Subgroups — #24916405

Order 10624.

356 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128255	C₂	1	1	yes	maximal, characteristic
2	#19127832	C₂	1	1	yes	maximal, characteristic, complemented
3	#19127831	C₂	1	1	yes	maximal, characteristic, complemented
4	#19127831	C₂	1	2	yes	maximal, complemented
5	#19127830	C₂	1	2	yes	maximal, complemented
6	#19127825	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127726	C₂	1	2	yes	maximal
8	C₄⋊C₈×D₈₃	C₂	1	1	yes	maximal, characteristic
9	#19127655	C₂	1	2	yes	maximal
10	C₄×#13050573	C₂	1	1	yes	maximal, characteristic
11	C₈×Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13050573	C₂²	1	1	yes	characteristic
13	C₂×#13050573	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₈₃	C₂²	1	2	yes	—
15	#14581791	C₄	1	1	yes	characteristic
16	#14581790	C₄	1	1	yes	characteristic
17	#14581782	C₄	1	2	yes	—
18	#14581772	C₄	1	1	yes	characteristic
19	#14581770	C₄	1	2	yes	—
20	#14581768	C₄	1	1	yes	characteristic
21	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
22	C₄×Dic₁₆₆	C₂²	1	2	yes	—
23	#14581749	C₂²	1	2	yes	—
24	C₄²×D₈₃	C₂²	1	1	yes	characteristic
25	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
27	Dic₈₃⋊C₈	C₂²	1	2	yes	—
28	Dic₈₃⋊C₈	C₂²	1	2	yes	—
29	Dic₈₃⋊C₈	C₂²	1	2	yes	—
30	#14581703	C₂²	1	2	yes	—
31	#14581702	C₂²	1	1	yes	characteristic
32	#14581702	C₂²	1	1	yes	characteristic
33	#14581702	C₂²	1	2	yes	—
34	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
35	C₈×Dic₈₃	C₂²	1	2	yes	—
36	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
39	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₈₃	C₂²	1	2	yes	—
41	C₂×Dic₁₆₆	C₂×C₄	1	1	yes	characteristic
42	C₂×Dic₁₆₆	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₈₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₈₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
46	#13050573	—	2	2	no	—
47	#13050573	Q₈	1	4	yes	complemented
48	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
49	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
50	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
51	C₈×D₈₃	—	2	4	no	—
52	C₄⋊Dic₈₃	C₂×C₄	1	1	yes	characteristic
53	C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
54	Dic₈₃⋊C₄	C₂×C₄	1	2	yes	—
55	Dic₈₃⋊C₄	C₂×C₄	1	2	yes	—
56	Dic₈₃⋊C₄	C₂×C₄	1	2	yes	—
57	Dic₈₃⋊C₄	C₂×C₄	1	2	yes	—
58	Dic₈₃⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₈₃)⋊C₄	C₂×C₄	1	2	yes	—
60	(C₂×D₈₃)⋊C₄	C₂×C₄	1	2	yes	—
61	C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₈₃	C₂³	1	2	yes	—
64	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
65	C₂×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₈×C₈₃	C₂³	1	2	yes	—
67	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
70	C₄⋊C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₈₃	C₂³	1	1	yes	characteristic
72	C₄²×C₈₃	C₂³	1	2	yes	—
73	C₂²×D₈₃	C₂²×C₄	1	1	yes	characteristic
74	Q₈×C₈₃	—	2	2	no	—
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	C₂×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₈₃	C₂²×C₄	1	2	yes	—
83	C₂×Dic₈₃	C₂²×C₄	1	2	yes	—
84	C₂×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
85	C₄×D₈₃	—	2	2	no	—
86	C₄×D₈₃	—	4	1	no	—
87	C₄×D₈₃	C₂×Q₈	1	2	yes	—
88	C₄×D₈₃	#41	1	2	yes	—
89	Dic₁₆₆	—	2	2	no	—
90	Dic₁₆₆	—	4	2	no	—
91	C₈×C₈₃	—	2	2	no	—
92	C₈×C₈₃	—	2	1	no	—
93	C₈×C₈₃	C₂×Q₈	1	2	yes	—
94	C₈₃⋊C₈	—	2	2	no	—
95	C₈₃⋊C₈	—	2	1	no	—
96	C₈₃⋊C₈	C₂×Q₈	1	2	yes	—
97	C₂²×C₈₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₈₃	—	2	1	no	—
99	C₂×D₈₃	—	2	1	no	—
100	C₂×D₈₃	C₄×Q₈	1	2	yes	—
101	C₄×C₈₃	—	2	2	no	—
102	C₄×C₈₃	—	2	2	no	—
103	C₄×C₈₃	—	2	1	no	—
104	C₄×C₈₃	#131	1	2	yes	—
105	C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
106	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
107	Dic₈₃	—	2	2	no	—
108	Dic₈₃	—	2	1	no	—
109	Dic₈₃	—	2	1	no	—
110	Dic₈₃	—	2	2	no	—
111	Dic₈₃	C₄×Q₈	1	2	yes	—
112	C₂×C₈₃	C₂×C₄×Q₈	1	1	yes	characteristic
113	C₂×C₈₃	C₂×#131	1	1	yes	characteristic
114	C₂×C₈₃	#568	1	1	yes	characteristic
115	D₈₃	—	2	2	no	—
116	#2962	—	83	1	no	maximal, Sylow
117	C₈₃	#2962	1	1	yes	characteristic, complemented, Sylow
118	#533	—	83	1	no	—
119	#446	—	83	1	no	—
120	#446	—	83	2	no	—
121	#446	—	83	1	no	—
122	#446	—	83	2	no	—
123	C₈×Q₈	—	83	1	no	—
124	C₈×Q₈	D₈₃	1	1	yes	characteristic, complemented
125	#424	—	83	2	no	—
126	C₂×C₄⋊C₈	—	83	1	no	—
127	#405	—	83	2	no	—
128	C₄×M₁₆	—	83	1	no	—
129	C₂×M₁₆	—	83	1	no	—
130	C₂×M₁₆	—	83	1	no	—
131	C₂²×C₈	—	83	2	no	—
132	C₄⋊Q₈	—	83	1	no	—
133	C₄⋊Q₈	—	83	1	no	—
134	#125	—	83	2	no	—
135	C₂²⋊Q₈	—	83	1	no	—
136	C₂²⋊Q₈	—	83	2	no	—
137	C₂²⋊Q₈	—	83	1	no	—
138	C₄×Q₈	—	83	1	no	—
139	C₄×Q₈	—	83	2	no	—
140	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
141	#117	—	83	2	no	—
142	C₂×C₄²	—	83	1	no	—
143	C₄⋊C₈	—	83	2	no	—
144	C₄⋊C₈	—	83	2	no	—
145	C₄⋊C₈	—	83	2	no	—
146	C₄⋊C₈	—	83	1	no	—
147	C₄⋊C₈	—	83	2	no	—
148	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
149	C₄⋊C₈	C₂×D₈₃	1	2	yes	—
150	#97	—	83	1	no	—
151	#97	—	83	1	no	—
152	#97	—	83	2	no	—
153	#97	—	83	2	no	—
154	C₄×C₈	—	83	2	no	—
155	C₄×C₈	—	83	1	no	—
156	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
157	C₄×C₈	C₂×D₈₃	1	2	yes	—
158	C₂×Q₈	—	83	2	no	—
159	C₂×Q₈	—	83	1	no	—
160	C₂×Q₈	C₄×D₈₃	1	1	yes	characteristic
161	C₂²×C₄	—	83	1	no	—
162	C₂²×C₄	—	83	1	no	—
163	C₂²×C₄	—	83	1	no	—
164	M₁₆	—	83	4	no	—
165	M₁₆	—	166	2	no	—
166	C₂×C₈	—	83	1	no	—
167	C₂×C₈	—	83	1	no	—
168	C₂×C₈	—	83	2	no	—
169	C₂×C₈	—	166	4	no	—
170	C₂×C₈	C₂²×D₈₃	1	2	yes	—
171	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
173	C₄⋊C₄	—	83	2	no	—
174	C₄⋊C₄	—	83	2	no	—
175	C₄⋊C₄	—	83	2	no	—
176	C₄⋊C₄	—	83	2	no	—
177	C₄⋊C₄	—	83	2	no	—
178	C₄⋊C₄	—	83	2	no	—
179	C₄⋊C₄	—	83	1	no	—
180	C₄⋊C₄	C₄×D₈₃	1	2	yes	—
181	C₄⋊C₄	C₄×D₈₃	1	1	yes	characteristic
182	C₂²⋊C₄	—	83	2	no	—
183	C₂²⋊C₄	—	83	2	no	—
184	C₄²	—	83	1	no	—
185	C₄²	—	83	1	no	—
186	C₄²	—	83	2	no	—
187	C₄²	—	83	1	no	—
188	C₄²	C₂²×D₈₃	1	1	yes	characteristic
189	C₄²	C₂²×D₈₃	1	2	yes	—
190	C₂³	—	83	1	no	—
191	Q₈	—	2	2	no	—
192	Q₈	—	332	2	no	—
193	Q₈	—	166	2	no	—
194	C₂×C₄	—	332	1	no	—
195	C₂×C₄	—	83	1	no	—
196	C₂×C₄	—	83	2	no	—
197	C₂×C₄	—	83	2	no	—
198	C₂×C₄	—	83	1	no	—
199	C₂×C₄	—	83	1	no	—
200	C₂×C₄	—	83	2	no	—
201	C₂×C₄	—	83	2	no	—
202	C₂×C₄	—	166	2	no	—
203	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×D₈₃	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₈₃	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
207	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	1	no	—
210	C₈	—	83	2	no	—
211	C₈	—	166	2	no	—
212	C₈	—	166	1	no	—
213	C₈	Q₈×D₈₃	1	2	yes	—
214	C₂²	—	83	2	no	—
215	C₂²	—	166	1	no	—
216	C₂²	—	166	1	no	—
217	C₂²	C₂²×C₄×D₈₃	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	2	no	—
221	C₄	—	83	2	no	—
222	C₄	—	166	1	no	—
223	C₄	—	166	2	no	—
224	C₄	—	166	1	no	—
225	C₄	—	166	2	no	—
226	C₄	#14581798	1	2	yes	—
227	C₄	C₂×Q₈×D₈₃	1	1	yes	characteristic
228	C₄	#41×D₈₃	1	1	yes	characteristic
229	C₂	—	166	2	no	—
230	C₂	C₄×Q₈×D₈₃	1	1	yes	characteristic
231	C₂	C₂×#14581798	1	1	yes	characteristic
232	C₂	#19128466	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₃
commutator, socle	C₂²×C₈₃	C₂³×C₄
Fitting	C₈×Q₈×C₈₃	C₂
prime radical	C₂²×D₈₃	C₂²×C₄
cosocle	C₂×C₄×C₈₃	C₂⁴
hypercenter	C₈×Q₈	D₈₃
hypocenter	C₈₃	#2962
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. #19127832	C₂	semidirect
3. #19127831	C₂	semidirect
4. #19127831	C₂	semidirect
5. #19127830	C₂	semidirect
6. #19127825	C₂	semidirect
11. C₈×Q₈×C₈₃	C₂	semidirect
47. #13050573	Q₈	semidirect
117. C₈₃	#2962	semidirect
124. C₈×Q₈	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128255	2
#19127832	2
#19127831	2
#19127830	2
#19127825	2
#19127726	2
C₄⋊C₈×D₈₃	2
#19127655	2
C₄×#13050573	2
C₈×Q₈×C₈₃	2
#2962	83