Subgroups — C₂×C₂₃×#2397 #19182170

Order 5888.

5150 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₂₃	C₂	1	6	yes	maximal, complemented
2	C₂₃×#2397	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₂₃×#399	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₃×#392	C₂	1	4	yes	maximal, complemented
5	C₂×C₂₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₄×C₂₃	C₂²	1	6	yes	—
7	C₂⁴×C₄×C₂₃	C₂²	1	4	yes	—
8	C₂×C₄×D₄×C₂₃	—	2	48	no	—
9	C₂×C₄×D₄×C₂₃	C₂²	1	48	yes	complemented
10	C₂²×C₄⋊C₄×C₂₃	C₂²	1	6	yes	—
11	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	12	yes	—
12	C₂²×C₄²×C₂₃	C₂²	1	3	yes	—
13	C₂₃×#399	C₂²	1	32	yes	complemented
14	C₂₃×#392	C₂²	1	32	yes	complemented
15	C₂₃×#376	C₂²	1	8	yes	complemented
16	C₂×C₂₃×#95	C₂²	1	4	yes	—
17	C₂⁵×C₂₃	C₂³	1	4	yes	—
18	C₂²×D₄×C₂₃	—	2	48	no	—
19	C₂²×D₄×C₂₃	—	2	24	no	—
20	C₂²×D₄×C₂₃	C₂³	1	24	yes	—
21	C₂³×C₄×C₂₃	—	2	48	no	—
22	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
23	C₂³×C₄×C₂₃	C₂³	1	16	yes	—
24	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
25	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₄×D₄×C₂₃	—	2	192	no	—
27	C₂×C₄⋊C₄×C₂₃	—	2	24	no	—
28	C₂×C₄⋊C₄×C₂₃	C₂³	1	24	yes	—
29	C₂×C₂²⋊C₄×C₂₃	—	2	48	no	—
30	C₂×C₂²⋊C₄×C₂₃	C₂³	1	48	yes	—
31	C₂×C₄²×C₂₃	D₄	1	24	yes	complemented
32	C₂×C₄²×C₂₃	C₂³	1	12	yes	—
33	C₂₃×#95	C₂³	1	16	yes	—
34	C₂⁴×C₂₃	—	2	24	no	—
35	C₂⁴×C₂₃	—	2	24	no	—
36	C₂⁴×C₂₃	#41	1	16	yes	—
37	C₂⁴×C₂₃	C₂⁴	1	8	yes	—
38	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
39	C₂×D₄×C₂₃	—	2	192	no	—
40	C₂×D₄×C₂₃	—	2	96	no	—
41	C₂×D₄×C₂₃	—	2	48	no	—
42	C₂²×C₄×C₂₃	—	2	24	no	—
43	C₂²×C₄×C₂₃	—	2	24	no	—
44	C₂²×C₄×C₂₃	—	2	96	no	—
45	C₂²×C₄×C₂₃	—	4	64	no	—
46	C₂²×C₄×C₂₃	C₂×D₄	1	24	yes	—
47	C₂²×C₄×C₂₃	C₂×D₄	1	12	yes	—
48	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
49	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
50	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
51	C₄⋊C₄×C₂₃	—	2	48	no	—
52	C₂²⋊C₄×C₂₃	—	2	96	no	—
53	C₄²×C₂₃	C₂×D₄	1	48	yes	complemented
54	C₂×#2397	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂³×C₂₃	—	2	96	no	—
56	C₂³×C₂₃	—	2	24	no	—
57	C₂³×C₂₃	—	2	24	no	—
58	C₂³×C₂₃	—	4	48	no	—
59	C₂³×C₂₃	—	4	16	no	—
60	C₂³×C₂₃	C₂²×D₄	1	6	yes	—
61	C₂³×C₂₃	C₂×#41	1	16	yes	—
62	C₂³×C₂₃	C₂×#41	1	8	yes	—
63	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
64	D₄×C₂₃	—	2	192	no	—
65	C₂×C₄×C₂₃	—	2	48	no	—
66	C₂×C₄×C₂₃	—	2	24	no	—
67	C₂×C₄×C₂₃	—	2	24	no	—
68	C₂×C₄×C₂₃	—	4	16	no	—
69	C₂×C₄×C₂₃	—	4	64	no	—
70	C₂×C₄×C₂₃	#127	1	16	yes	—
71	C₂×C₄×C₂₃	C₂²×D₄	1	24	yes	—
72	C₂×C₄×C₂₃	C₂²×D₄	1	12	yes	—
73	C₂²×C₄×D₄	C₂×C₂₃	1	6	yes	complemented
74	#2397	C₂×C₂₃	1	16	yes	direct product factor
75	C₂×#399	C₂×C₂₃	1	4	yes	complemented
76	C₂×#392	C₂×C₂₃	1	4	yes	complemented
77	C₂×#376	C₂×C₂₃	1	1	yes	characteristic, complemented
78	C₂²×C₂₃	—	2	48	no	—
79	C₂²×C₂₃	—	4	64	no	—
80	C₂²×C₂₃	—	4	24	no	—
81	C₂²×C₂₃	—	4	8	no	—
82	C₂²×C₂₃	#525	1	24	yes	—
83	C₂²×C₂₃	C₂×#127	1	4	yes	—
84	C₂²×C₂₃	C₂³×D₄	1	3	yes	—
85	C₂²×C₂₃	C₂²×#41	1	4	yes	—
86	C₄×C₂₃	—	2	24	no	—
87	C₄×C₂₃	—	4	8	no	—
88	C₄×C₂₃	C₂×#127	1	8	yes	—
89	C₂³×D₄	C₂²×C₂₃	1	6	yes	—
90	C₂⁴×C₄	C₂²×C₂₃	1	4	yes	—
91	C₂×C₄×D₄	—	2	48	no	—
92	C₂×C₄×D₄	C₂²×C₂₃	1	48	yes	complemented
93	C₂²×C₄⋊C₄	C₂²×C₂₃	1	6	yes	—
94	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	12	yes	—
95	C₂²×C₄²	C₂²×C₂₃	1	3	yes	—
96	#399	C₂²×C₂₃	1	32	yes	complemented
97	#392	C₂²×C₂₃	1	32	yes	complemented
98	#376	C₂²×C₂₃	1	8	yes	complemented
99	C₂×#95	C₂²×C₂₃	1	4	yes	—
100	C₂×C₂₃	—	4	16	no	—
101	C₂×C₂₃	#2397	1	8	yes	direct product factor
102	C₂×C₂₃	C₂×#525	1	6	yes	—
103	C₂×C₂₃	C₂²×#127	1	1	yes	characteristic
104	C₂⁵	C₂³×C₂₃	1	4	yes	—
105	C₂²×D₄	—	2	48	no	—
106	C₂²×D₄	—	2	24	no	—
107	C₂²×D₄	C₂³×C₂₃	1	24	yes	—
108	C₂³×C₄	—	2	48	no	—
109	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
110	C₂³×C₄	C₂³×C₂₃	1	16	yes	—
111	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
112	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
113	C₄×D₄	—	2	192	no	—
114	C₂×C₄⋊C₄	—	2	24	no	—
115	C₂×C₄⋊C₄	C₂³×C₂₃	1	24	yes	—
116	C₂×C₂²⋊C₄	—	2	48	no	—
117	C₂×C₂²⋊C₄	C₂³×C₂₃	1	48	yes	—
118	C₂×C₄²	D₄×C₂₃	1	24	yes	complemented
119	C₂×C₄²	C₂³×C₂₃	1	12	yes	—
120	#95	C₂³×C₂₃	1	16	yes	—
121	C₂₃	C₂×#2397	1	1	yes	characteristic, direct product factor, Sylow
122	C₂⁴	—	2	24	no	—
123	C₂⁴	—	2	24	no	—
124	C₂⁴	#41×C₂₃	1	16	yes	—
125	C₂⁴	C₂⁴×C₂₃	1	8	yes	—
126	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
127	C₂×D₄	—	2	192	no	—
128	C₂×D₄	—	2	96	no	—
129	C₂×D₄	—	2	48	no	—
130	C₂²×C₄	—	2	24	no	—
131	C₂²×C₄	—	2	24	no	—
132	C₂²×C₄	—	2	96	no	—
133	C₂²×C₄	—	4	64	no	—
134	C₂²×C₄	C₂×D₄×C₂₃	1	24	yes	—
135	C₂²×C₄	C₂×D₄×C₂₃	1	12	yes	—
136	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
137	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
138	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
139	C₄⋊C₄	—	2	48	no	—
140	C₂²⋊C₄	—	2	96	no	—
141	C₄²	C₂×D₄×C₂₃	1	48	yes	complemented
142	C₂³	—	2	24	no	—
143	C₂³	—	2	24	no	—
144	C₂³	—	2	96	no	—
145	C₂³	—	4	48	no	—
146	C₂³	—	4	16	no	—
147	C₂³	C₂²×D₄×C₂₃	1	6	yes	—
148	C₂³	C₂×#41×C₂₃	1	16	yes	—
149	C₂³	C₂×#41×C₂₃	1	8	yes	—
150	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
151	D₄	—	2	192	no	—
152	C₂×C₄	—	2	24	no	—
153	C₂×C₄	—	2	48	no	—
154	C₂×C₄	—	2	24	no	—
155	C₂×C₄	—	4	16	no	—
156	C₂×C₄	—	4	64	no	—
157	C₂×C₄	C₂₃×#127	1	16	yes	—
158	C₂×C₄	C₂²×D₄×C₂₃	1	24	yes	—
159	C₂×C₄	C₂²×D₄×C₂₃	1	12	yes	—
160	C₂²	—	2	48	no	—
161	C₂²	—	4	64	no	—
162	C₂²	—	4	24	no	—
163	C₂²	—	4	8	no	—
164	C₂²	C₂₃×#525	1	24	yes	—
165	C₂²	C₂×C₂₃×#127	1	4	yes	—
166	C₂²	C₂³×D₄×C₂₃	1	3	yes	—
167	C₂²	C₂²×#41×C₂₃	1	4	yes	—
168	C₄	—	2	24	no	—
169	C₄	—	4	8	no	—
170	C₄	C₂×C₂₃×#127	1	8	yes	—
171	C₂	—	4	16	no	—
172	C₂	C₂₃×#2397	1	8	yes	direct product factor
173	C₂	C₂×C₂₃×#525	1	6	yes	—
174	C₂	C₂²×C₂₃×#127	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₂₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₂₃
socle	C₂⁴×C₂₃	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄×D₄×C₂₃	C₂	semidirect
2. C₂₃×#2397	C₂	direct (N × H)
3. C₂×C₂₃×#399	C₂	semidirect
4. C₂×C₂₃×#392	C₂	semidirect
5. C₂×C₂₃×#376	C₂	semidirect
9. C₂×C₄×D₄×C₂₃	C₂²	semidirect
13. C₂₃×#399	C₂²	semidirect
14. C₂₃×#392	C₂²	semidirect
15. C₂₃×#376	C₂²	semidirect
31. C₂×C₄²×C₂₃	D₄	semidirect
53. C₄²×C₂₃	C₂×D₄	semidirect
54. C₂×#2397	C₂₃	direct (N × H)
73. C₂²×C₄×D₄	C₂×C₂₃	semidirect
74. #2397	C₂×C₂₃	direct (N × H)
75. C₂×#399	C₂×C₂₃	semidirect
76. C₂×#392	C₂×C₂₃	semidirect
77. C₂×#376	C₂×C₂₃	semidirect
92. C₂×C₄×D₄	C₂²×C₂₃	semidirect
96. #399	C₂²×C₂₃	semidirect
97. #392	C₂²×C₂₃	semidirect
98. #376	C₂²×C₂₃	semidirect
101. C₂×C₂₃	#2397	direct (N × H)
118. C₂×C₄²	D₄×C₂₃	semidirect
121. C₂₃	C₂×#2397	direct (N × H)
141. C₄²	C₂×D₄×C₂₃	semidirect
172. C₂	C₂₃×#2397	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×C₄×D₄×C₂₃	2
C₂₃×#2397	2
C₂×C₂₃×#399	2
C₂×C₂₃×#392	2
C₂×C₂₃×#376	2
C₂×#2397	23

Subgroups — C2×C23×#2397 #19182170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2397 #19182170