Finite Group Databasebeta

← #24915611

Subgroups — #24915611

Order 10624.

408 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#19127631C211yesmaximal, characteristic, complemented
2#19127630C212yesmaximal, complemented
3#19127627C212yesmaximal, complemented
4#19127623C212yesmaximal, complemented
5#19127619C212yesmaximal
6#19127612C212yesmaximal, complemented
7#19127602C211yesmaximal, characteristic
8#19127583C212yesmaximal, complemented
9#395×C83C211yesmaximal, characteristic, complemented
10C22×D332C2211yescharacteristic
11C2×C4⋊Dic83C2211yescharacteristic
12C2×Dic83⋊C4C2212yes—
13C2×Dic83⋊C4C2212yes—
14C2×(C2×D83)⋊C4C2212yes—
15C2×(C2×D83)⋊C4C2212yes—
16C2×(C2×D83)⋊C4C2212yes—
17C2×(C2×D83)⋊C4C2212yes—
18C2×(C2×D83)⋊C4C2212yes—
19C2×(C2×D83)⋊C4C2212yes—
20C2×C4×Dic83C2211yescharacteristic
21C83⋊#95C2212yes—
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2211yescharacteristic
26C22×Q8×C83C2211yescharacteristic
27C2×C4⋊C4×C83C2211yescharacteristic
28C2×C4⋊C4×C83C2212yes—
29#95×C83C2212yes—
30#95×C83C2211yescharacteristic
31C23×D83C2312yes—
32C2×D332—22no—
33C2×D332—41no—
34C2×D332—41no—
35C22×Dic83C2312yes—
36C22×Dic83C2312yes—
37C22×Dic83C2312yes—
38C4⋊Dic83—41no—
39Dic83⋊C4—24no—
40Dic83⋊C4—42no—
41(C2×D83)⋊C4—42no—
42(C2×D83)⋊C4—42no—
43(C2×D83)⋊C4—42no—
44(C2×D83)⋊C4—42no—
45(C2×D83)⋊C4—42no—
46(C2×D83)⋊C4—42no—
47C4×Dic83—41no—
48C2×Q8×C83—41no—
49C2×Q8×C83—42no—
50C22×C4×C83C2312yes—
51C22×C4×C83C2312yes—
52C22×C4×C83C2311yescharacteristic
53C22×C4×C83C2311yescharacteristic
54C22×C4×C83C2311yescharacteristic
55C4⋊C4×C83—41no—
56C4⋊C4×C83—42no—
57C23×C83C2411yescharacteristic
58C22×D83—22no—
59C22×D83—22no—
60C22×D83—22no—
61C22×D83—22no—
62C22×D83—22no—
63C22×D83—22no—
64C22×D83—22no—
65Q8×C83—44no—
66C2×C4×C83—21no—
67C2×C4×C83—21no—
68C2×C4×C83—21no—
69C2×C4×C83—22no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83—21no—
73C2×C4×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83#4112yes—
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83—22no—
82C2×Dic83—22no—
83C2×Dic83—22no—
84C2×Dic83—22no—
85C2×Dic83—22no—
86C2×Dic83—22no—
87C2×Dic83—22no—
88D332—44no—
89C22×C83C2×#4111yescharacteristic
90C22×C83H(5,𝔽2)11yescharacteristic
91C22×C83H(5,𝔽2)12yes—
92C22×C83H(5,𝔽2)11yescharacteristic
93C22×C832−1+412yes—
94C2×D83—42no—
95C2×D83—42no—
96C2×D83—42no—
97C2×D83—42no—
98C2×D83—42no—
99C2×D83—42no—
100C2×D83—42no—
101C4×C83—22no—
102C4×C83—41no—
103C4×C83—41no—
104C4×C83—42no—
105C4×C83—42no—
106Dic83—42no—
107Dic83—42no—
108Dic83—42no—
109C2×C83#53811yescharacteristic
110C2×C83#54212yes—
111C2×C83#56011yescharacteristic
112C2×C83#56212yes—
113C2×C83#56311yescharacteristic
114D83—82no—
115#2830—831nomaximal, Sylow
116C83#283011yescharacteristic, complemented, Sylow
117#398—831no—
118#397—832no—
119#397—832no—
120#396—832no—
121#395D8311yescharacteristic, complemented
122#394—832no—
123#393—832no—
124#390—831no—
125#387—832no—
126C22×Q8C2×D8311yescharacteristic
127C22×D4—831no—
128C2×C4⋊C4—832no—
129C2×C4⋊C4—831no—
130C2×C4⋊C4—832no—
131C2×C4⋊C4C2×D8311yescharacteristic
132C2×C4⋊C4C2×D8312yes—
133C2×C22⋊C4—832no—
134C2×C22⋊C4—832no—
135C2×C22⋊C4—832no—
136C2×C22⋊C4—832no—
137C2×C22⋊C4—832no—
138C2×C22⋊C4—832no—
139C2×C42—831no—
140#95—832no—
141#95—832no—
142#95—832no—
143#95—832no—
144#95—831no—
145#95C2×D8312yes—
146#95C2×D8311yescharacteristic
147C24—832no—
148C2×Q8—42no—
149C2×Q8—41no—
150C2×D4—3321no—
151C2×D4—3321no—
152C2×D4—1662no—
153C22×C4—832no—
154C22×C4—832no—
155C22×C4—832no—
156C22×C4C22×D8312yes—
157C22×C4C22×D8311yescharacteristic
158C22×C4C22×D8312yes—
159C22×C4C22×D8311yescharacteristic
160C22×C4C22×D8311yescharacteristic
161C4⋊C4—41no—
162C4⋊C4—42no—
163C4⋊C4—3322no—
164C4⋊C4—3321no—
165C4⋊C4—1664no—
166C22⋊C4—3322no—
167C22⋊C4—3322no—
168C22⋊C4—3322no—
169C22⋊C4—3322no—
170C22⋊C4—3322no—
171C22⋊C4—3322no—
172C42—3321no—
173C23—1662no—
174C23—1662no—
175C23—1662no—
176C23—1662no—
177C23—1662no—
178C23—1662no—
179C23—1662no—
180C23C23×D8311yescharacteristic
181Q8—44no—
182D4—3324no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—1662no—
196C2×C4—1662no—
197C2×C4—1662no—
198C2×C4—1662no—
199C2×C4—1662no—
200C2×C4—1662no—
201C2×C4—1662no—
202C2×C4—1662no—
203C2×C4—1662no—
204C2×C4#1305059112yes—
205C22—3322no—
206C22—3322no—
207C22—3322no—
208C22—3322no—
209C22—3322no—
210C22—3322no—
211C22—3322no—
212C22C2×#1305059111yescharacteristic
213C22#1458183712yes—
214C22#1458183711yescharacteristic
215C22#1458183811yescharacteristic
216C22#1458184012yes—
217C4—22no—
218C4—41no—
219C4—42no—
220C4—41no—
221C4—42no—
222C4—3322no—
223C4—3322no—
224C4—3322no—
225C2—6642no—
226C2#1912829111yescharacteristic
227C2#1912832512yes—
228C2#1912844011yescharacteristic
229C2#1912844712yes—
230C2#1912845011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#395×C83C2
prime radicalC22×D332C22
hypercenter#395D83
hypocenterC83#2830
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. #19127631C2semidirect
2. #19127630C2semidirect
3. #19127627C2semidirect
4. #19127623C2semidirect
6. #19127612C2semidirect
8. #19127583C2semidirect
9. #395×C83C2semidirect
116. C83#2830semidirect
121. #395D83semidirect

Maximal subgroups

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

subgroupindex
#191276312
#191276302
#191276272
#191276232
#191276192
#191276122
#191276022
#191275832
#395×C832
#283083