Subgroups — C₂×C₂₃×#3556 #19210386

Order 5888.

6338 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3556	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₂₃×#564	C₂	1	4	yes	maximal, complemented
3	C₂×C₂₃×#557	C₂	1	12	yes	maximal, complemented
4	C₂×C₂₃×#541	C₂	1	12	yes	maximal
5	C₂×C₂₃×#533	C₂	1	3	yes	maximal
6	C₂₃×#564	C₂²	1	64	yes	complemented
7	C₂₃×#557	C₂²	1	192	yes	complemented
8	C₂₃×#541	C₂²	1	192	yes	—
9	C₂₃×#533	C₂²	1	48	yes	—
10	C₂×C₂₃×C₄⋊Q₈	C₂²	1	6	yes	—
11	C₂×C₂₃×C₄⋊Q₈	C₂²	1	12	yes	—
12	C₂×C₂₃×#126	C₂²	1	16	yes	—
13	C₂×C₂₃×#125	C₂²	1	6	yes	—
14	C₂×C₂₃×#125	C₂²	1	12	yes	—
15	C₂×C₂₃×#125	C₂²	1	48	yes	—
16	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	12	yes	—
17	C₂×C₄×Q₈×C₂₃	C₂²	1	12	yes	—
18	C₂×C₄×Q₈×C₂₃	C₂²	1	24	yes	—
19	C₂×C₂₃×#117	C₂²	1	6	yes	—
20	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
21	C₂²×Q₈×C₂₃	C₂³	1	12	yes	—
22	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
23	C₂₃×C₄⋊Q₈	C₂³	1	48	yes	—
24	C₂₃×C₄⋊Q₈	C₂³	1	96	yes	—
25	C₂₃×#126	C₂³	1	128	yes	—
26	C₂₃×#125	C₂³	1	96	yes	—
27	C₂₃×#125	C₂³	1	384	yes	—
28	C₂₃×#125	C₂³	1	48	yes	—
29	C₂₃×C₂²⋊Q₈	C₂³	1	96	yes	—
30	C₄×Q₈×C₂₃	C₂³	1	96	yes	—
31	C₄×Q₈×C₂₃	C₂³	1	192	yes	—
32	C₂₃×#117	C₂³	1	48	yes	—
33	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
34	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
35	C₂×C₄⋊C₄×C₂₃	C₂³	1	24	yes	—
36	C₂×C₄⋊C₄×C₂₃	C₂³	1	48	yes	—
37	C₂×C₂²⋊C₄×C₂₃	C₂³	1	12	yes	—
38	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
39	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
41	C₂×C₄²×C₂₃	C₂³	1	12	yes	—
42	C₂×C₄²×C₂₃	C₂³	1	16	yes	—
43	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
44	C₂×Q₈×C₂₃	—	2	48	no	—
45	C₂×Q₈×C₂₃	C₂⁴	1	48	yes	—
46	C₂²×C₄×C₂₃	—	2	12	no	—
47	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
48	C₂²×C₄×C₂₃	C₂⁴	1	24	yes	—
49	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
50	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
51	C₄⋊C₄×C₂₃	C₂⁴	1	48	yes	—
52	C₄⋊C₄×C₂₃	C₂⁴	1	48	yes	—
53	C₄⋊C₄×C₂₃	C₂⁴	1	96	yes	—
54	C₄⋊C₄×C₂₃	C₂⁴	1	192	yes	—
55	C₂²⋊C₄×C₂₃	C₂⁴	1	48	yes	—
56	C₄²×C₂₃	C₂⁴	1	12	yes	—
57	C₄²×C₂₃	C₂⁴	1	4	yes	—
58	C₄²×C₂₃	C₂⁴	1	48	yes	—
59	C₄²×C₂₃	C₂⁴	1	64	yes	—
60	C₂×#3556	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₂₃	—	2	6	no	—
62	C₂³×C₂₃	C₂⁵	1	2	yes	—
63	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
64	Q₈×C₂₃	—	2	96	no	—
65	C₂×C₄×C₂₃	—	2	6	no	—
66	C₂×C₄×C₂₃	—	2	48	no	—
67	C₂×C₄×C₂₃	—	2	24	no	—
68	C₂×C₄×C₂₃	2₋¹⁺⁴	1	12	yes	—
69	C₂×C₄×C₂₃	C₂⁵	1	48	yes	—
70	C₂×C₄×C₂₃	C₂⁵	1	6	yes	—
71	C₂×C₄×C₂₃	C₂⁵	1	6	yes	—
72	#3556	C₂×C₂₃	1	32	yes	direct product factor
73	C₂×#564	C₂×C₂₃	1	4	yes	complemented
74	C₂×#557	C₂×C₂₃	1	12	yes	complemented
75	C₂×#541	C₂×C₂₃	1	12	yes	—
76	C₂×#533	C₂×C₂₃	1	3	yes	—
77	C₂²×C₂₃	—	2	6	no	—
78	C₂²×C₂₃	—	4	4	no	—
79	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	6	yes	—
80	C₂²×C₂₃	C₂⁶	1	1	yes	characteristic
81	C₄×C₂₃	—	2	6	no	—
82	C₄×C₂₃	—	2	48	no	—
83	C₄×C₂₃	C₂×2₋¹⁺⁴	1	12	yes	—
84	#564	C₂²×C₂₃	1	64	yes	complemented
85	#557	C₂²×C₂₃	1	192	yes	complemented
86	#541	C₂²×C₂₃	1	192	yes	—
87	#533	C₂²×C₂₃	1	48	yes	—
88	C₂×C₄⋊Q₈	C₂²×C₂₃	1	6	yes	—
89	C₂×C₄⋊Q₈	C₂²×C₂₃	1	12	yes	—
90	C₂×#126	C₂²×C₂₃	1	16	yes	—
91	C₂×#125	C₂²×C₂₃	1	6	yes	—
92	C₂×#125	C₂²×C₂₃	1	12	yes	—
93	C₂×#125	C₂²×C₂₃	1	48	yes	—
94	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	12	yes	—
95	C₂×C₄×Q₈	C₂²×C₂₃	1	12	yes	—
96	C₂×C₄×Q₈	C₂²×C₂₃	1	24	yes	—
97	C₂×#117	C₂²×C₂₃	1	6	yes	—
98	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
99	C₂×C₂₃	—	4	2	no	—
100	C₂×C₂₃	#3556	1	4	yes	direct product factor
101	C₂×C₂₃	C₂²×2₋¹⁺⁴	1	3	yes	—
102	C₂²×Q₈	C₂³×C₂₃	1	12	yes	—
103	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
104	C₄⋊Q₈	C₂³×C₂₃	1	48	yes	—
105	C₄⋊Q₈	C₂³×C₂₃	1	96	yes	—
106	#126	C₂³×C₂₃	1	128	yes	—
107	#125	C₂³×C₂₃	1	48	yes	—
108	#125	C₂³×C₂₃	1	96	yes	—
109	#125	C₂³×C₂₃	1	384	yes	—
110	C₂²⋊Q₈	C₂³×C₂₃	1	96	yes	—
111	C₄×Q₈	C₂³×C₂₃	1	96	yes	—
112	C₄×Q₈	C₂³×C₂₃	1	192	yes	—
113	#117	C₂³×C₂₃	1	48	yes	—
114	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
115	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
116	C₂×C₄⋊C₄	C₂³×C₂₃	1	24	yes	—
117	C₂×C₄⋊C₄	C₂³×C₂₃	1	48	yes	—
118	C₂×C₂²⋊C₄	C₂³×C₂₃	1	12	yes	—
119	C₂×C₄²	C₂³×C₂₃	1	12	yes	—
120	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
121	C₂×C₄²	C₂³×C₂₃	1	16	yes	—
122	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
123	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
124	C₂₃	C₂×#3556	1	1	yes	characteristic, direct product factor, Sylow
125	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
126	C₂×Q₈	—	2	48	no	—
127	C₂×Q₈	C₂⁴×C₂₃	1	48	yes	—
128	C₂²×C₄	—	2	12	no	—
129	C₂²×C₄	C₂⁴×C₂₃	1	24	yes	—
130	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
131	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
132	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
133	C₄⋊C₄	C₂⁴×C₂₃	1	48	yes	—
134	C₄⋊C₄	C₂⁴×C₂₃	1	48	yes	—
135	C₄⋊C₄	C₂⁴×C₂₃	1	96	yes	—
136	C₄⋊C₄	C₂⁴×C₂₃	1	192	yes	—
137	C₂²⋊C₄	C₂⁴×C₂₃	1	48	yes	—
138	C₄²	C₂⁴×C₂₃	1	12	yes	—
139	C₄²	C₂⁴×C₂₃	1	4	yes	—
140	C₄²	C₂⁴×C₂₃	1	48	yes	—
141	C₄²	C₂⁴×C₂₃	1	64	yes	—
142	C₂³	—	2	6	no	—
143	C₂³	C₂⁵×C₂₃	1	2	yes	—
144	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
145	Q₈	—	2	96	no	—
146	C₂×C₄	—	2	6	no	—
147	C₂×C₄	—	2	48	no	—
148	C₂×C₄	—	2	24	no	—
149	C₂×C₄	C₂₃×2₋¹⁺⁴	1	12	yes	—
150	C₂×C₄	C₂⁵×C₂₃	1	48	yes	—
151	C₂×C₄	C₂⁵×C₂₃	1	6	yes	—
152	C₂×C₄	C₂⁵×C₂₃	1	6	yes	—
153	C₂²	—	2	6	no	—
154	C₂²	—	4	4	no	—
155	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	6	yes	—
156	C₂²	C₂⁶×C₂₃	1	1	yes	characteristic
157	C₄	—	2	6	no	—
158	C₄	—	2	48	no	—
159	C₄	C₂×C₂₃×2₋¹⁺⁴	1	12	yes	—
160	C₂	—	4	2	no	—
161	C₂	C₂₃×#3556	1	4	yes	direct product factor
162	C₂	C₂²×C₂₃×2₋¹⁺⁴	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×C₂₃
prime radical	C₂⁴×C₂₃	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3556	C₂	direct (N × H)
2. C₂×C₂₃×#564	C₂	semidirect
3. C₂×C₂₃×#557	C₂	semidirect
6. C₂₃×#564	C₂²	semidirect
7. C₂₃×#557	C₂²	semidirect
60. C₂×#3556	C₂₃	direct (N × H)
72. #3556	C₂×C₂₃	direct (N × H)
73. C₂×#564	C₂×C₂₃	semidirect
74. C₂×#557	C₂×C₂₃	semidirect
84. #564	C₂²×C₂₃	semidirect
85. #557	C₂²×C₂₃	semidirect
100. C₂×C₂₃	#3556	direct (N × H)
124. C₂₃	C₂×#3556	direct (N × H)
161. C₂	C₂₃×#3556	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×#3556	2
C₂×C₂₃×#564	2
C₂×C₂₃×#557	2
C₂×C₂₃×#541	2
C₂×C₂₃×#533	2
C₂×#3556	23

Subgroups — C2×C23×#3556 #19210386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3556 #19210386