Subgroups — C₂₃×#34173 #19181799

Order 5888.

5126 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#519	C₂	1	2	yes	maximal, complemented
3	Q₈×C₂²⋊C₄×C₂₃	C₂	1	16	yes	maximal, complemented
4	C₂₃×#2310	C₂	1	6	yes	maximal, complemented
5	C₂₃×#2280	C₂	1	6	yes	maximal
6	C₂³×Q₈×C₂₃	—	2	16	no	—
7	C₂³×Q₈×C₂₃	C₄	1	16	yes	complemented
8	C₂³×Q₈×C₂₃	C₂²	1	3	yes	—
9	C₂³×Q₈×C₂₃	C₂²	1	8	yes	—
10	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₂₃	C₂²	1	3	yes	—
12	C₂₃×#519	—	2	16	no	—
13	C₂×C₄×Q₈×C₂₃	C₂²	1	16	yes	complemented
14	C₂×C₂₃×#117	C₂²	1	12	yes	—
15	C₂₃×#391	C₂²	1	48	yes	complemented
16	C₂₃×#381	C₂²	1	12	yes	—
17	C₂₃×#380	C₂²	1	24	yes	—
18	C₂₃×#379	C₂²	1	4	yes	—
19	C₄×C₂²⋊C₄×C₂₃	C₂²	1	24	yes	—
20	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
21	C₂²×Q₈×C₂₃	—	2	128	no	—
22	C₂²×Q₈×C₂₃	—	2	64	no	—
23	C₂²×Q₈×C₂₃	—	2	24	no	—
24	C₂²×Q₈×C₂₃	C₂×C₄	1	64	yes	—
25	C₂²×Q₈×C₂₃	C₂×C₄	1	24	yes	—
26	C₂²×Q₈×C₂₃	C₂×C₄	1	8	yes	—
27	C₂²×Q₈×C₂₃	D₄	1	16	yes	—
28	C₂²×Q₈×C₂₃	C₂³	1	12	yes	—
29	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
30	C₂³×C₄×C₂₃	—	2	24	no	—
31	C₂³×C₄×C₂₃	C₂×C₄	1	24	yes	—
32	C₂³×C₄×C₂₃	C₂³	1	12	yes	—
33	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
34	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
35	C₄×Q₈×C₂₃	—	2	64	no	—
36	C₂₃×#117	—	2	48	no	—
37	C₂×C₄⋊C₄×C₂₃	C₂³	1	24	yes	—
38	C₂×C₂²⋊C₄×C₂₃	C₂³	1	16	yes	—
39	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
40	C₂×C₄²×C₂₃	C₂³	1	24	yes	—
41	C₂₃×#95	C₂³	1	48	yes	—
42	C₂⁴×C₂₃	—	2	4	no	—
43	C₂⁴×C₂₃	—	2	8	no	—
44	C₂⁴×C₂₃	C₂²×C₄	1	4	yes	—
45	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
46	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
47	C₂×Q₈×C₂₃	—	2	256	no	—
48	C₂×Q₈×C₂₃	—	2	96	no	—
49	C₂×Q₈×C₂₃	—	2	48	no	—
50	C₂×Q₈×C₂₃	C₂²⋊C₄	1	64	yes	complemented
51	C₂×Q₈×C₂₃	C₂²×C₄	1	48	yes	—
52	C₂×Q₈×C₂₃	C₂²×C₄	1	16	yes	—
53	C₂×Q₈×C₂₃	C₂×D₄	1	32	yes	—
54	C₂²×C₄×C₂₃	—	2	12	no	—
55	C₂²×C₄×C₂₃	—	2	96	no	—
56	C₂²×C₄×C₂₃	—	2	48	no	—
57	C₂²×C₄×C₂₃	C₂²×C₄	1	12	yes	—
58	C₂²×C₄×C₂₃	C₂²×C₄	1	48	yes	—
59	C₂²×C₄×C₂₃	C₂²×C₄	1	12	yes	—
60	C₂²×C₄×C₂₃	C₂×D₄	1	24	yes	—
61	C₂²×C₄×C₂₃	C₂⁴	1	16	yes	—
62	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
63	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
64	C₄⋊C₄×C₂₃	—	2	48	no	—
65	C₂²⋊C₄×C₂₃	—	2	16	no	—
66	C₂²⋊C₄×C₂₃	—	2	32	no	—
67	C₄²×C₂₃	—	2	48	no	—
68	#34173	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₂₃	—	2	32	no	—
70	C₂³×C₂₃	—	2	8	no	—
71	C₂³×C₂₃	—	2	16	no	—
72	C₂³×C₂₃	—	2	4	no	—
73	C₂³×C₂₃	—	2	4	no	—
74	C₂³×C₂₃	—	2	4	no	—
75	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
76	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
77	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
78	C₂³×C₂₃	2₋¹⁺⁴	1	8	yes	—
79	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
80	Q₈×C₂₃	—	2	96	no	—
81	Q₈×C₂₃	C₂×C₂²⋊C₄	1	64	yes	—
82	C₂×C₄×C₂₃	—	2	24	no	—
83	C₂×C₄×C₂₃	—	2	12	no	—
84	C₂×C₄×C₂₃	—	2	96	no	—
85	C₂×C₄×C₂₃	—	2	16	no	—
86	C₂×C₄×C₂₃	—	2	32	no	—
87	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	48	yes	—
88	C₂×C₄×C₂₃	C₂³×C₄	1	12	yes	—
89	C₂×C₄×C₂₃	C₂³×C₄	1	12	yes	—
90	C₂×C₄×C₂₃	C₂²×D₄	1	24	yes	—
91	C₂⁴×Q₈	C₂×C₂₃	1	1	yes	characteristic
92	C₂×#519	C₂×C₂₃	1	2	yes	complemented
93	Q₈×C₂²⋊C₄	C₂×C₂₃	1	16	yes	complemented
94	#2310	C₂×C₂₃	1	6	yes	complemented
95	#2280	C₂×C₂₃	1	6	yes	—
96	C₂²×C₂₃	—	2	32	no	—
97	C₂²×C₂₃	—	2	4	no	—
98	C₂²×C₂₃	—	2	4	no	—
99	C₂²×C₂₃	—	2	4	no	—
100	C₂²×C₂₃	—	2	8	no	—
101	C₂²×C₂₃	—	2	16	no	—
102	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	4	yes	—
103	C₂²×C₂₃	#519	1	8	yes	complemented
104	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
105	C₂²×C₂₃	C₂³×D₄	1	2	yes	—
106	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	4	yes	—
107	C₄×C₂₃	—	2	12	no	—
108	C₄×C₂₃	—	4	16	no	—
109	C₄×C₂₃	C₂²×C₂²⋊C₄	1	24	yes	—
110	C₂³×Q₈	—	2	16	no	—
111	C₂³×Q₈	C₄×C₂₃	1	16	yes	complemented
112	C₂³×Q₈	C₂²×C₂₃	1	3	yes	—
113	C₂³×Q₈	C₂²×C₂₃	1	8	yes	—
114	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
115	C₂⁴×C₄	C₂²×C₂₃	1	3	yes	—
116	#519	—	2	16	no	—
117	C₂×C₄×Q₈	C₂²×C₂₃	1	16	yes	complemented
118	C₂×#117	C₂²×C₂₃	1	12	yes	—
119	#391	C₂²×C₂₃	1	48	yes	complemented
120	#381	C₂²×C₂₃	1	12	yes	—
121	#380	C₂²×C₂₃	1	24	yes	—
122	#379	C₂²×C₂₃	1	4	yes	—
123	C₄×C₂²⋊C₄	C₂²×C₂₃	1	24	yes	—
124	C₂×C₂₃	—	2	4	no	—
125	C₂×C₂₃	—	2	8	no	—
126	C₂×C₂₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₂₃	C₂×#519	1	2	yes	—
128	C₂×C₂₃	#3486	1	4	yes	—
129	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
130	C₂²×Q₈	—	2	128	no	—
131	C₂²×Q₈	—	2	64	no	—
132	C₂²×Q₈	—	2	24	no	—
133	C₂²×Q₈	C₂×C₄×C₂₃	1	64	yes	—
134	C₂²×Q₈	C₂×C₄×C₂₃	1	24	yes	—
135	C₂²×Q₈	C₂×C₄×C₂₃	1	8	yes	—
136	C₂²×Q₈	D₄×C₂₃	1	16	yes	—
137	C₂²×Q₈	C₂³×C₂₃	1	12	yes	—
138	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
139	C₂³×C₄	—	2	24	no	—
140	C₂³×C₄	C₂×C₄×C₂₃	1	24	yes	—
141	C₂³×C₄	C₂³×C₂₃	1	12	yes	—
142	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
143	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
144	C₄×Q₈	—	2	64	no	—
145	#117	—	2	48	no	—
146	C₂×C₄⋊C₄	C₂³×C₂₃	1	24	yes	—
147	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
148	C₂×C₂²⋊C₄	C₂³×C₂₃	1	16	yes	—
149	C₂×C₄²	C₂³×C₂₃	1	24	yes	—
150	#95	C₂³×C₂₃	1	48	yes	—
151	C₂₃	#34173	1	1	yes	characteristic, direct product factor, Sylow
152	C₂⁴	—	2	4	no	—
153	C₂⁴	—	2	8	no	—
154	C₂⁴	C₂²×C₄×C₂₃	1	4	yes	—
155	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
156	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
157	C₂×Q₈	—	2	256	no	—
158	C₂×Q₈	—	2	96	no	—
159	C₂×Q₈	—	2	48	no	—
160	C₂×Q₈	C₂²⋊C₄×C₂₃	1	64	yes	complemented
161	C₂×Q₈	C₂²×C₄×C₂₃	1	48	yes	—
162	C₂×Q₈	C₂²×C₄×C₂₃	1	16	yes	—
163	C₂×Q₈	C₂×D₄×C₂₃	1	32	yes	—
164	C₂²×C₄	—	2	12	no	—
165	C₂²×C₄	—	2	96	no	—
166	C₂²×C₄	—	2	48	no	—
167	C₂²×C₄	C₂²×C₄×C₂₃	1	12	yes	—
168	C₂²×C₄	C₂²×C₄×C₂₃	1	48	yes	—
169	C₂²×C₄	C₂²×C₄×C₂₃	1	12	yes	—
170	C₂²×C₄	C₂×D₄×C₂₃	1	24	yes	—
171	C₂²×C₄	C₂⁴×C₂₃	1	16	yes	—
172	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
173	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
174	C₄⋊C₄	—	2	48	no	—
175	C₂²⋊C₄	—	2	16	no	—
176	C₂²⋊C₄	—	2	32	no	—
177	C₄²	—	2	48	no	—
178	C₂³	—	2	32	no	—
179	C₂³	—	2	4	no	—
180	C₂³	—	2	4	no	—
181	C₂³	—	2	4	no	—
182	C₂³	—	2	8	no	—
183	C₂³	—	2	16	no	—
184	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
185	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
186	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
187	C₂³	C₂₃×2₋¹⁺⁴	1	8	yes	—
188	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
189	Q₈	—	2	96	no	—
190	Q₈	C₂×C₂²⋊C₄×C₂₃	1	64	yes	—
191	C₂×C₄	—	2	12	no	—
192	C₂×C₄	—	2	24	no	—
193	C₂×C₄	—	2	96	no	—
194	C₂×C₄	—	2	16	no	—
195	C₂×C₄	—	2	32	no	—
196	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	48	yes	—
197	C₂×C₄	C₂³×C₄×C₂₃	1	12	yes	—
198	C₂×C₄	C₂³×C₄×C₂₃	1	12	yes	—
199	C₂×C₄	C₂²×D₄×C₂₃	1	24	yes	—
200	C₂²	—	2	32	no	—
201	C₂²	—	2	4	no	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	8	no	—
204	C₂²	—	2	16	no	—
205	C₂²	—	2	4	no	—
206	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	4	yes	—
207	C₂²	C₂₃×#519	1	8	yes	complemented
208	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
209	C₂²	C₂³×D₄×C₂₃	1	2	yes	—
210	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	4	yes	—
211	C₄	—	2	12	no	—
212	C₄	—	4	16	no	—
213	C₄	C₂²×C₂²⋊C₄×C₂₃	1	24	yes	—
214	C₂	—	2	4	no	—
215	C₂	—	2	8	no	—
216	C₂	C₂³×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
217	C₂	C₂×C₂₃×#519	1	2	yes	—
218	C₂	C₂₃×#3486	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
prime radical	C₂⁵×C₂₃	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#519	C₂	semidirect
3. Q₈×C₂²⋊C₄×C₂₃	C₂	semidirect
4. C₂₃×#2310	C₂	semidirect
7. C₂³×Q₈×C₂₃	C₄	semidirect
13. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
15. C₂₃×#391	C₂²	semidirect
50. C₂×Q₈×C₂₃	C₂²⋊C₄	semidirect
68. #34173	C₂₃	direct (N × H)
92. C₂×#519	C₂×C₂₃	semidirect
93. Q₈×C₂²⋊C₄	C₂×C₂₃	semidirect
94. #2310	C₂×C₂₃	semidirect
103. C₂²×C₂₃	#519	semidirect
111. C₂³×Q₈	C₄×C₂₃	semidirect
117. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
119. #391	C₂²×C₂₃	semidirect
151. C₂₃	#34173	direct (N × H)
160. C₂×Q₈	C₂²⋊C₄×C₂₃	semidirect
207. C₂²	C₂₃×#519	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₂₃	2
C₂×C₂₃×#519	2
Q₈×C₂²⋊C₄×C₂₃	2
C₂₃×#2310	2
C₂₃×#2280	2
#34173	23

Subgroups — C23×#34173 #19181799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#34173 #19181799