Subgroups — #43252677

Order 28544.

188 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₂₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#28383450	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₂₂₃⋊C₈	—	2	1	no	—
5	C₂×C₄⋊Dic₂₂₃	—	2	1	no	—
6	C₂×C₄⋊Dic₂₂₃	—	2	1	no	—
7	C₂×C₄⋊Dic₂₂₃	C₄	1	2	yes	—
8	C₄⋊(C₂₂₃⋊C₈)	—	2	1	no	—
9	C₄⋊(C₂₂₃⋊C₈)	C₄	1	2	yes	complemented
10	C₂²×C₈×C₂₂₃	—	2	1	no	—
11	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₂₂₃	—	2	1	no	—
13	C₄⋊C₈×C₂₂₃	C₄	1	2	yes	complemented
14	C₂²×Dic₂₂₃	—	2	2	no	—
15	C₂×C₂₂₃⋊C₈	—	2	2	no	—
16	C₂×C₂₂₃⋊C₈	—	4	1	no	—
17	C₂×C₂₂₃⋊C₈	—	4	1	no	—
18	C₄⋊Dic₂₂₃	—	4	2	no	—
19	C₄⋊Dic₂₂₃	—	4	1	no	—
20	C₄⋊Dic₂₂₃	—	4	1	no	—
21	C₄⋊Dic₂₂₃	—	4	1	no	—
22	C₄⋊Dic₂₂₃	—	4	1	no	—
23	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₂₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₂₃	D₄	1	1	yes	characteristic
26	C₂×C₈×C₂₂₃	—	2	2	no	—
27	C₂×C₈×C₂₂₃	—	4	1	no	—
28	C₂×C₈×C₂₂₃	—	4	1	no	—
29	C₄²×C₂₂₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₂₂₃	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₂₂₃	D₄	1	1	yes	characteristic
32	C₄²×C₂₂₃	Q₈	1	1	yes	characteristic
33	C₂×Dic₂₂₃	—	4	2	no	—
34	C₂×Dic₂₂₃	—	4	2	no	—
35	C₂×Dic₂₂₃	—	4	2	no	—
36	C₂₂₃⋊C₈	—	4	2	no	—
37	C₂³×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₂₃	—	2	1	no	—
39	C₂×C₄×C₂₂₃	—	2	1	no	—
40	C₂×C₄×C₂₂₃	C₄²	1	1	yes	characteristic
41	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₂₃	D₈	1	1	yes	characteristic
47	C₂×C₄×C₂₂₃	D₈	1	1	yes	characteristic
48	C₂×C₄×C₂₂₃	SD₁₆	1	1	yes	characteristic
49	C₂×C₄×C₂₂₃	SD₁₆	1	1	yes	characteristic
50	C₂×C₄×C₂₂₃	SD₁₆	1	1	yes	characteristic
51	C₂×C₄×C₂₂₃	SD₁₆	1	1	yes	characteristic
52	C₂×C₄×C₂₂₃	Q₁₆	1	1	yes	characteristic
53	C₂×C₄×C₂₂₃	Q₁₆	1	1	yes	characteristic
54	C₈×C₂₂₃	—	4	2	no	—
55	C₂²×C₂₂₃	#95	1	1	yes	characteristic
56	C₂²×C₂₂₃	#100	1	1	yes	characteristic
57	C₂²×C₂₂₃	#101	1	1	yes	characteristic
58	C₂²×C₂₂₃	#102	1	1	yes	characteristic
59	C₂²×C₂₂₃	#102	1	1	yes	characteristic
60	C₂²×C₂₂₃	#103	1	1	yes	characteristic
61	C₂²×C₂₂₃	#103	1	1	yes	characteristic
62	C₄×C₂₂₃	—	2	1	no	—
63	C₄×C₂₂₃	—	2	1	no	—
64	C₄×C₂₂₃	#102	1	1	yes	characteristic
65	C₄×C₂₂₃	#102	1	1	yes	characteristic
66	C₄×C₂₂₃	#103	1	1	yes	characteristic
67	C₄×C₂₂₃	#103	1	1	yes	characteristic
68	C₄×C₂₂₃	#106	1	1	yes	characteristic
69	C₄×C₂₂₃	#106	1	1	yes	characteristic
70	C₄×C₂₂₃	#107	1	1	yes	characteristic
71	C₄×C₂₂₃	#107	1	1	yes	characteristic
72	Dic₂₂₃	—	8	2	no	—
73	C₂×C₂₂₃	#331	1	1	yes	characteristic
74	C₂×C₂₂₃	#332	1	1	yes	characteristic
75	C₂×C₂₂₃	#332	1	1	yes	characteristic
76	C₂×C₂₂₃	#333	1	1	yes	characteristic
77	C₂×C₂₂₃	#340	1	1	yes	characteristic
78	C₂×C₂₂₃	#340	1	1	yes	characteristic
79	C₂×C₂₂₃	#343	1	1	yes	characteristic
80	C₂₂₃	#1296	1	1	yes	characteristic, complemented, Sylow
81	#1296	—	223	1	no	maximal, Sylow
82	C₂×C₄⋊C₈	—	223	1	no	—
83	C₂×C₄⋊C₈	D₂₂₃	1	1	yes	characteristic
84	#384	—	223	1	no	—
85	C₂²×C₈	—	2	1	no	—
86	C₂²×C₈	—	446	1	no	—
87	C₂×C₄⋊C₄	—	446	1	no	—
88	C₂×C₄⋊C₄	—	446	1	no	—
89	C₂×C₄⋊C₄	—	223	2	no	—
90	C₂×C₄²	C₂×D₂₂₃	1	1	yes	characteristic
91	C₄⋊C₈	—	2	1	no	—
92	C₄⋊C₈	—	446	1	no	—
93	C₄⋊C₈	—	223	2	no	—
94	C₄⋊C₈	Dic₂₂₃	1	2	yes	complemented
95	C₂²×C₄	—	446	2	no	—
96	C₂²×C₄	C₄×D₂₂₃	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
98	C₂²×C₄	D₈₉₂	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	4	1	no	—
101	C₂×C₈	—	4	1	no	—
102	C₂×C₈	—	892	1	no	—
103	C₂×C₈	—	892	1	no	—
104	C₂×C₈	—	446	2	no	—
105	C₄⋊C₄	—	892	2	no	—
106	C₄⋊C₄	—	892	1	no	—
107	C₄⋊C₄	—	892	1	no	—
108	C₄⋊C₄	—	892	1	no	—
109	C₄⋊C₄	—	892	1	no	—
110	C₄²	C₄×D₂₂₃	1	1	yes	characteristic
111	C₄²	C₂×Dic₂₂₃	1	1	yes	characteristic
112	C₄²	C₂²⋊D₂₂₃	1	1	yes	characteristic
113	C₄²	Dic₄₄₆	1	1	yes	characteristic
114	C₂³	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	892	2	no	—
118	C₂×C₄	—	892	2	no	—
119	C₂×C₄	—	892	2	no	—
120	C₂×C₄	C₄×Dic₂₂₃	1	1	yes	characteristic
121	C₂×C₄	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
122	C₂×C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
123	C₂×C₄	Dic₂₂₃⋊C₄	1	1	yes	characteristic
124	C₂×C₄	Dic₂₂₃⋊C₄	1	1	yes	characteristic
125	C₂×C₄	C₄⋊Dic₂₂₃	1	1	yes	characteristic
126	C₂×C₄	#16813155	1	1	yes	characteristic
127	C₂×C₄	D₁₇₈₄	1	1	yes	characteristic
128	C₂×C₄	#16813157	1	1	yes	characteristic
129	C₂×C₄	#16813158	1	1	yes	characteristic
130	C₂×C₄	#16813159	1	1	yes	characteristic
131	C₂×C₄	#16813159	1	1	yes	characteristic
132	C₂×C₄	#16813160	1	1	yes	characteristic
133	C₂×C₄	Dic₈₉₂	1	1	yes	characteristic
134	C₈	—	4	2	no	—
135	C₈	—	892	2	no	—
136	C₂²	C₂₂₃⋊#95	1	1	yes	characteristic
137	C₂²	#20297220	1	1	yes	characteristic
138	C₂²	#20297222	1	1	yes	characteristic
139	C₂²	#20297224	1	1	yes	characteristic
140	C₂²	#20297226	1	1	yes	characteristic
141	C₂²	#20297227	1	1	yes	characteristic
142	C₂²	#20297229	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	1784	2	no	—
146	C₄	#20297225	1	1	yes	characteristic
147	C₄	#20297226	1	1	yes	characteristic
148	C₄	#20297228	1	1	yes	characteristic
149	C₄	#20297229	1	1	yes	characteristic
150	C₄	#20297235	1	1	yes	characteristic
151	C₄	#20297236	1	1	yes	characteristic
152	C₄	#20297237	1	1	yes	characteristic
153	C₄	#20297238	1	1	yes	characteristic
154	C₂	#28383323	1	1	yes	characteristic
155	C₂	#28383325	1	1	yes	characteristic
156	C₂	#28383327	1	1	yes	characteristic
157	C₂	#28383328	1	1	yes	characteristic
158	C₂	#28383342	1	1	yes	characteristic
159	C₂	#28383343	1	1	yes	characteristic
160	C₂	#28383348	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	(C₂×D₂₂₃)⋊C₄
commutator	C₂×C₄×C₂₂₃	C₄²
Frattini	C₂×C₄²	C₂×D₂₂₃
Fitting	C₂×C₄⋊C₈×C₂₂₃	C₂
socle, prime radical	C₂³×C₂₂₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₂₂₃	C₂²
hypercenter	C₂×C₄⋊C₈	D₂₂₃
hypocenter	C₂₂₃	#1296
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
9. C₄⋊(C₂₂₃⋊C₈)	C₄	semidirect
13. C₄⋊C₈×C₂₂₃	C₄	semidirect
80. C₂₂₃	#1296	semidirect
94. C₄⋊C₈	Dic₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₂₂₃⋊C₈)	2
#28383450	2
C₂×C₄⋊C₈×C₂₂₃	2
#1296	223