Subgroups — C₂×C₂₃×#1295 #19155653

Order 5888.

830 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	C₂²×C₂₃×#97	C₂	1	2	yes	maximal
2	C₂×C₂₃×#382	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1295	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₈×C₂₃	—	2	2	no	—
5	C₂²×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
6	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×#97	—	2	2	no	—
8	C₂×C₂₃×#97	—	2	4	no	—
9	C₂×C₂₃×#97	C₄	1	8	yes	complemented
10	C₂×C₂₃×#97	C₂²	1	4	yes	—
11	C₂₃×#382	—	2	2	no	—
12	C₂₃×#382	C₄	1	2	yes	—
13	C₂₃×#382	C₂²	1	2	yes	—
14	C₂×C₂₃×#95	—	2	2	no	—
15	C₂³×C₄×C₂₃	—	2	2	no	—
16	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₂₃	D₄	1	2	yes	—
18	C₂²×C₈×C₂₃	—	2	16	no	—
19	C₂²×C₈×C₂₃	—	2	8	no	—
20	C₂²×C₈×C₂₃	—	2	4	no	—
21	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
22	C₂×C₄⋊C₄×C₂₃	—	4	4	no	—
23	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₂₃	D₄	1	2	yes	—
27	C₂×C₄²×C₂₃	D₄	1	2	yes	—
28	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
29	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
30	C₂₃×#97	—	2	8	no	—
31	C₂₃×#97	—	2	4	no	—
32	C₂₃×#97	C₂×C₄	1	8	yes	complemented
33	C₂₃×#95	—	2	4	no	—
34	C₂₃×#95	—	2	4	no	—
35	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	—	2	2	no	—
37	C₂²×C₄×C₂₃	—	2	2	no	—
38	C₂²×C₄×C₂₃	—	2	8	no	—
39	C₂²×C₄×C₂₃	—	2	4	no	—
40	C₂²×C₄×C₂₃	—	4	4	no	—
41	C₂²×C₄×C₂₃	—	4	8	no	—
42	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
46	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
47	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
50	C₂×C₈×C₂₃	—	2	32	no	—
51	C₂×C₈×C₂₃	—	2	16	no	—
52	C₂×C₈×C₂₃	—	2	8	no	—
53	C₄⋊C₄×C₂₃	—	4	4	no	—
54	C₄⋊C₄×C₂₃	—	4	4	no	—
55	C₄²×C₂₃	C₄²	1	2	yes	—
56	C₄²×C₂₃	C₂²⋊C₄	1	4	yes	—
57	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
58	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
59	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
60	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
62	C₂×#1295	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₂₃	#95	1	2	yes	—
64	C₂³×C₂₃	#100	1	2	yes	—
65	C₂³×C₂₃	#101	1	2	yes	—
66	C₂³×C₂₃	#104	1	8	yes	—
67	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₂₃	—	2	8	no	—
69	C₂×C₄×C₂₃	—	2	8	no	—
70	C₂×C₄×C₂₃	—	2	4	no	—
71	C₂×C₄×C₂₃	—	2	4	no	—
72	C₂×C₄×C₂₃	—	2	1	no	—
73	C₂×C₄×C₂₃	—	2	1	no	—
74	C₂×C₄×C₂₃	—	2	1	no	—
75	C₂×C₄×C₂₃	—	2	1	no	—
76	C₂×C₄×C₂₃	—	2	4	no	—
77	C₂×C₄×C₂₃	—	2	4	no	—
78	C₂×C₄×C₂₃	—	4	4	no	—
79	C₂×C₄×C₂₃	—	4	8	no	—
80	C₂×C₄×C₂₃	—	8	8	no	—
81	C₂×C₄×C₂₃	#95	1	4	yes	—
82	C₂×C₄×C₂₃	#95	1	2	yes	—
83	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
85	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
86	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₈×C₂₃	—	2	16	no	—
88	C₂²×#97	C₂×C₂₃	1	2	yes	—
89	C₂×#382	C₂×C₂₃	1	1	yes	characteristic
90	#1295	C₂×C₂₃	1	4	yes	direct product factor
91	C₂²×C₂₃	#329	1	8	yes	—
92	C₂²×C₂₃	#330	1	8	yes	—
93	C₂²×C₂₃	#339	1	8	yes	—
94	C₂²×C₂₃	#343	1	4	yes	—
95	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
96	C₂²×C₂₃	C₂×#100	1	1	yes	characteristic
97	C₂²×C₂₃	C₂×#101	1	1	yes	characteristic
98	C₂²×C₂₃	C₂×#104	1	4	yes	—
99	C₄×C₂₃	—	2	2	no	—
100	C₄×C₂₃	—	2	4	no	—
101	C₄×C₂₃	—	2	2	no	—
102	C₄×C₂₃	—	2	2	no	—
103	C₄×C₂₃	—	2	1	no	—
104	C₄×C₂₃	—	2	1	no	—
105	C₄×C₂₃	—	8	4	no	—
106	C₂³×C₈	—	2	2	no	—
107	C₂²×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
108	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
109	C₂×#97	—	2	2	no	—
110	C₂×#97	—	2	4	no	—
111	C₂×#97	C₄×C₂₃	1	8	yes	complemented
112	C₂×#97	C₂²×C₂₃	1	4	yes	—
113	#382	—	2	2	no	—
114	#382	C₄×C₂₃	1	2	yes	—
115	#382	C₂²×C₂₃	1	2	yes	—
116	C₂×#95	—	2	2	no	—
117	C₂×C₂₃	#1295	1	8	yes	direct product factor
118	C₂×C₂₃	C₂×#329	1	2	yes	—
119	C₂×C₂₃	C₂×#330	1	2	yes	—
120	C₂×C₂₃	C₂×#339	1	2	yes	—
121	C₂×C₂₃	C₂×#343	1	1	yes	characteristic
122	C₂³×C₄	—	2	2	no	—
123	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
124	C₂³×C₄	D₄×C₂₃	1	2	yes	—
125	C₂²×C₈	—	2	16	no	—
126	C₂²×C₈	—	2	8	no	—
127	C₂²×C₈	—	2	4	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	4	4	no	—
130	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
131	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
132	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
133	C₂×C₄²	D₄×C₂₃	1	2	yes	—
134	C₂×C₄²	D₄×C₂₃	1	2	yes	—
135	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
136	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
137	#97	—	2	8	no	—
138	#97	—	2	4	no	—
139	#97	C₂×C₄×C₂₃	1	8	yes	complemented
140	#95	—	2	4	no	—
141	#95	—	2	4	no	—
142	C₂₃	C₂×#1295	1	1	yes	characteristic, direct product factor, Sylow
143	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	—	2	8	no	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	4	4	no	—
149	C₂²×C₄	—	4	8	no	—
150	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
151	C₂²×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
152	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
153	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
154	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
155	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
156	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
157	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
158	C₂×C₈	—	2	32	no	—
159	C₂×C₈	—	2	16	no	—
160	C₂×C₈	—	2	8	no	—
161	C₄⋊C₄	—	4	4	no	—
162	C₄⋊C₄	—	4	4	no	—
163	C₄²	C₄²×C₂₃	1	2	yes	—
164	C₄²	C₂²⋊C₄×C₂₃	1	4	yes	—
165	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
166	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
167	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
168	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
169	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
170	C₂³	C₂₃×#95	1	2	yes	—
171	C₂³	C₂₃×#100	1	2	yes	—
172	C₂³	C₂₃×#101	1	2	yes	—
173	C₂³	C₂₃×#104	1	8	yes	—
174	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
175	C₂×C₄	—	2	8	no	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	4	8	no	—
187	C₂×C₄	—	8	8	no	—
188	C₂×C₄	C₂₃×#95	1	4	yes	—
189	C₂×C₄	C₂₃×#95	1	2	yes	—
190	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
191	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
192	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
193	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
194	C₈	—	2	16	no	—
195	C₂²	C₂₃×#329	1	8	yes	—
196	C₂²	C₂₃×#330	1	8	yes	—
197	C₂²	C₂₃×#339	1	8	yes	—
198	C₂²	C₂₃×#343	1	4	yes	—
199	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
200	C₂²	C₂×C₂₃×#100	1	1	yes	characteristic
201	C₂²	C₂×C₂₃×#101	1	1	yes	characteristic
202	C₂²	C₂×C₂₃×#104	1	4	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	4	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	2	2	no	—
209	C₄	—	8	4	no	—
210	C₂	C₂₃×#1295	1	8	yes	direct product factor
211	C₂	C₂×C₂₃×#329	1	2	yes	—
212	C₂	C₂×C₂₃×#330	1	2	yes	—
213	C₂	C₂×C₂₃×#339	1	2	yes	—
214	C₂	C₂×C₂₃×#343	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₂₃	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1295	C₂	direct (N × H)
9. C₂×C₂₃×#97	C₄	semidirect
32. C₂₃×#97	C₂×C₄	semidirect
62. C₂×#1295	C₂₃	direct (N × H)
90. #1295	C₂×C₂₃	direct (N × H)
111. C₂×#97	C₄×C₂₃	semidirect
117. C₂×C₂₃	#1295	direct (N × H)
139. #97	C₂×C₄×C₂₃	semidirect
142. C₂₃	C₂×#1295	direct (N × H)
210. C₂	C₂₃×#1295	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#97	2
C₂×C₂₃×#382	2
C₂₃×#1295	2
C₂×#1295	23

Subgroups — C2×C23×#1295 #19155653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1295 #19155653