Subgroups — C₄×C₂₃×#379 #19161395

Order 5888.

6734 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#379	C₂	1	2	yes	maximal
3	C₂×C₄×C₂²⋊C₄×C₂₃	C₂	1	6	yes	maximal, complemented
4	C₂₃×C₂²⋊#95	C₂	1	6	yes	maximal, complemented
5	C₂⁶×C₂₃	C₂²	1	1	yes	characteristic
6	C₂⁴×C₄×C₂₃	—	2	24	no	—
7	C₂⁴×C₄×C₂₃	C₄	1	8	yes	complemented
8	C₂⁴×C₄×C₂₃	C₂²	1	3	yes	—
9	C₂⁴×C₄×C₂₃	C₂²	1	3	yes	—
10	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	12	yes	—
11	C₂²×C₄²×C₂₃	C₂²	1	4	yes	complemented
12	C₂₃×#379	C₄	1	16	yes	direct product factor
13	C₄×C₂²⋊C₄×C₂₃	—	2	24	no	—
14	C₂×C₂₃×#95	C₂²	1	12	yes	complemented
15	C₂⁵×C₂₃	—	2	12	no	—
16	C₂⁵×C₂₃	—	2	12	no	—
17	C₂⁵×C₂₃	C₂×C₄	1	4	yes	—
18	C₂⁵×C₂₃	C₂×C₄	1	8	yes	—
19	C₂⁵×C₂₃	C₂³	1	3	yes	—
20	C₂³×C₄×C₂₃	—	2	128	no	—
21	C₂³×C₄×C₂₃	—	2	12	no	—
22	C₂³×C₄×C₂₃	—	2	96	no	—
23	C₂³×C₄×C₂₃	—	2	24	no	—
24	C₂³×C₄×C₂₃	—	2	24	no	—
25	C₂³×C₄×C₂₃	C₂×C₄	1	12	yes	—
26	C₂³×C₄×C₂₃	C₂×C₄	1	12	yes	—
27	C₂³×C₄×C₂₃	D₄	1	24	yes	—
28	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
29	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
30	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
31	C₂×C₂²⋊C₄×C₂₃	—	2	24	no	—
32	C₂×C₂²⋊C₄×C₂₃	—	2	24	no	—
33	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	48	yes	complemented
34	C₂×C₄²×C₂₃	—	2	24	no	—
35	C₂₃×#95	—	2	24	no	—
36	C₂⁴×C₂₃	—	2	32	no	—
37	C₂⁴×C₂₃	—	2	96	no	—
38	C₂⁴×C₂₃	—	2	24	no	—
39	C₂⁴×C₂₃	—	2	24	no	—
40	C₂⁴×C₂₃	—	2	96	no	—
41	C₂⁴×C₂₃	—	2	12	no	—
42	C₂⁴×C₂₃	—	2	12	no	—
43	C₂⁴×C₂₃	C₄²	1	16	yes	complemented
44	C₂⁴×C₂₃	C₂²×C₄	1	6	yes	—
45	C₂⁴×C₂₃	C₂²×C₄	1	12	yes	—
46	C₂⁴×C₂₃	C₂×D₄	1	12	yes	—
47	C₂⁴×C₂₃	#41	1	12	yes	—
48	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	—	2	96	no	—
50	C₂²×C₄×C₂₃	—	2	256	no	—
51	C₂²×C₄×C₂₃	—	2	24	no	—
52	C₂²×C₄×C₂₃	—	2	12	no	—
53	C₂²×C₄×C₂₃	—	2	48	no	—
54	C₂²×C₄×C₂₃	—	2	48	no	—
55	C₂²×C₄×C₂₃	—	2	96	no	—
56	C₂²×C₄×C₂₃	—	2	24	no	—
57	C₂²×C₄×C₂₃	—	2	24	no	—
58	C₂²×C₄×C₂₃	C₂²⋊C₄	1	48	yes	complemented
59	C₂²×C₄×C₂₃	C₂²×C₄	1	6	yes	—
60	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₂₃	C₂²×C₄	1	16	yes	complemented
62	C₂²×C₄×C₂₃	C₂×D₄	1	12	yes	—
63	C₂²×C₄×C₂₃	C₂×D₄	1	12	yes	—
64	C₂²⋊C₄×C₂₃	—	2	96	no	—
65	C₄²×C₂₃	—	4	16	no	—
66	C₄×#379	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₂₃	—	2	32	no	—
68	C₂³×C₂₃	—	2	96	no	—
69	C₂³×C₂₃	—	2	256	no	—
70	C₂³×C₂₃	—	2	12	no	—
71	C₂³×C₂₃	—	2	12	no	—
72	C₂³×C₂₃	—	2	48	no	—
73	C₂³×C₂₃	—	2	24	no	—
74	C₂³×C₂₃	—	2	24	no	—
75	C₂³×C₂₃	—	2	48	no	—
76	C₂³×C₂₃	—	2	96	no	—
77	C₂³×C₂₃	C₂×C₄²	1	12	yes	—
78	C₂³×C₂₃	C₂×C₂²⋊C₄	1	12	yes	—
79	C₂³×C₂₃	#117	1	12	yes	—
80	C₂³×C₂₃	C₄×D₄	1	48	yes	—
81	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
82	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
83	C₂³×C₂₃	C₂²×D₄	1	6	yes	—
84	C₂³×C₂₃	C₂×#41	1	6	yes	—
85	C₂×C₄×C₂₃	—	2	96	no	—
86	C₂×C₄×C₂₃	—	2	12	no	—
87	C₂×C₄×C₂₃	—	2	24	no	—
88	C₂×C₄×C₂₃	—	2	96	no	—
89	C₂×C₄×C₂₃	—	2	48	no	—
90	C₂×C₄×C₂₃	—	4	8	no	—
91	C₂×C₄×C₂₃	—	4	24	no	—
92	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	12	yes	—
93	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	12	yes	—
94	C₂×C₄×C₂₃	#120	1	16	yes	—
95	C₂⁵×C₄	C₂×C₂₃	1	1	yes	characteristic
96	C₂×#379	C₂×C₂₃	1	2	yes	—
97	C₂×C₄×C₂²⋊C₄	C₂×C₂₃	1	6	yes	complemented
98	C₂²⋊#95	C₂×C₂₃	1	6	yes	complemented
99	C₂²×C₂₃	—	2	128	no	—
100	C₂²×C₂₃	—	2	24	no	—
101	C₂²×C₂₃	—	2	12	no	—
102	C₂²×C₂₃	—	2	12	no	—
103	C₂²×C₂₃	—	2	24	no	—
104	C₂²×C₂₃	—	2	96	no	—
105	C₂²×C₂₃	C₄×C₂²⋊C₄	1	24	yes	complemented
106	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
107	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	3	yes	—
108	C₂²×C₂₃	C₂×#117	1	3	yes	—
109	C₂²×C₂₃	C₂×C₄×D₄	1	12	yes	—
110	C₂²×C₂₃	C₂×#120	1	4	yes	—
111	C₂²×C₂₃	#525	1	12	yes	—
112	C₄×C₂₃	—	2	12	no	—
113	C₄×C₂₃	—	4	16	no	—
114	C₄×C₂₃	#379	1	8	yes	direct product factor
115	C₂⁶	C₂²×C₂₃	1	1	yes	characteristic
116	C₂⁴×C₄	—	2	24	no	—
117	C₂⁴×C₄	C₄×C₂₃	1	8	yes	complemented
118	C₂⁴×C₄	C₂²×C₂₃	1	3	yes	—
119	C₂⁴×C₄	C₂²×C₂₃	1	3	yes	—
120	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	12	yes	—
121	C₂²×C₄²	C₂²×C₂₃	1	4	yes	complemented
122	#379	C₄×C₂₃	1	16	yes	direct product factor
123	C₄×C₂²⋊C₄	—	2	24	no	—
124	C₂×#95	C₂²×C₂₃	1	12	yes	complemented
125	C₂×C₂₃	—	2	24	no	—
126	C₂×C₂₃	C₂×C₄×C₂²⋊C₄	1	3	yes	—
127	C₂×C₂₃	C₂×#379	1	1	yes	characteristic
128	C₂×C₂₃	#2280	1	3	yes	—
129	C₂×C₂₃	C₄×#120	1	8	yes	—
130	C₂⁵	—	2	12	no	—
131	C₂⁵	—	2	12	no	—
132	C₂⁵	C₂×C₄×C₂₃	1	4	yes	—
133	C₂⁵	C₂×C₄×C₂₃	1	8	yes	—
134	C₂⁵	C₂³×C₂₃	1	3	yes	—
135	C₂³×C₄	—	2	128	no	—
136	C₂³×C₄	—	2	12	no	—
137	C₂³×C₄	—	2	96	no	—
138	C₂³×C₄	—	2	24	no	—
139	C₂³×C₄	—	2	24	no	—
140	C₂³×C₄	C₂×C₄×C₂₃	1	12	yes	—
141	C₂³×C₄	C₂×C₄×C₂₃	1	12	yes	—
142	C₂³×C₄	D₄×C₂₃	1	24	yes	—
143	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
144	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
145	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
146	C₂×C₂²⋊C₄	—	2	24	no	—
147	C₂×C₂²⋊C₄	—	2	24	no	—
148	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	48	yes	complemented
149	C₂×C₄²	—	2	24	no	—
150	#95	—	2	24	no	—
151	C₂₃	C₄×#379	1	1	yes	characteristic, direct product factor, Sylow
152	C₂⁴	—	2	32	no	—
153	C₂⁴	—	2	96	no	—
154	C₂⁴	—	2	24	no	—
155	C₂⁴	—	2	24	no	—
156	C₂⁴	—	2	96	no	—
157	C₂⁴	—	2	12	no	—
158	C₂⁴	—	2	12	no	—
159	C₂⁴	C₄²×C₂₃	1	16	yes	complemented
160	C₂⁴	C₂²×C₄×C₂₃	1	6	yes	—
161	C₂⁴	C₂²×C₄×C₂₃	1	12	yes	—
162	C₂⁴	C₂×D₄×C₂₃	1	12	yes	—
163	C₂⁴	#41×C₂₃	1	12	yes	—
164	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
165	C₂²×C₄	—	2	96	no	—
166	C₂²×C₄	—	2	256	no	—
167	C₂²×C₄	—	2	24	no	—
168	C₂²×C₄	—	2	12	no	—
169	C₂²×C₄	—	2	48	no	—
170	C₂²×C₄	—	2	48	no	—
171	C₂²×C₄	—	2	96	no	—
172	C₂²×C₄	—	2	24	no	—
173	C₂²×C₄	—	2	24	no	—
174	C₂²×C₄	C₂²⋊C₄×C₂₃	1	48	yes	complemented
175	C₂²×C₄	C₂²×C₄×C₂₃	1	6	yes	—
176	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
177	C₂²×C₄	C₂²×C₄×C₂₃	1	16	yes	complemented
178	C₂²×C₄	C₂×D₄×C₂₃	1	12	yes	—
179	C₂²×C₄	C₂×D₄×C₂₃	1	12	yes	—
180	C₂²⋊C₄	—	2	96	no	—
181	C₄²	—	4	16	no	—
182	C₂³	—	2	32	no	—
183	C₂³	—	2	96	no	—
184	C₂³	—	2	256	no	—
185	C₂³	—	2	96	no	—
186	C₂³	—	2	12	no	—
187	C₂³	—	2	12	no	—
188	C₂³	—	2	48	no	—
189	C₂³	—	2	24	no	—
190	C₂³	—	2	24	no	—
191	C₂³	—	2	48	no	—
192	C₂³	C₂×C₄²×C₂₃	1	12	yes	—
193	C₂³	C₂×C₂²⋊C₄×C₂₃	1	12	yes	—
194	C₂³	C₂₃×#117	1	12	yes	—
195	C₂³	C₄×D₄×C₂₃	1	48	yes	—
196	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
197	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
198	C₂³	C₂²×D₄×C₂₃	1	6	yes	—
199	C₂³	C₂×#41×C₂₃	1	6	yes	—
200	C₂×C₄	—	2	96	no	—
201	C₂×C₄	—	2	12	no	—
202	C₂×C₄	—	2	24	no	—
203	C₂×C₄	—	2	96	no	—
204	C₂×C₄	—	2	48	no	—
205	C₂×C₄	—	4	8	no	—
206	C₂×C₄	—	4	24	no	—
207	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	12	yes	—
208	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	12	yes	—
209	C₂×C₄	C₂₃×#120	1	16	yes	—
210	C₂²	—	2	128	no	—
211	C₂²	—	2	24	no	—
212	C₂²	—	2	12	no	—
213	C₂²	—	2	12	no	—
214	C₂²	—	2	24	no	—
215	C₂²	—	2	96	no	—
216	C₂²	C₄×C₂²⋊C₄×C₂₃	1	24	yes	complemented
217	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
218	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	3	yes	—
219	C₂²	C₂×C₂₃×#117	1	3	yes	—
220	C₂²	C₂×C₄×D₄×C₂₃	1	12	yes	—
221	C₂²	C₂×C₂₃×#120	1	4	yes	—
222	C₂²	C₂₃×#525	1	12	yes	—
223	C₄	—	2	12	no	—
224	C₄	—	4	16	no	—
225	C₄	C₂₃×#379	1	8	yes	direct product factor
226	C₂	—	2	24	no	—
227	C₂	C₂×C₄×C₂²⋊C₄×C₂₃	1	3	yes	—
228	C₂	C₂×C₂₃×#379	1	1	yes	characteristic
229	C₂	C₂₃×#2280	1	3	yes	—
230	C₂	C₄×C₂₃×#120	1	8	yes	—

	subgroup	G/H
center	C₂³×C₄×C₂₃	C₂³
commutator	C₂²	C₂²×C₄²×C₂₃
Frattini, cosocle	C₂⁴	C₂⁴×C₂₃
socle	C₂⁴×C₂₃	C₂⁴
prime radical	C₂⁶×C₂₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×C₂²⋊C₄×C₂₃	C₂	semidirect
4. C₂₃×C₂²⋊#95	C₂	semidirect
7. C₂⁴×C₄×C₂₃	C₄	semidirect
11. C₂²×C₄²×C₂₃	C₂²	semidirect
12. C₂₃×#379	C₄	direct (N × H)
14. C₂×C₂₃×#95	C₂²	semidirect
33. C₂×C₂²⋊C₄×C₂₃	C₂×C₄	semidirect
43. C₂⁴×C₂₃	C₄²	semidirect
58. C₂²×C₄×C₂₃	C₂²⋊C₄	semidirect
61. C₂²×C₄×C₂₃	C₂²×C₄	semidirect
66. C₄×#379	C₂₃	direct (N × H)
97. C₂×C₄×C₂²⋊C₄	C₂×C₂₃	semidirect
98. C₂²⋊#95	C₂×C₂₃	semidirect
105. C₂²×C₂₃	C₄×C₂²⋊C₄	semidirect
114. C₄×C₂₃	#379	direct (N × H)
117. C₂⁴×C₄	C₄×C₂₃	semidirect
121. C₂²×C₄²	C₂²×C₂₃	semidirect
122. #379	C₄×C₂₃	direct (N × H)
124. C₂×#95	C₂²×C₂₃	semidirect
148. C₂×C₂²⋊C₄	C₂×C₄×C₂₃	semidirect
151. C₂₃	C₄×#379	direct (N × H)
159. C₂⁴	C₄²×C₂₃	semidirect
174. C₂²×C₄	C₂²⋊C₄×C₂₃	semidirect
177. C₂²×C₄	C₂²×C₄×C₂₃	semidirect
216. C₂²	C₄×C₂²⋊C₄×C₂₃	semidirect
225. C₄	C₂₃×#379	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₄×C₂₃	2
C₂×C₂₃×#379	2
C₂×C₄×C₂²⋊C₄×C₂₃	2
C₂₃×C₂²⋊#95	2
C₄×#379	23

Subgroups — C4×C23×#379 #19161395

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂₃×#379 #19161395