Subgroups — #24916457

Order 10624.

372 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128235	C₂	1	1	yes	maximal, characteristic
2	#19127829	C₂	1	2	yes	maximal, complemented
3	#19127774	C₂	1	2	yes	maximal, complemented
4	#19127772	C₂	1	2	yes	maximal, complemented
5	C₈×C₂²⋊D₈₃	C₂	1	2	yes	maximal, complemented
6	#19127763	C₂	1	2	yes	maximal, complemented
7	#19127754	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127658	C₂	1	2	yes	maximal, complemented
9	C₂×#97×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13050573	C₂²	1	2	yes	—
11	C₂×C₈×D₈₃	C₂²	1	2	yes	—
12	#14581786	C₄	1	2	yes	—
13	#14581784	C₄	1	1	yes	characteristic
14	#14581780	C₄	1	1	yes	characteristic
15	#14581775	C₄	1	2	yes	—
16	#14581771	C₄	1	1	yes	characteristic
17	#14581766	C₄	1	1	yes	characteristic
18	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
19	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
20	C₄×C₂²⋊D₈₃	C₂²	1	2	yes	—
21	#14581749	C₂²	1	2	yes	—
22	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
23	Dic₈₃⋊C₈	C₂²	1	2	yes	—
24	Dic₈₃⋊C₈	C₂²	1	2	yes	—
25	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
26	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
27	(C₂×D₈₃)⋊C₈	C₂²	1	2	yes	—
28	(C₂×D₈₃)⋊C₈	C₂²	1	2	yes	—
29	#14581702	C₂²	1	2	yes	—
30	C₄×C₈₃⋊C₈	C₂²	1	2	yes	—
31	C₈×Dic₈₃	C₂²	1	2	yes	—
32	C₂²×C₈×C₈₃	C₂²	1	2	yes	—
33	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
34	#97×C₈₃	C₂²	1	2	yes	—
35	#97×C₈₃	C₂²	1	1	yes	characteristic
36	#97×C₈₃	C₂²	1	1	yes	characteristic
37	C₂×Dic₁₆₆	C₂×C₄	1	1	yes	characteristic
38	C₂×D₃₃₂	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂²⋊D₈₃	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₈₃	C₂³	1	2	yes	—
41	#13050573	—	2	4	no	—
42	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
43	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
44	C₈×D₈₃	—	2	4	no	—
45	C₄⋊Dic₈₃	C₂×C₄	1	1	yes	characteristic
46	C₄⋊Dic₈₃	C₂×C₄	1	1	yes	characteristic
47	Dic₈₃⋊C₄	C₂×C₄	1	2	yes	—
48	Dic₈₃⋊C₄	C₂×C₄	1	2	yes	—
49	Dic₈₃⋊C₄	C₂×C₄	1	2	yes	—
50	Dic₈₃⋊C₄	C₂×C₄	1	2	yes	—
51	C₂²⋊Dic₈₃	C₂×C₄	1	2	yes	—
52	(C₂×D₈₃)⋊C₄	C₂×C₄	1	2	yes	—
53	(C₂×D₈₃)⋊C₄	C₂×C₄	1	2	yes	—
54	(C₂×D₈₃)⋊C₄	C₂×C₄	1	2	yes	—
55	(C₂×D₈₃)⋊C₄	C₂×C₄	1	2	yes	—
56	C₄×Dic₈₃	C₂³	1	2	yes	—
57	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₈₃	—	2	4	no	—
61	C₂×C₈×C₈₃	C₂³	1	2	yes	—
62	C₂×C₈×C₈₃	C₂³	1	2	yes	—
63	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
64	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
65	C₄²×C₈₃	C₂³	1	1	yes	characteristic
66	C₄²×C₈₃	C₂³	1	1	yes	characteristic
67	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₈₃	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₈₃	—	2	2	no	—
70	C₂×C₄×C₈₃	—	2	2	no	—
71	C₂×C₄×C₈₃	—	4	1	no	—
72	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
77	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
78	C₂²⋊D₈₃	—	2	4	no	—
79	C₂×Dic₈₃	C₂²×C₄	1	2	yes	—
80	C₂×Dic₈₃	C₂²×C₄	1	2	yes	—
81	C₂×Dic₈₃	C₂²×C₄	1	2	yes	—
82	D₃₃₂	—	4	1	no	—
83	C₄×D₈₃	—	2	2	no	—
84	C₄×D₈₃	—	2	2	no	—
85	Dic₁₆₆	—	4	1	no	—
86	C₈×C₈₃	—	2	2	no	—
87	C₈×C₈₃	#41	1	4	yes	—
88	C₈₃⋊C₈	—	2	2	no	—
89	C₈₃⋊C₈	—	2	2	no	—
90	C₂²×C₈₃	—	2	1	no	—
91	C₂²×C₈₃	—	2	1	no	—
92	C₂²×C₈₃	#131	1	2	yes	—
93	C₂²×C₈₃	C₂³×C₄	1	1	yes	characteristic
94	C₂×D₈₃	—	2	2	no	—
95	C₂×D₈₃	—	2	2	no	—
96	C₂×D₈₃	—	2	2	no	—
97	C₄×C₈₃	—	2	1	no	—
98	C₄×C₈₃	—	2	1	no	—
99	C₄×C₈₃	—	2	1	no	—
100	C₄×C₈₃	—	2	1	no	—
101	C₄×C₈₃	—	2	1	no	—
102	C₄×C₈₃	C₂×#41	1	2	yes	—
103	Dic₈₃	—	2	2	no	—
104	Dic₈₃	—	2	2	no	—
105	Dic₈₃	—	2	2	no	—
106	C₂×C₈₃	—	2	2	no	—
107	C₂×C₈₃	C₄×#41	1	1	yes	characteristic
108	C₂×C₈₃	C₂×#131	1	1	yes	characteristic
109	C₂×C₈₃	#568	1	1	yes	characteristic
110	D₈₃	—	4	2	no	—
111	#2968	—	83	1	no	maximal, Sylow
112	C₈₃	#2968	1	1	yes	characteristic, complemented, Sylow
113	#529	—	83	1	no	—
114	#446	—	83	2	no	—
115	#435	—	83	2	no	—
116	#435	—	83	2	no	—
117	C₈×D₄	—	83	2	no	—
118	#433	—	83	2	no	—
119	#431	—	83	1	no	—
120	#405	—	83	2	no	—
121	C₂×#97	D₈₃	1	1	yes	characteristic, complemented
122	C₂×M₁₆	—	83	2	no	—
123	C₂²×C₈	—	83	2	no	—
124	C₂²×C₈	C₂×D₈₃	1	2	yes	—
125	#126	—	83	2	no	—
126	#125	—	83	1	no	—
127	#124	—	83	1	no	—
128	#123	—	83	2	no	—
129	C₂²⋊Q₈	—	83	1	no	—
130	#121	—	83	1	no	—
131	C₄×Q₈	—	83	1	no	—
132	C₄×D₄	—	83	2	no	—
133	C₄×D₄	—	83	1	no	—
134	#117	—	83	2	no	—
135	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
136	C₄⋊C₈	—	83	2	no	—
137	C₄⋊C₈	—	83	2	no	—
138	C₄⋊C₈	—	83	2	no	—
139	C₂²⋊C₈	—	83	1	no	—
140	C₂²⋊C₈	—	83	1	no	—
141	C₂²⋊C₈	—	83	2	no	—
142	C₂²⋊C₈	—	83	2	no	—
143	#97	—	83	2	no	—
144	#97	C₂×D₈₃	1	1	yes	characteristic
145	#97	C₂×D₈₃	1	1	yes	characteristic
146	#97	C₂×D₈₃	1	2	yes	—
147	C₄×C₈	—	83	2	no	—
148	C₄×C₈	—	83	2	no	—
149	C₂×Q₈	—	83	1	no	—
150	C₂×D₄	—	83	2	no	—
151	C₂×D₄	—	83	1	no	—
152	C₂²×C₄	—	83	2	no	—
153	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
154	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
155	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
156	M₁₆	—	166	4	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	83	2	no	—
159	C₂×C₈	—	83	2	no	—
160	C₂×C₈	—	166	4	no	—
161	C₂×C₈	C₂²×D₈₃	1	2	yes	—
162	C₂×C₈	C₂²×D₈₃	1	2	yes	—
163	C₄⋊C₄	—	83	2	no	—
164	C₄⋊C₄	—	83	2	no	—
165	C₄⋊C₄	—	83	1	no	—
166	C₄⋊C₄	—	83	2	no	—
167	C₄⋊C₄	—	83	1	no	—
168	C₄⋊C₄	—	83	2	no	—
169	C₂²⋊C₄	—	83	2	no	—
170	C₂²⋊C₄	—	83	2	no	—
171	C₂²⋊C₄	—	83	2	no	—
172	C₂²⋊C₄	—	83	2	no	—
173	C₂²⋊C₄	—	83	2	no	—
174	C₄²	—	83	2	no	—
175	C₄²	C₄×D₈₃	1	1	yes	characteristic
176	C₄²	C₄×D₈₃	1	1	yes	characteristic
177	C₄²	C₂²×D₈₃	1	1	yes	characteristic
178	C₄²	C₂²×D₈₃	1	1	yes	characteristic
179	C₂³	—	83	2	no	—
180	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
181	Q₈	—	332	1	no	—
182	D₄	—	332	1	no	—
183	D₄	—	166	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	83	2	no	—
188	C₂×C₄	—	83	2	no	—
189	C₂×C₄	—	83	2	no	—
190	C₂×C₄	—	166	2	no	—
191	C₂×C₄	—	166	2	no	—
192	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
195	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
197	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	—	166	2	no	—
200	C₈	—	166	2	no	—
201	C₈	#13050590	1	4	yes	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	166	2	no	—
205	C₂²	—	166	2	no	—
206	C₂²	—	166	2	no	—
207	C₂²	#14581800	1	2	yes	—
208	C₂²	C₂²×C₄×D₈₃	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	166	2	no	—
215	C₄	—	166	2	no	—
216	C₄	—	166	2	no	—
217	C₄	C₂×#13050590	1	2	yes	—
218	C₂	—	2	2	no	—
219	C₂	—	332	2	no	—
220	C₂	C₄×#13050590	1	1	yes	characteristic
221	C₂	C₂×#14581800	1	1	yes	characteristic
222	C₂	#19128467	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₃
commutator, socle	C₂²×C₈₃	C₂³×C₄
Fitting	C₂×#97×C₈₃	C₂
prime radical	#14581766	C₄
cosocle	C₂×C₄×C₈₃	C₂⁴
hypercenter	C₂×#97	D₈₃
hypocenter	C₈₃	#2968
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. #19127829	C₂	semidirect
3. #19127774	C₂	semidirect
4. #19127772	C₂	semidirect
5. C₈×C₂²⋊D₈₃	C₂	semidirect
6. #19127763	C₂	semidirect
7. #19127754	C₂	semidirect
8. #19127658	C₂	semidirect
9. C₂×#97×C₈₃	C₂	semidirect
112. C₈₃	#2968	semidirect
121. C₂×#97	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128235	2
#19127829	2
#19127774	2
#19127772	2
C₈×C₂²⋊D₈₃	2
#19127763	2
#19127754	2
#19127658	2
C₂×#97×C₈₃	2
#2968	83