Subgroups — #24919866

Order 10624.

366 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128397	C₂	1	2	yes	maximal, complemented
2	#19128103	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128026	C₂	1	2	yes	maximal, complemented
4	#19128020	C₂	1	2	yes	maximal, complemented
5	#19128002	C₂	1	2	yes	maximal, complemented
6	#19127937	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127937	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₆₆₄	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127739	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14581726	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₆₆₄	C₂²	1	1	yes	characteristic
13	#14581787	—	2	2	no	—
14	#14581784	C₂²	1	2	yes	—
15	#14581774	—	2	2	no	—
16	#14581766	C₂²	1	2	yes	—
17	#14581766	C₂²	1	2	yes	—
18	#14581764	—	2	2	no	—
19	C₄×D₃₃₂	C₂²	1	2	yes	—
20	(C₂²⋊D₈₃)⋊C₄	—	2	2	no	—
21	#14581751	C₂²	1	2	yes	—
22	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
23	#14581726	C₂²	1	1	yes	characteristic
24	#14581726	C₂²	1	1	yes	characteristic
25	#14581726	C₂²	1	1	yes	characteristic
26	#14581726	C₂²	1	2	yes	—
27	#14581724	C₂²	1	2	yes	—
28	#14581724	C₂²	1	2	yes	—
29	#14581714	C₂²	1	2	yes	—
30	#14581714	C₂²	1	2	yes	—
31	#14581714	C₂²	1	2	yes	—
32	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
33	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
34	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
36	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
37	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
39	C₂×D₃₃₂	C₂³	1	2	yes	—
40	C₂×C₂²⋊D₈₃	—	2	2	no	—
41	C₂×C₂²⋊D₈₃	—	2	2	no	—
42	C₂²×Dic₈₃	—	2	2	no	—
43	C₂×C₄×D₈₃	—	2	2	no	—
44	D₆₆₄	—	2	2	no	—
45	C₄⋊Dic₈₃	—	2	2	no	—
46	C₄⋊Dic₈₃	C₂³	1	2	yes	—
47	C₄⋊Dic₈₃	C₂³	1	2	yes	—
48	C₄⋊Dic₈₃	C₂³	1	2	yes	—
49	Dic₈₃⋊C₄	—	2	2	no	—
50	Dic₈₃⋊C₄	—	2	2	no	—
51	C₂²⋊Dic₈₃	—	2	2	no	—
52	(C₂×D₈₃)⋊C₄	—	2	2	no	—
53	(C₂×D₈₃)⋊C₄	—	2	2	no	—
54	(C₂×D₈₃)⋊C₄	—	2	2	no	—
55	C₄×Dic₈₃	—	2	2	no	—
56	C₂×D₄×C₈₃	D₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₈₃	—	2	2	no	—
60	C₂×C₈×C₈₃	—	2	2	no	—
61	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
66	C₂²⋊C₄×C₈₃	D₄	1	1	yes	characteristic
67	C₂²⋊C₄×C₈₃	D₄	1	1	yes	characteristic
68	C₄²×C₈₃	C₂³	1	1	yes	characteristic
69	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
70	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×D₈₃	—	2	2	no	—
72	D₄×C₈₃	—	2	2	no	—
73	C₂×C₄×C₈₃	—	2	1	no	—
74	C₂×C₄×C₈₃	—	2	1	no	—
75	C₂×C₄×C₈₃	—	2	1	no	—
76	C₂×C₄×C₈₃	—	2	1	no	—
77	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
82	C₂²⋊D₈₃	—	4	4	no	—
83	C₂²⋊D₈₃	—	8	2	no	—
84	C₂×Dic₈₃	—	2	2	no	—
85	C₂×Dic₈₃	—	2	2	no	—
86	C₂×Dic₈₃	—	2	2	no	—
87	C₂×Dic₈₃	—	8	2	no	—
88	D₃₃₂	—	2	2	no	—
89	D₃₃₂	—	2	2	no	—
90	C₄×D₈₃	—	8	2	no	—
91	C₈×C₈₃	—	2	1	no	—
92	C₈×C₈₃	—	2	1	no	—
93	C₈×C₈₃	—	2	1	no	—
94	C₈×C₈₃	#41	1	2	yes	—
95	C₂²×C₈₃	—	2	1	no	—
96	C₂²×C₈₃	—	2	1	no	—
97	C₂²×C₈₃	—	2	1	no	—
98	C₂²×C₈₃	—	2	1	no	—
99	C₂²×C₈₃	—	2	1	no	—
100	C₂²×C₈₃	#135	1	2	yes	—
101	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
102	C₂×D₈₃	—	4	2	no	—
103	C₂×D₈₃	—	4	2	no	—
104	C₂×D₈₃	—	4	2	no	—
105	C₄×C₈₃	—	2	1	no	—
106	C₄×C₈₃	—	2	1	no	—
107	C₄×C₈₃	—	2	1	no	—
108	C₄×C₈₃	—	2	1	no	—
109	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
110	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
111	Dic₈₃	—	4	2	no	—
112	Dic₈₃	—	4	2	no	—
113	Dic₈₃	—	4	2	no	—
114	C₂×C₈₃	—	2	2	no	—
115	C₂×C₈₃	—	4	1	no	—
116	C₂×C₈₃	#547	1	1	yes	characteristic
117	C₂×C₈₃	C₂×#135	1	1	yes	characteristic
118	C₂×C₈₃	#576	1	1	yes	characteristic
119	D₈₃	—	8	2	no	—
120	#3339	—	83	1	no	maximal, Sylow
121	C₈₃	#3339	1	1	yes	characteristic, complemented, Sylow
122	#553	—	83	2	no	—
123	#499	—	83	1	no	—
124	#482	—	83	2	no	—
125	#481	—	83	2	no	—
126	#478	—	83	2	no	—
127	#466	—	83	1	no	—
128	#466	—	83	1	no	—
129	C₄×D₈	—	83	1	no	—
130	C₈×D₄	D₈₃	1	1	yes	characteristic, complemented
131	#427	—	83	1	no	—
132	C₂×#107	—	83	1	no	—
133	C₂×D₈	—	83	1	no	—
134	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
135	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
136	#126	—	166	2	no	—
137	#125	—	83	2	no	—
138	#123	—	166	2	no	—
139	#121	—	83	2	no	—
140	#121	—	83	2	no	—
141	#121	—	166	2	no	—
142	C₄×D₄	—	83	2	no	—
143	C₄×D₄	—	166	2	no	—
144	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
145	#117	—	83	2	no	—
146	C₂×C₄⋊C₄	—	83	2	no	—
147	#107	—	83	1	no	—
148	#107	—	83	1	no	—
149	#107	—	83	2	no	—
150	#107	—	83	1	no	—
151	#106	—	83	2	no	—
152	#106	—	83	2	no	—
153	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
154	#102	—	83	2	no	—
155	#102	—	83	2	no	—
156	#102	—	83	2	no	—
157	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
158	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
159	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
160	C₂×D₄	—	83	2	no	—
161	C₂×D₄	—	166	2	no	—
162	C₂×D₄	—	166	2	no	—
163	C₂×D₄	D₃₃₂	1	1	yes	characteristic
164	C₂²×C₄	—	166	2	no	—
165	C₂²×C₄	—	166	2	no	—
166	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
167	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
168	D₈	—	166	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
174	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
175	C₄⋊C₄	—	83	2	no	—
176	C₄⋊C₄	—	83	2	no	—
177	C₄⋊C₄	—	83	2	no	—
178	C₄⋊C₄	—	166	2	no	—
179	C₄⋊C₄	—	166	2	no	—
180	C₄⋊C₄	—	166	2	no	—
181	C₄⋊C₄	D₃₃₂	1	1	yes	characteristic
182	C₂²⋊C₄	—	166	2	no	—
183	C₂²⋊C₄	—	166	2	no	—
184	C₂²⋊C₄	—	166	2	no	—
185	C₂²⋊C₄	—	166	2	no	—
186	C₂²⋊C₄	D₃₃₂	1	1	yes	characteristic
187	C₂²⋊C₄	D₃₃₂	1	1	yes	characteristic
188	C₄²	—	166	2	no	—
189	C₄²	C₂²×D₈₃	1	1	yes	characteristic
190	C₂³	—	166	2	no	—
191	C₂³	C₂×D₃₃₂	1	1	yes	characteristic
192	C₂³	C₂×D₃₃₂	1	1	yes	characteristic
193	D₄	—	2	2	no	—
194	D₄	—	332	4	no	—
195	D₄	—	664	2	no	—
196	D₄	—	166	2	no	—
197	D₄	—	166	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	664	2	no	—
203	C₂×C₄	—	664	2	no	—
204	C₂×C₄	—	166	2	no	—
205	C₂×C₄	—	166	2	no	—
206	C₂×C₄	—	166	2	no	—
207	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
208	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
209	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
210	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
211	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
212	C₈	—	2	1	no	—
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	#13050589	1	2	yes	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	332	2	no	—
222	C₂²	—	332	2	no	—
223	C₂²	—	332	2	no	—
224	C₂²	#14581814	1	2	yes	—
225	C₂²	C₂²×D₃₃₂	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	332	2	no	—
231	C₄	—	332	2	no	—
232	C₄	—	332	2	no	—
233	C₄	C₂×#13050589	1	1	yes	characteristic
234	C₄	#14581839	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	—	4	1	no	—
237	C₂	—	664	2	no	—
238	C₂	#19128355	1	1	yes	characteristic
239	C₂	C₂×#14581814	1	1	yes	characteristic
240	C₂	#19128506	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₃₂
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	C₈×D₄×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
hypercenter	C₈×D₄	D₈₃
hypocenter	C₈₃	#3339
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #19128397	C₂	semidirect
2. #19128103	C₂	semidirect
3. #19128026	C₂	semidirect
4. #19128020	C₂	semidirect
5. #19128002	C₂	semidirect
6. #19127937	C₂	semidirect
7. #19127937	C₂	semidirect
8. C₄×D₆₆₄	C₂	semidirect
9. #19127739	C₂	semidirect
10. C₂×#14581726	C₂	semidirect
11. C₈×D₄×C₈₃	C₂	semidirect
121. C₈₃	#3339	semidirect
130. C₈×D₄	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128397	2
#19128103	2
#19128026	2
#19128020	2
#19128002	2
#19127937	2
C₄×D₆₆₄	2
#19127739	2
C₂×#14581726	2
C₈×D₄×C₈₃	2
#3339	83