Subgroups — #2931×C₂₂₃ #43251946

Order 28544.

500 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#131×C₂₂₃	C₂	1	2	yes	maximal, complemented
2	#532×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#436×C₂₂₃	C₂	1	4	yes	maximal, complemented
4	C₈×D₄×C₂₂₃	C₂	1	4	yes	maximal, complemented
5	#423×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#408×C₂₂₃	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₂₂₃	C₂²	1	2	yes	—
9	#131×C₂₂₃	—	2	8	no	—
10	C₂×M₁₆×C₂₂₃	C₂²	1	2	yes	—
11	C₂×M₁₆×C₂₂₃	C₂²	1	4	yes	—
12	C₂²×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₂₂₃	C₂²	1	4	yes	—
14	C₂²×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₂₂₃	C₄	1	1	yes	characteristic
16	#127×C₂₂₃	C₄	1	1	yes	characteristic
17	#124×C₂₂₃	C₄	1	2	yes	—
18	#121×C₂₂₃	C₄	1	4	yes	—
19	C₄×D₄×C₂₂₃	C₂²	1	4	yes	—
20	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₂₂₃	C₂²	1	4	yes	—
22	C₂²⋊C₈×C₂₂₃	C₂²	1	8	yes	—
23	C₄×C₈×C₂₂₃	C₂²	1	2	yes	—
24	C₄×C₈×C₂₂₃	C₂²	1	2	yes	—
25	#41×C₂₂₃	—	2	4	no	—
26	#41×C₂₂₃	—	2	4	no	—
27	C₂×Q₈×C₂₂₃	C₂×C₄	1	2	yes	—
28	C₂×D₄×C₂₂₃	C₂×C₄	1	4	yes	—
29	C₂×D₄×C₂₂₃	C₂×C₄	1	4	yes	—
30	C₂×D₄×C₂₂₃	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₂₂₃	C₂³	1	4	yes	—
33	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
34	M₁₆×C₂₂₃	—	2	4	no	—
35	M₁₆×C₂₂₃	—	2	8	no	—
36	C₂×C₈×C₂₂₃	—	2	2	no	—
37	C₂×C₈×C₂₂₃	—	2	8	no	—
38	C₂×C₈×C₂₂₃	D₄	1	4	yes	complemented
39	C₂×C₈×C₂₂₃	C₂³	1	2	yes	—
40	C₂×C₈×C₂₂₃	C₂³	1	4	yes	—
41	C₂×C₈×C₂₂₃	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₂₂₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₂₂₃	C₂×C₄	1	4	yes	—
44	C₂²⋊C₄×C₂₂₃	C₂×C₄	1	8	yes	—
45	C₄²×C₂₂₃	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₂₂₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₂₂₃	C₂³	1	2	yes	—
48	C₂³×C₂₂₃	C₂²×C₄	1	4	yes	—
49	C₂³×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
50	Q₈×C₂₂₃	—	2	4	no	—
51	D₄×C₂₂₃	—	2	8	no	—
52	D₄×C₂₂₃	—	2	4	no	—
53	D₄×C₂₂₃	—	2	8	no	—
54	C₂×C₄×C₂₂₃	—	2	4	no	—
55	C₂×C₄×C₂₂₃	—	2	4	no	—
56	C₂×C₄×C₂₂₃	—	2	4	no	—
57	C₂×C₄×C₂₂₃	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₂₂₃	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₂₂₃	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₂₂₃	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₂₂₃	#41	1	2	yes	—
63	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
64	C₈×C₂₂₃	—	2	4	no	—
65	C₈×C₂₂₃	—	2	2	no	—
66	C₈×C₂₂₃	C₂×D₄	1	2	yes	—
67	C₈×C₂₂₃	C₂×D₄	1	2	yes	—
68	C₂²×C₂₂₃	—	2	2	no	—
69	C₂²×C₂₂₃	—	2	4	no	—
70	C₂²×C₂₂₃	—	2	8	no	—
71	C₂²×C₂₂₃	C₄×D₄	1	2	yes	—
72	C₂²×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
73	C₄×C₂₂₃	—	2	4	no	—
74	C₄×C₂₂₃	—	2	2	no	—
75	C₄×C₂₂₃	C₄×D₄	1	2	yes	—
76	C₄×C₂₂₃	#131	1	4	yes	complemented
77	C₄×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₂₂₃	C₂×#41	1	1	yes	characteristic
79	C₂×C₂₂₃	—	2	2	no	—
80	C₂×C₂₂₃	—	4	4	no	—
81	C₂×C₂₂₃	C₂×C₄×D₄	1	1	yes	characteristic
82	C₂×C₂₂₃	C₂×#131	1	2	yes	—
83	C₂₂₃	#2931	1	1	yes	characteristic, direct product factor, Sylow
84	#2931	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂×#131	C₂×C₂₂₃	1	2	yes	complemented
86	#532	C₂×C₂₂₃	1	1	yes	characteristic
87	#436	C₂×C₂₂₃	1	4	yes	complemented
88	C₈×D₄	C₂×C₂₂₃	1	4	yes	complemented
89	#423	C₂×C₂₂₃	1	1	yes	characteristic, complemented
90	#408	C₂×C₂₂₃	1	2	yes	complemented
91	C₂×C₄×C₈	C₂×C₂₂₃	1	1	yes	characteristic, complemented
92	C₂×#41	C₂²×C₂₂₃	1	2	yes	—
93	#131	—	2	8	no	—
94	C₂×M₁₆	C₂²×C₂₂₃	1	2	yes	—
95	C₂×M₁₆	C₂²×C₂₂₃	1	4	yes	—
96	C₂²×C₈	C₂²×C₂₂₃	1	4	yes	—
97	C₂²×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
98	C₂²×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
99	C₄⋊Q₈	C₄×C₂₂₃	1	1	yes	characteristic
100	#127	C₄×C₂₂₃	1	1	yes	characteristic
101	#124	C₄×C₂₂₃	1	2	yes	—
102	#121	C₄×C₂₂₃	1	4	yes	—
103	C₄×D₄	C₂²×C₂₂₃	1	4	yes	—
104	C₂×C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
105	C₄⋊C₈	C₂²×C₂₂₃	1	4	yes	—
106	C₂²⋊C₈	C₂²×C₂₂₃	1	8	yes	—
107	C₄×C₈	C₂²×C₂₂₃	1	2	yes	—
108	C₄×C₈	C₂²×C₂₂₃	1	2	yes	—
109	#41	—	2	4	no	—
110	#41	—	2	4	no	—
111	C₂×Q₈	C₂×C₄×C₂₂₃	1	2	yes	—
112	C₂×D₄	C₂×C₄×C₂₂₃	1	4	yes	—
113	C₂×D₄	C₂×C₄×C₂₂₃	1	4	yes	—
114	C₂×D₄	C₂×C₄×C₂₂₃	1	2	yes	—
115	C₂²×C₄	C₂×C₄×C₂₂₃	1	2	yes	—
116	C₂²×C₄	C₂³×C₂₂₃	1	4	yes	—
117	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
118	M₁₆	—	2	4	no	—
119	M₁₆	—	2	8	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	2	8	no	—
122	C₂×C₈	D₄×C₂₂₃	1	4	yes	complemented
123	C₂×C₈	C₂³×C₂₂₃	1	2	yes	—
124	C₂×C₈	C₂³×C₂₂₃	1	4	yes	—
125	C₂×C₈	C₂³×C₂₂₃	1	1	yes	characteristic
126	C₂×C₈	C₂³×C₂₂₃	1	1	yes	characteristic
127	C₄⋊C₄	C₂×C₄×C₂₂₃	1	4	yes	—
128	C₂²⋊C₄	C₂×C₄×C₂₂₃	1	8	yes	—
129	C₄²	C₂×C₄×C₂₂₃	1	1	yes	characteristic
130	C₄²	C₂×C₄×C₂₂₃	1	1	yes	characteristic
131	C₄²	C₂³×C₂₂₃	1	2	yes	—
132	C₂³	C₂²×C₄×C₂₂₃	1	4	yes	—
133	C₂³	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
134	Q₈	—	2	4	no	—
135	D₄	—	2	8	no	—
136	D₄	—	2	4	no	—
137	D₄	—	2	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	C₂²×C₄×C₂₂₃	1	2	yes	—
142	C₂×C₄	C₂²×C₄×C₂₂₃	1	4	yes	—
143	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
144	C₂×C₄	C₂²×C₄×C₂₂₃	1	2	yes	—
145	C₂×C₄	C₂×D₄×C₂₂₃	1	2	yes	—
146	C₂×C₄	#41×C₂₂₃	1	2	yes	—
147	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
148	C₈	—	2	4	no	—
149	C₈	—	2	2	no	—
150	C₈	C₂×D₄×C₂₂₃	1	2	yes	—
151	C₈	C₂×D₄×C₂₂₃	1	2	yes	—
152	C₂²	—	2	2	no	—
153	C₂²	—	2	8	no	—
154	C₂²	—	2	4	no	—
155	C₂²	C₄×D₄×C₂₂₃	1	2	yes	—
156	C₂²	C₂³×C₄×C₂₂₃	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	4	no	—
159	C₄	C₄×D₄×C₂₂₃	1	2	yes	—
160	C₄	#131×C₂₂₃	1	4	yes	complemented
161	C₄	C₂²×D₄×C₂₂₃	1	1	yes	characteristic
162	C₄	C₂×#41×C₂₂₃	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	—	4	4	no	—
165	C₂	C₂×C₄×D₄×C₂₂₃	1	1	yes	characteristic
166	C₂	C₂×#131×C₂₂₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₂₂₃	C₂³
commutator	C₂²	C₂³×C₄×C₂₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₂₃
socle	C₂²×C₂₂₃	C₂³×C₄
prime radical	#532×C₂₂₃	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#131×C₂₂₃	C₂	semidirect
3. #436×C₂₂₃	C₂	semidirect
4. C₈×D₄×C₂₂₃	C₂	semidirect
5. #423×C₂₂₃	C₂	semidirect
6. #408×C₂₂₃	C₂	semidirect
7. C₂×C₄×C₈×C₂₂₃	C₂	semidirect
38. C₂×C₈×C₂₂₃	D₄	semidirect
76. C₄×C₂₂₃	#131	semidirect
83. C₂₂₃	#2931	direct (N × H)
84. #2931	C₂₂₃	direct (N × H)
85. C₂×#131	C₂×C₂₂₃	semidirect
87. #436	C₂×C₂₂₃	semidirect
88. C₈×D₄	C₂×C₂₂₃	semidirect
89. #423	C₂×C₂₂₃	semidirect
90. #408	C₂×C₂₂₃	semidirect
91. C₂×C₄×C₈	C₂×C₂₂₃	semidirect
122. C₂×C₈	D₄×C₂₂₃	semidirect
160. C₄	#131×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#131×C₂₂₃	2
#532×C₂₂₃	2
#436×C₂₂₃	2
C₈×D₄×C₂₂₃	2
#423×C₂₂₃	2
#408×C₂₂₃	2
C₂×C₄×C₈×C₂₂₃	2
#2931	223

Subgroups — #2931×C223 #43251946

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2931×C₂₂₃ #43251946