Subgroups — C₂₃×#2625 #15670244

Order 2944.

464 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×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#400	C₂	1	2	yes	maximal, complemented
4	C₂₃×#398	C₂	1	2	yes	maximal, complemented
5	C₂₃×#397	C₂	1	4	yes	maximal
6	C₂₃×#391	C₂	1	2	yes	maximal, complemented
7	C₂₃×#389	C₂	1	2	yes	maximal, complemented
8	C₂₃×#380	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×C₂²⋊Q₈	—	2	4	no	—
12	C₂₃×#117	—	2	4	no	—
13	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	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×C₄²×C₂₃	C₂²	1	2	yes	—
23	C₂₃×#95	C₂²	1	2	yes	—
24	C₂₃×#95	C₂²	1	4	yes	—
25	C₂₃×#95	C₂²	1	4	yes	—
26	C₂₃×#95	C₂²	1	4	yes	—
27	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
28	C₂×Q₈×C₂₃	—	2	2	no	—
29	C₂×Q₈×C₂₃	—	2	4	no	—
30	C₂²×C₄×C₂₃	—	2	4	no	—
31	C₂²×C₄×C₂₃	D₄	1	4	yes	—
32	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
34	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
35	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
36	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₂₃	—	2	2	no	—
40	C₄⋊C₄×C₂₃	—	2	2	no	—
41	C₄⋊C₄×C₂₃	—	2	2	no	—
42	C₄⋊C₄×C₂₃	—	2	4	no	—
43	C₄⋊C₄×C₂₃	—	2	4	no	—
44	C₄⋊C₄×C₂₃	—	4	4	no	—
45	C₂²⋊C₄×C₂₃	—	2	4	no	—
46	C₂²⋊C₄×C₂₃	—	2	4	no	—
47	C₂²⋊C₄×C₂₃	—	4	2	no	—
48	C₄²×C₂₃	—	2	4	no	—
49	C₂³×C₂₃	—	2	2	no	—
50	C₂³×C₂₃	—	2	1	no	—
51	C₂³×C₂₃	—	2	2	no	—
52	C₂³×C₂₃	—	2	1	no	—
53	C₂³×C₂₃	C₂×D₄	1	2	yes	—
54	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
55	Q₈×C₂₃	—	4	4	no	—
56	C₂×C₄×C₂₃	—	2	2	no	—
57	C₂×C₄×C₂₃	—	2	8	no	—
58	C₂×C₄×C₂₃	—	2	2	no	—
59	C₂×C₄×C₂₃	—	2	2	no	—
60	C₂×C₄×C₂₃	—	2	2	no	—
61	C₂×C₄×C₂₃	—	2	2	no	—
62	C₂×C₄×C₂₃	—	2	4	no	—
63	C₂×C₄×C₂₃	—	2	4	no	—
64	C₂×C₄×C₂₃	—	2	4	no	—
65	C₂×C₄×C₂₃	—	2	2	no	—
66	C₂×C₄×C₂₃	—	2	2	no	—
67	C₂×C₄×C₂₃	—	2	4	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	2	4	no	—
71	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₂₃	#41	1	4	yes	—
74	#2625	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂²×C₂₃	—	2	2	no	—
76	C₂²×C₂₃	—	2	2	no	—
77	C₂²×C₂₃	—	2	2	no	—
78	C₂²×C₂₃	—	4	2	no	—
79	C₂²×C₂₃	—	4	2	no	—
80	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
81	C₂²×C₂₃	C₂×#41	1	2	yes	—
82	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
83	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
84	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
85	C₄×C₂₃	—	2	2	no	—
86	C₄×C₂₃	—	4	2	no	—
87	C₄×C₂₃	—	4	2	no	—
88	C₄×C₂₃	—	4	2	no	—
89	C₄×C₂₃	—	4	2	no	—
90	C₄×C₂₃	—	4	4	no	—
91	C₄×C₂₃	#123	1	4	yes	—
92	C₂×C₂²⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
93	C₂×#117	C₂×C₂₃	1	1	yes	characteristic
94	#400	C₂×C₂₃	1	2	yes	complemented
95	#398	C₂×C₂₃	1	2	yes	complemented
96	#397	C₂×C₂₃	1	4	yes	—
97	#391	C₂×C₂₃	1	2	yes	complemented
98	#389	C₂×C₂₃	1	2	yes	complemented
99	#380	C₂×C₂₃	1	1	yes	characteristic
100	C₂×C₂₃	—	4	2	no	—
101	C₂×C₂₃	C₂×#123	1	1	yes	characteristic
102	C₂×C₂₃	#536	1	1	yes	characteristic
103	C₂×C₂₃	#537	1	1	yes	characteristic
104	C₂×C₂₃	#541	1	2	yes	—
105	C₂×C₂₃	#542	1	2	yes	—
106	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
107	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
108	C₂²⋊Q₈	—	2	4	no	—
109	#117	—	2	4	no	—
110	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
111	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
112	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
113	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
114	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
115	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
116	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
117	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
118	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
119	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
120	#95	C₂²×C₂₃	1	2	yes	—
121	#95	C₂²×C₂₃	1	4	yes	—
122	#95	C₂²×C₂₃	1	4	yes	—
123	#95	C₂²×C₂₃	1	4	yes	—
124	C₂₃	#2625	1	1	yes	characteristic, direct product factor, Sylow
125	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
126	C₂×Q₈	—	2	2	no	—
127	C₂×Q₈	—	2	4	no	—
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	D₄×C₂₃	1	4	yes	—
130	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
131	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
132	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
133	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
134	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
135	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
136	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
137	C₄⋊C₄	—	2	2	no	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄⋊C₄	—	2	2	no	—
140	C₄⋊C₄	—	2	4	no	—
141	C₄⋊C₄	—	2	4	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₂²⋊C₄	—	2	4	no	—
144	C₂²⋊C₄	—	2	4	no	—
145	C₂²⋊C₄	—	4	2	no	—
146	C₄²	—	2	4	no	—
147	C₂³	—	2	2	no	—
148	C₂³	—	2	1	no	—
149	C₂³	—	2	2	no	—
150	C₂³	—	2	1	no	—
151	C₂³	C₂×D₄×C₂₃	1	2	yes	—
152	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
153	Q₈	—	4	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	8	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
170	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
171	C₂×C₄	#41×C₂₃	1	4	yes	—
172	C₂²	—	2	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	4	2	no	—
176	C₂²	—	4	2	no	—
177	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
178	C₂²	C₂×#41×C₂₃	1	2	yes	—
179	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
180	C₂²	C₂₃×2₋¹⁺⁴	1	2	yes	—
181	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	4	2	no	—
185	C₄	—	4	2	no	—
186	C₄	—	4	2	no	—
187	C₄	—	4	4	no	—
188	C₄	C₂₃×#123	1	4	yes	—
189	C₂	—	4	2	no	—
190	C₂	C₂×C₂₃×#123	1	1	yes	characteristic
191	C₂	C₂₃×#536	1	1	yes	characteristic
192	C₂	C₂₃×#537	1	1	yes	characteristic
193	C₂	C₂₃×#541	1	2	yes	—
194	C₂	C₂₃×#542	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂³
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
3. C₂₃×#400	C₂	semidirect
4. C₂₃×#398	C₂	semidirect
6. C₂₃×#391	C₂	semidirect
7. C₂₃×#389	C₂	semidirect
74. #2625	C₂₃	direct (N × H)
94. #400	C₂×C₂₃	semidirect
95. #398	C₂×C₂₃	semidirect
97. #391	C₂×C₂₃	semidirect
98. #389	C₂×C₂₃	semidirect
124. C₂₃	#2625	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×C₂²⋊Q₈	2
C₂×C₂₃×#117	2
C₂₃×#400	2
C₂₃×#398	2
C₂₃×#397	2
C₂₃×#391	2
C₂₃×#389	2
C₂₃×#380	2
#2625	23

Subgroups — C23×#2625 #15670244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2625 #15670244