Subgroups — C₂₃×#13812 #19161438

Order 5888.

1546 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄⋊C₄×C₂₃	C₂	1	3	yes	maximal
2	C₂₃×C₄⋊#95	C₂	1	3	yes	maximal
3	C₂₃×#1434	C₂	1	6	yes	maximal
4	C₄×C₂₃×#95	C₂	1	1	yes	maximal, characteristic
5	C₄×C₂₃×#95	C₂	1	2	yes	maximal
6	C₂²×C₄⋊C₄×C₂₃	C₂²	1	3	yes	—
7	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₂₃	C₂²	1	3	yes	—
9	C₂²×C₄²×C₂₃	C₂²	1	6	yes	—
10	C₄×C₄⋊C₄×C₂₃	C₄	1	24	yes	complemented
11	C₂×C₂₃×#95	C₂²	1	3	yes	—
12	C₂×C₂₃×#95	C₂²	1	6	yes	—
13	C₂×C₂₃×#95	C₂²	1	6	yes	—
14	C₂×C₂₃×#95	C₂²	1	6	yes	—
15	C₂×C₂₃×#95	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
17	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
18	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
19	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
20	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₂₃	—	2	6	no	—
22	C₂×C₄⋊C₄×C₂₃	—	2	6	no	—
23	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	12	yes	—
24	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	12	yes	—
25	C₂×C₄²×C₂₃	—	2	12	no	—
26	C₂×C₄²×C₂₃	—	4	12	no	—
27	C₂×C₄²×C₂₃	C₂×C₄	1	12	yes	—
28	C₂×C₄²×C₂₃	C₂×C₄	1	24	yes	—
29	C₂₃×#95	—	2	6	no	—
30	C₂₃×#95	—	2	12	no	—
31	C₂₃×#95	—	4	6	no	—
32	C₂₃×#95	C₂×C₄	1	24	yes	—
33	C₂₃×#95	Q₈	1	4	yes	—
34	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	—	2	6	no	—
36	C₂²×C₄×C₂₃	—	2	6	no	—
37	C₂²×C₄×C₂₃	—	2	6	no	—
38	C₂²×C₄×C₂₃	—	2	2	no	—
39	C₂²×C₄×C₂₃	—	2	6	no	—
40	C₂²×C₄×C₂₃	—	2	6	no	—
41	C₂²×C₄×C₂₃	—	2	6	no	—
42	C₂²×C₄×C₂₃	—	2	6	no	—
43	C₂²×C₄×C₂₃	—	2	12	no	—
44	C₂²×C₄×C₂₃	—	2	12	no	—
45	C₂²×C₄×C₂₃	—	2	3	no	—
46	C₂²×C₄×C₂₃	—	2	3	no	—
47	C₂²×C₄×C₂₃	C₂²×C₄	1	12	yes	—
48	C₂²×C₄×C₂₃	C₂²×C₄	1	6	yes	—
49	C₂²×C₄×C₂₃	C₂²×C₄	1	12	yes	—
50	C₂²×C₄×C₂₃	C₂²×C₄	1	6	yes	—
51	C₂²×C₄×C₂₃	C₂²×C₄	1	6	yes	—
52	C₂²×C₄×C₂₃	C₂×Q₈	1	6	yes	—
53	C₂²×C₄×C₂₃	#41	1	6	yes	—
54	C₂²×C₄×C₂₃	#41	1	6	yes	—
55	C₂²×C₄×C₂₃	#41	1	2	yes	—
56	C₄⋊C₄×C₂₃	—	2	24	no	—
57	C₄⋊C₄×C₂₃	C₄²	1	16	yes	—
58	C₄²×C₂₃	—	2	24	no	—
59	C₄²×C₂₃	—	4	24	no	—
60	#13812	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₂₃	C₂³×C₄	1	3	yes	—
62	C₂³×C₂₃	C₂²×Q₈	1	1	yes	characteristic
63	C₂³×C₂₃	C₂×#41	1	3	yes	—
64	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
65	C₂³×C₂₃	C₂×#41	1	3	yes	—
66	C₂³×C₂₃	H(5,𝔽₂)	1	3	yes	—
67	C₂³×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
68	C₂×C₄×C₂₃	—	2	6	no	—
69	C₂×C₄×C₂₃	—	2	6	no	—
70	C₂×C₄×C₂₃	—	2	12	no	—
71	C₂×C₄×C₂₃	—	2	12	no	—
72	C₂×C₄×C₂₃	—	2	12	no	—
73	C₂×C₄×C₂₃	—	2	12	no	—
74	C₂×C₄×C₂₃	—	2	6	no	—
75	C₂×C₄×C₂₃	—	2	12	no	—
76	C₂×C₄×C₂₃	—	2	6	no	—
77	C₂×C₄×C₂₃	—	2	12	no	—
78	C₂×C₄×C₂₃	—	2	12	no	—
79	C₂×C₄×C₂₃	—	2	6	no	—
80	C₂×C₄×C₂₃	—	4	6	no	—
81	C₂×C₄×C₂₃	—	4	12	no	—
82	C₂×C₄×C₂₃	—	4	6	no	—
83	C₂×C₄×C₂₃	—	4	12	no	—
84	C₂×C₄×C₂₃	C₂×C₄²	1	8	yes	—
85	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
86	C₂×C₄×C₂₃	#117	1	12	yes	—
87	C₂×C₄×C₂₃	#117	1	12	yes	—
88	C₂×C₄×C₂₃	C₄×Q₈	1	12	yes	—
89	C₂×C₄×C₄⋊C₄	C₂×C₂₃	1	3	yes	—
90	C₄⋊#95	C₂×C₂₃	1	3	yes	—
91	#1434	C₂×C₂₃	1	6	yes	—
92	C₄×#95	C₂×C₂₃	1	1	yes	characteristic
93	C₄×#95	C₂×C₂₃	1	2	yes	—
94	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
95	C₂²×C₂₃	C₂×#117	1	3	yes	—
96	C₂²×C₂₃	C₂×#117	1	3	yes	—
97	C₂²×C₂₃	C₂×C₄×Q₈	1	3	yes	—
98	C₂²×C₂₃	C₄×#41	1	3	yes	—
99	C₂²×C₂₃	#518	1	3	yes	—
100	C₂²×C₂₃	#520	1	3	yes	—
101	C₂²×C₂₃	#553	1	6	yes	—
102	C₂²×C₂₃	#554	1	3	yes	—
103	C₂²×C₂₃	#556	1	3	yes	—
104	C₂²×C₂₃	#557	1	1	yes	characteristic
105	C₂²×C₂₃	#559	1	3	yes	—
106	C₄×C₂₃	—	2	12	no	—
107	C₄×C₂₃	—	2	8	no	—
108	C₄×C₂₃	—	4	6	no	—
109	C₄×C₂₃	—	4	12	no	—
110	C₄×C₂₃	#376	1	8	yes	—
111	C₂²×C₄⋊C₄	C₂²×C₂₃	1	3	yes	—
112	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
113	C₂²×C₄²	C₂²×C₂₃	1	3	yes	—
114	C₂²×C₄²	C₂²×C₂₃	1	6	yes	—
115	C₄×C₄⋊C₄	C₄×C₂₃	1	24	yes	complemented
116	C₂×#95	C₂²×C₂₃	1	3	yes	—
117	C₂×#95	C₂²×C₂₃	1	6	yes	—
118	C₂×#95	C₂²×C₂₃	1	6	yes	—
119	C₂×#95	C₂²×C₂₃	1	6	yes	—
120	C₂×#95	C₂²×C₂₃	1	1	yes	characteristic
121	C₂×C₂₃	C₂×#376	1	1	yes	characteristic
122	C₂×C₂₃	C₄²×Q₈	1	1	yes	characteristic
123	C₂×C₂₃	#2275	1	1	yes	characteristic
124	C₂×C₂₃	#2345	1	3	yes	—
125	C₂×C₂₃	#2347	1	3	yes	—
126	C₂×C₂₃	#2355	1	3	yes	—
127	C₂×C₂₃	#2357	1	3	yes	—
128	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
129	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
130	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
131	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
132	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
133	C₂×C₄⋊C₄	—	2	6	no	—
134	C₂×C₄⋊C₄	—	2	6	no	—
135	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	12	yes	—
136	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	12	yes	—
137	C₂×C₄²	—	2	12	no	—
138	C₂×C₄²	—	4	12	no	—
139	C₂×C₄²	C₂×C₄×C₂₃	1	12	yes	—
140	C₂×C₄²	C₂×C₄×C₂₃	1	24	yes	—
141	#95	—	2	6	no	—
142	#95	—	2	12	no	—
143	#95	—	4	6	no	—
144	#95	C₂×C₄×C₂₃	1	24	yes	—
145	#95	Q₈×C₂₃	1	4	yes	—
146	C₂₃	#13812	1	1	yes	characteristic, direct product factor, Sylow
147	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
148	C₂²×C₄	—	2	6	no	—
149	C₂²×C₄	—	2	6	no	—
150	C₂²×C₄	—	2	6	no	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	2	6	no	—
153	C₂²×C₄	—	2	12	no	—
154	C₂²×C₄	—	2	6	no	—
155	C₂²×C₄	—	2	6	no	—
156	C₂²×C₄	—	2	6	no	—
157	C₂²×C₄	—	2	3	no	—
158	C₂²×C₄	—	2	12	no	—
159	C₂²×C₄	—	2	3	no	—
160	C₂²×C₄	C₂²×C₄×C₂₃	1	6	yes	—
161	C₂²×C₄	C₂²×C₄×C₂₃	1	12	yes	—
162	C₂²×C₄	C₂²×C₄×C₂₃	1	12	yes	—
163	C₂²×C₄	C₂²×C₄×C₂₃	1	6	yes	—
164	C₂²×C₄	C₂²×C₄×C₂₃	1	6	yes	—
165	C₂²×C₄	C₂×Q₈×C₂₃	1	6	yes	—
166	C₂²×C₄	#41×C₂₃	1	6	yes	—
167	C₂²×C₄	#41×C₂₃	1	6	yes	—
168	C₂²×C₄	#41×C₂₃	1	2	yes	—
169	C₄⋊C₄	—	2	24	no	—
170	C₄⋊C₄	C₄²×C₂₃	1	16	yes	—
171	C₄²	—	2	24	no	—
172	C₄²	—	4	24	no	—
173	C₂³	C₂³×C₄×C₂₃	1	3	yes	—
174	C₂³	C₂²×Q₈×C₂₃	1	1	yes	characteristic
175	C₂³	C₂×#41×C₂₃	1	3	yes	—
176	C₂³	C₂×#41×C₂₃	1	3	yes	—
177	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
178	C₂³	C₂₃×H(5,𝔽₂)	1	3	yes	—
179	C₂³	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
180	C₂×C₄	—	2	12	no	—
181	C₂×C₄	—	2	12	no	—
182	C₂×C₄	—	2	6	no	—
183	C₂×C₄	—	2	6	no	—
184	C₂×C₄	—	2	12	no	—
185	C₂×C₄	—	2	12	no	—
186	C₂×C₄	—	2	6	no	—
187	C₂×C₄	—	2	12	no	—
188	C₂×C₄	—	2	6	no	—
189	C₂×C₄	—	2	12	no	—
190	C₂×C₄	—	2	12	no	—
191	C₂×C₄	—	2	6	no	—
192	C₂×C₄	—	4	6	no	—
193	C₂×C₄	—	4	12	no	—
194	C₂×C₄	—	4	6	no	—
195	C₂×C₄	—	4	12	no	—
196	C₂×C₄	C₂×C₄²×C₂₃	1	8	yes	—
197	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
198	C₂×C₄	C₂₃×#117	1	12	yes	—
199	C₂×C₄	C₂₃×#117	1	12	yes	—
200	C₂×C₄	C₄×Q₈×C₂₃	1	12	yes	—
201	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
202	C₂²	C₂×C₂₃×#117	1	3	yes	—
203	C₂²	C₂×C₂₃×#117	1	3	yes	—
204	C₂²	C₂×C₄×Q₈×C₂₃	1	3	yes	—
205	C₂²	C₄×#41×C₂₃	1	3	yes	—
206	C₂²	C₂₃×#518	1	3	yes	—
207	C₂²	C₂₃×#520	1	3	yes	—
208	C₂²	C₂₃×#553	1	6	yes	—
209	C₂²	C₂₃×#554	1	3	yes	—
210	C₂²	C₂₃×#556	1	3	yes	—
211	C₂²	C₂₃×#557	1	1	yes	characteristic
212	C₂²	C₂₃×#559	1	3	yes	—
213	C₄	—	2	8	no	—
214	C₄	—	2	12	no	—
215	C₄	—	4	6	no	—
216	C₄	—	4	12	no	—
217	C₄	C₂₃×#376	1	8	yes	—
218	C₂	C₂×C₂₃×#376	1	1	yes	characteristic
219	C₂	C₄²×Q₈×C₂₃	1	1	yes	characteristic
220	C₂	C₂₃×#2275	1	1	yes	characteristic
221	C₂	C₂₃×#2345	1	3	yes	—
222	C₂	C₂₃×#2347	1	3	yes	—
223	C₂	C₂₃×#2355	1	3	yes	—
224	C₂	C₂₃×#2357	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₂₃	C₂⁴
commutator	C₂²	C₂²×C₄²×C₂₃
Frattini, cosocle	C₂⁴	C₂⁴×C₂₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₄×C₄⋊C₄×C₂₃	C₄	semidirect
60. #13812	C₂₃	direct (N × H)
115. C₄×C₄⋊C₄	C₄×C₂₃	semidirect
146. C₂₃	#13812	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×C₄×C₄⋊C₄×C₂₃	2
C₂₃×C₄⋊#95	2
C₂₃×#1434	2
C₄×C₂₃×#95	2
#13812	23

Subgroups — C23×#13812 #19161438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13812 #19161438