Subgroups — C₂²×C₂₃×#396 #19182162

Order 5888.

5838 conjugacy classes of subgroups.

The subgroup lattice has 192 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	3	yes	maximal, complemented
2	C₂³×C₂²⋊C₄×C₂₃	C₂	1	3	yes	maximal
3	C₂×C₂₃×#396	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₂₃×#95	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁶×C₂₃	C₂²	1	1	yes	characteristic
6	C₂⁴×C₄×C₂₃	C₂²	1	3	yes	—
7	C₂⁴×C₄×C₂₃	C₂²	1	3	yes	—
8	C₂²×C₄⋊C₄×C₂₃	—	2	24	no	—
9	C₂²×C₄⋊C₄×C₂₃	C₂²	1	36	yes	complemented
10	C₂²×C₂²⋊C₄×C₂₃	—	2	24	no	—
11	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	36	yes	—
12	C₂₃×#396	C₂²	1	64	yes	direct product factor
13	C₂×C₂₃×#95	C₂²	1	12	yes	complemented
14	C₂⁵×C₂₃	—	2	12	no	—
15	C₂⁵×C₂₃	—	2	12	no	—
16	C₂⁵×C₂₃	Q₈	1	8	yes	—
17	C₂⁵×C₂₃	C₂³	1	6	yes	—
18	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
19	C₂³×C₄×C₂₃	—	2	12	no	—
20	C₂³×C₄×C₂₃	—	2	12	no	—
21	C₂³×C₄×C₂₃	—	2	12	no	—
22	C₂³×C₄×C₂₃	—	2	12	no	—
23	C₂³×C₄×C₂₃	—	2	12	no	—
24	C₂³×C₄×C₂₃	D₄	1	24	yes	—
25	C₂³×C₄×C₂₃	C₂³	1	18	yes	—
26	C₂³×C₄×C₂₃	C₂³	1	18	yes	—
27	C₂×C₄⋊C₄×C₂₃	—	2	144	no	—
28	C₂×C₄⋊C₄×C₂₃	C₂³	1	48	yes	complemented
29	C₂×C₂²⋊C₄×C₂₃	—	2	144	no	—
30	C₂×C₂²⋊C₄×C₂₃	C₂³	1	48	yes	—
31	C₂₃×#95	C₂³	1	16	yes	complemented
32	C₂⁴×C₂₃	—	2	96	no	—
33	C₂⁴×C₂₃	—	2	36	no	—
34	C₂⁴×C₂₃	—	2	36	no	—
35	C₂⁴×C₂₃	—	4	48	no	—
36	C₂⁴×C₂₃	—	4	16	no	—
37	C₂⁴×C₂₃	C₂×D₄	1	12	yes	—
38	C₂⁴×C₂₃	C₂×Q₈	1	24	yes	—
39	C₂⁴×C₂₃	C₂×Q₈	1	4	yes	—
40	C₂⁴×C₂₃	#41	1	12	yes	—
41	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
42	C₂⁴×C₂₃	C₂⁴	1	3	yes	—
43	C₂²×C₄×C₂₃	—	2	96	no	—
44	C₂²×C₄×C₂₃	—	2	36	no	—
45	C₂²×C₄×C₂₃	—	2	36	no	—
46	C₂²×C₄×C₂₃	—	2	36	no	—
47	C₂²×C₄×C₂₃	—	2	36	no	—
48	C₂²×C₄×C₂₃	—	2	36	no	—
49	C₂²×C₄×C₂₃	—	4	48	no	—
50	C₂²×C₄×C₂₃	C₂×D₄	1	72	yes	—
51	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
52	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
53	C₄⋊C₄×C₂₃	—	2	96	no	—
54	C₂²⋊C₄×C₂₃	—	2	96	no	—
55	C₂²×#396	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂³×C₂₃	—	2	144	no	—
57	C₂³×C₂₃	—	2	12	no	—
58	C₂³×C₂₃	—	2	12	no	—
59	C₂³×C₂₃	—	4	72	no	—
60	C₂³×C₂₃	—	4	24	no	—
61	C₂³×C₂₃	—	4	128	no	—
62	C₂³×C₂₃	#121	1	48	yes	—
63	C₂³×C₂₃	C₂²⋊Q₈	1	48	yes	—
64	C₂³×C₂₃	#126	1	16	yes	—
65	C₂³×C₂₃	C₂²×D₄	1	18	yes	—
66	C₂³×C₂₃	C₂²×Q₈	1	8	yes	—
67	C₂³×C₂₃	C₂²×Q₈	1	6	yes	—
68	C₂³×C₂₃	C₂×#41	1	18	yes	—
69	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
70	C₂×C₄×C₂₃	—	2	144	no	—
71	C₂×C₄×C₂₃	—	2	12	no	—
72	C₂×C₄×C₂₃	—	2	12	no	—
73	C₂×C₄×C₂₃	—	2	12	no	—
74	C₂×C₄×C₂₃	—	2	12	no	—
75	C₂×C₄×C₂₃	—	2	12	no	—
76	C₂×C₄×C₂₃	—	4	72	no	—
77	C₂×C₄×C₂₃	C₂²×D₄	1	24	yes	—
78	C₂³×C₄⋊C₄	C₂×C₂₃	1	3	yes	complemented
79	C₂³×C₂²⋊C₄	C₂×C₂₃	1	3	yes	—
80	C₂×#396	C₂×C₂₃	1	24	yes	direct product factor
81	C₂²×#95	C₂×C₂₃	1	1	yes	characteristic, complemented
82	C₂²×C₂₃	—	2	24	no	—
83	C₂²×C₂₃	—	4	96	no	—
84	C₂²×C₂₃	—	4	12	no	—
85	C₂²×C₂₃	—	4	4	no	—
86	C₂²×C₂₃	#396	1	64	yes	direct product factor
87	C₂²×C₂₃	C₂×#121	1	36	yes	—
88	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	36	yes	—
89	C₂²×C₂₃	C₂×#126	1	12	yes	—
90	C₂²×C₂₃	C₂³×D₄	1	3	yes	—
91	C₂²×C₂₃	C₂³×Q₈	1	1	yes	characteristic
92	C₂²×C₂₃	C₂²×#41	1	3	yes	—
93	C₄×C₂₃	—	2	24	no	—
94	C₄×C₂₃	—	4	12	no	—
95	C₂⁶	C₂²×C₂₃	1	1	yes	characteristic
96	C₂⁴×C₄	C₂²×C₂₃	1	3	yes	—
97	C₂⁴×C₄	C₂²×C₂₃	1	3	yes	—
98	C₂²×C₄⋊C₄	—	2	24	no	—
99	C₂²×C₄⋊C₄	C₂²×C₂₃	1	36	yes	complemented
100	C₂²×C₂²⋊C₄	—	2	24	no	—
101	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	36	yes	—
102	#396	C₂²×C₂₃	1	64	yes	direct product factor
103	C₂×#95	C₂²×C₂₃	1	12	yes	complemented
104	C₂×C₂₃	—	4	8	no	—
105	C₂×C₂₃	C₂×#396	1	24	yes	direct product factor
106	C₂×C₂₃	C₂²×#121	1	3	yes	—
107	C₂×C₂₃	C₂²×C₂²⋊Q₈	1	3	yes	—
108	C₂×C₂₃	C₂²×#126	1	1	yes	characteristic
109	C₂⁵	—	2	12	no	—
110	C₂⁵	—	2	12	no	—
111	C₂⁵	Q₈×C₂₃	1	8	yes	—
112	C₂⁵	C₂³×C₂₃	1	6	yes	—
113	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
114	C₂³×C₄	—	2	12	no	—
115	C₂³×C₄	—	2	12	no	—
116	C₂³×C₄	—	2	12	no	—
117	C₂³×C₄	—	2	12	no	—
118	C₂³×C₄	—	2	12	no	—
119	C₂³×C₄	D₄×C₂₃	1	24	yes	—
120	C₂³×C₄	C₂³×C₂₃	1	18	yes	—
121	C₂³×C₄	C₂³×C₂₃	1	18	yes	—
122	C₂×C₄⋊C₄	—	2	144	no	—
123	C₂×C₄⋊C₄	C₂³×C₂₃	1	48	yes	complemented
124	C₂×C₂²⋊C₄	—	2	144	no	—
125	C₂×C₂²⋊C₄	C₂³×C₂₃	1	48	yes	—
126	#95	C₂³×C₂₃	1	16	yes	complemented
127	C₂₃	C₂²×#396	1	1	yes	characteristic, direct product factor, Sylow
128	C₂⁴	—	2	96	no	—
129	C₂⁴	—	2	36	no	—
130	C₂⁴	—	2	36	no	—
131	C₂⁴	—	4	48	no	—
132	C₂⁴	—	4	16	no	—
133	C₂⁴	C₂×D₄×C₂₃	1	12	yes	—
134	C₂⁴	C₂×Q₈×C₂₃	1	24	yes	—
135	C₂⁴	C₂×Q₈×C₂₃	1	4	yes	—
136	C₂⁴	#41×C₂₃	1	12	yes	—
137	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
138	C₂⁴	C₂⁴×C₂₃	1	3	yes	—
139	C₂²×C₄	—	2	96	no	—
140	C₂²×C₄	—	2	36	no	—
141	C₂²×C₄	—	2	36	no	—
142	C₂²×C₄	—	2	36	no	—
143	C₂²×C₄	—	2	36	no	—
144	C₂²×C₄	—	2	36	no	—
145	C₂²×C₄	—	4	48	no	—
146	C₂²×C₄	C₂×D₄×C₂₃	1	72	yes	—
147	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
148	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
149	C₄⋊C₄	—	2	96	no	—
150	C₂²⋊C₄	—	2	96	no	—
151	C₂³	—	2	12	no	—
152	C₂³	—	2	12	no	—
153	C₂³	—	2	144	no	—
154	C₂³	—	4	128	no	—
155	C₂³	—	4	72	no	—
156	C₂³	—	4	24	no	—
157	C₂³	C₂₃×#121	1	48	yes	—
158	C₂³	C₂₃×C₂²⋊Q₈	1	48	yes	—
159	C₂³	C₂₃×#126	1	16	yes	—
160	C₂³	C₂²×D₄×C₂₃	1	18	yes	—
161	C₂³	C₂²×Q₈×C₂₃	1	8	yes	—
162	C₂³	C₂²×Q₈×C₂₃	1	6	yes	—
163	C₂³	C₂×#41×C₂₃	1	18	yes	—
164	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
165	C₂×C₄	—	2	12	no	—
166	C₂×C₄	—	2	12	no	—
167	C₂×C₄	—	2	144	no	—
168	C₂×C₄	—	2	12	no	—
169	C₂×C₄	—	2	12	no	—
170	C₂×C₄	—	2	12	no	—
171	C₂×C₄	—	4	72	no	—
172	C₂×C₄	C₂²×D₄×C₂₃	1	24	yes	—
173	C₂²	—	2	24	no	—
174	C₂²	—	4	12	no	—
175	C₂²	—	4	4	no	—
176	C₂²	—	4	96	no	—
177	C₂²	C₂₃×#396	1	64	yes	direct product factor
178	C₂²	C₂×C₂₃×#121	1	36	yes	—
179	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	36	yes	—
180	C₂²	C₂×C₂₃×#126	1	12	yes	—
181	C₂²	C₂³×D₄×C₂₃	1	3	yes	—
182	C₂²	C₂³×Q₈×C₂₃	1	1	yes	characteristic
183	C₂²	C₂²×#41×C₂₃	1	3	yes	—
184	C₄	—	2	24	no	—
185	C₄	—	4	12	no	—
186	C₂	—	4	8	no	—
187	C₂	C₂×C₂₃×#396	1	24	yes	direct product factor
188	C₂	C₂²×C₂₃×#121	1	3	yes	—
189	C₂	C₂²×C₂₃×C₂²⋊Q₈	1	3	yes	—
190	C₂	C₂²×C₂₃×#126	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵×C₂₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₂₃
prime radical	C₂⁶×C₂₃	C₂²
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
1. C₂³×C₄⋊C₄×C₂₃	C₂	semidirect
3. C₂×C₂₃×#396	C₂	direct (N × H)
4. C₂²×C₂₃×#95	C₂	semidirect
9. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
12. C₂₃×#396	C₂²	direct (N × H)
13. C₂×C₂₃×#95	C₂²	semidirect
28. C₂×C₄⋊C₄×C₂₃	C₂³	semidirect
31. C₂₃×#95	C₂³	semidirect
55. C₂²×#396	C₂₃	direct (N × H)
78. C₂³×C₄⋊C₄	C₂×C₂₃	semidirect
80. C₂×#396	C₂×C₂₃	direct (N × H)
81. C₂²×#95	C₂×C₂₃	semidirect
86. C₂²×C₂₃	#396	direct (N × H)
99. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
102. #396	C₂²×C₂₃	direct (N × H)
103. C₂×#95	C₂²×C₂₃	semidirect
105. C₂×C₂₃	C₂×#396	direct (N × H)
123. C₂×C₄⋊C₄	C₂³×C₂₃	semidirect
126. #95	C₂³×C₂₃	semidirect
127. C₂₃	C₂²×#396	direct (N × H)
177. C₂²	C₂₃×#396	direct (N × H)
187. C₂	C₂×C₂₃×#396	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
C₂³×C₂²⋊C₄×C₂₃	2
C₂×C₂₃×#396	2
C₂²×C₂₃×#95	2
C₂²×#396	23

Subgroups — C22×C23×#396 #19182162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#396 #19182162