Subgroups — C₂₃×#7181 #19154807

Order 5888.

240 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#364	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1772	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1450	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×M₃₂	—	2	1	no	—
5	C₂₃×#431	—	2	1	no	—
6	C₂×C₂₃×#108	C₄	1	1	yes	characteristic
7	C₂₃×#427	C₄	1	1	yes	characteristic
8	C₂₃×#405	—	2	1	no	—
9	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×#364	—	2	1	no	—
11	C₂₃×#364	C₄	1	2	yes	complemented
12	C₂₃×#339	—	2	1	no	—
13	C₂₃×#322	C₄	1	2	yes	complemented
14	C₈²×C₂₃	—	2	1	no	—
15	C₂×M₁₆×C₂₃	—	2	1	no	—
16	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
18	C₂₃×#117	—	2	1	no	—
19	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
20	C₂₃×M₃₂	—	2	2	no	—
21	C₂₃×M₃₂	—	4	1	no	—
22	C₂×C₁₆×C₂₃	—	4	1	no	—
23	C₂₃×#108	—	2	1	no	—
24	C₂₃×#108	C₈	1	2	yes	complemented
25	C₂₃×#107	—	2	1	no	—
26	C₂₃×#106	C₈	1	2	yes	complemented
27	C₂₃×C₄⋊C₈	—	2	2	no	—
28	C₂₃×C₂²⋊C₈	—	2	2	no	—
29	C₂₃×#97	—	4	1	no	—
30	C₄×C₈×C₂₃	—	2	2	no	—
31	C₄×C₈×C₂₃	—	4	1	no	—
32	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
35	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	—	2	1	no	—
37	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
38	M₁₆×C₂₃	—	2	2	no	—
39	M₁₆×C₂₃	—	4	1	no	—
40	C₂×C₈×C₂₃	—	2	4	no	—
41	C₂×C₈×C₂₃	—	4	1	no	—
42	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₂₃	M₁₆	1	1	yes	characteristic
49	C₂×C₈×C₂₃	M₁₆	1	1	yes	characteristic
50	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
51	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₂₃	Q₁₆	1	1	yes	characteristic
54	C₄⋊C₄×C₂₃	—	2	2	no	—
55	C₂²⋊C₄×C₂₃	—	4	1	no	—
56	C₄²×C₂₃	—	2	1	no	—
57	C₄²×C₂₃	—	4	1	no	—
58	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
60	C₁₆×C₂₃	—	4	2	no	—
61	#7181	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂³×C₂₃	#104	1	1	yes	characteristic
63	C₂×C₄×C₂₃	—	2	1	no	—
64	C₂×C₄×C₂₃	—	2	1	no	—
65	C₂×C₄×C₂₃	—	4	2	no	—
66	C₂×C₄×C₂₃	—	4	2	no	—
67	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
68	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
70	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
71	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
72	C₂×C₄×C₂₃	#104	1	1	yes	characteristic
73	C₈×C₂₃	—	4	4	no	—
74	C₈×C₂₃	—	4	2	no	—
75	C₈×C₂₃	C₄×C₈	1	1	yes	characteristic
76	C₈×C₂₃	#97	1	1	yes	characteristic
77	C₈×C₂₃	#102	1	1	yes	characteristic
78	C₈×C₂₃	#103	1	1	yes	characteristic
79	C₈×C₂₃	C₄⋊C₈	1	1	yes	characteristic
80	C₈×C₂₃	C₄⋊C₈	1	1	yes	characteristic
81	C₈×C₂₃	#106	1	1	yes	characteristic
82	C₈×C₂₃	#107	1	1	yes	characteristic
83	C₂×#364	C₂×C₂₃	1	1	yes	characteristic
84	#1772	C₂×C₂₃	1	1	yes	characteristic
85	#1450	C₂×C₂₃	1	1	yes	characteristic
86	C₂²×C₂₃	—	4	1	no	—
87	C₂²×C₂₃	D₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₂₃	Q₈⋊C₈	1	1	yes	characteristic
89	C₂²×C₂₃	#339	1	1	yes	characteristic
90	C₄×C₂₃	—	2	2	no	—
91	C₄×C₂₃	—	4	1	no	—
92	C₄×C₂₃	—	8	2	no	—
93	C₄×C₂₃	D₄⋊C₈	1	1	yes	characteristic
94	C₄×C₂₃	Q₈⋊C₈	1	1	yes	characteristic
95	C₄×C₂₃	#336	1	1	yes	characteristic
96	C₄×C₂₃	#340	1	1	yes	characteristic
97	C₂×M₃₂	—	2	1	no	—
98	#431	—	2	1	no	—
99	C₂×#108	C₄×C₂₃	1	1	yes	characteristic
100	#427	C₄×C₂₃	1	1	yes	characteristic
101	#405	—	2	1	no	—
102	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
103	#364	—	2	1	no	—
104	#364	C₄×C₂₃	1	2	yes	complemented
105	#339	—	2	1	no	—
106	#322	C₄×C₂₃	1	2	yes	complemented
107	C₈²	—	2	1	no	—
108	C₂×C₂₃	—	2	1	no	—
109	C₂×C₂₃	—	2	1	no	—
110	C₂×C₂₃	—	2	1	no	—
111	C₂×C₂₃	#1279	1	1	yes	characteristic
112	C₂×M₁₆	—	2	1	no	—
113	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
114	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
115	#117	—	2	1	no	—
116	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
117	M₃₂	—	2	2	no	—
118	M₃₂	—	4	1	no	—
119	C₂×C₁₆	—	4	1	no	—
120	#108	—	2	1	no	—
121	#108	C₈×C₂₃	1	2	yes	complemented
122	#107	—	2	1	no	—
123	#106	C₈×C₂₃	1	2	yes	complemented
124	C₄⋊C₈	—	2	2	no	—
125	C₂²⋊C₈	—	2	2	no	—
126	#97	—	4	1	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	4	1	no	—
129	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
130	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
131	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
132	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
133	C₂₃	#7181	1	1	yes	characteristic, direct product factor, Sylow
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
136	M₁₆	—	2	2	no	—
137	M₁₆	—	4	1	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
141	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
142	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
143	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
144	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
145	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
146	C₂×C₈	M₁₆×C₂₃	1	1	yes	characteristic
147	C₂×C₈	M₁₆×C₂₃	1	1	yes	characteristic
148	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
149	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
150	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
151	C₂×C₈	Q₁₆×C₂₃	1	1	yes	characteristic
152	C₄⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	4	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	4	1	no	—
156	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
157	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
158	C₁₆	—	4	2	no	—
159	C₂³	C₂₃×#104	1	1	yes	characteristic
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
165	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
166	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
167	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
168	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
169	C₂×C₄	C₂₃×#104	1	1	yes	characteristic
170	C₈	—	4	4	no	—
171	C₈	—	4	2	no	—
172	C₈	C₄×C₈×C₂₃	1	1	yes	characteristic
173	C₈	C₂₃×#97	1	1	yes	characteristic
174	C₈	C₂₃×#102	1	1	yes	characteristic
175	C₈	C₂₃×#103	1	1	yes	characteristic
176	C₈	C₂₃×C₄⋊C₈	1	1	yes	characteristic
177	C₈	C₂₃×C₄⋊C₈	1	1	yes	characteristic
178	C₈	C₂₃×#106	1	1	yes	characteristic
179	C₈	C₂₃×#107	1	1	yes	characteristic
180	C₂²	—	4	1	no	—
181	C₂²	C₂₃×D₄⋊C₈	1	1	yes	characteristic
182	C₂²	C₂₃×Q₈⋊C₈	1	1	yes	characteristic
183	C₂²	C₂₃×#339	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	4	1	no	—
186	C₄	—	8	2	no	—
187	C₄	C₂₃×D₄⋊C₈	1	1	yes	characteristic
188	C₄	C₂₃×Q₈⋊C₈	1	1	yes	characteristic
189	C₄	C₂₃×#336	1	1	yes	characteristic
190	C₄	C₂₃×#340	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	—	2	1	no	—
194	C₂	C₂₃×#1279	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₂₃	#102
commutator	C₈	C₄×C₈×C₂₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₂₃
socle	C₂×C₂₃	#1279
prime radical	C₂³×C₂₃	#104
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₂₃×#364	C₄	semidirect
13. C₂₃×#322	C₄	semidirect
24. C₂₃×#108	C₈	semidirect
26. C₂₃×#106	C₈	semidirect
61. #7181	C₂₃	direct (N × H)
104. #364	C₄×C₂₃	semidirect
106. #322	C₄×C₂₃	semidirect
121. #108	C₈×C₂₃	semidirect
123. #106	C₈×C₂₃	semidirect
133. C₂₃	#7181	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#364	2
C₂₃×#1772	2
C₂₃×#1450	2
#7181	23

Subgroups — C23×#7181 #19154807

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7181 #19154807