Group #19155277 — subgroups

Subgroups — C₂₃×#7651 #19155277

Order 5888.

624 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×#127	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1831	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1270	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₃×#127	C₄	1	1	yes	characteristic
5	C₂×C₄×D₄×C₂₃	—	2	2	no	—
6	C₂×C₄×D₄×C₂₃	C₄	1	2	yes	—
7	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
8	C₂₃×#390	—	2	2	no	—
9	C₂₃×#390	C₂²	1	2	yes	—
10	C₂₃×#384	C₄	1	1	yes	characteristic
11	C₄³×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#336	—	2	2	no	—
13	C₂₃×#336	C₂²	1	4	yes	—
14	C₂²×D₄×C₂₃	—	2	2	no	—
15	C₂²×D₄×C₂₃	C₂×C₄	1	2	yes	—
16	C₂³×C₄×C₂₃	—	2	2	no	—
17	C₂²×C₈×C₂₃	—	2	4	no	—
18	C₂₃×#127	—	2	4	no	—
19	C₄×D₄×C₂₃	—	4	4	no	—
20	C₄×D₄×C₂₃	—	4	4	no	—
21	C₄×D₄×C₂₃	—	4	4	no	—
22	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
23	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
24	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
25	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
26	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₂₃	D₄	1	2	yes	—
29	C₂×C₄²×C₂₃	D₄	1	2	yes	—
30	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
31	C₂₃×C₄⋊C₈	—	4	4	no	—
32	C₂⁴×C₂₃	—	2	2	no	—
33	C₂×D₄×C₂₃	—	2	8	no	—
34	C₂×D₄×C₂₃	—	2	2	no	—
35	C₂×D₄×C₂₃	—	4	2	no	—
36	C₂×D₄×C₂₃	—	4	2	no	—
37	C₂×D₄×C₂₃	—	4	4	no	—
38	C₂×D₄×C₂₃	—	4	4	no	—
39	C₂×D₄×C₂₃	M₁₆	1	4	yes	—
40	C₂²×C₄×C₂₃	—	2	2	no	—
41	C₂²×C₄×C₂₃	—	4	4	no	—
42	C₂²×C₄×C₂₃	—	4	8	no	—
43	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
48	C₂×C₈×C₂₃	—	4	4	no	—
49	C₂×C₈×C₂₃	—	4	4	no	—
50	C₂×C₈×C₂₃	—	4	4	no	—
51	C₄⋊C₄×C₂₃	—	4	2	no	—
52	C₄⋊C₄×C₂₃	—	4	4	no	—
53	C₂²⋊C₄×C₂₃	—	4	4	no	—
54	C₂²⋊C₄×C₂₃	—	4	4	no	—
55	C₂²⋊C₄×C₂₃	—	4	4	no	—
56	C₄²×C₂₃	—	2	2	no	—
57	C₄²×C₂₃	—	2	2	no	—
58	C₄²×C₂₃	—	4	2	no	—
59	C₄²×C₂₃	—	4	1	no	—
60	C₄²×C₂₃	D₈	1	4	yes	—
61	C₄²×C₂₃	SD₁₆	1	4	yes	—
62	#7651	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₂₃	—	2	4	no	—
64	C₂³×C₂₃	—	4	2	no	—
65	C₂³×C₂₃	—	4	2	no	—
66	C₂³×C₂₃	—	4	4	no	—
67	C₂³×C₂₃	—	4	4	no	—
68	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	D₄×C₂₃	—	4	4	no	—
70	D₄×C₂₃	—	4	4	no	—
71	D₄×C₂₃	—	4	16	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	1	no	—
75	C₂×C₄×C₂₃	—	2	1	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	2	1	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	4	2	no	—
81	C₂×C₄×C₂₃	—	4	4	no	—
82	C₂×C₄×C₂₃	—	8	8	no	—
83	C₂×C₄×C₂₃	#100	1	2	yes	—
84	C₂×C₄×C₂₃	#102	1	4	yes	—
85	C₂×C₄×C₂₃	#102	1	4	yes	—
86	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
87	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
89	C₈×C₂₃	—	8	4	no	—
90	C₄×#127	C₂×C₂₃	1	1	yes	characteristic
91	#1831	C₂×C₂₃	1	2	yes	complemented
92	#1270	C₂×C₂₃	1	4	yes	complemented
93	C₂²×C₂₃	—	4	4	no	—
94	C₂²×C₂₃	—	4	8	no	—
95	C₂²×C₂₃	—	8	2	no	—
96	C₂²×C₂₃	—	8	2	no	—
97	C₂²×C₂₃	—	8	4	no	—
98	C₂²×C₂₃	#407	1	1	yes	characteristic
99	C₂²×C₂₃	#410	1	1	yes	characteristic
100	C₂²×C₂₃	C₂×#100	1	1	yes	characteristic
101	C₂²×C₂₃	C₂×#102	1	2	yes	—
102	C₂²×C₂₃	#421	1	2	yes	—
103	C₄×C₂₃	—	2	2	no	—
104	C₄×C₂₃	—	2	4	no	—
105	C₄×C₂₃	—	2	2	no	—
106	C₄×C₂₃	—	2	2	no	—
107	C₄×C₂₃	—	4	1	no	—
108	C₄×C₂₃	—	8	2	no	—
109	C₄×C₂₃	#331	1	4	yes	—
110	C₂×#127	C₄×C₂₃	1	1	yes	characteristic
111	C₂×C₄×D₄	—	2	2	no	—
112	C₂×C₄×D₄	C₄×C₂₃	1	2	yes	—
113	C₂×C₄⋊C₈	—	2	2	no	—
114	#390	—	2	2	no	—
115	#390	C₂²×C₂₃	1	2	yes	—
116	#384	C₄×C₂₃	1	1	yes	characteristic
117	C₄³	C₂²×C₂₃	1	1	yes	characteristic
118	#336	—	2	2	no	—
119	#336	C₂²×C₂₃	1	4	yes	—
120	C₂×C₂₃	—	8	4	no	—
121	C₂×C₂₃	#1466	1	1	yes	characteristic
122	C₂×C₂₃	#1486	1	2	yes	—
123	C₂×C₂₃	#1517	1	2	yes	—
124	C₂×C₂₃	#1536	1	1	yes	characteristic
125	C₂×C₂₃	C₂×#331	1	1	yes	characteristic
126	C₂²×D₄	—	2	2	no	—
127	C₂²×D₄	C₂×C₄×C₂₃	1	2	yes	—
128	C₂³×C₄	—	2	2	no	—
129	C₂²×C₈	—	2	4	no	—
130	#127	—	2	4	no	—
131	C₄×D₄	—	4	4	no	—
132	C₄×D₄	—	4	4	no	—
133	C₄×D₄	—	4	4	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
136	C₂×C₂²⋊C₄	—	2	2	no	—
137	C₂×C₂²⋊C₄	—	2	4	no	—
138	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
139	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
140	C₂×C₄²	D₄×C₂₃	1	2	yes	—
141	C₂×C₄²	D₄×C₂₃	1	2	yes	—
142	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
143	C₄⋊C₈	—	4	4	no	—
144	C₂₃	#7651	1	1	yes	characteristic, direct product factor, Sylow
145	C₂⁴	—	2	2	no	—
146	C₂×D₄	—	2	8	no	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	—	4	2	no	—
149	C₂×D₄	—	4	2	no	—
150	C₂×D₄	—	4	4	no	—
151	C₂×D₄	—	4	4	no	—
152	C₂×D₄	M₁₆×C₂₃	1	4	yes	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	4	4	no	—
155	C₂²×C₄	—	4	8	no	—
156	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
157	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
158	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
159	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
160	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
161	C₂×C₈	—	4	4	no	—
162	C₂×C₈	—	4	4	no	—
163	C₂×C₈	—	4	4	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₂²⋊C₄	—	4	4	no	—
167	C₂²⋊C₄	—	4	4	no	—
168	C₂²⋊C₄	—	4	4	no	—
169	C₄²	—	2	2	no	—
170	C₄²	—	2	2	no	—
171	C₄²	—	4	2	no	—
172	C₄²	—	4	1	no	—
173	C₄²	D₈×C₂₃	1	4	yes	—
174	C₄²	SD₁₆×C₂₃	1	4	yes	—
175	C₂³	—	2	4	no	—
176	C₂³	—	4	2	no	—
177	C₂³	—	4	2	no	—
178	C₂³	—	4	4	no	—
179	C₂³	—	4	4	no	—
180	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
181	D₄	—	4	4	no	—
182	D₄	—	4	4	no	—
183	D₄	—	4	16	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	—	8	8	no	—
195	C₂×C₄	C₂₃×#100	1	2	yes	—
196	C₂×C₄	C₂₃×#102	1	4	yes	—
197	C₂×C₄	C₂₃×#102	1	4	yes	—
198	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
199	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
200	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
201	C₈	—	8	4	no	—
202	C₂²	—	4	4	no	—
203	C₂²	—	4	8	no	—
204	C₂²	—	8	4	no	—
205	C₂²	—	8	2	no	—
206	C₂²	—	8	2	no	—
207	C₂²	C₂₃×#407	1	1	yes	characteristic
208	C₂²	C₂₃×#410	1	1	yes	characteristic
209	C₂²	C₂×C₂₃×#100	1	1	yes	characteristic
210	C₂²	C₂×C₂₃×#102	1	2	yes	—
211	C₂²	C₂₃×#421	1	2	yes	—
212	C₄	—	2	2	no	—
213	C₄	—	2	4	no	—
214	C₄	—	2	2	no	—
215	C₄	—	2	2	no	—
216	C₄	—	4	1	no	—
217	C₄	—	8	2	no	—
218	C₄	C₂₃×#331	1	4	yes	—
219	C₂	—	8	4	no	—
220	C₂	C₂₃×#1466	1	1	yes	characteristic
221	C₂	C₂₃×#1486	1	2	yes	—
222	C₂	C₂₃×#1517	1	2	yes	—
223	C₂	C₂₃×#1536	1	1	yes	characteristic
224	C₂	C₂×C₂₃×#331	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₂₃×#127

C₂

yes

maximal, characteristic

C₂₃×#1831

C₂

yes

maximal, complemented

C₂₃×#1270

C₂

yes

maximal, complemented

C₂×C₂₃×#127

C₄

yes

characteristic

C₂×C₄×D₄×C₂₃

—

C₂×C₄×D₄×C₂₃

C₄

yes

—

C₂×C₂₃×C₄⋊C₈

—

C₂₃×#390

—

C₂₃×#390

C₂²

yes

—

C₂₃×#384

C₄

yes

characteristic

C₄³×C₂₃

C₂²

yes

characteristic

C₂₃×#336

—

C₂₃×#336

C₂²

yes

—

C₂²×D₄×C₂₃

—

C₂²×D₄×C₂₃

C₂×C₄

yes

—

C₂³×C₄×C₂₃

—

C₂²×C₈×C₂₃

—

C₂₃×#127

—

C₄×D₄×C₂₃

—

C₄×D₄×C₂₃

—

C₄×D₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₂×C₂²⋊C₄×C₂₃

—

C₂×C₂²⋊C₄×C₂₃

—

C₂×C₄²×C₂₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₂₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₂₃

D₄

yes

—

C₂×C₄²×C₂₃

D₄

yes

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×C₄⋊C₈

—

C₂⁴×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

M₁₆

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

D₈

yes

—

C₄²×C₂₃

SD₁₆

yes

—

#7651

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#100

yes

—

C₂×C₄×C₂₃

#102

yes

—

C₂×C₄×C₂₃

#102

yes

—

C₂×C₄×C₂₃

C₂×M₁₆

yes

—

C₂×C₄×C₂₃

C₂×D₈

yes

—

C₂×C₄×C₂₃

C₂×SD₁₆

yes

—

C₈×C₂₃

—

C₄×#127

C₂×C₂₃

yes

characteristic

#1831

C₂×C₂₃

yes

complemented

#1270

C₂×C₂₃

yes

complemented

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

yes

C₂²×C₂₃

yes

100

C₂²×C₂₃

C₂×#100

yes

characteristic

101

C₂²×C₂₃

C₂×#102

yes

—

102

C₂²×C₂₃

#421

yes

—

103

C₄×C₂₃

—

104

C₄×C₂₃

—

105

C₄×C₂₃

—

106

C₄×C₂₃

—

107

C₄×C₂₃

—

108

C₄×C₂₃

—

109

C₄×C₂₃

#331

yes

—

110

C₂×#127

C₄×C₂₃

yes

characteristic

111

C₂×C₄×D₄

—

112

C₂×C₄×D₄

C₄×C₂₃

yes

—

113

C₂×C₄⋊C₈

—

114

#390

—

115

#390

C₂²×C₂₃

yes

—

116

#384

C₄×C₂₃

yes

characteristic

117

C₄³

C₂²×C₂₃

yes

characteristic

118

#336

—

119

#336

C₂²×C₂₃

yes

—

120

C₂×C₂₃

—

121

C₂×C₂₃

#1466

yes

characteristic

122

C₂×C₂₃

#1486

yes

—

123

C₂×C₂₃

#1517

yes

—

124

C₂×C₂₃

#1536

yes

characteristic

125

C₂×C₂₃

C₂×#331

yes

characteristic

126

C₂²×D₄

—

127

C₂²×D₄

C₂×C₄×C₂₃

yes

—

128

C₂³×C₄

—

129

C₂²×C₈

—

130

#127

—

131

C₄×D₄

—

132

C₄×D₄

—

133

C₄×D₄

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄⋊C₄

C₂×C₄×C₂₃

yes

—

136

C₂×C₂²⋊C₄

—

137

C₂×C₂²⋊C₄

—

138

C₂×C₄²

C₂×C₄×C₂₃

yes

characteristic

139

C₂×C₄²

C₂×C₄×C₂₃

yes

characteristic

140

C₂×C₄²

D₄×C₂₃

yes

—

141

C₂×C₄²

D₄×C₂₃

yes

—

142

C₂×C₄²

C₂³×C₂₃

yes

characteristic

143

C₄⋊C₈

—

144

C₂₃

#7651

yes

characteristic, direct product factor, Sylow

145

C₂⁴

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

M₁₆×C₂₃

yes

—

153

C₂²×C₄

—

154

C₂²×C₄

—

155

C₂²×C₄

—

156

C₂²×C₄

C₂²⋊C₄×C₂₃

yes

characteristic

157

C₂²×C₄

C₂²⋊C₄×C₂₃

yes

—

158

C₂²×C₄

C₂²⋊C₄×C₂₃

yes

characteristic

159

C₂²×C₄

C₂²×C₄×C₂₃

yes

characteristic

160

C₂²×C₄

C₂×D₄×C₂₃

yes

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₂²⋊C₄

—

167

C₂²⋊C₄

—

168

C₂²⋊C₄

—

169

C₄²

—

170

C₄²

—

171

C₄²

—

172

C₄²

—

173

C₄²

D₈×C₂₃

yes

—

174

C₄²

SD₁₆×C₂₃

yes

—

175

C₂³

—

176

C₂³

—

177

C₂³

—

178

C₂³

—

179

C₂³

—

180

C₂³

C₂×C₂²⋊C₄×C₂₃

yes

characteristic

181

D₄

—

182

D₄

—

183

D₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

C₂₃×#100

yes

—

196

C₂×C₄

C₂₃×#102

yes

—

197

C₂×C₄

C₂₃×#102

yes

—

198

C₂×C₄

C₂×M₁₆×C₂₃

yes

—

199

C₂×C₄

C₂×D₈×C₂₃

yes

—

200

C₂×C₄

C₂×SD₁₆×C₂₃

yes

—

201

C₈

—

202

C₂²

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

C₂₃×#407

yes

characteristic

208

C₂²

C₂₃×#410

yes

characteristic

209

C₂²

C₂×C₂₃×#100

yes

characteristic

210

C₂²

C₂×C₂₃×#102

yes

—

211

C₂²

C₂₃×#421

yes

—

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

C₂₃×#331

yes

—

219

C₂

—

220

C₂

C₂₃×#1466

yes

characteristic

221

C₂

C₂₃×#1486

yes

—

222

C₂

C₂₃×#1517

yes

—

223

C₂

C₂₃×#1536

yes

characteristic

224

C₂

C₂×C₂₃×#331

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂×C₂₃×#127	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₂₃

C₂×C₂²⋊C₄

commutator

C₂²×C₄

C₂²×C₄×C₂₃

Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

prime radical

C₂×C₂₃×#127

C₄

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

N	H ≅ G/N	type
2. C₂₃×#1831	C₂	semidirect
3. C₂₃×#1270	C₂	semidirect
62. #7651	C₂₃	direct (N × H)
91. #1831	C₂×C₂₃	semidirect
92. #1270	C₂×C₂₃	semidirect
144. C₂₃	#7651	direct (N × H)

H ≅ G/N

type

2. C₂₃×#1831

C₂

semidirect

3. C₂₃×#1270

C₂

semidirect

62. #7651

C₂₃

direct (N × H)

91. #1831

C₂×C₂₃

semidirect

92. #1270

C₂×C₂₃

semidirect

144. C₂₃

#7651

direct (N × H)

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

subgroup	index
C₄×C₂₃×#127	2
C₂₃×#1831	2
C₂₃×#1270	2
#7651	23

subgroup

index

C₄×C₂₃×#127

C₂₃×#1831

C₂₃×#1270

#7651

Subgroups — C₂₃×#7651 #19155277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#7651 #19155277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7651 #19155277