Subgroups — C₂₃×#33317 #19180943

Order 5888.

822 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#501	C₂	1	4	yes	maximal
2	C₂₃×#2862	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2859	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1847	C₂	1	4	yes	maximal
5	C₂₃×#1846	C₂	1	4	yes	maximal
6	C₄×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
7	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₂₃×#125	C₂²	1	2	yes	—
9	C₂₃×#501	—	2	16	no	—
10	C₂×C₂₃×#107	C₂²	1	8	yes	—
11	C₂×C₂₃×#106	C₂²	1	8	yes	—
12	C₂×C₂₃×#97	C₂²	1	2	yes	—
13	C₂×C₂₃×#97	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
15	C₂₃×#391	—	2	4	no	—
16	C₂₃×#389	—	2	4	no	—
17	C₂₃×#384	C₂²	1	2	yes	—
18	C₂₃×#382	C₂²	1	2	yes	—
19	C₂₃×#382	C₂²	1	2	yes	—
20	C₂₃×#382	C₂²	1	2	yes	—
21	C₄³×C₂₃	C₂²	1	1	yes	characteristic
22	C₂²×Q₈×C₂₃	—	2	2	no	—
23	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
24	C₂₃×C₄⋊Q₈	—	2	8	no	—
25	C₂₃×#125	—	2	8	no	—
26	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
27	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
28	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
30	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
31	C₂×C₄²×C₂₃	D₄	1	4	yes	—
32	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
35	C₂₃×#107	—	2	16	no	—
36	C₂₃×#106	—	2	16	no	—
37	C₂₃×#97	D₄	1	8	yes	—
38	C₂₃×#97	Q₈	1	8	yes	complemented
39	C₄×C₈×C₂₃	Q₈	1	8	yes	complemented
40	C₂₃×#95	—	2	4	no	—
41	C₂₃×#95	—	2	4	no	—
42	C₂₃×#95	—	2	4	no	—
43	C₂×Q₈×C₂₃	—	4	8	no	—
44	C₂×Q₈×C₂₃	—	4	4	no	—
45	C₂²×C₄×C₂₃	—	2	4	no	—
46	C₂²×C₄×C₂₃	—	2	4	no	—
47	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
51	C₂×C₈×C₂₃	C₂×D₄	1	8	yes	—
52	C₂×C₈×C₂₃	C₂×Q₈	1	8	yes	—
53	C₂×C₈×C₂₃	C₂×Q₈	1	8	yes	—
54	C₄⋊C₄×C₂₃	—	2	4	no	—
55	C₄⋊C₄×C₂₃	—	2	8	no	—
56	C₄⋊C₄×C₂₃	—	2	4	no	—
57	C₄⋊C₄×C₂₃	—	4	8	no	—
58	C₄⋊C₄×C₂₃	—	4	8	no	—
59	C₄⋊C₄×C₂₃	—	4	4	no	—
60	C₄²×C₂₃	—	4	4	no	—
61	C₄²×C₂₃	C₂×D₄	1	4	yes	—
62	C₄²×C₂₃	C₂×Q₈	1	4	yes	—
63	C₄²×C₂₃	C₂×Q₈	1	4	yes	—
64	#33317	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
66	Q₈×C₂₃	—	8	8	no	—
67	C₂×C₄×C₂₃	—	2	4	no	—
68	C₂×C₄×C₂₃	—	2	4	no	—
69	C₂×C₄×C₂₃	—	2	4	no	—
70	C₂×C₄×C₂₃	—	4	4	no	—
71	C₂×C₄×C₂₃	—	4	4	no	—
72	C₂×C₄×C₂₃	—	4	8	no	—
73	C₂×C₄×C₂₃	—	4	8	no	—
74	C₂×C₄×C₂₃	#127	1	4	yes	—
75	C₂×C₄×C₂₃	C₄⋊Q₈	1	4	yes	—
76	C₂×C₄×C₂₃	C₄⋊Q₈	1	4	yes	—
77	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
78	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
79	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
80	C₈×C₂₃	C₄⋊Q₈	1	16	yes	—
81	C₂×#501	C₂×C₂₃	1	4	yes	—
82	#2862	C₂×C₂₃	1	1	yes	characteristic
83	#2859	C₂×C₂₃	1	1	yes	characteristic
84	#1847	C₂×C₂₃	1	4	yes	—
85	#1846	C₂×C₂₃	1	4	yes	—
86	C₄×#97	C₂×C₂₃	1	1	yes	characteristic
87	C₂²×C₂₃	C₂×#127	1	1	yes	characteristic
88	C₂²×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
89	C₂²×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
90	C₂²×C₂₃	#575	1	4	yes	—
91	C₄×C₂₃	—	2	4	no	—
92	C₄×C₂₃	—	2	4	no	—
93	C₄×C₂₃	—	2	4	no	—
94	C₄×C₂₃	—	8	4	no	—
95	C₄×C₂₃	—	8	4	no	—
96	C₄×C₂₃	C₂×C₄⋊Q₈	1	4	yes	—
97	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
98	C₂×#125	C₂²×C₂₃	1	2	yes	—
99	#501	—	2	16	no	—
100	C₂×#107	C₂²×C₂₃	1	8	yes	—
101	C₂×#106	C₂²×C₂₃	1	8	yes	—
102	C₂×#97	C₂²×C₂₃	1	2	yes	—
103	C₂×#97	C₂²×C₂₃	1	2	yes	—
104	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
105	#391	—	2	4	no	—
106	#389	—	2	4	no	—
107	#384	C₂²×C₂₃	1	2	yes	—
108	#382	C₂²×C₂₃	1	2	yes	—
109	#382	C₂²×C₂₃	1	2	yes	—
110	#382	C₂²×C₂₃	1	2	yes	—
111	C₄³	C₂²×C₂₃	1	1	yes	characteristic
112	C₂×C₂₃	#2868	1	1	yes	characteristic
113	C₂×C₂₃	#3153	1	2	yes	—
114	C₂×C₂₃	#3162	1	2	yes	—
115	C₂×C₂₃	#3166	1	2	yes	—
116	C₂²×Q₈	—	2	2	no	—
117	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
118	C₄⋊Q₈	—	2	8	no	—
119	#125	—	2	8	no	—
120	C₂×C₄⋊C₄	—	2	4	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	2	4	no	—
123	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
124	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
125	C₂×C₄²	D₄×C₂₃	1	4	yes	—
126	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
127	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
128	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
129	#107	—	2	16	no	—
130	#106	—	2	16	no	—
131	#97	D₄×C₂₃	1	8	yes	—
132	#97	Q₈×C₂₃	1	8	yes	complemented
133	C₄×C₈	Q₈×C₂₃	1	8	yes	complemented
134	#95	—	2	4	no	—
135	#95	—	2	4	no	—
136	#95	—	2	4	no	—
137	C₂₃	#33317	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×Q₈	—	4	8	no	—
139	C₂×Q₈	—	4	4	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
143	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
144	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
145	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
146	C₂×C₈	C₂×D₄×C₂₃	1	8	yes	—
147	C₂×C₈	C₂×Q₈×C₂₃	1	8	yes	—
148	C₂×C₈	C₂×Q₈×C₂₃	1	8	yes	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	2	8	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	4	8	no	—
153	C₄⋊C₄	—	4	8	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄²	—	4	4	no	—
156	C₄²	C₂×D₄×C₂₃	1	4	yes	—
157	C₄²	C₂×Q₈×C₂₃	1	4	yes	—
158	C₄²	C₂×Q₈×C₂₃	1	4	yes	—
159	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
160	Q₈	—	8	8	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	4	8	no	—
165	C₂×C₄	—	4	8	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	C₂₃×#127	1	4	yes	—
169	C₂×C₄	C₂₃×C₄⋊Q₈	1	4	yes	—
170	C₂×C₄	C₂₃×C₄⋊Q₈	1	4	yes	—
171	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
172	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
173	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
174	C₈	C₂₃×C₄⋊Q₈	1	16	yes	—
175	C₂²	C₂×C₂₃×#127	1	1	yes	characteristic
176	C₂²	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
177	C₂²	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
178	C₂²	C₂₃×#575	1	4	yes	—
179	C₄	—	2	4	no	—
180	C₄	—	2	4	no	—
181	C₄	—	2	4	no	—
182	C₄	—	8	4	no	—
183	C₄	—	8	4	no	—
184	C₄	C₂×C₂₃×C₄⋊Q₈	1	4	yes	—
185	C₂	C₂₃×#2868	1	1	yes	characteristic
186	C₂	C₂₃×#3153	1	2	yes	—
187	C₂	C₂₃×#3162	1	2	yes	—
188	C₂	C₂₃×#3166	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
38. C₂₃×#97	Q₈	semidirect
39. C₄×C₈×C₂₃	Q₈	semidirect
64. #33317	C₂₃	direct (N × H)
132. #97	Q₈×C₂₃	semidirect
133. C₄×C₈	Q₈×C₂₃	semidirect
137. C₂₃	#33317	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×#501	2
C₂₃×#2862	2
C₂₃×#2859	2
C₂₃×#1847	2
C₂₃×#1846	2
C₄×C₂₃×#97	2
#33317	23

Subgroups — C23×#33317 #19180943

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33317 #19180943