Subgroups — C₂₃×#34174 #19181800

Order 5888.

5318 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#519	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2341	C₂	1	16	yes	maximal, complemented
4	C₂₃×#2309	C₂	1	6	yes	maximal, complemented
5	C₂₃×#2280	C₂	1	6	yes	maximal, complemented
6	C₂²×#41×C₂₃	—	2	16	no	—
7	C₂²×#41×C₂₃	C₄	1	16	yes	complemented
8	C₂²×#41×C₂₃	C₂²	1	8	yes	—
9	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂³×D₄×C₂₃	C₂²	1	3	yes	—
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×#390	C₂²	1	48	yes	complemented
16	C₂₃×#381	C₂²	1	4	yes	—
17	C₂₃×#380	C₂²	1	24	yes	complemented
18	C₂₃×#379	C₂²	1	12	yes	—
19	C₄×C₂²⋊C₄×C₂₃	C₂²	1	24	yes	complemented
20	C₂⁵×C₂₃	C₂³	1	3	yes	—
21	C₂×#41×C₂₃	—	2	128	no	—
22	C₂×#41×C₂₃	—	2	64	no	—
23	C₂×#41×C₂₃	C₂×C₄	1	64	yes	—
24	C₂²×Q₈×C₂₃	C₂×C₄	1	8	yes	—
25	C₂²×Q₈×C₂₃	D₄	1	16	yes	—
26	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
27	C₂²×D₄×C₂₃	—	2	24	no	—
28	C₂²×D₄×C₂₃	C₂×C₄	1	24	yes	—
29	C₂²×D₄×C₂₃	C₂³	1	12	yes	—
30	C₂³×C₄×C₂₃	—	2	24	no	—
31	C₂³×C₄×C₂₃	C₂×C₄	1	24	yes	—
32	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₂₃	C₂³	1	12	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	8	yes	—
39	C₂×C₂²⋊C₄×C₂₃	C₂³	1	48	yes	—
40	C₂×C₄²×C₂₃	C₂³	1	24	yes	—
41	C₂₃×#95	C₂³	1	16	yes	—
42	C₂⁴×C₂₃	—	2	12	no	—
43	C₂⁴×C₂₃	—	4	12	no	—
44	C₂⁴×C₂₃	C₂²×C₄	1	12	yes	—
45	C₂⁴×C₂₃	C₂⁴	1	6	yes	—
46	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
47	#41×C₂₃	—	2	256	no	—
48	C₂×Q₈×C₂₃	C₂²⋊C₄	1	64	yes	complemented
49	C₂×Q₈×C₂₃	C₂²×C₄	1	16	yes	—
50	C₂×Q₈×C₂₃	C₂×D₄	1	32	yes	—
51	C₂×D₄×C₂₃	—	2	96	no	—
52	C₂×D₄×C₂₃	—	2	48	no	—
53	C₂×D₄×C₂₃	C₂²×C₄	1	48	yes	—
54	C₂²×C₄×C₂₃	—	2	4	no	—
55	C₂²×C₄×C₂₃	—	2	96	no	—
56	C₂²×C₄×C₂₃	—	2	48	no	—
57	C₂²×C₄×C₂₃	C₂²×C₄	1	4	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	2	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	—	4	48	no	—
67	C₄²×C₂₃	—	2	48	no	—
68	#34174	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₂₃	—	2	4	no	—
70	C₂³×C₂₃	—	2	24	no	—
71	C₂³×C₂₃	—	2	12	no	—
72	C₂³×C₂₃	—	2	24	no	—
73	C₂³×C₂₃	—	4	24	no	—
74	C₂³×C₂₃	—	4	48	no	—
75	C₂³×C₂₃	C₂³×C₄	1	12	yes	—
76	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
77	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
78	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
79	Q₈×C₂₃	C₂×C₂²⋊C₄	1	64	yes	—
80	D₄×C₂₃	—	2	96	no	—
81	C₂×C₄×C₂₃	—	2	8	no	—
82	C₂×C₄×C₂₃	—	2	4	no	—
83	C₂×C₄×C₂₃	—	2	96	no	—
84	C₂×C₄×C₂₃	—	2	16	no	—
85	C₂×C₄×C₂₃	—	2	32	no	—
86	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	48	yes	—
87	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
88	C₂×C₄×C₂₃	C₂³×C₄	1	12	yes	—
89	C₂×C₄×C₂₃	C₂²×D₄	1	24	yes	—
90	C₂³×#41	C₂×C₂₃	1	1	yes	characteristic
91	C₂×#519	C₂×C₂₃	1	2	yes	complemented
92	#2341	C₂×C₂₃	1	16	yes	complemented
93	#2309	C₂×C₂₃	1	6	yes	complemented
94	#2280	C₂×C₂₃	1	6	yes	complemented
95	C₂²×C₂₃	—	2	4	no	—
96	C₂²×C₂₃	—	2	8	no	—
97	C₂²×C₂₃	—	2	12	no	—
98	C₂²×C₂₃	—	2	24	no	—
99	C₂²×C₂₃	—	4	48	no	—
100	C₂²×C₂₃	—	4	24	no	—
101	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	4	yes	—
102	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
103	C₂²×C₂₃	C₂³×D₄	1	2	yes	—
104	C₂²×C₂₃	#585	1	4	yes	—
105	C₄×C₂₃	—	2	4	no	—
106	C₄×C₂₃	—	4	16	no	—
107	C₄×C₂₃	C₂²×C₂²⋊C₄	1	24	yes	—
108	C₂²×#41	—	2	16	no	—
109	C₂²×#41	C₄×C₂₃	1	16	yes	complemented
110	C₂²×#41	C₂²×C₂₃	1	8	yes	—
111	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
112	C₂³×D₄	C₂²×C₂₃	1	3	yes	—
113	C₂⁴×C₄	C₂²×C₂₃	1	3	yes	—
114	#519	—	2	16	no	—
115	C₂×C₄×Q₈	C₂²×C₂₃	1	16	yes	complemented
116	C₂×#117	C₂²×C₂₃	1	12	yes	—
117	#390	C₂²×C₂₃	1	48	yes	complemented
118	#381	C₂²×C₂₃	1	4	yes	—
119	#380	C₂²×C₂₃	1	24	yes	complemented
120	#379	C₂²×C₂₃	1	12	yes	—
121	C₄×C₂²⋊C₄	C₂²×C₂₃	1	24	yes	complemented
122	C₂×C₂₃	—	2	4	no	—
123	C₂×C₂₃	—	4	12	no	—
124	C₂×C₂₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₂₃	#3428	1	2	yes	—
126	C₂×C₂₃	#3487	1	4	yes	—
127	C₂⁵	C₂³×C₂₃	1	3	yes	—
128	C₂×#41	—	2	128	no	—
129	C₂×#41	—	2	64	no	—
130	C₂×#41	C₂×C₄×C₂₃	1	64	yes	—
131	C₂²×Q₈	C₂×C₄×C₂₃	1	8	yes	—
132	C₂²×Q₈	D₄×C₂₃	1	16	yes	—
133	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
134	C₂²×D₄	—	2	24	no	—
135	C₂²×D₄	C₂×C₄×C₂₃	1	24	yes	—
136	C₂²×D₄	C₂³×C₂₃	1	12	yes	—
137	C₂³×C₄	—	2	24	no	—
138	C₂³×C₄	C₂×C₄×C₂₃	1	24	yes	—
139	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
140	C₂³×C₄	C₂³×C₂₃	1	12	yes	—
141	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
142	C₄×Q₈	—	2	64	no	—
143	#117	—	2	48	no	—
144	C₂×C₄⋊C₄	C₂³×C₂₃	1	24	yes	—
145	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
146	C₂×C₂²⋊C₄	C₂³×C₂₃	1	48	yes	—
147	C₂×C₄²	C₂³×C₂₃	1	24	yes	—
148	#95	C₂³×C₂₃	1	16	yes	—
149	C₂₃	#34174	1	1	yes	characteristic, direct product factor, Sylow
150	C₂⁴	—	2	12	no	—
151	C₂⁴	—	4	12	no	—
152	C₂⁴	C₂²×C₄×C₂₃	1	12	yes	—
153	C₂⁴	C₂⁴×C₂₃	1	6	yes	—
154	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
155	#41	—	2	256	no	—
156	C₂×Q₈	C₂²⋊C₄×C₂₃	1	64	yes	complemented
157	C₂×Q₈	C₂²×C₄×C₂₃	1	16	yes	—
158	C₂×Q₈	C₂×D₄×C₂₃	1	32	yes	—
159	C₂×D₄	—	2	96	no	—
160	C₂×D₄	—	2	48	no	—
161	C₂×D₄	C₂²×C₄×C₂₃	1	48	yes	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	96	no	—
164	C₂²×C₄	—	2	48	no	—
165	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
166	C₂²×C₄	C₂²×C₄×C₂₃	1	48	yes	—
167	C₂²×C₄	C₂²×C₄×C₂₃	1	12	yes	—
168	C₂²×C₄	C₂×D₄×C₂₃	1	24	yes	—
169	C₂²×C₄	C₂⁴×C₂₃	1	16	yes	—
170	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
171	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
172	C₄⋊C₄	—	2	48	no	—
173	C₂²⋊C₄	—	2	16	no	—
174	C₂²⋊C₄	—	4	48	no	—
175	C₄²	—	2	48	no	—
176	C₂³	—	2	4	no	—
177	C₂³	—	2	12	no	—
178	C₂³	—	2	24	no	—
179	C₂³	—	2	24	no	—
180	C₂³	—	4	24	no	—
181	C₂³	—	4	48	no	—
182	C₂³	C₂³×C₄×C₂₃	1	12	yes	—
183	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
184	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
185	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
186	Q₈	C₂×C₂²⋊C₄×C₂₃	1	64	yes	—
187	D₄	—	2	96	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	8	no	—
190	C₂×C₄	—	2	96	no	—
191	C₂×C₄	—	2	16	no	—
192	C₂×C₄	—	2	32	no	—
193	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	48	yes	—
194	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
195	C₂×C₄	C₂³×C₄×C₂₃	1	12	yes	—
196	C₂×C₄	C₂²×D₄×C₂₃	1	24	yes	—
197	C₂²	—	2	4	no	—
198	C₂²	—	2	8	no	—
199	C₂²	—	2	24	no	—
200	C₂²	—	2	12	no	—
201	C₂²	—	4	48	no	—
202	C₂²	—	4	24	no	—
203	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	4	yes	—
204	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
205	C₂²	C₂³×D₄×C₂₃	1	2	yes	—
206	C₂²	C₂₃×#585	1	4	yes	—
207	C₄	—	2	4	no	—
208	C₄	—	4	16	no	—
209	C₄	C₂²×C₂²⋊C₄×C₂₃	1	24	yes	—
210	C₂	—	2	4	no	—
211	C₂	—	4	12	no	—
212	C₂	C₂³×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
213	C₂	C₂₃×#3428	1	2	yes	—
214	C₂	C₂₃×#3487	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
prime radical	C₂³×#41×C₂₃	C₂
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. C₂₃×#2341	C₂	semidirect
4. C₂₃×#2309	C₂	semidirect
5. C₂₃×#2280	C₂	semidirect
7. C₂²×#41×C₂₃	C₄	semidirect
13. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
15. C₂₃×#390	C₂²	semidirect
17. C₂₃×#380	C₂²	semidirect
19. C₄×C₂²⋊C₄×C₂₃	C₂²	semidirect
48. C₂×Q₈×C₂₃	C₂²⋊C₄	semidirect
68. #34174	C₂₃	direct (N × H)
91. C₂×#519	C₂×C₂₃	semidirect
92. #2341	C₂×C₂₃	semidirect
93. #2309	C₂×C₂₃	semidirect
94. #2280	C₂×C₂₃	semidirect
109. C₂²×#41	C₄×C₂₃	semidirect
115. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
117. #390	C₂²×C₂₃	semidirect
119. #380	C₂²×C₂₃	semidirect
121. C₄×C₂²⋊C₄	C₂²×C₂₃	semidirect
149. C₂₃	#34174	direct (N × H)
156. C₂×Q₈	C₂²⋊C₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂³×#41×C₂₃	2
C₂×C₂₃×#519	2
C₂₃×#2341	2
C₂₃×#2309	2
C₂₃×#2280	2
#34174	23

Subgroups — C23×#34174 #19181800

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#34174 #19181800