
59	C₂×C₄×C₂₃	—	2	144	no	—
60	C₂×C₄×C₂₃	—	2	12	no	—
61	C₂×C₄×C₂₃	—	2	24	no	—
62	C₂×C₄×C₂₃	—	2	12	no	—
63	C₂×C₄×C₂₃	—	2	12	no	—
64	C₂×C₄×C₂₃	—	4	24	no	—
65	C₂×C₄×C₂₃	C₂²×D₄	1	24	yes	—
66	C₂×C₄×C₂₃	C₂²×Q₈	1	24	yes	complemented
67	C₂⁴×Q₈	C₂×C₂₃	1	1	yes	characteristic
68	C₂³×C₄⋊C₄	C₂×C₂₃	1	3	yes	—
69	C₂×#395	C₂×C₂₃	1	24	yes	direct product factor
70	C₂²×#95	C₂×C₂₃	1	3	yes	—
71	C₂²×C₂₃	#395	1	64	yes	direct product factor
72	C₂²×C₂₃	C₂×#120	1	12	yes	—
73	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	36	yes	—
74	C₂²×C₂₃	C₂×C₄⋊Q₈	1	36	yes	—
75	C₂²×C₂₃	C₂³×D₄	1	3	yes	—
76	C₂²×C₂₃	C₂³×Q₈	1	3	yes	—
77	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
78	C₄×C₂₃	—	2	24	no	—
79	C₄×C₂₃	—	2	24	no	—
80	C₄×C₂₃	—	4	4	no	—
81	C₂³×Q₈	—	2	24	no	—
82	C₂³×Q₈	C₂²×C₂₃	1	12	yes	—
83	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
84	C₂⁴×C₄	C₂²×C₂₃	1	3	yes	—
85	C₂⁴×C₄	C₂²×C₂₃	1	3	yes	—
86	C₂²×C₄⋊C₄	—	2	24	no	—
87	C₂²×C₄⋊C₄	C₂²×C₂₃	1	36	yes	—
88	#395	C₂²×C₂₃	1	64	yes	direct product factor
89	C₂×#95	C₂²×C₂₃	1	36	yes	—
90	C₂×C₂₃	C₂×#395	1	24	yes	direct product factor
91	C₂×C₂₃	C₂²×#120	1	1	yes	characteristic
92	C₂×C₂₃	C₂²×C₂²⋊Q₈	1	3	yes	—
93	C₂×C₂₃	C₂²×C₄⋊Q₈	1	3	yes	—
94	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
95	C₂²×Q₈	—	2	128	no	—
96	C₂²×Q₈	—	2	144	no	—
97	C₂²×Q₈	C₂³×C₂₃	1	16	yes	—
98	C₂³×C₄	—	2	12	no	—
99	C₂³×C₄	—	2	24	no	—
100	C₂³×C₄	—	2	12	no	—
101	C₂³×C₄	—	2	12	no	—
102	C₂³×C₄	D₄×C₂₃	1	24	yes	—
103	C₂³×C₄	Q₈×C₂₃	1	24	yes	complemented
104	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
105	C₂³×C₄	C₂³×C₂₃	1	18	yes	—
106	C₂³×C₄	C₂³×C₂₃	1	18	yes	—
107	C₂×C₄⋊C₄	—	2	144	no	—
108	C₂×C₄⋊C₄	C₂³×C₂₃	1	48	yes	—
109	#95	C₂³×C₂₃	1	48	yes	—
110	C₂₃	C₂²×#395	1	1	yes	characteristic, direct product factor, Sylow
111	C₂⁴	C₂×D₄×C₂₃	1	12	yes	—
112	C₂⁴	C₂×Q₈×C₂₃	1	12	yes	—
113	C₂⁴	#41×C₂₃	1	4	yes	—
114	C₂⁴	C₂⁴×C₂₃	1	3	yes	—
115	C₂×Q₈	—	2	384	no	—
116	C₂×Q₈	—	2	96	no	—
117	C₂²×C₄	—	2	96	no	—
118	C₂²×C₄	—	2	96	no	—
119	C₂²×C₄	—	2	36	no	—
120	C₂²×C₄	—	2	72	no	—
121	C₂²×C₄	—	2	36	no	—
122	C₂²×C₄	—	2	36	no	—
123	C₂²×C₄	—	4	16	no	—
124	C₂²×C₄	C₂×D₄×C₂₃	1	72	yes	—
125	C₂²×C₄	C₂×Q₈×C₂₃	1	72	yes	complemented
126	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
127	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
128	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
129	C₄⋊C₄	—	2	96	no	—
130	C₂³	C₂₃×#120	1	16	yes	—
131	C₂³	C₂₃×C₂²⋊Q₈	1	48	yes	—
132	C₂³	C₂₃×C₄⋊Q₈	1	48	yes	—
133	C₂³	C₂²×D₄×C₂₃	1	18	yes	—
134	C₂³	C₂²×Q₈×C₂₃	1	18	yes	—
135	C₂³	C₂×#41×C₂₃	1	6	yes	—
136	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
137	Q₈	—	2	128	no	—
138	C₂×C₄	—	2	12	no	—
139	C₂×C₄	—	2	24	no	—
140	C₂×C₄	—	2	12	no	—
141	C₂×C₄	—	2	144	no	—
142	C₂×C₄	—	2	144	no	—
143	C₂×C₄	—	2	12	no	—
144	C₂×C₄	—	4	24	no	—
145	C₂×C₄	C₂²×D₄×C₂₃	1	24	yes	—
146	C₂×C₄	C₂²×Q₈×C₂₃	1	24	yes	complemented
147	C₂²	C₂₃×#395	1	64	yes	direct product factor
148	C₂²	C₂×C₂₃×#120	1	12	yes	—
149	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	36	yes	—
150	C₂²	C₂×C₂₃×C₄⋊Q₈	1	36	yes	—
151	C₂²	C₂³×D₄×C₂₃	1	3	yes	—
152	C₂²	C₂³×Q₈×C₂₃	1	3	yes	—
153	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
154	C₄	—	2	24	no	—
155	C₄	—	2	24	no	—
156	C₄	—	4	4	no	—
157	C₂	C₂×C₂₃×#395	1	24	yes	direct product factor
158	C₂	C₂²×C₂₃×#120	1	1	yes	characteristic
159	C₂	C₂²×C₂₃×C₂²⋊Q₈	1	3	yes	—
160	C₂	C₂²×C₂₃×C₄⋊Q₈	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₂₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₂₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₂₃×#395	C₂	direct (N × H)
12. C₂₃×#395	C₂²	direct (N × H)
23. C₂³×C₄×C₂₃	Q₈	semidirect
44. C₂²×C₄×C₂₃	C₂×Q₈	semidirect
49. C₂²×#395	C₂₃	direct (N × H)
66. C₂×C₄×C₂₃	C₂²×Q₈	semidirect
69. C₂×#395	C₂×C₂₃	direct (N × H)
71. C₂²×C₂₃	#395	direct (N × H)
88. #395	C₂²×C₂₃	direct (N × H)
90. C₂×C₂₃	C₂×#395	direct (N × H)
103. C₂³×C₄	Q₈×C₂₃	semidirect
110. C₂₃	C₂²×#395	direct (N × H)
125. C₂²×C₄	C₂×Q₈×C₂₃	semidirect
146. C₂×C₄	C₂²×Q₈×C₂₃	semidirect
147. C₂²	C₂₃×#395	direct (N × H)
157. C₂	C₂×C₂₃×#395	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₂₃	2
C₂³×C₄⋊C₄×C₂₃	2
C₂×C₂₃×#395	2
C₂²×C₂₃×#95	2
C₂²×#395	23

Subgroups — C22×C23×#395 #19182160

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#395 #19182160