Subgroups — C₂²×C₂₃×#400 #19182167

Order 5888.

3550 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	C₂³×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#400	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₂₃×#95	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₂₃×#95	C₂	1	2	yes	maximal
6	C₂²×C₂₃×#95	C₂	1	2	yes	maximal
7	C₂⁴×C₄×C₂₃	C₂²	1	2	yes	—
8	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂⁴×C₄×C₂₃	C₂²	1	2	yes	—
10	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₂₃	—	2	8	no	—
13	C₂²×C₄⋊C₄×C₂₃	—	2	8	no	—
14	C₂²×C₄⋊C₄×C₂₃	C₂²	1	12	yes	—
15	C₂²×C₄⋊C₄×C₂₃	C₂²	1	12	yes	—
16	C₂₃×#400	C₂²	1	64	yes	direct product factor
17	C₂×C₂₃×#95	C₂²	1	12	yes	—
18	C₂×C₂₃×#95	C₂²	1	24	yes	—
19	C₂×C₂₃×#95	C₂²	1	24	yes	—
20	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
21	C₂³×C₄×C₂₃	—	2	4	no	—
22	C₂³×C₄×C₂₃	—	2	4	no	—
23	C₂³×C₄×C₂₃	—	2	8	no	—
24	C₂³×C₄×C₂₃	—	2	4	no	—
25	C₂³×C₄×C₂₃	—	2	8	no	—
26	C₂³×C₄×C₂₃	—	2	8	no	—
27	C₂³×C₄×C₂₃	—	2	16	no	—
28	C₂³×C₄×C₂₃	—	2	8	no	—
29	C₂³×C₄×C₂₃	—	2	8	no	—
30	C₂³×C₄×C₂₃	—	2	8	no	—
31	C₂³×C₄×C₂₃	D₄	1	8	yes	—
32	C₂³×C₄×C₂₃	Q₈	1	8	yes	—
33	C₂³×C₄×C₂₃	C₂³	1	12	yes	—
34	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
35	C₂³×C₄×C₂₃	C₂³	1	12	yes	—
36	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
37	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
38	C₂×C₄⋊C₄×C₂₃	—	2	48	no	—
39	C₂×C₄⋊C₄×C₂₃	—	2	48	no	—
40	C₂×C₄⋊C₄×C₂₃	C₂³	1	16	yes	—
41	C₂×C₄⋊C₄×C₂₃	C₂³	1	16	yes	—
42	C₂₃×#95	C₂³	1	32	yes	—
43	C₂₃×#95	C₂³	1	32	yes	—
44	C₂₃×#95	C₂³	1	16	yes	—
45	C₂⁴×C₂₃	C₂×D₄	1	4	yes	—
46	C₂⁴×C₂₃	C₂×Q₈	1	4	yes	—
47	C₂⁴×C₂₃	#41	1	4	yes	—
48	C₂⁴×C₂₃	#41	1	8	yes	—
49	C₂⁴×C₂₃	#41	1	8	yes	—
50	C₂⁴×C₂₃	C₂⁴	1	3	yes	—
51	C₂²×C₄×C₂₃	—	2	32	no	—
52	C₂²×C₄×C₂₃	—	2	32	no	—
53	C₂²×C₄×C₂₃	—	2	12	no	—
54	C₂²×C₄×C₂₃	—	2	12	no	—
55	C₂²×C₄×C₂₃	—	2	24	no	—
56	C₂²×C₄×C₂₃	—	2	12	no	—
57	C₂²×C₄×C₂₃	—	2	24	no	—
58	C₂²×C₄×C₂₃	—	2	24	no	—
59	C₂²×C₄×C₂₃	—	2	48	no	—
60	C₂²×C₄×C₂₃	—	2	24	no	—
61	C₂²×C₄×C₂₃	—	2	24	no	—
62	C₂²×C₄×C₂₃	—	2	24	no	—
63	C₂²×C₄×C₂₃	—	4	16	no	—
64	C₂²×C₄×C₂₃	—	4	32	no	—
65	C₂²×C₄×C₂₃	—	4	32	no	—
66	C₂²×C₄×C₂₃	C₂×D₄	1	24	yes	—
67	C₂²×C₄×C₂₃	C₂×Q₈	1	24	yes	—
68	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
69	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
70	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
71	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
72	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
73	C₄⋊C₄×C₂₃	—	2	32	no	—
74	C₄⋊C₄×C₂₃	—	2	32	no	—
75	C₂²×#400	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₂₃	C₂²⋊Q₈	1	16	yes	—
77	C₂³×C₂₃	#123	1	16	yes	—
78	C₂³×C₂₃	#124	1	16	yes	—
79	C₂³×C₂₃	#125	1	32	yes	—
80	C₂³×C₂₃	#126	1	32	yes	—
81	C₂³×C₂₃	C₂²×D₄	1	6	yes	—
82	C₂³×C₂₃	C₂²×Q₈	1	6	yes	—
83	C₂³×C₂₃	C₂×#41	1	6	yes	—
84	C₂³×C₂₃	C₂×#41	1	12	yes	—
85	C₂³×C₂₃	C₂×#41	1	12	yes	—
86	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
87	C₂×C₄×C₂₃	—	2	48	no	—
88	C₂×C₄×C₂₃	—	2	48	no	—
89	C₂×C₄×C₂₃	—	2	4	no	—
90	C₂×C₄×C₂₃	—	2	4	no	—
91	C₂×C₄×C₂₃	—	2	8	no	—
92	C₂×C₄×C₂₃	—	2	4	no	—
93	C₂×C₄×C₂₃	—	2	8	no	—
94	C₂×C₄×C₂₃	—	2	8	no	—
95	C₂×C₄×C₂₃	—	2	16	no	—
96	C₂×C₄×C₂₃	—	2	8	no	—
97	C₂×C₄×C₂₃	—	2	8	no	—
98	C₂×C₄×C₂₃	—	2	8	no	—
99	C₂×C₄×C₂₃	—	4	24	no	—
100	C₂×C₄×C₂₃	—	4	48	no	—
101	C₂×C₄×C₂₃	—	4	48	no	—
102	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
103	C₂×C₄×C₂₃	C₂²×Q₈	1	8	yes	—
104	C₂³×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
105	C₂³×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
106	C₂×#400	C₂×C₂₃	1	24	yes	direct product factor
107	C₂²×#95	C₂×C₂₃	1	2	yes	—
108	C₂²×#95	C₂×C₂₃	1	2	yes	—
109	C₂²×#95	C₂×C₂₃	1	1	yes	characteristic
110	C₂²×C₂₃	#400	1	64	yes	direct product factor
111	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	12	yes	—
112	C₂²×C₂₃	C₂×#123	1	12	yes	—
113	C₂²×C₂₃	C₂×#124	1	12	yes	—
114	C₂²×C₂₃	C₂×#125	1	24	yes	—
115	C₂²×C₂₃	C₂×#126	1	24	yes	—
116	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
117	C₂²×C₂₃	C₂³×Q₈	1	1	yes	characteristic
118	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
119	C₂²×C₂₃	C₂²×#41	1	2	yes	—
120	C₂²×C₂₃	C₂²×#41	1	2	yes	—
121	C₄×C₂₃	—	2	8	no	—
122	C₄×C₂₃	—	2	8	no	—
123	C₄×C₂₃	—	4	4	no	—
124	C₄×C₂₃	—	4	8	no	—
125	C₄×C₂₃	—	4	8	no	—
126	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
127	C₂⁴×C₄	C₂²×C₂₃	1	2	yes	—
128	C₂⁴×C₄	C₂²×C₂₃	1	2	yes	—
129	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
130	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
131	C₂²×C₄⋊C₄	—	2	8	no	—
132	C₂²×C₄⋊C₄	—	2	8	no	—
133	C₂²×C₄⋊C₄	C₂²×C₂₃	1	12	yes	—
134	C₂²×C₄⋊C₄	C₂²×C₂₃	1	12	yes	—
135	#400	C₂²×C₂₃	1	64	yes	direct product factor
136	C₂×#95	C₂²×C₂₃	1	24	yes	—
137	C₂×#95	C₂²×C₂₃	1	24	yes	—
138	C₂×#95	C₂²×C₂₃	1	12	yes	—
139	C₂×C₂₃	C₂×#400	1	24	yes	direct product factor
140	C₂×C₂₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
141	C₂×C₂₃	C₂²×#123	1	1	yes	characteristic
142	C₂×C₂₃	C₂²×#124	1	1	yes	characteristic
143	C₂×C₂₃	C₂²×#125	1	2	yes	—
144	C₂×C₂₃	C₂²×#126	1	2	yes	—
145	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
146	C₂³×C₄	—	2	4	no	—
147	C₂³×C₄	—	2	4	no	—
148	C₂³×C₄	—	2	8	no	—
149	C₂³×C₄	—	2	4	no	—
150	C₂³×C₄	—	2	8	no	—
151	C₂³×C₄	—	2	8	no	—
152	C₂³×C₄	—	2	16	no	—
153	C₂³×C₄	—	2	8	no	—
154	C₂³×C₄	—	2	8	no	—
155	C₂³×C₄	—	2	8	no	—
156	C₂³×C₄	D₄×C₂₃	1	8	yes	—
157	C₂³×C₄	Q₈×C₂₃	1	8	yes	—
158	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
159	C₂³×C₄	C₂³×C₂₃	1	12	yes	—
160	C₂³×C₄	C₂³×C₂₃	1	12	yes	—
161	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
162	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
163	C₂×C₄⋊C₄	—	2	48	no	—
164	C₂×C₄⋊C₄	—	2	48	no	—
165	C₂×C₄⋊C₄	C₂³×C₂₃	1	16	yes	—
166	C₂×C₄⋊C₄	C₂³×C₂₃	1	16	yes	—
167	#95	C₂³×C₂₃	1	16	yes	—
168	#95	C₂³×C₂₃	1	32	yes	—
169	#95	C₂³×C₂₃	1	32	yes	—
170	C₂₃	C₂²×#400	1	1	yes	characteristic, direct product factor, Sylow
171	C₂⁴	C₂×D₄×C₂₃	1	4	yes	—
172	C₂⁴	C₂×Q₈×C₂₃	1	4	yes	—
173	C₂⁴	#41×C₂₃	1	4	yes	—
174	C₂⁴	#41×C₂₃	1	8	yes	—
175	C₂⁴	#41×C₂₃	1	8	yes	—
176	C₂⁴	C₂⁴×C₂₃	1	3	yes	—
177	C₂²×C₄	—	2	32	no	—
178	C₂²×C₄	—	2	32	no	—
179	C₂²×C₄	—	2	12	no	—
180	C₂²×C₄	—	2	12	no	—
181	C₂²×C₄	—	2	24	no	—
182	C₂²×C₄	—	2	12	no	—
183	C₂²×C₄	—	2	24	no	—
184	C₂²×C₄	—	2	24	no	—
185	C₂²×C₄	—	2	48	no	—
186	C₂²×C₄	—	2	24	no	—
187	C₂²×C₄	—	2	24	no	—
188	C₂²×C₄	—	2	24	no	—
189	C₂²×C₄	—	4	16	no	—
190	C₂²×C₄	—	4	32	no	—
191	C₂²×C₄	—	4	32	no	—
192	C₂²×C₄	C₂×D₄×C₂₃	1	24	yes	—
193	C₂²×C₄	C₂×Q₈×C₂₃	1	24	yes	—
194	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
195	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
196	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
197	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
198	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
199	C₄⋊C₄	—	2	32	no	—
200	C₄⋊C₄	—	2	32	no	—
201	C₂³	C₂₃×C₂²⋊Q₈	1	16	yes	—
202	C₂³	C₂₃×#123	1	16	yes	—
203	C₂³	C₂₃×#124	1	16	yes	—
204	C₂³	C₂₃×#125	1	32	yes	—
205	C₂³	C₂₃×#126	1	32	yes	—
206	C₂³	C₂²×D₄×C₂₃	1	6	yes	—
207	C₂³	C₂²×Q₈×C₂₃	1	6	yes	—
208	C₂³	C₂×#41×C₂₃	1	6	yes	—
209	C₂³	C₂×#41×C₂₃	1	12	yes	—
210	C₂³	C₂×#41×C₂₃	1	12	yes	—
211	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	48	no	—
216	C₂×C₄	—	2	48	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	2	16	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	4	24	no	—
225	C₂×C₄	—	4	48	no	—
226	C₂×C₄	—	4	48	no	—
227	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
228	C₂×C₄	C₂²×Q₈×C₂₃	1	8	yes	—
229	C₂²	C₂₃×#400	1	64	yes	direct product factor
230	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	12	yes	—
231	C₂²	C₂×C₂₃×#123	1	12	yes	—
232	C₂²	C₂×C₂₃×#124	1	12	yes	—
233	C₂²	C₂×C₂₃×#125	1	24	yes	—
234	C₂²	C₂×C₂₃×#126	1	24	yes	—
235	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
236	C₂²	C₂³×Q₈×C₂₃	1	1	yes	characteristic
237	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
238	C₂²	C₂²×#41×C₂₃	1	2	yes	—
239	C₂²	C₂²×#41×C₂₃	1	2	yes	—
240	C₄	—	2	8	no	—
241	C₄	—	2	8	no	—
242	C₄	—	4	4	no	—
243	C₄	—	4	8	no	—
244	C₄	—	4	8	no	—
245	C₂	C₂×C₂₃×#400	1	24	yes	direct product factor
246	C₂	C₂²×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
247	C₂	C₂²×C₂₃×#123	1	1	yes	characteristic
248	C₂	C₂²×C₂₃×#124	1	1	yes	characteristic
249	C₂	C₂²×C₂₃×#125	1	2	yes	—
250	C₂	C₂²×C₂₃×#126	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂×C₂₃×#400	C₂	direct (N × H)
16. C₂₃×#400	C₂²	direct (N × H)
75. C₂²×#400	C₂₃	direct (N × H)
106. C₂×#400	C₂×C₂₃	direct (N × H)
110. C₂²×C₂₃	#400	direct (N × H)
135. #400	C₂²×C₂₃	direct (N × H)
139. C₂×C₂₃	C₂×#400	direct (N × H)
170. C₂₃	C₂²×#400	direct (N × H)
229. C₂²	C₂₃×#400	direct (N × H)
245. C₂	C₂×C₂₃×#400	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄⋊C₄×C₂₃	2
C₂×C₂₃×#400	2
C₂²×C₂₃×#95	2
C₂²×#400	23

Subgroups — C22×C23×#400 #19182167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#400 #19182167