Subgroups — C₂₃×#7178 #19154804

Order 5888.

274 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#346	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×#107	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#334	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆×C₂₃	—	2	1	no	—
5	C₂×C₂₃×#107	C₄	1	2	yes	—
6	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
7	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
9	C₂₃×#346	—	2	1	no	—
10	C₂₃×#346	C₄	1	2	yes	complemented
11	C₂₃×#340	—	2	1	no	—
12	C₂₃×#334	—	2	1	no	—
13	C₂₃×#334	C₄	1	2	yes	complemented
14	C₂²×C₈×C₂₃	—	4	1	no	—
15	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
18	C₂×C₄²×C₂₃	—	4	1	no	—
19	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
20	C₂×C₁₆×C₂₃	—	2	4	no	—
21	C₂×C₁₆×C₂₃	—	2	2	no	—
22	C₂₃×#107	—	2	2	no	—
23	C₂₃×#107	C₈	1	4	yes	complemented
24	C₂₃×C₄⋊C₈	—	2	2	no	—
25	C₂₃×C₄⋊C₈	—	4	1	no	—
26	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
30	C₂₃×#95	—	4	1	no	—
31	C₂²×C₄×C₂₃	—	4	2	no	—
32	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
35	C₂×C₈×C₂₃	—	2	2	no	—
36	C₂×C₈×C₂₃	—	4	2	no	—
37	C₂×C₈×C₂₃	—	8	2	no	—
38	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
44	C₄⋊C₄×C₂₃	—	2	4	no	—
45	C₄⋊C₄×C₂₃	—	4	2	no	—
46	C₄²×C₂₃	—	8	2	no	—
47	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
50	C₄²×C₂₃	Q₁₆	1	1	yes	characteristic
51	C₁₆×C₂₃	—	2	4	no	—
52	#7178	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂³×C₂₃	#102	1	1	yes	characteristic
54	C₂×C₄×C₂₃	—	2	2	no	—
55	C₂×C₄×C₂₃	—	2	1	no	—
56	C₂×C₄×C₂₃	—	2	1	no	—
57	C₂×C₄×C₂₃	—	4	4	no	—
58	C₂×C₄×C₂₃	—	8	2	no	—
59	C₂×C₄×C₂₃	—	8	2	no	—
60	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
61	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
62	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
63	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
64	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
65	C₂×C₄×C₂₃	#104	1	2	yes	—
66	C₂×C₄×C₂₃	#106	1	1	yes	characteristic
67	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
68	C₈×C₂₃	—	2	1	no	—
69	C₈×C₂₃	—	2	1	no	—
70	C₈×C₂₃	—	8	2	no	—
71	C₈×C₂₃	C₄×C₈	1	1	yes	characteristic
72	C₈×C₂₃	#97	1	1	yes	characteristic
73	C₈×C₂₃	C₄⋊C₈	1	2	yes	—
74	C₂×#346	C₂×C₂₃	1	1	yes	characteristic
75	C₄×#107	C₂×C₂₃	1	1	yes	characteristic
76	C₂×#334	C₂×C₂₃	1	1	yes	characteristic
77	C₂²×C₂₃	D₄⋊C₈	1	2	yes	—
78	C₂²×C₂₃	#340	1	1	yes	characteristic
79	C₂²×C₂₃	#360	1	2	yes	—
80	C₂²×C₂₃	#361	1	1	yes	characteristic
81	C₂²×C₂₃	#362	1	1	yes	characteristic
82	C₄×C₂₃	—	2	1	no	—
83	C₄×C₂₃	—	2	1	no	—
84	C₄×C₂₃	—	2	1	no	—
85	C₄×C₂₃	—	2	1	no	—
86	C₄×C₂₃	—	8	4	no	—
87	C₄×C₂₃	Q₈⋊C₈	1	2	yes	—
88	C₄×C₂₃	#336	1	1	yes	characteristic
89	C₄×C₂₃	#339	1	1	yes	characteristic
90	C₂²×C₁₆	—	2	1	no	—
91	C₂×#107	C₄×C₂₃	1	2	yes	—
92	C₂×C₄⋊C₈	—	2	1	no	—
93	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
94	C₄×C₄⋊C₄	—	2	1	no	—
95	#346	—	2	1	no	—
96	#346	C₄×C₂₃	1	2	yes	complemented
97	#340	—	2	1	no	—
98	#334	—	2	1	no	—
99	#334	C₄×C₂₃	1	2	yes	complemented
100	C₂×C₂₃	#1279	1	1	yes	characteristic
101	C₂×C₂₃	#1333	1	2	yes	—
102	C₂×C₂₃	#1334	1	2	yes	—
103	C₂×C₂₃	#1386	1	1	yes	characteristic
104	C₂×C₂₃	#1387	1	1	yes	characteristic
105	C₂²×C₈	—	4	1	no	—
106	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
107	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄²	—	4	1	no	—
110	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
111	C₂×C₁₆	—	2	4	no	—
112	C₂×C₁₆	—	2	2	no	—
113	#107	—	2	2	no	—
114	#107	C₈×C₂₃	1	4	yes	complemented
115	C₄⋊C₈	—	2	2	no	—
116	C₄⋊C₈	—	4	1	no	—
117	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
118	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
119	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
120	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
121	#95	—	4	1	no	—
122	C₂₃	#7178	1	1	yes	characteristic, direct product factor, Sylow
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
125	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
126	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	8	2	no	—
130	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
131	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
132	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
133	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
134	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
135	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
136	C₄⋊C₄	—	2	4	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄²	—	8	2	no	—
139	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
140	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
141	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
142	C₄²	Q₁₆×C₂₃	1	1	yes	characteristic
143	C₁₆	—	2	4	no	—
144	C₂³	C₂₃×#102	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	4	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
152	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
153	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
154	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
155	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
156	C₂×C₄	C₂₃×#104	1	2	yes	—
157	C₂×C₄	C₂₃×#106	1	1	yes	characteristic
158	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
159	C₈	—	2	1	no	—
160	C₈	—	2	1	no	—
161	C₈	—	8	2	no	—
162	C₈	C₄×C₈×C₂₃	1	1	yes	characteristic
163	C₈	C₂₃×#97	1	1	yes	characteristic
164	C₈	C₂₃×C₄⋊C₈	1	2	yes	—
165	C₂²	C₂₃×D₄⋊C₈	1	2	yes	—
166	C₂²	C₂₃×#340	1	1	yes	characteristic
167	C₂²	C₂₃×#360	1	2	yes	—
168	C₂²	C₂₃×#361	1	1	yes	characteristic
169	C₂²	C₂₃×#362	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	8	4	no	—
175	C₄	C₂₃×Q₈⋊C₈	1	2	yes	—
176	C₄	C₂₃×#336	1	1	yes	characteristic
177	C₄	C₂₃×#339	1	1	yes	characteristic
178	C₂	C₂₃×#1279	1	1	yes	characteristic
179	C₂	C₂₃×#1333	1	2	yes	—
180	C₂	C₂₃×#1334	1	2	yes	—
181	C₂	C₂₃×#1386	1	1	yes	characteristic
182	C₂	C₂₃×#1387	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	#102
commutator	C₈	C₄×C₈×C₂₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₂₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
10. C₂₃×#346	C₄	semidirect
13. C₂₃×#334	C₄	semidirect
23. C₂₃×#107	C₈	semidirect
52. #7178	C₂₃	direct (N × H)
96. #346	C₄×C₂₃	semidirect
99. #334	C₄×C₂₃	semidirect
114. #107	C₈×C₂₃	semidirect
122. C₂₃	#7178	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×#346	2
C₄×C₂₃×#107	2
C₂×C₂₃×#334	2
#7178	23

Subgroups — C23×#7178 #19154804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7178 #19154804