Subgroups — #529×C₂₂₃ #28383241

Order 14272.

234 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#126×C₂₂₃	C₂	1	2	yes	maximal, complemented
2	#125×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#124×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#123×C₂₂₃	C₂	1	2	yes	maximal, complemented
5	C₂²⋊Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#121×C₂₂₃	C₂	1	1	yes	maximal, characteristic
7	C₄×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×D₄×C₂₂₃	C₂	1	2	yes	maximal, complemented
10	#117×C₂₂₃	C₂	1	2	yes	maximal, complemented
11	C₂×C₄²×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₂×D₄×C₂₂₃	C₂²	1	2	yes	—
14	C₂×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₂₂₃	C₂²	1	2	yes	—
19	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
20	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
21	C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
23	C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
24	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
25	C₂²⋊C₄×C₂₂₃	C₂²	1	2	yes	—
26	C₂²⋊C₄×C₂₂₃	C₂²	1	2	yes	—
27	C₂²⋊C₄×C₂₂₃	C₂²	1	2	yes	—
28	C₂²⋊C₄×C₂₂₃	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₂₂₃	C₂²	1	2	yes	—
30	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
31	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
32	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
33	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
34	C₄²×C₂₂₃	C₂²	1	2	yes	—
35	C₂³×C₂₂₃	C₂³	1	2	yes	—
36	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
37	Q₈×C₂₂₃	—	2	2	no	—
38	D₄×C₂₂₃	—	2	4	no	—
39	D₄×C₂₂₃	—	2	2	no	—
40	C₂×C₄×C₂₂₃	—	2	2	no	—
41	C₂×C₄×C₂₂₃	—	2	2	no	—
42	C₂×C₄×C₂₂₃	—	2	4	no	—
43	C₂×C₄×C₂₂₃	—	2	2	no	—
44	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
49	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
50	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
51	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
52	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
53	C₂²×C₂₂₃	—	2	1	no	—
54	C₂²×C₂₂₃	—	2	1	no	—
55	C₂²×C₂₂₃	—	2	2	no	—
56	C₂²×C₂₂₃	—	2	2	no	—
57	C₂²×C₂₂₃	—	2	2	no	—
58	C₂²×C₂₂₃	#41	1	2	yes	—
59	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
60	C₄×C₂₂₃	—	2	1	no	—
61	C₄×C₂₂₃	—	2	1	no	—
62	C₄×C₂₂₃	—	2	1	no	—
63	C₄×C₂₂₃	—	2	1	no	—
64	C₄×C₂₂₃	—	2	2	no	—
65	C₄×C₂₂₃	—	2	2	no	—
66	C₄×C₂₂₃	—	2	2	no	—
67	C₄×C₂₂₃	#41	1	2	yes	—
68	C₄×C₂₂₃	#41	1	2	yes	—
69	C₂×C₂₂₃	—	2	2	no	—
70	C₂×C₂₂₃	—	4	2	no	—
71	C₂×C₂₂₃	C₂×#41	1	1	yes	characteristic
72	C₂×C₂₂₃	C₂×#41	1	1	yes	characteristic
73	C₂×C₂₂₃	C₂×#41	1	1	yes	characteristic
74	C₂₂₃	#529	1	1	yes	characteristic, direct product factor, Sylow
75	#529	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	#126	C₂×C₂₂₃	1	2	yes	complemented
77	#125	C₂×C₂₂₃	1	1	yes	characteristic, complemented
78	#124	C₂×C₂₂₃	1	1	yes	characteristic, complemented
79	#123	C₂×C₂₂₃	1	2	yes	complemented
80	C₂²⋊Q₈	C₂×C₂₂₃	1	1	yes	characteristic, complemented
81	#121	C₂×C₂₂₃	1	1	yes	characteristic
82	C₄×Q₈	C₂×C₂₂₃	1	1	yes	characteristic, complemented
83	C₄×D₄	C₂×C₂₂₃	1	1	yes	characteristic, complemented
84	C₄×D₄	C₂×C₂₂₃	1	2	yes	complemented
85	#117	C₂×C₂₂₃	1	2	yes	complemented
86	C₂×C₄²	C₂×C₂₂₃	1	1	yes	characteristic, complemented
87	C₂×Q₈	C₂²×C₂₂₃	1	1	yes	characteristic
88	C₂×D₄	C₂²×C₂₂₃	1	2	yes	—
89	C₂×D₄	C₂²×C₂₂₃	1	1	yes	characteristic
90	C₂²×C₄	C₂²×C₂₂₃	1	1	yes	characteristic
91	C₂²×C₄	C₂²×C₂₂₃	1	1	yes	characteristic
92	C₂²×C₄	C₂²×C₂₂₃	1	1	yes	characteristic
93	C₂²×C₄	C₂²×C₂₂₃	1	2	yes	—
94	C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	—
95	C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	—
96	C₄⋊C₄	C₂²×C₂₂₃	1	1	yes	characteristic
97	C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	—
98	C₄⋊C₄	C₂²×C₂₂₃	1	1	yes	characteristic
99	C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	—
100	C₂²⋊C₄	C₂²×C₂₂₃	1	2	yes	—
101	C₂²⋊C₄	C₂²×C₂₂₃	1	2	yes	—
102	C₂²⋊C₄	C₂²×C₂₂₃	1	2	yes	—
103	C₂²⋊C₄	C₂²×C₂₂₃	1	2	yes	—
104	C₂²⋊C₄	C₂²×C₂₂₃	1	2	yes	—
105	C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
106	C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
107	C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
108	C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
109	C₄²	C₂²×C₂₂₃	1	2	yes	—
110	C₂³	C₂³×C₂₂₃	1	2	yes	—
111	C₂³	C₂³×C₂₂₃	1	1	yes	characteristic
112	Q₈	—	2	2	no	—
113	D₄	—	2	4	no	—
114	D₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
120	C₂×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
121	C₂×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
122	C₂×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
123	C₂×C₄	C₂³×C₂₂₃	1	2	yes	—
124	C₂×C₄	C₂³×C₂₂₃	1	2	yes	—
125	C₂×C₄	C₂³×C₂₂₃	1	2	yes	—
126	C₂×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
127	C₂×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	2	no	—
132	C₂²	—	2	2	no	—
133	C₂²	#41×C₂₂₃	1	2	yes	—
134	C₂²	C₂⁴×C₂₂₃	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	#41×C₂₂₃	1	2	yes	—
143	C₄	#41×C₂₂₃	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	—	4	2	no	—
146	C₂	C₂×#41×C₂₂₃	1	1	yes	characteristic
147	C₂	C₂×#41×C₂₂₃	1	1	yes	characteristic
148	C₂	C₂×#41×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₂₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₂₂₃
socle	C₂²×C₂₂₃	C₂⁴
prime radical	#121×C₂₂₃	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. #126×C₂₂₃	C₂	semidirect
2. #125×C₂₂₃	C₂	semidirect
3. #124×C₂₂₃	C₂	semidirect
4. #123×C₂₂₃	C₂	semidirect
5. C₂²⋊Q₈×C₂₂₃	C₂	semidirect
7. C₄×Q₈×C₂₂₃	C₂	semidirect
8. C₄×D₄×C₂₂₃	C₂	semidirect
9. C₄×D₄×C₂₂₃	C₂	semidirect
10. #117×C₂₂₃	C₂	semidirect
11. C₂×C₄²×C₂₂₃	C₂	semidirect
74. C₂₂₃	#529	direct (N × H)
75. #529	C₂₂₃	direct (N × H)
76. #126	C₂×C₂₂₃	semidirect
77. #125	C₂×C₂₂₃	semidirect
78. #124	C₂×C₂₂₃	semidirect
79. #123	C₂×C₂₂₃	semidirect
80. C₂²⋊Q₈	C₂×C₂₂₃	semidirect
82. C₄×Q₈	C₂×C₂₂₃	semidirect
83. C₄×D₄	C₂×C₂₂₃	semidirect
84. C₄×D₄	C₂×C₂₂₃	semidirect
85. #117	C₂×C₂₂₃	semidirect
86. C₂×C₄²	C₂×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#126×C₂₂₃	2
#125×C₂₂₃	2
#124×C₂₂₃	2
#123×C₂₂₃	2
C₂²⋊Q₈×C₂₂₃	2
#121×C₂₂₃	2
C₄×Q₈×C₂₂₃	2
C₄×D₄×C₂₂₃	2
#117×C₂₂₃	2
C₂×C₄²×C₂₂₃	2
#529	223

Subgroups — #529×C223 #28383241

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #529×C₂₂₃ #28383241