Subgroups — #24916477

Order 10624.

406 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19127832	C₂	1	2	yes	maximal, complemented
2	#19127773	C₂	1	4	yes	maximal, complemented
3	D₄×C₈₃⋊C₈	C₂	1	2	yes	maximal, complemented
4	#19127758	C₂	1	2	yes	maximal, complemented
5	#19127758	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13050574	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14581703	C₂	1	2	yes	maximal, complemented
8	C₄×#41×C₈₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13050574	C₂²	1	2	yes	—
10	C₂²×C₈₃⋊C₈	C₂²	1	4	yes	—
11	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
12	C₄⋊(C₈₃⋊C₈)	C₂²	1	4	yes	—
13	C₂²⋊(C₈₃⋊C₈)	C₂²	1	4	yes	—
14	C₂²⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
15	#14581703	C₂²	1	4	yes	—
16	#14581703	C₂²	1	1	yes	characteristic
17	#14581703	C₂²	1	2	yes	—
18	#14581703	C₂²	1	1	yes	characteristic
19	C₄×C₈₃⋊C₈	C₂²	1	2	yes	—
20	C₂×#41×C₈₃	C₄	1	1	yes	characteristic
21	C₄×Q₈×C₈₃	C₄	1	1	yes	characteristic
22	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₈₃	C₄	1	2	yes	—
24	C₄×D₄×C₈₃	C₄	1	1	yes	characteristic
25	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₈₃	C₂²	1	2	yes	—
27	#117×C₈₃	C₄	1	1	yes	characteristic
28	#117×C₈₃	C₄	1	2	yes	—
29	C₂×C₄²×C₈₃	C₂²	1	2	yes	—
30	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
31	#13050574	—	2	4	no	—
32	C₂×C₈₃⋊C₈	—	2	8	no	—
33	C₂×C₈₃⋊C₈	C₂³	1	4	yes	—
34	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
35	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
36	#41×C₈₃	—	2	4	no	—
37	C₂×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
38	C₂×D₄×C₈₃	C₂×C₄	1	2	yes	—
39	C₂×D₄×C₈₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₈₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
47	C₄⋊C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
48	C₄⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
49	C₄⋊C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
50	C₂²⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
51	C₂²⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
52	C₂²⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
53	C₄²×C₈₃	C₂×C₄	1	2	yes	—
54	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₈₃	C₂×C₄	1	2	yes	—
56	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₈₃	C₂³	1	1	yes	characteristic
58	C₄²×C₈₃	C₂³	1	1	yes	characteristic
59	C₄²×C₈₃	C₂³	1	2	yes	—
60	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
61	C₂³×C₈₃	C₂²×C₄	1	2	yes	—
62	Q₈×C₈₃	—	2	2	no	—
63	D₄×C₈₃	—	2	4	no	—
64	D₄×C₈₃	M₁₆	1	4	yes	complemented
65	C₂×C₄×C₈₃	—	2	4	no	—
66	C₂×C₄×C₈₃	—	2	2	no	—
67	C₂×C₄×C₈₃	—	2	4	no	—
68	C₂×C₄×C₈₃	—	2	2	no	—
69	C₂×C₄×C₈₃	—	4	2	no	—
70	C₂×C₄×C₈₃	—	4	1	no	—
71	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
80	C₈₃⋊C₈	—	2	4	no	—
81	C₈₃⋊C₈	—	2	2	no	—
82	C₈₃⋊C₈	#41	1	4	yes	complemented
83	C₂²×C₈₃	—	2	2	no	—
84	C₂²×C₈₃	—	2	2	no	—
85	C₂²×C₈₃	—	2	1	no	—
86	C₂²×C₈₃	—	2	1	no	—
87	C₂²×C₈₃	—	2	1	no	—
88	C₂²×C₈₃	C₂×M₁₆	1	4	yes	—
89	C₂²×C₈₃	C₂³×C₄	1	1	yes	characteristic
90	C₄×C₈₃	—	2	2	no	—
91	C₄×C₈₃	—	2	1	no	—
92	C₄×C₈₃	—	2	1	no	—
93	C₄×C₈₃	—	2	2	no	—
94	C₄×C₈₃	—	2	1	no	—
95	C₄×C₈₃	—	2	2	no	—
96	C₄×C₈₃	—	2	1	no	—
97	C₄×C₈₃	C₂×M₁₆	1	2	yes	—
98	C₄×C₈₃	C₂×#41	1	2	yes	—
99	C₂×C₈₃	—	2	4	no	—
100	C₂×C₈₃	—	4	1	no	—
101	C₂×C₈₃	C₄×#41	1	1	yes	characteristic
102	C₂×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
103	C₂×C₈₃	#568	1	1	yes	characteristic
104	#2970	—	83	1	no	maximal, Sylow
105	C₈₃	#2970	1	1	yes	characteristic, complemented, Sylow
106	C₄×#41	D₈₃	1	1	yes	characteristic
107	#446	—	83	2	no	—
108	#435	—	83	4	no	—
109	C₈×D₄	—	83	2	no	—
110	#432	—	83	1	no	—
111	#432	—	83	2	no	—
112	C₄×M₁₆	—	83	1	no	—
113	C₂×#97	—	83	2	no	—
114	C₂×#41	Dic₈₃	1	1	yes	characteristic
115	C₂×M₁₆	—	83	2	no	—
116	C₂²×C₈	—	83	4	no	—
117	C₄×Q₈	Dic₈₃	1	1	yes	characteristic
118	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
119	C₄×D₄	Dic₈₃	1	2	yes	—
120	C₄×D₄	Dic₈₃	1	1	yes	characteristic
121	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
122	C₄×D₄	C₂×D₈₃	1	2	yes	—
123	#117	Dic₈₃	1	1	yes	characteristic
124	#117	Dic₈₃	1	2	yes	—
125	C₂×C₄²	C₂×D₈₃	1	2	yes	—
126	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
127	C₄⋊C₈	—	83	2	no	—
128	C₄⋊C₈	—	83	4	no	—
129	C₂²⋊C₈	—	83	4	no	—
130	C₂²⋊C₈	—	83	2	no	—
131	#97	—	83	1	no	—
132	#97	—	83	2	no	—
133	#97	—	83	4	no	—
134	#97	—	83	1	no	—
135	C₄×C₈	—	83	2	no	—
136	#41	—	2	4	no	—
137	C₂×Q₈	C₂×Dic₈₃	1	1	yes	characteristic
138	C₂×D₄	C₂×Dic₈₃	1	2	yes	—
139	C₂×D₄	C₂×Dic₈₃	1	1	yes	characteristic
140	C₂²×C₄	C₂×Dic₈₃	1	2	yes	—
141	C₂²×C₄	C₂×Dic₈₃	1	1	yes	characteristic
142	C₂²×C₄	C₂×Dic₈₃	1	1	yes	characteristic
143	C₂²×C₄	C₂×Dic₈₃	1	2	yes	—
144	C₂²×C₄	C₂²×D₈₃	1	2	yes	—
145	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
146	M₁₆	—	166	4	no	—
147	C₂×C₈	—	83	2	no	—
148	C₂×C₈	—	83	2	no	—
149	C₂×C₈	—	83	4	no	—
150	C₂×C₈	—	166	8	no	—
151	C₄⋊C₄	C₂×Dic₈₃	1	2	yes	—
152	C₄⋊C₄	C₂×Dic₈₃	1	1	yes	characteristic
153	C₄⋊C₄	C₂×Dic₈₃	1	2	yes	—
154	C₄⋊C₄	C₂×Dic₈₃	1	1	yes	characteristic
155	C₂²⋊C₄	C₂×Dic₈₃	1	2	yes	—
156	C₂²⋊C₄	C₂×Dic₈₃	1	2	yes	—
157	C₂²⋊C₄	C₂×Dic₈₃	1	2	yes	—
158	C₄²	C₂×Dic₈₃	1	2	yes	—
159	C₄²	C₂×Dic₈₃	1	1	yes	characteristic
160	C₄²	C₂×Dic₈₃	1	2	yes	—
161	C₄²	C₂×Dic₈₃	1	1	yes	characteristic
162	C₄²	C₂²×D₈₃	1	1	yes	characteristic
163	C₄²	C₂²×D₈₃	1	1	yes	characteristic
164	C₄²	C₂²×D₈₃	1	2	yes	—
165	C₂³	C₂²×Dic₈₃	1	1	yes	characteristic
166	C₂³	C₂²×Dic₈₃	1	2	yes	—
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	#13050574	1	4	yes	complemented
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
177	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
178	C₂×C₄	C₂²×Dic₈₃	1	2	yes	—
179	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
180	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
181	C₂×C₄	C₂²×Dic₈₃	1	2	yes	—
182	C₂×C₄	C₂²×Dic₈₃	1	2	yes	—
183	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
184	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
185	C₈	—	83	4	no	—
186	C₈	—	166	4	no	—
187	C₈	—	166	2	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	C₂×#13050574	1	4	yes	—
194	C₂²	C₂³×Dic₈₃	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	1	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	1	no	—
202	C₄	C₂×#13050574	1	2	yes	—
203	C₄	#41×D₈₃	1	2	yes	—
204	C₂	—	2	4	no	—
205	C₂	—	4	1	no	—
206	C₂	#41×Dic₈₃	1	1	yes	characteristic
207	C₂	C₂²×#13050574	1	1	yes	characteristic
208	C₂	#19128468	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₃
commutator, socle	C₂²×C₈₃	C₂³×C₄
Fitting	C₄×#41×C₈₃	C₂
prime radical	C₂×#41×C₈₃	C₄
cosocle	C₂×C₄×C₈₃	C₂⁴
hypercenter	C₄×#41	D₈₃
hypocenter	C₈₃	#2970
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #19127832	C₂	semidirect
2. #19127773	C₂	semidirect
3. D₄×C₈₃⋊C₈	C₂	semidirect
4. #19127758	C₂	semidirect
5. #19127758	C₂	semidirect
6. C₄×#13050574	C₂	semidirect
7. C₂×#14581703	C₂	semidirect
64. D₄×C₈₃	M₁₆	semidirect
82. C₈₃⋊C₈	#41	semidirect
105. C₈₃	#2970	semidirect
169. D₄	#13050574	semidirect

Maximal subgroups

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

subgroup	index
#19127832	2
#19127773	2
D₄×C₈₃⋊C₈	2
#19127758	2
C₄×#13050574	2
C₂×#14581703	2
C₄×#41×C₈₃	2
#2970	83