Subgroups — C₂₃×#7176 #19154802

Order 5888.

246 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×#108	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×M₃₂	—	2	1	no	—
5	C₂×C₂₃×#108	C₄	1	2	yes	—
6	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
7	C₄×M₁₆×C₂₃	—	2	1	no	—
8	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#346	C₄	1	2	yes	—
10	C₄×C₁₆×C₂₃	—	2	1	no	—
11	C₂₃×#341	—	2	1	no	—
12	C₂₃×#335	—	2	1	no	—
13	C₂₃×#335	C₄	1	2	yes	—
14	C₂×M₁₆×C₂₃	—	2	2	no	—
15	C₂²×C₈×C₂₃	—	4	1	no	—
16	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
19	C₂₃×M₃₂	—	2	4	no	—
20	C₂×C₁₆×C₂₃	—	2	2	no	—
21	C₂₃×#108	—	2	2	no	—
22	C₂₃×#108	C₈	1	4	yes	complemented
23	C₂₃×C₄⋊C₈	—	2	2	no	—
24	C₂₃×C₄⋊C₈	—	4	1	no	—
25	C₂₃×#97	—	4	1	no	—
26	C₄×C₈×C₂₃	—	4	1	no	—
27	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	Q₁₆	1	1	yes	characteristic
34	M₁₆×C₂₃	—	2	4	no	—
35	M₁₆×C₂₃	—	4	2	no	—
36	C₂×C₈×C₂₃	—	2	2	no	—
37	C₂×C₈×C₂₃	—	4	2	no	—
38	C₂×C₈×C₂₃	—	4	2	no	—
39	C₂×C₈×C₂₃	—	8	2	no	—
40	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
46	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₂₃	D₈	1	1	yes	characteristic
49	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
50	C₁₆×C₂₃	—	2	4	no	—
51	#7176	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂³×C₂₃	#103	1	1	yes	characteristic
53	C₂×C₄×C₂₃	—	2	1	no	—
54	C₂×C₄×C₂₃	—	2	1	no	—
55	C₂×C₄×C₂₃	—	4	1	no	—
56	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
57	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
58	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
59	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
60	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
61	C₂×C₄×C₂₃	#104	1	2	yes	—
62	C₂×C₄×C₂₃	#106	1	1	yes	characteristic
63	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
64	C₈×C₂₃	—	2	1	no	—
65	C₈×C₂₃	—	2	1	no	—
66	C₈×C₂₃	—	4	4	no	—
67	C₈×C₂₃	—	8	2	no	—
68	C₈×C₂₃	C₄×C₈	1	1	yes	characteristic
69	C₈×C₂₃	#97	1	1	yes	characteristic
70	C₈×C₂₃	C₄⋊C₈	1	2	yes	—
71	C₄×M₃₂	C₂×C₂₃	1	1	yes	characteristic
72	C₄×#108	C₂×C₂₃	1	1	yes	characteristic
73	C₂×#335	C₂×C₂₃	1	1	yes	characteristic
74	C₂²×C₂₃	—	2	1	no	—
75	C₂²×C₂₃	—	2	1	no	—
76	C₂²×C₂₃	Q₈⋊C₈	1	2	yes	—
77	C₂²×C₂₃	#340	1	1	yes	characteristic
78	C₄×C₂₃	—	2	1	no	—
79	C₄×C₂₃	—	2	1	no	—
80	C₄×C₂₃	—	2	1	no	—
81	C₄×C₂₃	D₄⋊C₈	1	2	yes	—
82	C₄×C₂₃	#336	1	1	yes	characteristic
83	C₄×C₂₃	#339	1	1	yes	characteristic
84	C₄×C₂₃	#361	1	1	yes	characteristic
85	C₄×C₂₃	#362	1	1	yes	characteristic
86	C₂×M₃₂	—	2	1	no	—
87	C₂×#108	C₄×C₂₃	1	2	yes	—
88	C₂×C₄⋊C₈	—	2	1	no	—
89	C₄×M₁₆	—	2	1	no	—
90	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
91	#346	C₄×C₂₃	1	2	yes	—
92	C₄×C₁₆	—	2	1	no	—
93	#341	—	2	1	no	—
94	#335	—	2	1	no	—
95	#335	C₄×C₂₃	1	2	yes	—
96	C₂×C₂₃	—	2	2	no	—
97	C₂×C₂₃	#1279	1	1	yes	characteristic
98	C₂×C₂₃	#1387	1	1	yes	characteristic
99	C₂×C₂₃	#1388	1	1	yes	characteristic
100	C₂×M₁₆	—	2	2	no	—
101	C₂²×C₈	—	4	1	no	—
102	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
103	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
104	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
105	M₃₂	—	2	4	no	—
106	C₂×C₁₆	—	2	2	no	—
107	#108	—	2	2	no	—
108	#108	C₈×C₂₃	1	4	yes	complemented
109	C₄⋊C₈	—	2	2	no	—
110	C₄⋊C₈	—	4	1	no	—
111	#97	—	4	1	no	—
112	C₄×C₈	—	4	1	no	—
113	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
114	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
115	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
116	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
117	C₂₃	#7176	1	1	yes	characteristic, direct product factor, Sylow
118	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
119	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
120	C₂²×C₄	Q₁₆×C₂₃	1	1	yes	characteristic
121	M₁₆	—	2	4	no	—
122	M₁₆	—	4	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	8	2	no	—
127	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
128	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
129	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
130	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
131	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
132	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
133	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
134	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
135	C₄²	D₈×C₂₃	1	1	yes	characteristic
136	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
137	C₁₆	—	2	4	no	—
138	C₂³	C₂₃×#103	1	1	yes	characteristic
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
143	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
144	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
145	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
146	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
147	C₂×C₄	C₂₃×#104	1	2	yes	—
148	C₂×C₄	C₂₃×#106	1	1	yes	characteristic
149	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	—	2	1	no	—
152	C₈	—	4	4	no	—
153	C₈	—	8	2	no	—
154	C₈	C₄×C₈×C₂₃	1	1	yes	characteristic
155	C₈	C₂₃×#97	1	1	yes	characteristic
156	C₈	C₂₃×C₄⋊C₈	1	2	yes	—
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	C₂₃×Q₈⋊C₈	1	2	yes	—
160	C₂²	C₂₃×#340	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	C₂₃×D₄⋊C₈	1	2	yes	—
165	C₄	C₂₃×#336	1	1	yes	characteristic
166	C₄	C₂₃×#339	1	1	yes	characteristic
167	C₄	C₂₃×#361	1	1	yes	characteristic
168	C₄	C₂₃×#362	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	C₂₃×#1279	1	1	yes	characteristic
171	C₂	C₂₃×#1387	1	1	yes	characteristic
172	C₂	C₂₃×#1388	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#102
commutator	C₈	C₄×C₈×C₂₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₂₃
socle	C₂²×C₂₃	#340
prime radical	C₂³×C₂₃	#103
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
22. C₂₃×#108	C₈	semidirect
51. #7176	C₂₃	direct (N × H)
108. #108	C₈×C₂₃	semidirect
117. C₂₃	#7176	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₃×M₃₂	2
C₄×C₂₃×#108	2
C₂×C₂₃×#335	2
#7176	23

Subgroups — C23×#7176 #19154802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7176 #19154802