Finite Group Databasebeta

Subgroups — #24909136

Order 10624.

238 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128271	C₂	1	1	yes	maximal, characteristic
2	#19127673	C₂	1	2	yes	maximal
3	#19127496	C₂	1	1	yes	maximal, characteristic, complemented
4	#19127495	C₂	1	2	yes	maximal, complemented
5	#354×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13050589	C₂²	1	1	yes	characteristic
7	C₂×Q₈×D₈₃	C₄	1	1	yes	characteristic, complemented
8	#14581792	C₄	1	1	yes	characteristic, complemented
9	#14581790	C₄	1	1	yes	characteristic, complemented
10	#14581779	C₂²	1	1	yes	characteristic
11	#14581774	—	2	2	no	—
12	#14581769	—	2	2	no	—
13	#14581749	C₄	1	1	yes	characteristic, complemented
14	#14581748	—	2	2	no	—
15	#14581707	C₂²	1	2	yes	—
16	#14581706	—	2	2	no	—
17	#124×C₈₃	C₂²	1	1	yes	characteristic
18	#99×C₈₃	C₂²	1	2	yes	—
19	#13050589	—	4	2	no	—
20	C₂×Dic₁₆₆	—	2	1	no	—
21	C₂×Dic₁₆₆	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₁₆₆	C₂×C₄	1	1	yes	characteristic
23	Q₈×D₈₃	—	4	2	no	—
24	C₂×C₂²⋊D₈₃	D₄	1	2	yes	—
25	C₂²×Dic₈₃	D₄	1	2	yes	—
26	C₂×C₄×D₈₃	—	2	1	no	—
27	C₂×C₄×D₈₃	C₂×C₄	1	1	yes	characteristic
28	C₄⋊Dic₈₃	—	2	2	no	—
29	Dic₈₃⋊C₄	—	2	1	no	—
30	Dic₈₃⋊C₄	—	2	2	no	—
31	Dic₈₃⋊C₄	—	4	2	no	—
32	C₂²⋊Dic₈₃	—	2	2	no	—
33	C₂²⋊Dic₈₃	—	2	2	no	—
34	(C₂×D₈₃)⋊C₄	—	2	1	no	—
35	C₄×Dic₈₃	—	4	2	no	—
36	C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
37	C₂×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
38	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
39	C₂²⋊C₄×C₈₃	—	2	2	no	—
40	C₂²⋊C₄×C₈₃	—	2	2	no	—
41	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₈₃	C₂×D₄	1	2	yes	—
43	C₂²×D₈₃	C₂²⋊C₄	1	1	yes	characteristic
44	Q₈×C₈₃	—	4	1	no	—
45	D₄×C₈₃	—	4	1	no	—
46	C₂×C₄×C₈₃	—	2	1	no	—
47	C₂×C₄×C₈₃	—	2	1	no	—
48	C₂×C₄×C₈₃	—	4	2	no	—
49	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²⋊D₈₃	—	4	2	no	—
51	C₂×Dic₈₃	—	2	2	no	—
52	C₂×Dic₈₃	—	2	2	no	—
53	C₂×Dic₈₃	—	2	1	no	—
54	C₂×Dic₈₃	—	2	1	no	—
55	C₂×Dic₈₃	—	4	2	no	—
56	C₂×Dic₈₃	C₂²⋊C₄	1	2	yes	—
57	C₂×Dic₈₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₄×D₈₃	—	2	2	no	—
59	C₄×D₈₃	—	4	2	no	—
60	Dic₁₆₆	—	4	1	no	—
61	Dic₁₆₆	—	4	2	no	—
62	Dic₁₆₆	—	4	1	no	—
63	C₂²×C₈₃	—	2	2	no	—
64	C₂²×C₈₃	—	4	2	no	—
65	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×D₈₃	—	2	1	no	—
67	C₂×D₈₃	—	4	1	no	—
68	C₄×C₈₃	—	2	1	no	—
69	C₄×C₈₃	—	4	1	no	—
70	C₄×C₈₃	—	4	1	no	—
71	C₄×C₈₃	—	8	2	no	—
72	Dic₈₃	—	2	2	no	—
73	Dic₈₃	—	4	1	no	—
74	Dic₈₃	—	4	1	no	—
75	Dic₈₃	—	8	2	no	—
76	Dic₈₃	#99	1	2	yes	—
77	C₂×C₈₃	—	2	1	no	—
78	C₂×C₈₃	—	4	2	no	—
79	C₂×C₈₃	C₂×#99	1	1	yes	characteristic
80	D₈₃	—	4	1	no	—
81	#2129	—	83	1	no	maximal, Sylow
82	C₈₃	#2129	1	1	yes	characteristic, complemented, Sylow
83	#536	—	83	1	no	—
84	#410	—	83	2	no	—
85	#354	—	83	1	no	—
86	#354	—	83	2	no	—
87	#354	D₈₃	1	1	yes	characteristic, complemented
88	C₂×#41	—	83	1	no	—
89	C₂²×Q₈	—	83	1	no	—
90	C₄⋊Q₈	—	83	1	no	—
91	C₄⋊Q₈	—	83	1	no	—
92	#124	—	83	1	no	—
93	#124	C₂×D₈₃	1	1	yes	characteristic
94	#123	—	166	2	no	—
95	C₂²⋊Q₈	—	166	2	no	—
96	#117	—	83	1	no	—
97	#117	—	166	2	no	—
98	#99	—	83	2	no	—
99	#99	—	166	2	no	—
100	#99	C₂×D₈₃	1	2	yes	—
101	#41	—	332	2	no	—
102	C₂×Q₈	—	332	2	no	—
103	C₂×Q₈	—	83	1	no	—
104	C₂×Q₈	—	83	1	no	—
105	C₂×Q₈	—	166	1	no	—
106	C₂×Q₈	C₄×D₈₃	1	1	yes	characteristic
107	C₂×D₄	—	83	2	no	—
108	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
109	C₂²×C₄	—	83	2	no	—
110	C₂²×C₄	—	83	1	no	—
111	C₂²×C₄	—	166	1	no	—
112	C₄⋊C₄	—	332	2	no	—
113	C₄⋊C₄	—	166	1	no	—
114	C₄⋊C₄	—	166	2	no	—
115	C₄⋊C₄	—	166	2	no	—
116	C₂²⋊C₄	—	2	2	no	—
117	C₂²⋊C₄	—	2	2	no	—
118	C₂²⋊C₄	—	166	1	no	—
119	C₂²⋊C₄	—	166	2	no	—
120	C₂²⋊C₄	—	166	2	no	—
121	C₄²	—	332	2	no	—
122	C₄²	—	83	1	no	—
123	C₄²	C₄×D₈₃	1	1	yes	characteristic
124	C₂³	—	83	1	no	—
125	C₂³	D₄×D₈₃	1	2	yes	—
126	Q₈	—	4	1	no	—
127	Q₈	—	332	1	no	—
128	Q₈	—	332	1	no	—
129	Q₈	—	332	2	no	—
130	D₄	—	4	1	no	—
131	D₄	—	332	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	332	2	no	—
136	C₂×C₄	—	332	2	no	—
137	C₂×C₄	—	83	1	no	—
138	C₂×C₄	—	83	2	no	—
139	C₂×C₄	—	166	2	no	—
140	C₂×C₄	—	166	1	no	—
141	C₂×C₄	—	166	1	no	—
142	C₂×C₄	—	166	2	no	—
143	C₂×C₄	—	166	2	no	—
144	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
145	C₂²	—	2	2	no	—
146	C₂²	—	4	2	no	—
147	C₂²	—	332	1	no	—
148	C₂²	—	166	1	no	—
149	C₂²	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	4	1	no	—
152	C₄	—	4	1	no	—
153	C₄	—	8	2	no	—
154	C₄	—	332	1	no	—
155	C₄	—	332	1	no	—
156	C₄	—	83	2	no	—
157	C₄	—	664	2	no	—
158	C₄	—	166	2	no	—
159	C₂	—	2	1	no	—
160	C₂	—	4	2	no	—
161	C₂	—	332	1	no	—
162	C₂	#99×D₈₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#99×D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₄
Frattini	C₂×D₄	C₂²×D₈₃
Fitting	#354×C₈₃	C₂
socle	C₂×C₈₃	C₂×#99
prime radical	C₂×#13050589	C₂²
cosocle	C₂×D₄×C₈₃	C₂³
hypercenter	#354	D₈₃
hypocenter	C₈₃	#2129
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #354

Complements

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

N	H ≅ G/N	type
3. #19127496	C₂	semidirect
4. #19127495	C₂	semidirect
5. #354×C₈₃	C₂	semidirect
7. C₂×Q₈×D₈₃	C₄	semidirect
8. #14581792	C₄	semidirect
9. #14581790	C₄	semidirect
13. #14581749	C₄	semidirect
82. C₈₃	#2129	semidirect
87. #354	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128271	2
#19127673	2
#19127496	2
#19127495	2
#354×C₈₃	2
#2129	83