Group #19158200 — subgroups

Subgroups — C₂₃×#10574 #19158200

Order 5888.

494 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 162 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	C₂₃×#2551	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1948	C₂	1	2	yes	maximal
3	C₂₃×#1844	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1296	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1279	C₂	1	2	yes	maximal
6	C₂×C₂₃×C₄⋊Q₈	—	2	2	no	—
7	C₂×C₂₃×#106	—	2	2	no	—
8	C₂×C₂₃×#106	—	2	2	no	—
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#395	—	2	2	no	—
12	C₂₃×#389	—	2	2	no	—
13	C₂₃×#384	C₂²	1	1	yes	characteristic
14	C₂₃×#384	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
16	C₂₃×#340	—	2	2	no	—
17	C₂²×Q₈×C₂₃	—	2	2	no	—
18	C₂²×C₈×C₂₃	—	2	1	no	—
19	C₂²×C₈×C₂₃	—	2	2	no	—
20	C₂₃×C₄⋊Q₈	—	4	4	no	—
21	C₂₃×C₄⋊Q₈	—	4	4	no	—
22	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
23	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
24	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
26	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
27	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
29	C₂×C₄²×C₂₃	—	2	2	no	—
30	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
31	C₂₃×#106	—	4	4	no	—
32	C₂₃×#106	—	4	4	no	—
33	C₂₃×C₄⋊C₈	—	2	2	no	—
34	C₂₃×C₄⋊C₈	Q₈	1	4	yes	complemented
35	C₄×C₈×C₂₃	—	2	1	no	—
36	C₄×C₈×C₂₃	Q₈	1	2	yes	complemented
37	C₂₃×#95	—	2	2	no	—
38	C₂₃×#95	—	2	2	no	—
39	C₂×Q₈×C₂₃	—	4	8	no	—
40	C₂×Q₈×C₂₃	—	8	2	no	—
41	C₂²×C₄×C₂₃	—	2	2	no	—
42	C₂²×C₄×C₂₃	—	2	2	no	—
43	C₂²×C₄×C₂₃	—	2	2	no	—
44	C₂²×C₄×C₂₃	—	4	1	no	—
45	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₂₃	—	2	2	no	—
48	C₂×C₈×C₂₃	—	2	2	no	—
49	C₂×C₈×C₂₃	—	2	2	no	—
50	C₂×C₈×C₂₃	—	2	4	no	—
51	C₂×C₈×C₂₃	—	4	2	no	—
52	C₂×C₈×C₂₃	—	4	2	no	—
53	C₄⋊C₄×C₂₃	—	2	2	no	—
54	C₄⋊C₄×C₂₃	—	2	2	no	—
55	C₄⋊C₄×C₂₃	—	4	2	no	—
56	C₄⋊C₄×C₂₃	—	4	8	no	—
57	C₄⋊C₄×C₂₃	—	4	2	no	—
58	C₄⋊C₄×C₂₃	—	4	2	no	—
59	C₄⋊C₄×C₂₃	—	4	2	no	—
60	C₄⋊C₄×C₂₃	—	4	2	no	—
61	C₄⋊C₄×C₂₃	—	4	2	no	—
62	C₄⋊C₄×C₂₃	SD₁₆	1	4	yes	—
63	C₄⋊C₄×C₂₃	SD₁₆	1	4	yes	—
64	C₄²×C₂₃	—	4	2	no	—
65	C₄²×C₂₃	—	4	2	no	—
66	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
67	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₂₃	#41	1	1	yes	characteristic
69	#10574	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₂₃	#120	1	1	yes	characteristic
71	Q₈×C₂₃	—	8	8	no	—
72	C₂×C₄×C₂₃	—	2	1	no	—
73	C₂×C₄×C₂₃	—	2	1	no	—
74	C₂×C₄×C₂₃	—	2	4	no	—
75	C₂×C₄×C₂₃	—	2	4	no	—
76	C₂×C₄×C₂₃	—	4	2	no	—
77	C₂×C₄×C₂₃	—	4	2	no	—
78	C₂×C₄×C₂₃	—	4	2	no	—
79	C₂×C₄×C₂₃	—	4	2	no	—
80	C₂×C₄×C₂₃	—	4	4	no	—
81	C₂×C₄×C₂₃	—	4	2	no	—
82	C₂×C₄×C₂₃	—	8	2	no	—
83	C₂×C₄×C₂₃	—	8	1	no	—
84	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
86	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
88	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
89	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
90	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
91	C₂×C₄×C₂₃	#137	1	2	yes	—
92	C₈×C₂₃	—	2	4	no	—
93	C₈×C₂₃	—	4	4	no	—
94	#2551	C₂×C₂₃	1	1	yes	characteristic
95	#1948	C₂×C₂₃	1	2	yes	—
96	#1844	C₂×C₂₃	1	1	yes	characteristic
97	#1296	C₂×C₂₃	1	1	yes	characteristic
98	#1279	C₂×C₂₃	1	2	yes	—
99	C₂²×C₂₃	#395	1	1	yes	characteristic
100	C₂²×C₂₃	#448	1	2	yes	—
101	C₂²×C₂₃	#450	1	2	yes	—
102	C₂²×C₂₃	#453	1	1	yes	characteristic
103	C₂²×C₂₃	#456	1	1	yes	characteristic
104	C₄×C₂₃	—	2	1	no	—
105	C₄×C₂₃	—	2	1	no	—
106	C₄×C₂₃	—	4	4	no	—
107	C₄×C₂₃	—	4	4	no	—
108	C₄×C₂₃	—	8	2	no	—
109	C₄×C₂₃	—	16	1	no	—
110	C₄×C₂₃	#475	1	2	yes	—
111	C₄×C₂₃	#476	1	2	yes	—
112	C₄×C₂₃	#498	1	2	yes	—
113	C₄×C₂₃	#501	1	2	yes	—
114	C₂×C₄⋊Q₈	—	2	2	no	—
115	C₂×#106	—	2	2	no	—
116	C₂×#106	—	2	2	no	—
117	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
118	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
119	#395	—	2	2	no	—
120	#389	—	2	2	no	—
121	#384	C₂²×C₂₃	1	1	yes	characteristic
122	#384	C₂²×C₂₃	1	1	yes	characteristic
123	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
124	#340	—	2	2	no	—
125	C₂×C₂₃	#1623	1	1	yes	characteristic
126	C₂×C₂₃	#1629	1	1	yes	characteristic
127	C₂×C₂₃	#1651	1	2	yes	—
128	C₂×C₂₃	#2058	1	2	yes	—
129	C₂×C₂₃	#2060	1	1	yes	characteristic
130	C₂²×Q₈	—	2	2	no	—
131	C₂²×C₈	—	2	1	no	—
132	C₂²×C₈	—	2	2	no	—
133	C₄⋊Q₈	—	4	4	no	—
134	C₄⋊Q₈	—	4	4	no	—
135	C₂×C₄⋊C₄	—	2	4	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
140	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
141	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
142	C₂×C₄²	—	2	2	no	—
143	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
144	#106	—	4	4	no	—
145	#106	—	4	4	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	Q₈×C₂₃	1	4	yes	complemented
148	C₄×C₈	—	2	1	no	—
149	C₄×C₈	Q₈×C₂₃	1	2	yes	complemented
150	#95	—	2	2	no	—
151	#95	—	2	2	no	—
152	C₂₃	#10574	1	1	yes	characteristic, direct product factor, Sylow
153	C₂×Q₈	—	4	8	no	—
154	C₂×Q₈	—	8	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	4	1	no	—
159	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
160	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	4	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	4	8	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	SD₁₆×C₂₃	1	4	yes	—
177	C₄⋊C₄	SD₁₆×C₂₃	1	4	yes	—
178	C₄²	—	4	2	no	—
179	C₄²	—	4	2	no	—
180	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
181	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
182	C₄²	#41×C₂₃	1	1	yes	characteristic
183	C₂³	C₂₃×#120	1	1	yes	characteristic
184	Q₈	—	8	8	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	—	8	1	no	—
197	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
198	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
199	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
200	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
201	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
202	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
203	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
204	C₂×C₄	C₂₃×#137	1	2	yes	—
205	C₈	—	2	4	no	—
206	C₈	—	4	4	no	—
207	C₂²	C₂₃×#395	1	1	yes	characteristic
208	C₂²	C₂₃×#448	1	2	yes	—
209	C₂²	C₂₃×#450	1	2	yes	—
210	C₂²	C₂₃×#453	1	1	yes	characteristic
211	C₂²	C₂₃×#456	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	4	no	—
215	C₄	—	4	4	no	—
216	C₄	—	8	2	no	—
217	C₄	—	16	1	no	—
218	C₄	C₂₃×#475	1	2	yes	—
219	C₄	C₂₃×#476	1	2	yes	—
220	C₄	C₂₃×#498	1	2	yes	—
221	C₄	C₂₃×#501	1	2	yes	—
222	C₂	C₂₃×#1623	1	1	yes	characteristic
223	C₂	C₂₃×#1629	1	1	yes	characteristic
224	C₂	C₂₃×#1651	1	2	yes	—
225	C₂	C₂₃×#2058	1	2	yes	—
226	C₂	C₂₃×#2060	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#2551

C₂

yes

maximal, characteristic

C₂₃×#1948

C₂

yes

maximal

C₂₃×#1844

C₂

yes

maximal, characteristic

C₂₃×#1296

C₂

yes

maximal, characteristic

C₂₃×#1279

C₂

yes

maximal

C₂×C₂₃×C₄⋊Q₈

—

C₂×C₂₃×#106

—

C₂×C₂₃×#106

—

C₂×C₂₃×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#395

—

C₂₃×#389

—

C₂₃×#384

C₂²

yes

characteristic

C₂₃×#384

C₂²

yes

characteristic

C₄×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂₃×#340

—

C₂²×Q₈×C₂₃

—

C₂²×C₈×C₂₃

—

C₂²×C₈×C₂₃

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

D₄

yes

—

C₂×C₄⋊C₄×C₂₃

D₄

yes

—

C₂×C₄⋊C₄×C₂₃

D₄

yes

—

C₂×C₄²×C₂₃

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×#106

—

C₂₃×#106

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

Q₈

yes

complemented

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

Q₈

yes

complemented

C₂₃×#95

—

C₂₃×#95

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

SD₁₆

yes

—

C₄⋊C₄×C₂₃

SD₁₆

yes

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

C₂×Q₈

yes

—

C₄²×C₂₃

C₂×Q₈

yes

C₄²×C₂₃

yes

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

#120

yes

characteristic

Q₈×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

—

C₂×C₄×C₂₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₂₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₂₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₂₃

C₄⋊Q₈

yes

—

C₂×C₄×C₂₃

C₂×SD₁₆

yes

—

C₂×C₄×C₂₃

C₂×SD₁₆

yes

—

C₂×C₄×C₂₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₂₃

#137

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

#2551

C₂×C₂₃

yes

characteristic

#1948

C₂×C₂₃

yes

—

#1844

C₂×C₂₃

yes

characteristic

#1296

C₂×C₂₃

yes

characteristic

#1279

C₂×C₂₃

yes

—

C₂²×C₂₃

#395

yes

characteristic

100

C₂²×C₂₃

#448

yes

—

101

C₂²×C₂₃

#450

yes

—

102

C₂²×C₂₃

#453

yes

characteristic

103

C₂²×C₂₃

#456

yes

characteristic

104

C₄×C₂₃

—

105

C₄×C₂₃

—

106

C₄×C₂₃

—

107

C₄×C₂₃

—

108

C₄×C₂₃

—

109

C₄×C₂₃

—

110

C₄×C₂₃

#475

yes

—

111

C₄×C₂₃

#476

yes

—

112

C₄×C₂₃

#498

yes

—

113

C₄×C₂₃

#501

yes

—

114

C₂×C₄⋊Q₈

—

115

C₂×#106

—

116

C₂×#106

—

117

C₂×C₄⋊C₈

C₂²×C₂₃

yes

—

118

C₂×C₄×C₈

C₂²×C₂₃

yes

characteristic

119

#395

—

120

#389

—

121

#384

C₂²×C₂₃

yes

characteristic

122

#384

C₂²×C₂₃

yes

characteristic

123

C₄×C₄⋊C₄

C₂²×C₂₃

yes

—

124

#340

—

125

C₂×C₂₃

#1623

yes

characteristic

126

C₂×C₂₃

#1629

yes

characteristic

127

C₂×C₂₃

#1651

yes

—

128

C₂×C₂₃

#2058

yes

—

129

C₂×C₂₃

#2060

yes

characteristic

130

C₂²×Q₈

—

131

C₂²×C₈

—

132

C₂²×C₈

—

133

C₄⋊Q₈

—

134

C₄⋊Q₈

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄⋊C₄

—

137

C₂×C₄⋊C₄

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄⋊C₄

D₄×C₂₃

yes

—

140

C₂×C₄⋊C₄

D₄×C₂₃

yes

—

141

C₂×C₄⋊C₄

D₄×C₂₃

yes

—

142

C₂×C₄²

—

143

C₂×C₄²

C₂³×C₂₃

yes

characteristic

144

#106

—

145

#106

—

146

C₄⋊C₈

—

147

C₄⋊C₈

Q₈×C₂₃

yes

complemented

148

C₄×C₈

—

149

C₄×C₈

Q₈×C₂₃

yes

complemented

150

#95

—

151

#95

—

152

C₂₃

#10574

yes

characteristic, direct product factor, Sylow

153

C₂×Q₈

—

154

C₂×Q₈

—

155

C₂²×C₄

—

156

C₂²×C₄

—

157

C₂²×C₄

—

158

C₂²×C₄

—

159

C₂²×C₄

C₂×D₄×C₂₃

yes

characteristic

160

C₂²×C₄

C₂×D₄×C₂₃

yes

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₄⋊C₄

SD₁₆×C₂₃

yes

—

177

C₄⋊C₄

SD₁₆×C₂₃

yes

—

178

C₄²

—

179

C₄²

—

180

C₄²

C₂×Q₈×C₂₃

yes

characteristic

181

C₄²

C₂×Q₈×C₂₃

yes

—

182

C₄²

#41×C₂₃

yes

characteristic

183

C₂³

C₂₃×#120

yes

characteristic

184

Q₈

—

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

—

196

C₂×C₄

—

197

C₂×C₄

C₂₃×C₂²⋊Q₈

yes

characteristic

198

C₂×C₄

C₂₃×C₂²⋊Q₈

yes

—

199

C₂×C₄

C₂₃×C₄⋊Q₈

yes

characteristic

200

C₂×C₄

C₂₃×C₄⋊Q₈

yes

—

201

C₂×C₄

C₂×SD₁₆×C₂₃

yes

—

202

C₂×C₄

C₂×SD₁₆×C₂₃

yes

—

203

C₂×C₄

C₂₃×AGL(1,ℤ/8)

yes

—

204

C₂×C₄

C₂₃×#137

yes

—

205

C₈

—

206

C₈

—

207

C₂²

C₂₃×#395

yes

characteristic

208

C₂²

C₂₃×#448

yes

—

209

C₂²

C₂₃×#450

yes

—

210

C₂²

C₂₃×#453

yes

characteristic

211

C₂²

C₂₃×#456

yes

characteristic

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

C₂₃×#475

yes

—

219

C₄

C₂₃×#476

yes

—

220

C₄

C₂₃×#498

yes

—

221

C₄

C₂₃×#501

yes

—

222

C₂

C₂₃×#1623

yes

characteristic

223

C₂

C₂₃×#1629

yes

characteristic

224

C₂

C₂₃×#1651

yes

—

225

C₂

C₂₃×#2058

yes

—

226

C₂

C₂₃×#2060

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₂₃

#120

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
34. C₂₃×C₄⋊C₈	Q₈	semidirect
36. C₄×C₈×C₂₃	Q₈	semidirect
69. #10574	C₂₃	direct (N × H)
147. C₄⋊C₈	Q₈×C₂₃	semidirect
149. C₄×C₈	Q₈×C₂₃	semidirect
152. C₂₃	#10574	direct (N × H)

H ≅ G/N

type

34. C₂₃×C₄⋊C₈

Q₈

semidirect

36. C₄×C₈×C₂₃

Q₈

semidirect

69. #10574

C₂₃

direct (N × H)

147. C₄⋊C₈

Q₈×C₂₃

semidirect

149. C₄×C₈

Q₈×C₂₃

semidirect

152. C₂₃

#10574

direct (N × H)

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

subgroup	index
C₂₃×#2551	2
C₂₃×#1948	2
C₂₃×#1844	2
C₂₃×#1296	2
C₂₃×#1279	2
#10574	23

subgroup

index

C₂₃×#2551

C₂₃×#1948

C₂₃×#1844

C₂₃×#1296

C₂₃×#1279

#10574

Subgroups — C₂₃×#10574 #19158200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#10574 #19158200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10574 #19158200