Subgroups — C₂₃×#8274 #19155900

Order 5888.

438 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1767	C₂	1	2	yes	maximal
2	C₄²×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#335	C₂	1	2	yes	maximal
4	C₂₃×#1277	C₂	1	2	yes	maximal
5	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
6	C₂×C₂₃×#97	—	2	2	no	—
7	C₂×C₄×C₈×C₂₃	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₂₃	C₄	1	2	yes	—
9	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₄³×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#336	—	2	2	no	—
13	C₂₃×#335	C₄	1	8	yes	complemented
14	C₂²×C₈×C₂₃	—	2	4	no	—
15	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
22	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
23	C₂₃×C₄⋊C₈	—	2	4	no	—
24	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
25	C₂₃×#97	—	4	4	no	—
26	C₄×C₈×C₂₃	—	2	4	no	—
27	C₄×C₈×C₂₃	—	2	4	no	—
28	C₄×C₈×C₂₃	—	2	2	no	—
29	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
31	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₂₃	—	2	2	no	—
36	C₂×C₈×C₂₃	—	2	2	no	—
37	C₂×C₈×C₂₃	—	2	2	no	—
38	C₂×C₈×C₂₃	—	2	8	no	—
39	C₂×C₈×C₂₃	—	4	4	no	—
40	C₂×C₈×C₂₃	—	4	4	no	—
41	C₂×C₈×C₂₃	C₄²	1	2	yes	—
42	C₂×C₈×C₂₃	C₄²	1	2	yes	—
43	C₂×C₈×C₂₃	M₁₆	1	4	yes	—
44	C₂×C₈×C₂₃	M₁₆	1	4	yes	—
45	C₄²×C₂₃	—	2	4	no	—
46	C₄²×C₂₃	—	2	2	no	—
47	C₄²×C₂₃	—	2	2	no	—
48	C₄²×C₂₃	—	2	2	no	—
49	C₄²×C₂₃	D₈	1	2	yes	—
50	C₄²×C₂₃	Q₁₆	1	2	yes	—
51	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
52	C₄²×C₂₃	#41	1	2	yes	—
53	#8274	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₂₃	—	2	2	no	—
56	C₂×C₄×C₂₃	—	2	2	no	—
57	C₂×C₄×C₂₃	—	2	2	no	—
58	C₂×C₄×C₂₃	—	2	2	no	—
59	C₂×C₄×C₂₃	—	2	2	no	—
60	C₂×C₄×C₂₃	#107	1	4	yes	—
61	C₂×C₄×C₂₃	#108	1	4	yes	—
62	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
64	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
65	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
66	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
67	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
68	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄×C₂₃	#135	1	2	yes	—
71	C₈×C₂₃	—	2	2	no	—
72	C₈×C₂₃	—	2	4	no	—
73	C₈×C₂₃	—	4	8	no	—
74	C₈×C₂₃	#97	1	4	yes	complemented
75	#1767	C₂×C₂₃	1	2	yes	—
76	C₄²×C₈	C₂×C₂₃	1	1	yes	characteristic
77	C₂×#335	C₂×C₂₃	1	2	yes	—
78	#1277	C₂×C₂₃	1	2	yes	—
79	C₂²×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₂₃	#423	1	2	yes	—
81	C₂²×C₂₃	C₂×#107	1	1	yes	characteristic
82	C₂²×C₂₃	#427	1	1	yes	characteristic
83	C₂²×C₂₃	C₂×#108	1	2	yes	—
84	C₄×C₂₃	—	2	4	no	—
85	C₄×C₂₃	—	2	4	no	—
86	C₄×C₂₃	—	2	2	no	—
87	C₄×C₂₃	C₂×#97	1	1	yes	characteristic
88	C₄×C₂₃	C₄×M₁₆	1	1	yes	characteristic
89	C₄×C₂₃	#432	1	2	yes	—
90	C₄×C₂₃	C₄×D₈	1	2	yes	—
91	C₄×C₂₃	C₄×Q₁₆	1	2	yes	—
92	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
93	C₂×#97	—	2	2	no	—
94	C₂×C₄×C₈	C₄×C₂₃	1	2	yes	—
95	C₂×C₄×C₈	C₄×C₂₃	1	2	yes	—
96	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
98	C₄³	C₂²×C₂₃	1	1	yes	characteristic
99	#336	—	2	2	no	—
100	#335	C₄×C₂₃	1	8	yes	complemented
101	C₂×C₂₃	#1567	1	2	yes	—
102	C₂×C₂₃	#1568	1	2	yes	—
103	C₂×C₂₃	#1767	1	1	yes	characteristic
104	C₂×C₂₃	C₄×#107	1	1	yes	characteristic
105	C₂×C₂₃	C₄×#108	1	1	yes	characteristic
106	C₂²×C₈	—	2	4	no	—
107	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
108	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
109	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
110	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
111	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
112	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
113	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
114	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
115	C₄⋊C₈	—	2	4	no	—
116	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
117	#97	—	4	4	no	—
118	C₄×C₈	—	2	4	no	—
119	C₄×C₈	—	2	4	no	—
120	C₄×C₈	—	2	2	no	—
121	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
122	C₂₃	#8274	1	1	yes	characteristic, direct product factor, Sylow
123	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
124	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
125	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
126	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
127	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	8	no	—
132	C₂×C₈	—	4	4	no	—
133	C₂×C₈	—	4	4	no	—
134	C₂×C₈	C₄²×C₂₃	1	2	yes	—
135	C₂×C₈	C₄²×C₂₃	1	2	yes	—
136	C₂×C₈	M₁₆×C₂₃	1	4	yes	—
137	C₂×C₈	M₁₆×C₂₃	1	4	yes	—
138	C₄²	—	2	4	no	—
139	C₄²	—	2	2	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₄²	D₈×C₂₃	1	2	yes	—
143	C₄²	Q₁₆×C₂₃	1	2	yes	—
144	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
145	C₄²	#41×C₂₃	1	2	yes	—
146	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	C₂₃×#107	1	4	yes	—
153	C₂×C₄	C₂₃×#108	1	4	yes	—
154	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
155	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
156	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
157	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
158	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
159	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
160	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
161	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
162	C₂×C₄	C₂₃×#135	1	2	yes	—
163	C₈	—	2	2	no	—
164	C₈	—	2	4	no	—
165	C₈	—	4	8	no	—
166	C₈	C₂₃×#97	1	4	yes	complemented
167	C₂²	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
168	C₂²	C₂₃×#423	1	2	yes	—
169	C₂²	C₂×C₂₃×#107	1	1	yes	characteristic
170	C₂²	C₂₃×#427	1	1	yes	characteristic
171	C₂²	C₂×C₂₃×#108	1	2	yes	—
172	C₄	—	2	2	no	—
173	C₄	—	2	4	no	—
174	C₄	—	2	4	no	—
175	C₄	C₂×C₂₃×#97	1	1	yes	characteristic
176	C₄	C₄×M₁₆×C₂₃	1	1	yes	characteristic
177	C₄	C₂₃×#432	1	2	yes	—
178	C₄	C₄×D₈×C₂₃	1	2	yes	—
179	C₄	C₄×Q₁₆×C₂₃	1	2	yes	—
180	C₂	C₂₃×#1567	1	2	yes	—
181	C₂	C₂₃×#1568	1	2	yes	—
182	C₂	C₂₃×#1767	1	1	yes	characteristic
183	C₂	C₄×C₂₃×#107	1	1	yes	characteristic
184	C₂	C₄×C₂₃×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle, prime radical	C₂³×C₂₃	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂₃×#335	C₄	semidirect
53. #8274	C₂₃	direct (N × H)
74. C₈×C₂₃	#97	semidirect
100. #335	C₄×C₂₃	semidirect
122. C₂₃	#8274	direct (N × H)
166. C₈	C₂₃×#97	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#1767	2
C₄²×C₈×C₂₃	2
C₂×C₂₃×#335	2
C₂₃×#1277	2
#8274	23

Subgroups — C23×#8274 #19155900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8274 #19155900