Subgroups — C₂₃×#3197 #15670816

Order 2944.

418 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#525	C₂	1	2	yes	maximal, complemented
2	C₂₃×#485	C₂	1	2	yes	maximal, complemented
3	C₂₃×#482	C₂	1	2	yes	maximal, complemented
4	C₂₃×#470	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#469	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#468	C₂	1	2	yes	maximal, complemented
7	C₂₃×#452	C₂	1	2	yes	maximal, complemented
8	C₂₃×#449	C₂	1	2	yes	maximal, complemented
9	C₂₃×#407	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₂₃	C₂²	1	2	yes	—
11	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
13	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	complemented
14	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
15	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#123	—	2	2	no	—
18	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₂₃×#121	C₂²	1	2	yes	—
21	C₂₃×#121	C₂²	1	2	yes	—
22	C₂₃×#120	—	2	2	no	—
23	C₄×D₄×C₂₃	—	2	2	no	—
24	C₄×D₄×C₂₃	—	2	2	no	—
25	C₂₃×#117	C₂²	1	2	yes	—
26	C₂₃×#107	C₂²	1	2	yes	complemented
27	C₂₃×#106	C₂²	1	1	yes	characteristic, complemented
28	C₂₃×#106	C₂²	1	1	yes	characteristic, complemented
29	C₂₃×#103	C₂²	1	2	yes	complemented
30	C₂₃×#103	C₂²	1	2	yes	complemented
31	C₂₃×#102	C₂²	1	2	yes	complemented
32	C₂₃×#102	C₂²	1	2	yes	complemented
33	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
34	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
35	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
36	#41×C₂₃	—	4	2	no	—
37	#41×C₂₃	—	4	2	no	—
38	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
39	C₂×D₄×C₂₃	—	2	2	no	—
40	C₂×D₄×C₂₃	—	2	2	no	—
41	C₂×D₄×C₂₃	—	2	2	no	—
42	C₂×D₄×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₃	—	2	2	no	—
44	C₂²×C₄×C₂₃	—	4	1	no	—
45	C₂²×C₄×C₂₃	—	4	1	no	—
46	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	D₄	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	Q₁₆×C₂₃	—	4	1	no	—
51	SD₁₆×C₂₃	—	4	2	no	—
52	D₈×C₂₃	—	4	1	no	—
53	M₁₆×C₂₃	—	4	1	no	—
54	M₁₆×C₂₃	—	4	1	no	—
55	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₂₃	C₂³	1	2	yes	—
57	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₂₃	—	2	2	no	—
59	C₄⋊C₄×C₂₃	—	2	2	no	—
60	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
61	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
62	C₂²⋊C₄×C₂₃	—	2	2	no	—
63	C₂²⋊C₄×C₂₃	—	2	2	no	—
64	C₂²⋊C₄×C₂₃	—	2	2	no	—
65	C₂²⋊C₄×C₂₃	—	2	2	no	—
66	C₂²⋊C₄×C₂₃	—	2	2	no	—
67	C₄²×C₂₃	—	2	2	no	—
68	C₂³×C₂₃	—	2	2	no	—
69	C₂³×C₂₃	—	4	1	no	—
70	C₂³×C₂₃	—	4	2	no	—
71	C₂³×C₂₃	C₂×D₄	1	2	yes	—
72	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
73	Q₈×C₂₃	—	2	2	no	—
74	Q₈×C₂₃	—	2	2	no	—
75	D₄×C₂₃	—	2	2	no	—
76	D₄×C₂₃	—	2	2	no	—
77	D₄×C₂₃	—	4	4	no	—
78	D₄×C₂₃	—	4	4	no	—
79	D₄×C₂₃	—	4	4	no	—
80	C₂×C₄×C₂₃	—	2	1	no	—
81	C₂×C₄×C₂₃	—	2	1	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	4	1	no	—
88	C₂×C₄×C₂₃	—	4	2	no	—
89	C₂×C₄×C₂₃	—	4	4	no	—
90	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
93	C₈×C₂₃	—	2	1	no	—
94	C₈×C₂₃	—	2	2	no	—
95	C₈×C₂₃	—	2	1	no	—
96	#3197	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂²×C₂₃	—	2	2	no	—
98	C₂²×C₂₃	—	2	1	no	—
99	C₂²×C₂₃	—	2	2	no	—
100	C₂²×C₂₃	—	2	2	no	—
101	C₂²×C₂₃	—	2	2	no	—
102	C₂²×C₂₃	—	4	4	no	—
103	C₂²×C₂₃	—	4	2	no	—
104	C₂²×C₂₃	—	4	2	no	—
105	C₂²×C₂₃	—	4	2	no	—
106	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
107	C₄×C₂₃	—	2	2	no	—
108	C₄×C₂₃	—	2	1	no	—
109	C₄×C₂₃	—	4	2	no	—
110	C₄×C₂₃	—	4	2	no	—
111	C₄×C₂₃	—	4	2	no	—
112	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
113	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
114	#525	C₂×C₂₃	1	2	yes	complemented
115	#485	C₂×C₂₃	1	2	yes	complemented
116	#482	C₂×C₂₃	1	2	yes	complemented
117	#470	C₂×C₂₃	1	1	yes	characteristic, complemented
118	#469	C₂×C₂₃	1	1	yes	characteristic, complemented
119	#468	C₂×C₂₃	1	2	yes	complemented
120	#452	C₂×C₂₃	1	2	yes	complemented
121	#449	C₂×C₂₃	1	2	yes	complemented
122	#407	C₂×C₂₃	1	1	yes	characteristic, complemented
123	C₂×C₂₃	—	4	2	no	—
124	C₂×C₂₃	—	4	1	no	—
125	C₂×C₂₃	—	8	2	no	—
126	C₂×C₂₃	#534	1	1	yes	characteristic
127	C₂×C₂₃	#575	1	2	yes	—
128	C₂×#41	C₂²×C₂₃	1	2	yes	—
129	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
130	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
131	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	complemented
132	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
133	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
134	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
135	#123	—	2	2	no	—
136	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
137	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
138	#121	C₂²×C₂₃	1	2	yes	—
139	#121	C₂²×C₂₃	1	2	yes	—
140	#120	—	2	2	no	—
141	C₄×D₄	—	2	2	no	—
142	C₄×D₄	—	2	2	no	—
143	#117	C₂²×C₂₃	1	2	yes	—
144	#107	C₂²×C₂₃	1	2	yes	complemented
145	#106	C₂²×C₂₃	1	1	yes	characteristic, complemented
146	#106	C₂²×C₂₃	1	1	yes	characteristic, complemented
147	#103	C₂²×C₂₃	1	2	yes	complemented
148	#103	C₂²×C₂₃	1	2	yes	complemented
149	#102	C₂²×C₂₃	1	2	yes	complemented
150	#102	C₂²×C₂₃	1	2	yes	complemented
151	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
152	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
153	C₂₃	#3197	1	1	yes	characteristic, direct product factor, Sylow
154	C₂⁴	D₄×C₂₃	1	1	yes	characteristic
155	#41	—	4	2	no	—
156	#41	—	4	2	no	—
157	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	C₂³×C₂₃	1	2	yes	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	—	4	1	no	—
164	C₂²×C₄	—	4	1	no	—
165	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
166	C₂²×C₄	D₄×C₂₃	1	2	yes	—
167	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
168	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
169	Q₁₆	—	4	1	no	—
170	SD₁₆	—	4	2	no	—
171	D₈	—	4	1	no	—
172	M₁₆	—	4	1	no	—
173	M₁₆	—	4	1	no	—
174	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
175	C₂×C₈	C₂³×C₂₃	1	2	yes	—
176	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
180	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₄²	—	2	2	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	4	1	no	—
189	C₂³	—	4	2	no	—
190	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
191	C₂³	C₂×D₄×C₂₃	1	2	yes	—
192	Q₈	—	2	2	no	—
193	Q₈	—	2	2	no	—
194	D₄	—	2	2	no	—
195	D₄	—	2	2	no	—
196	D₄	—	4	4	no	—
197	D₄	—	4	4	no	—
198	D₄	—	4	4	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
210	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
211	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
212	C₈	—	2	1	no	—
213	C₈	—	2	2	no	—
214	C₈	—	2	1	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	4	4	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	2	1	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	2	no	—
229	C₄	—	4	2	no	—
230	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
231	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
232	C₂	—	4	1	no	—
233	C₂	—	4	2	no	—
234	C₂	—	8	2	no	—
235	C₂	C₂₃×#534	1	1	yes	characteristic
236	C₂	C₂₃×#575	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#525	C₂	semidirect
2. C₂₃×#485	C₂	semidirect
3. C₂₃×#482	C₂	semidirect
4. C₂₃×#470	C₂	semidirect
5. C₂₃×#469	C₂	semidirect
6. C₂₃×#468	C₂	semidirect
7. C₂₃×#452	C₂	semidirect
8. C₂₃×#449	C₂	semidirect
9. C₂₃×#407	C₂	semidirect
12. C₂×Q₁₆×C₂₃	C₂²	semidirect
13. C₂×SD₁₆×C₂₃	C₂²	semidirect
14. C₂×D₈×C₂₃	C₂²	semidirect
26. C₂₃×#107	C₂²	semidirect
27. C₂₃×#106	C₂²	semidirect
28. C₂₃×#106	C₂²	semidirect
29. C₂₃×#103	C₂²	semidirect
30. C₂₃×#103	C₂²	semidirect
31. C₂₃×#102	C₂²	semidirect
32. C₂₃×#102	C₂²	semidirect
96. #3197	C₂₃	direct (N × H)
114. #525	C₂×C₂₃	semidirect
115. #485	C₂×C₂₃	semidirect
116. #482	C₂×C₂₃	semidirect
117. #470	C₂×C₂₃	semidirect
118. #469	C₂×C₂₃	semidirect
119. #468	C₂×C₂₃	semidirect
120. #452	C₂×C₂₃	semidirect
121. #449	C₂×C₂₃	semidirect
122. #407	C₂×C₂₃	semidirect
130. C₂×Q₁₆	C₂²×C₂₃	semidirect
131. C₂×SD₁₆	C₂²×C₂₃	semidirect
132. C₂×D₈	C₂²×C₂₃	semidirect
144. #107	C₂²×C₂₃	semidirect
145. #106	C₂²×C₂₃	semidirect
146. #106	C₂²×C₂₃	semidirect
147. #103	C₂²×C₂₃	semidirect
148. #103	C₂²×C₂₃	semidirect
149. #102	C₂²×C₂₃	semidirect
150. #102	C₂²×C₂₃	semidirect
153. C₂₃	#3197	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#525	2
C₂₃×#485	2
C₂₃×#482	2
C₂₃×#470	2
C₂₃×#469	2
C₂₃×#468	2
C₂₃×#452	2
C₂₃×#449	2
C₂₃×#407	2
#3197	23

Subgroups — C23×#3197 #15670816

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3197 #15670816