Finite Group Databasebeta

← #24913837

Subgroups — #24913837

Order 10624.

592 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14581770C212yesmaximal, complemented
2C2×#14581768C212yesmaximal, complemented
3#19127625C212yesmaximal, complemented
4#19127623C212yesmaximal, complemented
5#19127621C211yesmaximal, characteristic, complemented
6#19127602C211yesmaximal, characteristic
7#19127598C212yesmaximal, complemented
8#19127579C212yesmaximal
9#395×C83C211yesmaximal, characteristic, complemented
10C22×Dic166C2211yescharacteristic
11C22×D332C2211yescharacteristic
12C22×C4×D83C2212yes—
13#14581770—28no—
14#14581768—28no—
15C2×C4⋊Dic83C2212yes—
16C2×Dic83⋊C4C2212yes—
17C2×Dic83⋊C4C2212yes—
18C2×Dic83⋊C4C2212yes—
19C2×Dic83⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2212yes—
23C2×(C2×D83)⋊C4C2212yes—
24C2×(C2×D83)⋊C4C2212yes—
25C2×C4×Dic83C2211yescharacteristic
26C83⋊#95C2212yes—
27C83⋊#95C2211yescharacteristic
28C22×Q8×C83C2211yescharacteristic
29C2×C4⋊C4×C83C2212yes—
30C2×C4⋊C4×C83C2211yescharacteristic
31#95×C83C2212yes—
32#95×C83C2211yescharacteristic
33C23×D83C2312yes—
34C2×Dic166—24no—
35C2×Dic166—41no—
36C2×D332—41no—
37C2×D332—41no—
38C2×D332Q814yescomplemented
39C22×Dic83C2312yes—
40C22×Dic83C2312yes—
41C22×Dic83C2312yes—
42C2×C4×D83—24no—
43C2×C4×D83—24no—
44C2×C4×D83—42no—
45C4⋊Dic83—24no—
46Dic83⋊C4—24no—
47Dic83⋊C4—24no—
48Dic83⋊C4—42no—
49Dic83⋊C4D418yescomplemented
50(C2×D83)⋊C4—24no—
51(C2×D83)⋊C4—24no—
52(C2×D83)⋊C4—24no—
53(C2×D83)⋊C4—24no—
54(C2×D83)⋊C4—42no—
55C4×Dic83—41no—
56C2×Q8×C83—24no—
57C2×Q8×C83—41no—
58C22×C4×C83C2311yescharacteristic
59C22×C4×C83C2312yes—
60C22×C4×C83C2311yescharacteristic
61C22×C4×C83C2312yes—
62C22×C4×C83C2311yescharacteristic
63C4⋊C4×C83—24no—
64C4⋊C4×C83—41no—
65C23×C83C2411yescharacteristic
66C22×D83—22no—
67C22×D83—22no—
68C22×D83—22no—
69C22×D83—22no—
70C22×D83—22no—
71C22×D83—22no—
72C22×D83C2×Q814yes—
73Q8×C83—44no—
74C2×C4×C83—22no—
75C2×C4×C83—21no—
76C2×C4×C83—22no—
77C2×C4×C83—21no—
78C2×C4×C83—22no—
79C2×C4×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—21no—
82C2×C4×C83—21no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83C2×D414yes—
86C2×C4×C83C2×Q812yes—
87C2×Dic83—22no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83C2×D414yes—
95C2×Dic83C2×D414yes—
96D332—44no—
97C4×D83—48no—
98Dic166—44no—
99C22×C83C22×D412yes—
100C22×C83C22×Q811yescharacteristic
101C22×C83H(5,𝔽2)11yescharacteristic
102C22×C83H(5,𝔽2)11yescharacteristic
103C22×C832−1+412yes—
104C2×D83—24no—
105C2×D83—24no—
106C2×D83—24no—
107C2×D83—42no—
108C2×D83—42no—
109C2×D83—42no—
110C2×D83—42no—
111C4×C83—22no—
112C4×C83—24no—
113C4×C83—42no—
114C4×C83—41no—
115C4×C83—41no—
116Dic83—24no—
117Dic83—24no—
118Dic83—42no—
119C2×C83#53712yes—
120C2×C83#54311yescharacteristic
121C2×C83D4211yescharacteristic
122C2×C83D4×Q812yes—
123C2×C83#56311yescharacteristic
124D83—44no—
125#2683—831nomaximal, Sylow
126C83#268311yescharacteristic, complemented, Sylow
127C2×C22⋊Q8—832no—
128C2×C22⋊Q8—832no—
129#396—832no—
130#396—832no—
131#395—831no—
132#395D8311yescharacteristic, complemented
133#390—831no—
134#389—832no—
135#386—832no—
136C22×Q8—831no—
137C22×Q8C2×D8311yescharacteristic
138C22×D4—831no—
139C23×C4—832no—
140C22⋊Q8—1668no—
141C22⋊Q8—1668no—
142C2×C4⋊C4—832no—
143C2×C4⋊C4—832no—
144C2×C4⋊C4—832no—
145C2×C4⋊C4—832no—
146C2×C4⋊C4—832no—
147C2×C4⋊C4C2×D8312yes—
148C2×C4⋊C4C2×D8311yescharacteristic
149C2×C22⋊C4—832no—
150C2×C22⋊C4—832no—
151C2×C22⋊C4—832no—
152C2×C22⋊C4—832no—
153C2×C22⋊C4—832no—
154C2×C42—831no—
155#95—832no—
156#95—831no—
157#95C2×D8312yes—
158#95C2×D8311yescharacteristic
159C24—832no—
160C2×Q8—24no—
161C2×Q8—41no—
162C2×Q8—3321no—
163C2×Q8—1664no—
164C2×D4—3321no—
165C2×D4—3321no—
166C2×D4—834no—
167C22×C4—3322no—
168C22×C4—832no—
169C22×C4—832no—
170C22×C4—832no—
171C22×C4—1664no—
172C22×C4—1664no—
173C22×C4C22×D8311yescharacteristic
174C22×C4C22×D8312yes—
175C22×C4C22×D8311yescharacteristic
176C22×C4C22×D8312yes—
177C22×C4C22×D8311yescharacteristic
178C4⋊C4—24no—
179C4⋊C4—41no—
180C4⋊C4—3322no—
181C4⋊C4—838no—
182C4⋊C4—1664no—
183C4⋊C4—1664no—
184C4⋊C4—1664no—
185C22⋊C4—3322no—
186C22⋊C4—1664no—
187C22⋊C4—1664no—
188C22⋊C4—1664no—
189C22⋊C4—1664no—
190C42—3321no—
191C23—834no—
192C23—1662no—
193C23—1662no—
194C23—1662no—
195C23—1662no—
196C23—1662no—
197C23—1662no—
198C23C23×D8311yescharacteristic
199Q8—44no—
200Q8—3324no—
201D4—3324no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—3328no—
214C2×C4—834no—
215C2×C4—834no—
216C2×C4—1662no—
217C2×C4—1662no—
218C2×C4—1662no—
219C2×C4—1662no—
220C2×C4—1662no—
221C2×C4—1662no—
222C2×C4—1662no—
223C2×C4D4×D8314yes—
224C2×C4Q8×D8312yes—
225C22—3322no—
226C22—3322no—
227C22—3322no—
228C22—3322no—
229C22—1664no—
230C22—1664no—
231C22—1664no—
232C22C2×D4×D8312yes—
233C22C2×Q8×D8311yescharacteristic
234C22#1458183711yescharacteristic
235C22#1458183811yescharacteristic
236C22#1458184012yes—
237C4—22no—
238C4—24no—
239C4—41no—
240C4—41no—
241C4—42no—
242C4—3322no—
243C4—1664no—
244C4—1664no—
245C2—3324no—
246C2#1912828112yes—
247C2#1912832611yescharacteristic
248C2#1912833311yescharacteristic
249C2#1912836612yes—
250C2#1912845011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#395×C83C2
prime radicalC22×D332C22
hypercenter#395D83
hypocenterC83#2683
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×#14581770C2semidirect
2. C2×#14581768C2semidirect
3. #19127625C2semidirect
4. #19127623C2semidirect
5. #19127621C2semidirect
7. #19127598C2semidirect
9. #395×C83C2semidirect
38. C2×D332Q8semidirect
49. Dic83⋊C4D4semidirect
126. C83#2683semidirect
132. #395D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817702
C2×#145817682
#191276252
#191276232
#191276212
#191276022
#191275982
#191275792
#395×C832
#268383