Subgroups — C₂×C₂₃×#2817 #19184275

Order 5888.

2478 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×C₂²⋊Q₈	C₂	1	2	yes	maximal
2	C₂₃×#2817	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₂₃×#400	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₃×#397	C₂	1	4	yes	maximal
5	C₂×C₂₃×#395	C₂	1	4	yes	maximal, complemented
6	C₂×C₂₃×#381	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₂₃	C₂²	1	2	yes	—
8	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×C₂²⋊Q₈	—	2	16	no	—
10	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	16	yes	—
11	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
12	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
13	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
14	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
15	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
16	C₂₃×#400	C₂²	1	32	yes	complemented
17	C₂₃×#397	C₂²	1	32	yes	—
18	C₂₃×#395	C₂²	1	32	yes	complemented
19	C₂₃×#381	C₂²	1	8	yes	—
20	C₂×C₂₃×#95	C₂²	1	4	yes	—
21	C₂×C₂₃×#95	C₂²	1	8	yes	—
22	C₂×C₂₃×#95	C₂²	1	4	yes	—
23	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
24	C₂²×Q₈×C₂₃	—	2	8	no	—
25	C₂²×Q₈×C₂₃	—	4	8	no	—
26	C₂²×Q₈×C₂₃	C₂³	1	8	yes	—
27	C₂³×C₄×C₂₃	—	4	4	no	—
28	C₂³×C₄×C₂₃	D₄	1	8	yes	—
29	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
30	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
31	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
32	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
33	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
34	C₂₃×C₂²⋊Q₈	—	2	64	no	—
35	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
36	C₂×C₄⋊C₄×C₂₃	—	2	16	no	—
37	C₂×C₄⋊C₄×C₂₃	—	4	8	no	—
38	C₂×C₄⋊C₄×C₂₃	C₂³	1	16	yes	—
39	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
40	C₂×C₄⋊C₄×C₂₃	C₂³	1	16	yes	—
41	C₂×C₂²⋊C₄×C₂₃	—	2	16	no	—
42	C₂×C₂²⋊C₄×C₂₃	—	4	4	no	—
43	C₂×C₂²⋊C₄×C₂₃	C₂³	1	16	yes	—
44	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
45	C₂₃×#95	C₂³	1	16	yes	—
46	C₂₃×#95	C₂³	1	32	yes	—
47	C₂₃×#95	C₂³	1	16	yes	—
48	C₂⁴×C₂₃	—	2	4	no	—
49	C₂⁴×C₂₃	—	2	8	no	—
50	C₂⁴×C₂₃	C₂×D₄	1	4	yes	—
51	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
52	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
53	C₂×Q₈×C₂₃	—	2	16	no	—
54	C₂×Q₈×C₂₃	—	4	16	no	—
55	C₂×Q₈×C₂₃	—	4	32	no	—
56	C₂²×C₄×C₂₃	—	2	8	no	—
57	C₂²×C₄×C₂₃	—	2	8	no	—
58	C₂²×C₄×C₂₃	—	2	8	no	—
59	C₂²×C₄×C₂₃	—	2	8	no	—
60	C₂²×C₄×C₂₃	—	2	16	no	—
61	C₂²×C₄×C₂₃	—	2	16	no	—
62	C₂²×C₄×C₂₃	—	2	16	no	—
63	C₂²×C₄×C₂₃	—	4	16	no	—
64	C₂²×C₄×C₂₃	—	4	8	no	—
65	C₂²×C₄×C₂₃	C₂×D₄	1	8	yes	—
66	C₂²×C₄×C₂₃	C₂×D₄	1	16	yes	—
67	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
68	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
69	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
70	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
71	C₄⋊C₄×C₂₃	—	2	16	no	—
72	C₄⋊C₄×C₂₃	—	2	32	no	—
73	C₄⋊C₄×C₂₃	—	4	16	no	—
74	C₂²⋊C₄×C₂₃	—	2	32	no	—
75	C₂²⋊C₄×C₂₃	—	4	8	no	—
76	C₂×#2817	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₂₃	—	2	8	no	—
78	C₂³×C₂₃	—	2	16	no	—
79	C₂³×C₂₃	—	2	4	no	—
80	C₂³×C₂₃	—	2	8	no	—
81	C₂³×C₂₃	—	4	16	no	—
82	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
83	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
84	C₂³×C₂₃	H(5,𝔽₂)	1	4	yes	—
85	C₂³×C₂₃	2₋¹⁺⁴	1	8	yes	—
86	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
87	Q₈×C₂₃	—	4	32	no	—
88	C₂×C₄×C₂₃	—	2	16	no	—
89	C₂×C₄×C₂₃	—	2	8	no	—
90	C₂×C₄×C₂₃	—	2	8	no	—
91	C₂×C₄×C₂₃	—	2	8	no	—
92	C₂×C₄×C₂₃	—	2	8	no	—
93	C₂×C₄×C₂₃	—	2	16	no	—
94	C₂×C₄×C₂₃	—	2	16	no	—
95	C₂×C₄×C₂₃	—	2	16	no	—
96	C₂×C₄×C₂₃	—	4	16	no	—
97	C₂×C₄×C₂₃	—	4	16	no	—
98	C₂×C₄×C₂₃	—	4	16	no	—
99	C₂×C₄×C₂₃	—	4	16	no	—
100	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
101	C₂²×C₂²⋊Q₈	C₂×C₂₃	1	2	yes	—
102	#2817	C₂×C₂₃	1	16	yes	direct product factor
103	C₂×#400	C₂×C₂₃	1	4	yes	complemented
104	C₂×#397	C₂×C₂₃	1	4	yes	—
105	C₂×#395	C₂×C₂₃	1	4	yes	complemented
106	C₂×#381	C₂×C₂₃	1	1	yes	characteristic
107	C₂²×C₂₃	—	2	4	no	—
108	C₂²×C₂₃	—	2	8	no	—
109	C₂²×C₂₃	—	4	16	no	—
110	C₂²×C₂₃	—	4	8	no	—
111	C₂²×C₂₃	#534	1	4	yes	—
112	C₂²×C₂₃	#536	1	8	yes	—
113	C₂²×C₂₃	#563	1	16	yes	—
114	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
115	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
116	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	4	yes	—
117	C₄×C₂₃	—	2	8	no	—
118	C₄×C₂₃	—	4	8	no	—
119	C₄×C₂₃	—	4	8	no	—
120	C₄×C₂₃	—	4	8	no	—
121	C₂³×Q₈	C₂²×C₂₃	1	2	yes	—
122	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
123	C₂×C₂²⋊Q₈	—	2	16	no	—
124	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	16	yes	—
125	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
126	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
127	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
128	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
129	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
130	#400	C₂²×C₂₃	1	32	yes	complemented
131	#397	C₂²×C₂₃	1	32	yes	—
132	#395	C₂²×C₂₃	1	32	yes	complemented
133	#381	C₂²×C₂₃	1	8	yes	—
134	C₂×#95	C₂²×C₂₃	1	8	yes	—
135	C₂×#95	C₂²×C₂₃	1	4	yes	—
136	C₂×#95	C₂²×C₂₃	1	4	yes	—
137	C₂×C₂₃	—	4	4	no	—
138	C₂×C₂₃	#2817	1	8	yes	direct product factor
139	C₂×C₂₃	C₂×#534	1	1	yes	characteristic
140	C₂×C₂₃	C₂×#536	1	2	yes	—
141	C₂×C₂₃	C₂×#563	1	4	yes	—
142	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
143	C₂²×Q₈	—	2	8	no	—
144	C₂²×Q₈	—	4	8	no	—
145	C₂²×Q₈	C₂³×C₂₃	1	8	yes	—
146	C₂³×C₄	—	4	4	no	—
147	C₂³×C₄	D₄×C₂₃	1	8	yes	—
148	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
149	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
150	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
151	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
152	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
153	C₂²⋊Q₈	—	2	64	no	—
154	C₂×C₄⋊C₄	—	2	8	no	—
155	C₂×C₄⋊C₄	—	2	16	no	—
156	C₂×C₄⋊C₄	—	4	8	no	—
157	C₂×C₄⋊C₄	C₂³×C₂₃	1	16	yes	—
158	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
159	C₂×C₄⋊C₄	C₂³×C₂₃	1	16	yes	—
160	C₂×C₂²⋊C₄	—	2	16	no	—
161	C₂×C₂²⋊C₄	—	4	4	no	—
162	C₂×C₂²⋊C₄	C₂³×C₂₃	1	16	yes	—
163	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
164	#95	C₂³×C₂₃	1	16	yes	—
165	#95	C₂³×C₂₃	1	32	yes	—
166	#95	C₂³×C₂₃	1	16	yes	—
167	C₂₃	C₂×#2817	1	1	yes	characteristic, direct product factor, Sylow
168	C₂⁴	—	2	4	no	—
169	C₂⁴	—	2	8	no	—
170	C₂⁴	C₂×D₄×C₂₃	1	4	yes	—
171	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
172	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
173	C₂×Q₈	—	2	16	no	—
174	C₂×Q₈	—	4	16	no	—
175	C₂×Q₈	—	4	32	no	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	2	8	no	—
178	C₂²×C₄	—	2	8	no	—
179	C₂²×C₄	—	2	8	no	—
180	C₂²×C₄	—	2	16	no	—
181	C₂²×C₄	—	2	16	no	—
182	C₂²×C₄	—	2	16	no	—
183	C₂²×C₄	—	4	16	no	—
184	C₂²×C₄	—	4	8	no	—
185	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	—
186	C₂²×C₄	C₂×D₄×C₂₃	1	16	yes	—
187	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
188	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
189	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
190	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
191	C₄⋊C₄	—	2	16	no	—
192	C₄⋊C₄	—	2	32	no	—
193	C₄⋊C₄	—	4	16	no	—
194	C₂²⋊C₄	—	2	32	no	—
195	C₂²⋊C₄	—	4	8	no	—
196	C₂³	—	2	4	no	—
197	C₂³	—	2	8	no	—
198	C₂³	—	2	8	no	—
199	C₂³	—	2	16	no	—
200	C₂³	—	4	16	no	—
201	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
202	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
203	C₂³	C₂₃×H(5,𝔽₂)	1	4	yes	—
204	C₂³	C₂₃×2₋¹⁺⁴	1	8	yes	—
205	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
206	Q₈	—	4	32	no	—
207	C₂×C₄	—	2	8	no	—
208	C₂×C₄	—	2	16	no	—
209	C₂×C₄	—	2	8	no	—
210	C₂×C₄	—	2	8	no	—
211	C₂×C₄	—	2	8	no	—
212	C₂×C₄	—	2	16	no	—
213	C₂×C₄	—	2	16	no	—
214	C₂×C₄	—	2	16	no	—
215	C₂×C₄	—	4	16	no	—
216	C₂×C₄	—	4	16	no	—
217	C₂×C₄	—	4	16	no	—
218	C₂×C₄	—	4	16	no	—
219	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
220	C₂²	—	2	4	no	—
221	C₂²	—	2	8	no	—
222	C₂²	—	4	8	no	—
223	C₂²	—	4	16	no	—
224	C₂²	C₂₃×#534	1	4	yes	—
225	C₂²	C₂₃×#536	1	8	yes	—
226	C₂²	C₂₃×#563	1	16	yes	—
227	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
228	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
229	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	4	yes	—
230	C₄	—	2	8	no	—
231	C₄	—	4	8	no	—
232	C₄	—	4	8	no	—
233	C₄	—	4	8	no	—
234	C₂	—	4	4	no	—
235	C₂	C₂₃×#2817	1	8	yes	direct product factor
236	C₂	C₂×C₂₃×#534	1	1	yes	characteristic
237	C₂	C₂×C₂₃×#536	1	2	yes	—
238	C₂	C₂×C₂₃×#563	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₂₃
prime radical	C₂⁵×C₂₃	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
2. C₂₃×#2817	C₂	direct (N × H)
3. C₂×C₂₃×#400	C₂	semidirect
5. C₂×C₂₃×#395	C₂	semidirect
16. C₂₃×#400	C₂²	semidirect
18. C₂₃×#395	C₂²	semidirect
76. C₂×#2817	C₂₃	direct (N × H)
102. #2817	C₂×C₂₃	direct (N × H)
103. C₂×#400	C₂×C₂₃	semidirect
105. C₂×#395	C₂×C₂₃	semidirect
130. #400	C₂²×C₂₃	semidirect
132. #395	C₂²×C₂₃	semidirect
138. C₂×C₂₃	#2817	direct (N × H)
167. C₂₃	C₂×#2817	direct (N × H)
235. C₂	C₂₃×#2817	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×C₂²⋊Q₈	2
C₂₃×#2817	2
C₂×C₂₃×#400	2
C₂×C₂₃×#397	2
C₂×C₂₃×#395	2
C₂×C₂₃×#381	2
C₂×#2817	23

Subgroups — C2×C23×#2817 #19184275

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2817 #19184275