Subgroups — C₂×C₂₃×#2858 #19184253

Order 5888.

4462 conjugacy classes of subgroups.

The subgroup lattice has 252 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, complemented
2	C₂²×C₂₃×#127	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂₃×#124	C₂	1	4	yes	maximal, complemented
4	C₂₃×#2858	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₂₃×#390	C₂	1	8	yes	maximal, complemented
6	C₂×C₄³×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Q₈×C₂₃	C₂²	1	2	yes	—
8	C₂³×D₄×C₂₃	C₂²	1	2	yes	—
9	C₂³×D₄×C₂₃	C₂²	1	4	yes	—
10	C₂×C₂₃×C₄⋊Q₈	—	2	8	no	—
11	C₂×C₂₃×C₄⋊Q₈	C₂²	1	8	yes	complemented
12	C₂×C₂₃×#127	—	2	8	no	—
13	C₂×C₂₃×#127	C₂²	1	8	yes	—
14	C₂×C₂₃×#124	—	2	32	no	—
15	C₂×C₂₃×#124	C₂²	1	32	yes	complemented
16	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
17	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	16	yes	—
18	C₂²×C₄²×C₂₃	C₂²	1	4	yes	—
19	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
21	C₂₃×#390	C₂²	1	64	yes	complemented
22	C₄³×C₂₃	C₂²	1	8	yes	complemented
23	C₂⁵×C₂₃	C₂³	1	4	yes	—
24	C₂²×Q₈×C₂₃	—	2	16	no	—
25	C₂²×Q₈×C₂₃	—	2	8	no	—
26	C₂²×Q₈×C₂₃	C₂³	1	8	yes	—
27	C₂²×D₄×C₂₃	—	2	16	no	—
28	C₂²×D₄×C₂₃	—	2	16	no	—
29	C₂²×D₄×C₂₃	—	2	8	no	—
30	C₂²×D₄×C₂₃	—	2	32	no	—
31	C₂²×D₄×C₂₃	C₂³	1	16	yes	—
32	C₂²×D₄×C₂₃	C₂³	1	8	yes	—
33	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
34	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
35	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
36	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
37	C₂₃×C₄⋊Q₈	—	2	32	no	—
38	C₂₃×#127	—	2	32	no	—
39	C₂₃×#124	—	2	128	no	—
40	C₂×C₄⋊C₄×C₂₃	—	2	16	no	—
41	C₂×C₄⋊C₄×C₂₃	C₂³	1	16	yes	—
42	C₂×C₂²⋊C₄×C₂₃	—	2	64	no	—
43	C₂×C₂²⋊C₄×C₂₃	C₂³	1	64	yes	—
44	C₂×C₄²×C₂₃	—	2	16	no	—
45	C₂×C₄²×C₂₃	D₄	1	8	yes	—
46	C₂×C₄²×C₂₃	D₄	1	16	yes	complemented
47	C₂×C₄²×C₂₃	C₂³	1	16	yes	—
48	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
49	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
50	C₂⁴×C₂₃	—	2	16	no	—
51	C₂⁴×C₂₃	—	2	8	no	—
52	C₂⁴×C₂₃	—	2	32	no	—
53	C₂⁴×C₂₃	C₂⁴	1	8	yes	—
54	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
55	C₂×Q₈×C₂₃	—	2	32	no	—
56	C₂×Q₈×C₂₃	—	2	16	no	—
57	C₂×Q₈×C₂₃	—	4	32	no	—
58	C₂×D₄×C₂₃	—	2	32	no	—
59	C₂×D₄×C₂₃	—	2	32	no	—
60	C₂×D₄×C₂₃	—	2	16	no	—
61	C₂×D₄×C₂₃	—	2	64	no	—
62	C₂×D₄×C₂₃	—	4	64	no	—
63	C₂×D₄×C₂₃	—	4	32	no	—
64	C₂²×C₄×C₂₃	—	2	16	no	—
65	C₂²×C₄×C₂₃	—	2	16	no	—
66	C₂²×C₄×C₂₃	—	2	8	no	—
67	C₂²×C₄×C₂₃	C₂×D₄	1	16	yes	—
68	C₂²×C₄×C₂₃	C₂×D₄	1	8	yes	—
69	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₂₃	C₂×D₄	1	8	yes	—
71	C₂²×C₄×C₂₃	#41	1	16	yes	—
72	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
73	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
74	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
75	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
76	C₄⋊C₄×C₂₃	—	2	32	no	—
77	C₂²⋊C₄×C₂₃	—	2	128	no	—
78	C₄²×C₂₃	—	2	32	no	—
79	C₄²×C₂₃	C₂×D₄	1	16	yes	—
80	C₄²×C₂₃	C₂×D₄	1	32	yes	complemented
81	C₂×#2858	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₂₃	—	2	16	no	—
83	C₂³×C₂₃	—	2	8	no	—
84	C₂³×C₂₃	—	2	32	no	—
85	C₂³×C₂₃	—	4	32	no	—
86	C₂³×C₂₃	—	4	16	no	—
87	C₂³×C₂₃	—	4	64	no	—
88	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
89	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
90	C₂³×C₂₃	C₂×#41	1	8	yes	—
91	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
92	Q₈×C₂₃	—	4	32	no	—
93	D₄×C₂₃	—	4	64	no	—
94	D₄×C₂₃	—	4	32	no	—
95	C₂×C₄×C₂₃	—	2	32	no	—
96	C₂×C₄×C₂₃	—	2	16	no	—
97	C₂×C₄×C₂₃	—	2	16	no	—
98	C₂×C₄×C₂₃	—	2	8	no	—
99	C₂×C₄×C₂₃	—	4	16	no	—
100	C₂×C₄×C₂₃	#124	1	32	yes	complemented
101	C₂×C₄×C₂₃	#127	1	16	yes	—
102	C₂×C₄×C₂₃	C₂²×D₄	1	16	yes	—
103	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
104	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
105	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
106	C₂×C₄×C₂₃	C₂×#41	1	16	yes	—
107	C₂²×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
108	C₂²×#127	C₂×C₂₃	1	1	yes	characteristic
109	C₂²×#124	C₂×C₂₃	1	4	yes	complemented
110	#2858	C₂×C₂₃	1	16	yes	direct product factor
111	C₂×#390	C₂×C₂₃	1	8	yes	complemented
112	C₂×C₄³	C₂×C₂₃	1	1	yes	characteristic, complemented
113	C₂²×C₂₃	—	4	16	no	—
114	C₂²×C₂₃	—	4	8	no	—
115	C₂²×C₂₃	—	4	32	no	—
116	C₂²×C₂₃	—	8	32	no	—
117	C₂²×C₂₃	C₂×#124	1	8	yes	—
118	C₂²×C₂₃	C₂×#127	1	4	yes	—
119	C₂²×C₂₃	#532	1	16	yes	—
120	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
121	C₂²×C₂₃	C₂³×D₄	1	2	yes	—
122	C₂²×C₂₃	C₂²×#41	1	4	yes	—
123	C₄×C₂₃	—	2	16	no	—
124	C₄×C₂₃	—	4	8	no	—
125	C₄×C₂₃	C₂×#124	1	16	yes	complemented
126	C₄×C₂₃	C₂×#127	1	8	yes	—
127	C₂³×Q₈	C₂²×C₂₃	1	2	yes	—
128	C₂³×D₄	C₂²×C₂₃	1	2	yes	—
129	C₂³×D₄	C₂²×C₂₃	1	4	yes	—
130	C₂×C₄⋊Q₈	—	2	8	no	—
131	C₂×C₄⋊Q₈	C₂²×C₂₃	1	8	yes	complemented
132	C₂×#127	—	2	8	no	—
133	C₂×#127	C₂²×C₂₃	1	8	yes	—
134	C₂×#124	—	2	32	no	—
135	C₂×#124	C₂²×C₂₃	1	32	yes	complemented
136	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
137	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	16	yes	—
138	C₂²×C₄²	C₂²×C₂₃	1	4	yes	—
139	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
140	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
141	#390	C₂²×C₂₃	1	64	yes	complemented
142	C₄³	C₂²×C₂₃	1	8	yes	complemented
143	C₂×C₂₃	—	8	8	no	—
144	C₂×C₂₃	#2858	1	8	yes	direct product factor
145	C₂×C₂₃	C₂²×#124	1	2	yes	—
146	C₂×C₂₃	C₂²×#127	1	1	yes	characteristic
147	C₂×C₂₃	C₂×#532	1	4	yes	—
148	C₂⁵	C₂³×C₂₃	1	4	yes	—
149	C₂²×Q₈	—	2	16	no	—
150	C₂²×Q₈	—	2	8	no	—
151	C₂²×Q₈	C₂³×C₂₃	1	8	yes	—
152	C₂²×D₄	—	2	16	no	—
153	C₂²×D₄	—	2	16	no	—
154	C₂²×D₄	—	2	8	no	—
155	C₂²×D₄	—	2	32	no	—
156	C₂²×D₄	C₂³×C₂₃	1	8	yes	—
157	C₂²×D₄	C₂³×C₂₃	1	16	yes	—
158	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
159	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
160	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
161	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
162	C₄⋊Q₈	—	2	32	no	—
163	#127	—	2	32	no	—
164	#124	—	2	128	no	—
165	C₂×C₄⋊C₄	—	2	16	no	—
166	C₂×C₄⋊C₄	C₂³×C₂₃	1	16	yes	—
167	C₂×C₂²⋊C₄	—	2	64	no	—
168	C₂×C₂²⋊C₄	C₂³×C₂₃	1	64	yes	—
169	C₂×C₄²	—	2	16	no	—
170	C₂×C₄²	D₄×C₂₃	1	8	yes	—
171	C₂×C₄²	D₄×C₂₃	1	16	yes	complemented
172	C₂×C₄²	C₂³×C₂₃	1	16	yes	—
173	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
174	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
175	C₂₃	C₂×#2858	1	1	yes	characteristic, direct product factor, Sylow
176	C₂⁴	—	2	16	no	—
177	C₂⁴	—	2	8	no	—
178	C₂⁴	—	2	32	no	—
179	C₂⁴	C₂⁴×C₂₃	1	8	yes	—
180	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
181	C₂×Q₈	—	2	32	no	—
182	C₂×Q₈	—	2	16	no	—
183	C₂×Q₈	—	4	32	no	—
184	C₂×D₄	—	2	32	no	—
185	C₂×D₄	—	2	32	no	—
186	C₂×D₄	—	2	16	no	—
187	C₂×D₄	—	2	64	no	—
188	C₂×D₄	—	4	64	no	—
189	C₂×D₄	—	4	32	no	—
190	C₂²×C₄	—	2	16	no	—
191	C₂²×C₄	—	2	16	no	—
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	C₂×D₄×C₂₃	1	16	yes	—
194	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	—
195	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
196	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	—
197	C₂²×C₄	#41×C₂₃	1	16	yes	—
198	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
199	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
200	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
201	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
202	C₄⋊C₄	—	2	32	no	—
203	C₂²⋊C₄	—	2	128	no	—
204	C₄²	—	2	32	no	—
205	C₄²	C₂×D₄×C₂₃	1	16	yes	—
206	C₄²	C₂×D₄×C₂₃	1	32	yes	complemented
207	C₂³	—	2	16	no	—
208	C₂³	—	2	8	no	—
209	C₂³	—	2	32	no	—
210	C₂³	—	4	32	no	—
211	C₂³	—	4	16	no	—
212	C₂³	—	4	64	no	—
213	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
214	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
215	C₂³	C₂×#41×C₂₃	1	8	yes	—
216	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
217	Q₈	—	4	32	no	—
218	D₄	—	4	64	no	—
219	D₄	—	4	32	no	—
220	C₂×C₄	—	2	16	no	—
221	C₂×C₄	—	2	32	no	—
222	C₂×C₄	—	2	16	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	4	16	no	—
225	C₂×C₄	C₂₃×#124	1	32	yes	complemented
226	C₂×C₄	C₂₃×#127	1	16	yes	—
227	C₂×C₄	C₂²×D₄×C₂₃	1	16	yes	—
228	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
229	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
230	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
231	C₂×C₄	C₂×#41×C₂₃	1	16	yes	—
232	C₂²	—	4	16	no	—
233	C₂²	—	4	8	no	—
234	C₂²	—	4	32	no	—
235	C₂²	—	8	32	no	—
236	C₂²	C₂×C₂₃×#124	1	8	yes	—
237	C₂²	C₂×C₂₃×#127	1	4	yes	—
238	C₂²	C₂₃×#532	1	16	yes	—
239	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
240	C₂²	C₂³×D₄×C₂₃	1	2	yes	—
241	C₂²	C₂²×#41×C₂₃	1	4	yes	—
242	C₄	—	2	16	no	—
243	C₄	—	4	8	no	—
244	C₄	C₂×C₂₃×#124	1	16	yes	complemented
245	C₄	C₂×C₂₃×#127	1	8	yes	—
246	C₂	—	8	8	no	—
247	C₂	C₂₃×#2858	1	8	yes	direct product factor
248	C₂	C₂²×C₂₃×#124	1	2	yes	—
249	C₂	C₂²×C₂₃×#127	1	1	yes	characteristic
250	C₂	C₂×C₂₃×#532	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₂₃
prime radical	C₂²×C₂₃×#127	C₂
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
1. C₂²×C₂₃×C₄⋊Q₈	C₂	semidirect
3. C₂²×C₂₃×#124	C₂	semidirect
4. C₂₃×#2858	C₂	direct (N × H)
5. C₂×C₂₃×#390	C₂	semidirect
6. C₂×C₄³×C₂₃	C₂	semidirect
11. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
15. C₂×C₂₃×#124	C₂²	semidirect
21. C₂₃×#390	C₂²	semidirect
22. C₄³×C₂₃	C₂²	semidirect
46. C₂×C₄²×C₂₃	D₄	semidirect
80. C₄²×C₂₃	C₂×D₄	semidirect
81. C₂×#2858	C₂₃	direct (N × H)
100. C₂×C₄×C₂₃	#124	semidirect
107. C₂²×C₄⋊Q₈	C₂×C₂₃	semidirect
109. C₂²×#124	C₂×C₂₃	semidirect
110. #2858	C₂×C₂₃	direct (N × H)
111. C₂×#390	C₂×C₂₃	semidirect
112. C₂×C₄³	C₂×C₂₃	semidirect
125. C₄×C₂₃	C₂×#124	semidirect
131. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
135. C₂×#124	C₂²×C₂₃	semidirect
141. #390	C₂²×C₂₃	semidirect
142. C₄³	C₂²×C₂₃	semidirect
144. C₂×C₂₃	#2858	direct (N × H)
171. C₂×C₄²	D₄×C₂₃	semidirect
175. C₂₃	C₂×#2858	direct (N × H)
206. C₄²	C₂×D₄×C₂₃	semidirect
225. C₂×C₄	C₂₃×#124	semidirect
244. C₄	C₂×C₂₃×#124	semidirect
247. C₂	C₂₃×#2858	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×#127	2
C₂²×C₂₃×#124	2
C₂₃×#2858	2
C₂×C₂₃×#390	2
C₂×C₄³×C₂₃	2
C₂×#2858	23

Subgroups — C2×C23×#2858 #19184253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2858 #19184253