Subgroups — C₂₃×#2784 #15670403

Order 2944.

472 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×#400	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#396	C₂	1	2	yes	maximal, complemented
3	C₂₃×#395	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#393	C₂	1	2	yes	maximal, complemented
5	C₂₃×#390	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#387	C₂	1	2	yes	maximal, complemented
7	C₂₃×#386	C₂	1	2	yes	maximal
8	C₂₃×#385	C₂	1	2	yes	maximal, complemented
9	C₂₃×#380	C₂	1	2	yes	maximal, complemented
10	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#95	C₂²	1	1	yes	characteristic
25	C₂₃×#95	C₂²	1	2	yes	—
26	C₂₃×#95	C₂²	1	1	yes	characteristic
27	C₂₃×#95	C₂²	1	2	yes	—
28	C₂₃×#95	C₂²	1	2	yes	—
29	C₂₃×#95	C₂²	1	2	yes	—
30	C₂₃×#95	C₂²	1	2	yes	—
31	C₂⁴×C₂₃	C₂³	1	2	yes	—
32	C₂×Q₈×C₂₃	—	4	2	no	—
33	C₂×Q₈×C₂₃	—	4	1	no	—
34	C₂×D₄×C₂₃	—	4	1	no	—
35	C₂×D₄×C₂₃	—	4	1	no	—
36	C₂×D₄×C₂₃	Q₈	1	4	yes	complemented
37	C₂²×C₄×C₂₃	—	2	4	no	—
38	C₂²×C₄×C₂₃	—	4	2	no	—
39	C₂²×C₄×C₂₃	—	4	2	no	—
40	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
48	C₄⋊C₄×C₂₃	—	2	4	no	—
49	C₄⋊C₄×C₂₃	—	4	2	no	—
50	C₄⋊C₄×C₂₃	—	4	2	no	—
51	C₄⋊C₄×C₂₃	—	4	1	no	—
52	C₄⋊C₄×C₂₃	—	4	1	no	—
53	C₂²⋊C₄×C₂₃	—	2	4	no	—
54	C₂²⋊C₄×C₂₃	—	4	2	no	—
55	C₂²⋊C₄×C₂₃	—	4	2	no	—
56	C₂²⋊C₄×C₂₃	—	4	2	no	—
57	C₂²⋊C₄×C₂₃	—	4	2	no	—
58	C₄²×C₂₃	—	4	1	no	—
59	C₂³×C₂₃	—	2	2	no	—
60	C₂³×C₂₃	—	2	2	no	—
61	C₂³×C₂₃	—	2	2	no	—
62	C₂³×C₂₃	—	2	2	no	—
63	C₂³×C₂₃	—	2	2	no	—
64	C₂³×C₂₃	—	2	2	no	—
65	C₂³×C₂₃	C₂×Q₈	1	4	yes	—
66	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
67	Q₈×C₂₃	—	4	4	no	—
68	D₄×C₂₃	—	4	4	no	—
69	C₂×C₄×C₂₃	—	2	1	no	—
70	C₂×C₄×C₂₃	—	2	2	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	1	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	2	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	1	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	4	8	no	—
90	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
91	C₂×C₄×C₂₃	#41	1	4	yes	—
92	#2784	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂²×C₂₃	—	2	4	no	—
94	C₂²×C₂₃	—	2	4	no	—
95	C₂²×C₂₃	—	2	4	no	—
96	C₂²×C₂₃	—	4	2	no	—
97	C₂²×C₂₃	—	4	2	no	—
98	C₂²×C₂₃	—	4	2	no	—
99	C₂²×C₂₃	—	4	2	no	—
100	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
101	C₂²×C₂₃	C₂×#41	1	2	yes	—
102	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
103	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
104	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
105	C₄×C₂₃	—	2	4	no	—
106	C₄×C₂₃	—	2	2	no	—
107	C₄×C₂₃	—	4	2	no	—
108	C₄×C₂₃	—	4	2	no	—
109	C₄×C₂₃	—	4	2	no	—
110	C₄×C₂₃	—	4	1	no	—
111	C₄×C₂₃	—	4	1	no	—
112	C₄×C₂₃	—	4	2	no	—
113	#400	C₂×C₂₃	1	1	yes	characteristic, complemented
114	#396	C₂×C₂₃	1	2	yes	complemented
115	#395	C₂×C₂₃	1	1	yes	characteristic, complemented
116	#393	C₂×C₂₃	1	2	yes	complemented
117	#390	C₂×C₂₃	1	1	yes	characteristic
118	#387	C₂×C₂₃	1	2	yes	complemented
119	#386	C₂×C₂₃	1	2	yes	—
120	#385	C₂×C₂₃	1	2	yes	complemented
121	#380	C₂×C₂₃	1	2	yes	complemented
122	C₂×C₂₃	—	4	4	no	—
123	C₂×C₂₃	#538	1	2	yes	—
124	C₂×C₂₃	#543	1	1	yes	characteristic
125	C₂×C₂₃	#554	1	2	yes	—
126	C₂×C₂₃	#555	1	1	yes	characteristic
127	C₂×C₂₃	#560	1	1	yes	characteristic
128	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
129	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
130	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
131	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
132	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
133	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
134	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
135	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
136	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
137	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
138	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
139	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
140	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
141	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
142	#95	C₂²×C₂₃	1	1	yes	characteristic
143	#95	C₂²×C₂₃	1	2	yes	—
144	#95	C₂²×C₂₃	1	1	yes	characteristic
145	#95	C₂²×C₂₃	1	2	yes	—
146	#95	C₂²×C₂₃	1	2	yes	—
147	#95	C₂²×C₂₃	1	2	yes	—
148	#95	C₂²×C₂₃	1	2	yes	—
149	C₂₃	#2784	1	1	yes	characteristic, direct product factor, Sylow
150	C₂⁴	C₂³×C₂₃	1	2	yes	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	1	no	—
153	C₂×D₄	—	4	1	no	—
154	C₂×D₄	—	4	1	no	—
155	C₂×D₄	Q₈×C₂₃	1	4	yes	complemented
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	—	4	2	no	—
159	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
160	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
161	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
162	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
164	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
166	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	4	1	no	—
171	C₄⋊C₄	—	4	1	no	—
172	C₂²⋊C₄	—	2	4	no	—
173	C₂²⋊C₄	—	4	2	no	—
174	C₂²⋊C₄	—	4	2	no	—
175	C₂²⋊C₄	—	4	2	no	—
176	C₂²⋊C₄	—	4	2	no	—
177	C₄²	—	4	1	no	—
178	C₂³	—	2	2	no	—
179	C₂³	—	2	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	C₂×Q₈×C₂₃	1	4	yes	—
185	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
186	Q₈	—	4	4	no	—
187	D₄	—	4	4	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	4	8	no	—
209	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
210	C₂×C₄	#41×C₂₃	1	4	yes	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	4	no	—
213	C₂²	—	2	4	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	C₂²×Q₈×C₂₃	1	1	yes	characteristic
219	C₂²	C₂×#41×C₂₃	1	2	yes	—
220	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
221	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
222	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
223	C₄	—	2	4	no	—
224	C₄	—	2	2	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	1	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	1	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	2	no	—
231	C₂	—	4	4	no	—
232	C₂	C₂₃×#538	1	2	yes	—
233	C₂	C₂₃×#543	1	1	yes	characteristic
234	C₂	C₂₃×#554	1	2	yes	—
235	C₂	C₂₃×#555	1	1	yes	characteristic
236	C₂	C₂₃×#560	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂₃×#400	C₂	semidirect
2. C₂₃×#396	C₂	semidirect
3. C₂₃×#395	C₂	semidirect
4. C₂₃×#393	C₂	semidirect
6. C₂₃×#387	C₂	semidirect
8. C₂₃×#385	C₂	semidirect
9. C₂₃×#380	C₂	semidirect
36. C₂×D₄×C₂₃	Q₈	semidirect
92. #2784	C₂₃	direct (N × H)
113. #400	C₂×C₂₃	semidirect
114. #396	C₂×C₂₃	semidirect
115. #395	C₂×C₂₃	semidirect
116. #393	C₂×C₂₃	semidirect
118. #387	C₂×C₂₃	semidirect
120. #385	C₂×C₂₃	semidirect
121. #380	C₂×C₂₃	semidirect
149. C₂₃	#2784	direct (N × H)
155. C₂×D₄	Q₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#400	2
C₂₃×#396	2
C₂₃×#395	2
C₂₃×#393	2
C₂₃×#390	2
C₂₃×#387	2
C₂₃×#386	2
C₂₃×#385	2
C₂₃×#380	2
#2784	23

Subgroups — C23×#2784 #15670403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2784 #15670403