Subgroups — C₂₃×#3398 #15671017

Order 2944.

350 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#556	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#497	C₂	1	2	yes	maximal
4	C₂₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#464	C₂	1	2	yes	maximal
6	C₂₃×#462	C₂	1	2	yes	maximal, complemented
7	C₂₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#461	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#441	C₂	1	2	yes	maximal, complemented
11	C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂×Q₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
17	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
20	C₂₃×#126	—	2	2	no	—
21	C₂₃×#125	—	2	1	no	—
22	C₂₃×#125	—	2	2	no	—
23	C₂₃×#124	C₂²	1	2	yes	—
24	C₂₃×C₂²⋊Q₈	—	2	1	no	—
25	C₄×Q₈×C₂₃	—	2	1	no	—
26	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
27	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
29	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
30	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×#117	—	2	1	no	—
32	C₂₃×#106	C₂²	1	1	yes	characteristic
33	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
35	C₂₃×#103	C₂²	1	1	yes	characteristic
36	C₂₃×#103	C₂²	1	2	yes	—
37	C₂₃×#103	C₂²	1	2	yes	—
38	C₂₃×#102	C₂²	1	1	yes	characteristic
39	C₂₃×#97	C₂²	1	2	yes	—
40	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
43	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
45	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	—	2	1	no	—
47	Q₁₆×C₂₃	—	2	4	no	—
48	Q₁₆×C₂₃	—	4	2	no	—
49	Q₁₆×C₂₃	—	4	2	no	—
50	SD₁₆×C₂₃	—	2	2	no	—
51	SD₁₆×C₂₃	—	4	2	no	—
52	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₂₃	C₂³	1	2	yes	—
54	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₂₃	—	2	1	no	—
56	C₄⋊C₄×C₂₃	—	2	2	no	—
57	C₄⋊C₄×C₂₃	—	2	1	no	—
58	C₄⋊C₄×C₂₃	—	2	2	no	—
59	C₄⋊C₄×C₂₃	—	2	2	no	—
60	C₄⋊C₄×C₂₃	—	2	1	no	—
61	C₄⋊C₄×C₂₃	—	2	2	no	—
62	C₄⋊C₄×C₂₃	—	2	1	no	—
63	C₄⋊C₄×C₂₃	—	2	1	no	—
64	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
66	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₂²⋊C₄×C₂₃	—	2	2	no	—
69	C₂²⋊C₄×C₂₃	—	2	1	no	—
70	C₂²⋊C₄×C₂₃	—	2	2	no	—
71	C₄²×C₂₃	—	2	2	no	—
72	C₄²×C₂₃	—	2	2	no	—
73	C₄²×C₂₃	—	2	1	no	—
74	C₄²×C₂₃	C₂³	1	1	yes	characteristic
75	C₄²×C₂₃	C₂³	1	2	yes	—
76	C₂³×C₂₃	—	2	1	no	—
77	Q₈×C₂₃	—	2	2	no	—
78	Q₈×C₂₃	—	2	1	no	—
79	Q₈×C₂₃	—	2	2	no	—
80	Q₈×C₂₃	—	2	2	no	—
81	Q₈×C₂₃	—	2	2	no	—
82	Q₈×C₂₃	—	2	2	no	—
83	Q₈×C₂₃	—	2	1	no	—
84	D₄×C₂₃	—	2	1	no	—
85	D₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	1	no	—
87	C₂×C₄×C₂₃	—	2	1	no	—
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	2	1	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	4	2	no	—
92	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
97	C₈×C₂₃	—	2	2	no	—
98	C₈×C₂₃	—	2	1	no	—
99	C₈×C₂₃	#41	1	2	yes	—
100	#3398	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂²×C₂₃	—	4	1	no	—
102	C₂²×C₂₃	—	4	1	no	—
103	C₂²×C₂₃	—	4	1	no	—
104	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₂₃	—	2	2	no	—
106	C₄×C₂₃	—	2	2	no	—
107	C₄×C₂₃	—	2	1	no	—
108	C₄×C₂₃	—	4	2	no	—
109	C₄×C₂₃	—	4	1	no	—
110	C₄×C₂₃	—	4	2	no	—
111	C₄×C₂₃	—	4	1	no	—
112	C₄×C₂₃	—	4	1	no	—
113	C₄×C₂₃	#137	1	2	yes	—
114	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
115	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
116	#557	C₂×C₂₃	1	1	yes	characteristic, complemented
117	#556	C₂×C₂₃	1	1	yes	characteristic
118	#497	C₂×C₂₃	1	2	yes	—
119	#494	C₂×C₂₃	1	1	yes	characteristic, complemented
120	#464	C₂×C₂₃	1	2	yes	—
121	#462	C₂×C₂₃	1	2	yes	complemented
122	#462	C₂×C₂₃	1	1	yes	characteristic, complemented
123	#461	C₂×C₂₃	1	1	yes	characteristic
124	#446	C₂×C₂₃	1	1	yes	characteristic, complemented
125	#441	C₂×C₂₃	1	2	yes	complemented
126	C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic
127	C₂×C₂₃	—	8	1	no	—
128	C₂×C₂₃	#550	1	1	yes	characteristic
129	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
130	C₂×C₂₃	#578	1	1	yes	characteristic
131	C₂×Q₁₆	C₂²×C₂₃	1	2	yes	—
132	C₂×Q₁₆	C₂²×C₂₃	1	2	yes	—
133	C₂×Q₁₆	C₂²×C₂₃	1	2	yes	—
134	C₂×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
135	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
136	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
137	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
138	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
139	#126	—	2	2	no	—
140	#125	—	2	1	no	—
141	#125	—	2	2	no	—
142	#124	C₂²×C₂₃	1	2	yes	—
143	C₂²⋊Q₈	—	2	1	no	—
144	C₄×Q₈	—	2	1	no	—
145	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
146	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
147	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
148	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
149	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
150	#117	—	2	1	no	—
151	#106	C₂²×C₂₃	1	1	yes	characteristic
152	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
153	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
154	#103	C₂²×C₂₃	1	1	yes	characteristic
155	#103	C₂²×C₂₃	1	2	yes	—
156	#103	C₂²×C₂₃	1	2	yes	—
157	#102	C₂²×C₂₃	1	1	yes	characteristic
158	#97	C₂²×C₂₃	1	2	yes	—
159	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
160	C₂₃	#3398	1	1	yes	characteristic, direct product factor, Sylow
161	C₂×Q₈	D₄×C₂₃	1	1	yes	characteristic
162	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
163	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
164	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
165	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
166	C₂²×C₄	—	2	1	no	—
167	Q₁₆	—	2	4	no	—
168	Q₁₆	—	4	2	no	—
169	Q₁₆	—	4	2	no	—
170	SD₁₆	—	2	2	no	—
171	SD₁₆	—	4	2	no	—
172	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
173	C₂×C₈	C₂³×C₂₃	1	2	yes	—
174	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	1	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
185	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
186	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
187	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	1	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₄²	—	2	2	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	1	no	—
194	C₄²	C₂³×C₂₃	1	1	yes	characteristic
195	C₄²	C₂³×C₂₃	1	2	yes	—
196	C₂³	—	2	1	no	—
197	Q₈	—	2	2	no	—
198	Q₈	—	2	1	no	—
199	Q₈	—	2	2	no	—
200	Q₈	—	2	2	no	—
201	Q₈	—	2	2	no	—
202	Q₈	—	2	2	no	—
203	Q₈	—	2	1	no	—
204	D₄	—	2	1	no	—
205	D₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
213	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
214	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
215	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
216	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
217	C₈	—	2	2	no	—
218	C₈	—	2	1	no	—
219	C₈	#41×C₂₃	1	2	yes	—
220	C₂²	—	4	1	no	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	1	no	—
223	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	1	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	1	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	1	no	—
231	C₄	—	4	1	no	—
232	C₄	C₂₃×#137	1	2	yes	—
233	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
234	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
235	C₂	—	8	1	no	—
236	C₂	C₂₃×#550	1	1	yes	characteristic
237	C₂	C₂×C₂₃×#137	1	1	yes	characteristic
238	C₂	C₂₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂×D₄×C₂₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#557	C₂	semidirect
4. C₂₃×#494	C₂	semidirect
6. C₂₃×#462	C₂	semidirect
7. C₂₃×#462	C₂	semidirect
9. C₂₃×#446	C₂	semidirect
10. C₂₃×#441	C₂	semidirect
100. #3398	C₂₃	direct (N × H)
116. #557	C₂×C₂₃	semidirect
119. #494	C₂×C₂₃	semidirect
121. #462	C₂×C₂₃	semidirect
122. #462	C₂×C₂₃	semidirect
124. #446	C₂×C₂₃	semidirect
125. #441	C₂×C₂₃	semidirect
160. C₂₃	#3398	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×#557	2
C₂₃×#556	2
C₂₃×#497	2
C₂₃×#494	2
C₂₃×#464	2
C₂₃×#462	2
C₂₃×#461	2
C₂₃×#446	2
C₂₃×#441	2
C₄×SD₁₆×C₂₃	2
#3398	23

Subgroups — C23×#3398 #15671017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3398 #15671017