Finite Group Databasebeta

← #24912634

Subgroups — #24912634

Order 10624.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.

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

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581791C211yesmaximal, characteristic, complemented
2C2×D4×Dic83C211yesmaximal, characteristic
3#19127613C214yesmaximal, complemented
4#19127604C211yesmaximal, characteristic
5#19127567C211yesmaximal, characteristic, complemented
6#19127566C211yesmaximal, characteristic, complemented
7#19127559C214yesmaximal, complemented
8C4×Dic83⋊C4C211yesmaximal, characteristic, complemented
9#390×C83C211yesmaximal, characteristic
10C22×Dic166C2212yes—
11C23×Dic83C2212yes—
12#14581791—24no—
13D4×Dic83—24no—
14C2×C4⋊Dic83C2211yescharacteristic
15C2×C4⋊Dic83C2211yescharacteristic
16C2×Dic83⋊C4C2212yes—
17C2×Dic83⋊C4C2212yes—
18C2×C22⋊Dic83C2212yes—
19C2×C22⋊Dic83C2214yes—
20C2×C4×Dic83C2211yescharacteristic
21C2×C4×Dic83C2211yescharacteristic
22C83⋊#95C2212yes—
23C83⋊#95C2214yes—
24C83⋊#95C2214yes—
25C22×D4×C83C2211yescharacteristic
26C2×C4⋊C4×C83C2211yescharacteristic
27C2×C22⋊C4×C83C2214yes—
28C2×C42×C83C2211yescharacteristic
29C2×Dic166—24no—
30C2×Dic166—44no—
31C22×Dic83—24no—
32C22×Dic83—28no—
33C22×Dic83C2312yes—
34C22×Dic83C2312yes—
35C22×Dic83C2312yes—
36C22×Dic83C2311yescharacteristic
37C22×Dic83C2311yescharacteristic
38C4⋊Dic83—22no—
39C4⋊Dic83—22no—
40Dic83⋊C4—24no—
41Dic83⋊C4—24no—
42C22⋊Dic83—24no—
43C22⋊Dic83—44no—
44C4×Dic83—22no—
45C4×Dic83D414yes—
46C24×C83C2312yes—
47C2×D4×C83—22no—
48C2×D4×C83—22no—
49C2×D4×C83Q814yescomplemented
50C22×C4×C83C2312yes—
51C22×C4×C83C2312yes—
52C22×C4×C83C2311yescharacteristic
53C4⋊C4×C83—22no—
54C22⋊C4×C83—44no—
55C42×C83—22no—
56C23×C83—22no—
57C23×C83—22no—
58C23×C83—24no—
59C23×C83—24no—
60C23×C83C2×Q814yes—
61C23×C83C2411yescharacteristic
62D4×C83—28no—
63C2×C4×C83—22no—
64C2×C4×C83—22no—
65C2×C4×C83—22no—
66C2×C4×C83—22no—
67C2×C4×C83—24no—
68C2×C4×C83C2×D412yes—
69C2×C4×C83C2×Q812yes—
70C2×C4×C83#4112yes—
71C2×Dic83—22no—
72C2×Dic83—22no—
73C2×Dic83—22no—
74C2×Dic83—22no—
75C2×Dic83—22no—
76C2×Dic83—24no—
77C2×Dic83—22no—
78C2×Dic83—24no—
79C2×Dic83—48no—
80C2×Dic83C2×D412yes—
81C2×Dic83C2×D412yes—
82C2×Dic83#4114yes—
83Dic166—48no—
84C22×C83—24no—
85C22×C83—28no—
86C22×C83—42no—
87C22×C83—42no—
88C22×C83—44no—
89C22×C83C22×D411yescharacteristic
90C22×C83C22×Q811yescharacteristic
91C22×C83C2×#4112yes—
92C22×C83C2×#4111yescharacteristic
93C22×C83H(5,𝔽2)12yes—
94C4×C83—42no—
95C4×C83—42no—
96C4×C83C22⋊Q814yes—
97Dic83—22no—
98Dic83—42no—
99Dic83—42no—
100Dic83—42no—
101Dic83#12414yes—
102C2×C83—44no—
103C2×C83C2×C22⋊Q811yescharacteristic
104C2×C83C2×#12411yescharacteristic
105C2×C83#53511yescharacteristic
106C2×C83#55412yes—
107C2×C83#55512yes—
108#2563—831nomaximal, Sylow
109C83#256311yescharacteristic, complemented, Sylow
110C2×C4⋊Q8—831no—
111C2×C4×D4—831no—
112#393—834no—
113#390—831no—
114#390D8311yescharacteristic
115#382—831no—
116#382—831no—
117#380—834no—
118C4×C4⋊C4—831no—
119C22×Q8—832no—
120C22×D4C2×D8311yescharacteristic
121C23×C4—832no—
122C4⋊Q8—1664no—
123C4×D4—1664no—
124C2×C4⋊C4—831no—
125C2×C4⋊C4—832no—
126C2×C4⋊C4—831no—
127C2×C4⋊C4—832no—
128C2×C4⋊C4C2×D8311yescharacteristic
129C2×C22⋊C4—834no—
130C2×C22⋊C4—832no—
131C2×C22⋊C4C2×D8314yes—
132C2×C42—831no—
133C2×C42—831no—
134C2×C42C2×D8311yescharacteristic
135#95—832no—
136#95—834no—
137#95—834no—
138C24C22×D8312yes—
139C2×Q8—3324no—
140C2×Q8—1664no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4Dic16614yescomplemented
144C22×C4—831no—
145C22×C4—831no—
146C22×C4—832no—
147C22×C4—832no—
148C22×C4—832no—
149C22×C4—1664no—
150C22×C4—1668no—
151C22×C4C22×D8312yes—
152C22×C4C22×D8312yes—
153C22×C4C22×D8311yescharacteristic
154C4⋊C4—22no—
155C4⋊C4—1664no—
156C4⋊C4—1664no—
157C4⋊C4—1662no—
158C4⋊C4—1662no—
159C22⋊C4—44no—
160C22⋊C4—3324no—
161C22⋊C4—1664no—
162C42—22no—
163C42—834no—
164C42—1662no—
165C23—22no—
166C23—24no—
167C23—22no—
168C23—24no—
169C23C2×Dic16614yes—
170C23C23×D8311yescharacteristic
171Q8—3328no—
172D4—28no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—3328no—
179C2×C4—832no—
180C2×C4—834no—
181C2×C4—832no—
182C2×C4—1662no—
183C2×C4—1662no—
184C2×C4—1662no—
185C2×C4—1664no—
186C2×C4—1662no—
187C2×C4—1662no—
188C2×C4—1664no—
189C2×C4—1662no—
190C2×C4D4×D8312yes—
191C2×C4C2×Dic16612yes—
192C2×C4#1305058912yes—
193C22—28no—
194C22—24no—
195C22—44no—
196C22—42no—
197C22—42no—
198C22C2×D4×D8311yescharacteristic
199C22C22×Dic16611yescharacteristic
200C22#41×D8312yes—
201C22C2×#1305058911yescharacteristic
202C22#1458183712yes—
203C4—42no—
204C4—42no—
205C4—3322no—
206C4—3322no—
207C4—3322no—
208C4—834no—
209C4—1662no—
210C4#1458176914yes—
211C2—44no—
212C2C2×#1458176911yescharacteristic
213C2#124×D8311yescharacteristic
214C2#1912826411yescharacteristic
215C2#1912841012yes—
216C2#1912841512yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#390×C83C2
prime radicalC22×D4×C83C22
hypercenter#390D83
hypocenterC83#2563
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#14581791C2semidirect
3. #19127613C2semidirect
5. #19127567C2semidirect
6. #19127566C2semidirect
7. #19127559C2semidirect
8. C4×Dic83⋊C4C2semidirect
49. C2×D4×C83Q8semidirect
109. C83#2563semidirect
143. C2×D4Dic166semidirect

Maximal subgroups

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

subgroupindex
C2×#145817912
C2×D4×Dic832
#191276132
#191276042
#191275672
#191275662
#191275592
C4×Dic83⋊C42
#390×C832
#256383