Subgroups — C₂×C₂₃×#2642 #19182254

Order 5888.

2286 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₃×#125	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2642	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#400	C₂	1	8	yes	maximal
5	C₂×C₂₃×#391	C₂	1	4	yes	maximal
6	C₂×C₂₃×#376	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₂₃	C₂²	1	2	yes	—
8	C₂×C₂₃×C₄⋊Q₈	—	2	8	no	—
9	C₂×C₂₃×C₄⋊Q₈	C₂²	1	8	yes	—
10	C₂×C₂₃×#125	—	2	8	no	—
11	C₂×C₂₃×#125	C₂²	1	8	yes	—
12	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
13	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
14	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
15	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
16	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#400	C₂²	1	64	yes	—
18	C₂₃×#391	C₂²	1	32	yes	—
19	C₂₃×#376	C₂²	1	8	yes	—
20	C₂×C₂₃×#95	C₂²	1	4	yes	—
21	C₂×C₂₃×#95	C₂²	1	16	yes	—
22	C₂²×Q₈×C₂₃	—	2	8	no	—
23	C₂²×Q₈×C₂₃	—	2	16	no	—
24	C₂²×Q₈×C₂₃	C₂³	1	8	yes	—
25	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
26	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
27	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
28	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
29	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
30	C₂₃×C₄⋊Q₈	—	2	32	no	—
31	C₂₃×#125	—	2	32	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	16	no	—
33	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
34	C₂×C₄⋊C₄×C₂₃	—	2	16	no	—
35	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
36	C₂×C₄⋊C₄×C₂₃	C₂³	1	16	yes	—
37	C₂×C₄⋊C₄×C₂₃	C₂³	1	16	yes	—
38	C₂×C₄²×C₂₃	—	2	8	no	—
39	C₂×C₄²×C₂₃	D₄	1	8	yes	—
40	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
41	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
42	C₂₃×#95	C₂³	1	16	yes	—
43	C₂₃×#95	C₂³	1	64	yes	—
44	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
45	C₂×Q₈×C₂₃	—	2	16	no	—
46	C₂×Q₈×C₂₃	—	2	32	no	—
47	C₂×Q₈×C₂₃	—	4	32	no	—
48	C₂²×C₄×C₂₃	—	2	8	no	—
49	C₂²×C₄×C₂₃	—	2	8	no	—
50	C₂²×C₄×C₂₃	—	2	8	no	—
51	C₂²×C₄×C₂₃	—	2	8	no	—
52	C₂²×C₄×C₂₃	—	2	16	no	—
53	C₂²×C₄×C₂₃	—	2	16	no	—
54	C₂²×C₄×C₂₃	—	2	16	no	—
55	C₂²×C₄×C₂₃	C₂×D₄	1	8	yes	—
56	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
57	C₂²×C₄×C₂₃	#41	1	8	yes	—
58	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
59	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
60	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
61	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
62	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
63	C₄⋊C₄×C₂₃	—	2	32	no	—
64	C₄⋊C₄×C₂₃	—	2	16	no	—
65	C₄⋊C₄×C₂₃	—	2	32	no	—
66	C₄²×C₂₃	—	2	16	no	—
67	C₄²×C₂₃	C₂×D₄	1	16	yes	—
68	C₂×#2642	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
70	C₂³×C₂₃	C₂×#41	1	4	yes	—
71	C₂³×C₂₃	2₋¹⁺⁴	1	8	yes	—
72	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
73	Q₈×C₂₃	—	4	32	no	—
74	C₂×C₄×C₂₃	—	2	16	no	—
75	C₂×C₄×C₂₃	—	2	8	no	—
76	C₂×C₄×C₂₃	—	2	8	no	—
77	C₂×C₄×C₂₃	—	2	8	no	—
78	C₂×C₄×C₂₃	—	2	8	no	—
79	C₂×C₄×C₂₃	—	2	16	no	—
80	C₂×C₄×C₂₃	—	2	16	no	—
81	C₂×C₄×C₂₃	—	2	16	no	—
82	C₂×C₄×C₂₃	—	4	16	no	—
83	C₂×C₄×C₂₃	—	4	16	no	—
84	C₂×C₄×C₂₃	—	4	16	no	—
85	C₂×C₄×C₂₃	#124	1	16	yes	—
86	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
87	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
88	C₂×C₄×C₂₃	C₂×#41	1	8	yes	—
89	C₂²×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
90	C₂²×#125	C₂×C₂₃	1	1	yes	characteristic
91	#2642	C₂×C₂₃	1	16	yes	direct product factor
92	C₂×#400	C₂×C₂₃	1	8	yes	—
93	C₂×#391	C₂×C₂₃	1	4	yes	—
94	C₂×#376	C₂×C₂₃	1	1	yes	characteristic
95	C₂²×C₂₃	C₂×#124	1	4	yes	—
96	C₂²×C₂₃	#536	1	8	yes	—
97	C₂²×C₂₃	#541	1	16	yes	—
98	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
99	C₂²×C₂₃	C₂²×#41	1	2	yes	—
100	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	4	yes	—
101	C₄×C₂₃	—	2	8	no	—
102	C₄×C₂₃	—	4	8	no	—
103	C₄×C₂₃	—	4	8	no	—
104	C₄×C₂₃	—	4	8	no	—
105	C₄×C₂₃	C₂×#124	1	8	yes	—
106	C₂³×Q₈	C₂²×C₂₃	1	2	yes	—
107	C₂×C₄⋊Q₈	—	2	8	no	—
108	C₂×C₄⋊Q₈	C₂²×C₂₃	1	8	yes	—
109	C₂×#125	—	2	8	no	—
110	C₂×#125	C₂²×C₂₃	1	8	yes	—
111	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
112	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
113	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
114	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
115	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
116	#400	C₂²×C₂₃	1	64	yes	—
117	#391	C₂²×C₂₃	1	32	yes	—
118	#376	C₂²×C₂₃	1	8	yes	—
119	C₂×#95	C₂²×C₂₃	1	4	yes	—
120	C₂×#95	C₂²×C₂₃	1	16	yes	—
121	C₂×C₂₃	#2642	1	8	yes	direct product factor
122	C₂×C₂₃	C₂²×#124	1	1	yes	characteristic
123	C₂×C₂₃	C₂×#536	1	2	yes	—
124	C₂×C₂₃	C₂×#541	1	4	yes	—
125	C₂²×Q₈	—	2	8	no	—
126	C₂²×Q₈	—	2	16	no	—
127	C₂²×Q₈	C₂³×C₂₃	1	8	yes	—
128	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
129	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
130	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
131	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
132	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
133	C₄⋊Q₈	—	2	32	no	—
134	#125	—	2	32	no	—
135	C₂×C₄⋊C₄	—	2	16	no	—
136	C₂×C₄⋊C₄	—	2	8	no	—
137	C₂×C₄⋊C₄	—	2	16	no	—
138	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
139	C₂×C₄⋊C₄	C₂³×C₂₃	1	16	yes	—
140	C₂×C₄⋊C₄	C₂³×C₂₃	1	16	yes	—
141	C₂×C₄²	—	2	8	no	—
142	C₂×C₄²	D₄×C₂₃	1	8	yes	—
143	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
144	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
145	#95	C₂³×C₂₃	1	16	yes	—
146	#95	C₂³×C₂₃	1	64	yes	—
147	C₂₃	C₂×#2642	1	1	yes	characteristic, direct product factor, Sylow
148	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
149	C₂×Q₈	—	2	16	no	—
150	C₂×Q₈	—	2	32	no	—
151	C₂×Q₈	—	4	32	no	—
152	C₂²×C₄	—	2	8	no	—
153	C₂²×C₄	—	2	8	no	—
154	C₂²×C₄	—	2	8	no	—
155	C₂²×C₄	—	2	8	no	—
156	C₂²×C₄	—	2	16	no	—
157	C₂²×C₄	—	2	16	no	—
158	C₂²×C₄	—	2	16	no	—
159	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	—
160	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
161	C₂²×C₄	#41×C₂₃	1	8	yes	—
162	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
163	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
164	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
165	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
166	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
167	C₄⋊C₄	—	2	32	no	—
168	C₄⋊C₄	—	2	16	no	—
169	C₄⋊C₄	—	2	32	no	—
170	C₄²	—	2	16	no	—
171	C₄²	C₂×D₄×C₂₃	1	16	yes	—
172	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
173	C₂³	C₂×#41×C₂₃	1	4	yes	—
174	C₂³	C₂₃×2₋¹⁺⁴	1	8	yes	—
175	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
176	Q₈	—	4	32	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	16	no	—
179	C₂×C₄	—	2	8	no	—
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	2	8	no	—
182	C₂×C₄	—	2	16	no	—
183	C₂×C₄	—	2	16	no	—
184	C₂×C₄	—	2	16	no	—
185	C₂×C₄	—	4	16	no	—
186	C₂×C₄	—	4	16	no	—
187	C₂×C₄	—	4	16	no	—
188	C₂×C₄	C₂₃×#124	1	16	yes	—
189	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
190	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
191	C₂×C₄	C₂×#41×C₂₃	1	8	yes	—
192	C₂²	C₂×C₂₃×#124	1	4	yes	—
193	C₂²	C₂₃×#536	1	8	yes	—
194	C₂²	C₂₃×#541	1	16	yes	—
195	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
196	C₂²	C₂²×#41×C₂₃	1	2	yes	—
197	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	4	yes	—
198	C₄	—	2	8	no	—
199	C₄	—	4	8	no	—
200	C₄	—	4	8	no	—
201	C₄	—	4	8	no	—
202	C₄	C₂×C₂₃×#124	1	8	yes	—
203	C₂	C₂₃×#2642	1	8	yes	direct product factor
204	C₂	C₂²×C₂₃×#124	1	1	yes	characteristic
205	C₂	C₂×C₂₃×#536	1	2	yes	—
206	C₂	C₂×C₂₃×#541	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2642	C₂	direct (N × H)
68. C₂×#2642	C₂₃	direct (N × H)
91. #2642	C₂×C₂₃	direct (N × H)
121. C₂×C₂₃	#2642	direct (N × H)
147. C₂₃	C₂×#2642	direct (N × H)
203. C₂	C₂₃×#2642	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×C₄⋊Q₈	2
C₂²×C₂₃×#125	2
C₂₃×#2642	2
C₂×C₂₃×#400	2
C₂×C₂₃×#391	2
C₂×C₂₃×#376	2
C₂×#2642	23

Subgroups — C2×C23×#2642 #19182254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2642 #19182254