Subgroups — #24921958

Order 10624.

890 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₃⋊#564	C₂	1	4	yes	maximal, complemented
2	#19128430	C₂	1	4	yes	maximal, complemented
3	#19128429	C₂	1	4	yes	maximal, complemented
4	#19128426	C₂	1	4	yes	maximal, complemented
5	#19128313	C₂	1	4	yes	maximal
6	#19128311	C₂	1	8	yes	maximal
7	#19128258	C₂	1	2	yes	maximal
8	#533×C₈₃	C₂	1	1	yes	maximal, characteristic
9	#14581792	C₂²	1	2	yes	—
10	#14581792	C₂²	1	2	yes	—
11	#14581792	C₂²	1	4	yes	—
12	#14581790	C₂²	1	8	yes	—
13	#14581785	C₂²	1	16	yes	—
14	#14581784	C₂²	1	4	yes	—
15	#14581784	C₂²	1	4	yes	—
16	#14581783	C₂²	1	8	yes	—
17	#14581783	C₂²	1	16	yes	—
18	#14581782	C₂²	1	16	yes	—
19	#14581782	C₂²	1	16	yes	—
20	#14581771	C₂²	1	4	yes	—
21	#14581771	C₂²	1	4	yes	—
22	Q₈×Dic₈₃	C₂²	1	8	yes	—
23	C₄×Dic₁₆₆	C₂²	1	4	yes	—
24	C₄×Dic₁₆₆	C₂²	1	4	yes	—
25	C₄×Dic₁₆₆	C₂²	1	8	yes	—
26	#14581757	C₂²	1	8	yes	—
27	#14581751	C₂²	1	4	yes	—
28	C₄⋊Q₈×C₈₃	C₂²	1	2	yes	—
29	#125×C₈₃	C₂²	1	2	yes	—
30	C₂²⋊Q₈×C₈₃	C₂²	1	4	yes	—
31	C₄×Q₈×C₈₃	C₂²	1	4	yes	—
32	#117×C₈₃	C₂²	1	2	yes	—
33	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
34	C₂×Dic₁₆₆	C₂³	1	4	yes	—
35	C₂×Dic₁₆₆	C₂³	1	4	yes	—
36	C₄⋊Dic₈₃	C₂³	1	4	yes	—
37	C₄⋊Dic₈₃	C₂³	1	4	yes	—
38	C₄⋊Dic₈₃	C₂³	1	4	yes	—
39	C₄⋊Dic₈₃	C₂³	1	16	yes	—
40	C₄⋊Dic₈₃	C₂³	1	4	yes	—
41	Dic₈₃⋊C₄	C₂³	1	8	yes	—
42	Dic₈₃⋊C₄	C₂³	1	8	yes	—
43	Dic₈₃⋊C₄	C₂³	1	16	yes	—
44	Dic₈₃⋊C₄	C₂³	1	16	yes	—
45	C₂²⋊Dic₈₃	C₂³	1	8	yes	—
46	C₄×Dic₈₃	C₂³	1	16	yes	—
47	C₄×Dic₈₃	C₂³	1	8	yes	—
48	C₂×Q₈×C₈₃	C₂³	1	4	yes	—
49	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₈₃	C₂³	1	2	yes	—
51	C₄⋊C₄×C₈₃	C₂³	1	4	yes	—
52	C₄⋊C₄×C₈₃	C₂³	1	8	yes	—
53	C₄⋊C₄×C₈₃	C₂³	1	4	yes	—
54	C₂²⋊C₄×C₈₃	C₂³	1	4	yes	—
55	C₄²×C₈₃	C₂³	1	1	yes	characteristic
56	C₄²×C₈₃	C₂³	1	4	yes	—
57	C₄²×C₈₃	C₂³	1	2	yes	—
58	C₄²×C₈₃	C₂³	1	1	yes	characteristic
59	C₂³×C₈₃	C₂⁴	1	1	yes	characteristic
60	Q₈×C₈₃	—	2	8	no	—
61	C₂×C₄×C₈₃	—	2	4	no	—
62	C₂×C₄×C₈₃	—	2	2	no	—
63	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
64	C₂×C₄×C₈₃	C₂⁴	1	2	yes	—
65	C₂×C₄×C₈₃	C₂⁴	1	8	yes	—
66	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
67	C₂×C₄×C₈₃	C₂⁴	1	2	yes	—
68	C₂×Dic₈₃	C₂⁴	1	8	yes	—
69	C₂×Dic₈₃	C₂⁴	1	8	yes	—
70	Dic₁₆₆	—	2	8	no	—
71	Dic₁₆₆	—	2	8	no	—
72	C₂²×C₈₃	—	2	2	no	—
73	C₂²×C₈₃	—	2	1	no	—
74	C₂²×C₈₃	C₂⁵	1	1	yes	characteristic
75	C₄×C₈₃	—	2	2	no	—
76	C₄×C₈₃	—	2	1	no	—
77	C₄×C₈₃	—	2	8	no	—
78	C₄×C₈₃	2₋¹⁺⁴	1	4	yes	—
79	C₄×C₈₃	2₋¹⁺⁴	1	2	yes	—
80	Dic₈₃	—	2	8	no	—
81	Dic₈₃	—	2	8	no	—
82	C₂×C₈₃	—	4	1	no	—
83	C₂×C₈₃	C₂×2₋¹⁺⁴	1	2	yes	—
84	C₂×C₈₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
85	#3556	—	83	1	no	maximal, Sylow
86	C₈₃	#3556	1	1	yes	characteristic, complemented, Sylow
87	#564	—	83	4	no	—
88	#557	—	83	4	no	—
89	#557	—	83	4	no	—
90	#557	—	83	4	no	—
91	#541	—	83	8	no	—
92	#541	—	83	4	no	—
93	#533	—	83	2	no	—
94	#533	D₈₃	1	1	yes	characteristic
95	C₄⋊Q₈	—	83	8	no	—
96	C₄⋊Q₈	—	83	2	no	—
97	C₄⋊Q₈	—	83	2	no	—
98	C₄⋊Q₈	—	83	4	no	—
99	C₄⋊Q₈	C₂×D₈₃	1	2	yes	—
100	#126	—	83	16	no	—
101	#125	—	83	16	no	—
102	#125	—	83	16	no	—
103	#125	—	83	8	no	—
104	#125	—	83	16	no	—
105	#125	—	83	4	no	—
106	#125	—	83	4	no	—
107	#125	C₂×D₈₃	1	2	yes	—
108	C₂²⋊Q₈	—	83	4	no	—
109	C₂²⋊Q₈	—	83	4	no	—
110	C₂²⋊Q₈	C₂×D₈₃	1	4	yes	—
111	C₄×Q₈	—	83	8	no	—
112	C₄×Q₈	—	83	8	no	—
113	C₄×Q₈	—	83	4	no	—
114	C₄×Q₈	—	83	4	no	—
115	C₄×Q₈	—	83	8	no	—
116	C₄×Q₈	C₂×D₈₃	1	4	yes	—
117	#117	—	83	4	no	—
118	#117	C₂×D₈₃	1	2	yes	—
119	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
120	C₂×Q₈	—	83	4	no	—
121	C₂×Q₈	—	83	4	no	—
122	C₂×Q₈	C₂²×D₈₃	1	4	yes	—
123	C₂²×C₄	C₂²×D₈₃	1	2	yes	—
124	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
125	C₄⋊C₄	—	83	4	no	—
126	C₄⋊C₄	—	83	4	no	—
127	C₄⋊C₄	—	83	4	no	—
128	C₄⋊C₄	—	83	4	no	—
129	C₄⋊C₄	—	83	8	no	—
130	C₄⋊C₄	—	83	8	no	—
131	C₄⋊C₄	—	83	16	no	—
132	C₄⋊C₄	—	83	16	no	—
133	C₄⋊C₄	—	83	16	no	—
134	C₄⋊C₄	C₂²×D₈₃	1	4	yes	—
135	C₄⋊C₄	C₂²×D₈₃	1	4	yes	—
136	C₄⋊C₄	C₂²×D₈₃	1	8	yes	—
137	C₂²⋊C₄	—	83	8	no	—
138	C₂²⋊C₄	C₂²×D₈₃	1	4	yes	—
139	C₄²	—	83	8	no	—
140	C₄²	—	83	16	no	—
141	C₄²	C₂²×D₈₃	1	1	yes	characteristic
142	C₄²	C₂²×D₈₃	1	2	yes	—
143	C₄²	C₂²×D₈₃	1	1	yes	characteristic
144	C₄²	C₂²×D₈₃	1	4	yes	—
145	C₂³	C₂³×D₈₃	1	1	yes	characteristic
146	Q₈	—	2	8	no	—
147	Q₈	—	166	8	no	—
148	Q₈	—	166	8	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	83	8	no	—
152	C₂×C₄	—	83	8	no	—
153	C₂×C₄	C₂³×D₈₃	1	8	yes	—
154	C₂×C₄	C₂³×D₈₃	1	2	yes	—
155	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
156	C₂×C₄	C₂³×D₈₃	1	2	yes	—
157	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
158	C₂²	—	2	2	no	—
159	C₂²	—	2	1	no	—
160	C₂²	C₂⁴×D₈₃	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	2	8	no	—
163	C₄	—	2	1	no	—
164	C₄	—	166	8	no	—
165	C₄	—	166	8	no	—
166	C₄	#14581839	1	4	yes	—
167	C₄	#14581840	1	2	yes	—
168	C₂	—	4	1	no	—
169	C₂	C₂×#14581839	1	2	yes	—
170	C₂	C₂×#14581840	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₈₃
commutator, socle, cosocle	C₂²×C₈₃	C₂⁵
Fitting	#533×C₈₃	C₂
prime radical	C₂³×C₈₃	C₂⁴
hypercenter	#533	D₈₃
hypocenter	C₈₃	#3556
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
1. C₈₃⋊#564	C₂	semidirect
2. #19128430	C₂	semidirect
3. #19128429	C₂	semidirect
4. #19128426	C₂	semidirect
86. C₈₃	#3556	semidirect

Maximal subgroups

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

subgroup	index
C₈₃⋊#564	2
#19128430	2
#19128429	2
#19128426	2
#19128313	2
#19128311	2
#19128258	2
#533×C₈₃	2
#3556	83