Finite Group Databasebeta

← #24912367

Subgroups — #24912367

Order 10624.

604 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 151 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×#14581769C212yesmaximal, complemented
2C2×#14581757C211yesmaximal, characteristic, complemented
3C2×D4×Dic83C211yesmaximal, characteristic
4#19127633C211yesmaximal, characteristic, complemented
5#19127631C211yesmaximal, characteristic, complemented
6#19127615C211yesmaximal, characteristic
7#19127595C212yesmaximal, complemented
8#19127594C212yesmaximal, complemented
9#19127565C211yesmaximal, characteristic, complemented
10#19127559C212yesmaximal, complemented
11#394×C83C211yesmaximal, characteristic
12C22×Dic166C2211yescharacteristic
13C23×Dic83C2212yes—
14#14581769—28no—
15#14581757—24no—
16D4×Dic83—24no—
17C2×C4⋊Dic83C2212yes—
18C2×C4⋊Dic83C2211yescharacteristic
19C2×Dic83⋊C4C2211yescharacteristic
20C2×Dic83⋊C4C2212yes—
21C2×Dic83⋊C4C2212yes—
22C2×Dic83⋊C4C2211yescharacteristic
23C2×C22⋊Dic83C2212yes—
24C2×C22⋊Dic83C2212yes—
25C2×C22⋊Dic83C2212yes—
26C2×C4×Dic83C2211yescharacteristic
27C2×C4×Dic83C2211yescharacteristic
28C2×C4×Dic83C2211yescharacteristic
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C83⋊#95C2211yescharacteristic
33C22×D4×C83C2211yescharacteristic
34C2×C4⋊C4×C83C2211yescharacteristic
35C2×C22⋊C4×C83C2212yes—
36C2×C22⋊C4×C83C2212yes—
37#95×C83C2211yescharacteristic
38C2×Dic166—22no—
39C2×Dic166—22no—
40C2×Dic166D414yescomplemented
41C22×Dic83—24no—
42C22×Dic83—24no—
43C22×Dic83—24no—
44C22×Dic83C2312yes—
45C22×Dic83C2312yes—
46C22×Dic83C2311yescharacteristic
47C22×Dic83C2311yescharacteristic
48C22×Dic83C2311yescharacteristic
49C22×Dic83C2311yescharacteristic
50C4⋊Dic83—22no—
51C4⋊Dic83—24no—
52Dic83⋊C4—24no—
53Dic83⋊C4—22no—
54Dic83⋊C4—24no—
55Dic83⋊C4—22no—
56C22⋊Dic83—24no—
57C22⋊Dic83—24no—
58C22⋊Dic83—24no—
59C4×Dic83—22no—
60C4×Dic83—22no—
61C4×Dic83D414yes—
62C24×C83C2312yes—
63C2×D4×C83—24no—
64C2×D4×C83—22no—
65C22×C4×C83C2312yes—
66C22×C4×C83C2311yescharacteristic
67C22×C4×C83C2311yescharacteristic
68C22×C4×C83C2311yescharacteristic
69C4⋊C4×C83—22no—
70C22⋊C4×C83—24no—
71C22⋊C4×C83—24no—
72C23×C83—22no—
73C23×C83—22no—
74C23×C83—22no—
75C23×C83—22no—
76C23×C83—22no—
77C23×C83—22no—
78C23×C83#4114yes—
79C23×C83C2411yescharacteristic
80D4×C83—28no—
81C2×C4×C83—21no—
82C2×C4×C83—22no—
83C2×C4×C83—21no—
84C2×C4×C83—22no—
85C2×C4×C83—21no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83C2×D412yes—
90C2×C4×C83C2×D412yes—
91C2×Dic83—21no—
92C2×Dic83—21no—
93C2×Dic83—22no—
94C2×Dic83—21no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83—22no—
102C2×Dic83—48no—
103C2×Dic83C2×D412yes—
104C2×Dic83C2×D412yes—
105C2×Dic83C2×D412yes—
106C2×Dic83C2×D412yes—
107C2×Dic83#4112yes—
108Dic166—28no—
109C22×C83—24no—
110C22×C83—24no—
111C22×C83—24no—
112C22×C83—42no—
113C22×C83—42no—
114C22×C83—42no—
115C22×C83—42no—
116C22×C83C22×D411yescharacteristic
117C22×C83C22×D411yescharacteristic
118C22×C83C2×#4111yescharacteristic
119C22×C83C2×#4112yes—
120C22×C832−1+412yes—
121C4×C83—22no—
122C4×C83—22no—
123C4×C83—41no—
124C4×C83—42no—
125Dic83—22no—
126Dic83—22no—
127Dic83—42no—
128Dic83—41no—
129Dic83—42no—
130Dic83#12114yes—
131C2×C83—44no—
132C2×C83C2×#12111yescharacteristic
133C2×C83#52511yescharacteristic
134C2×C83#53611yescharacteristic
135C2×C83#54812yes—
136C2×C83#55212yes—
137#2536—831nomaximal, Sylow
138C83#253611yescharacteristic, complemented, Sylow
139C2×C22⋊Q8—832no—
140C2×C4×Q8—831no—
141C2×C4×D4—831no—
142#398—831no—
143#398—831no—
144#394—831no—
145#394D8311yescharacteristic
146#388—832no—
147#388—832no—
148#382—831no—
149#380—832no—
150C22×Q8—831no—
151C22×D4C2×D8311yescharacteristic
152C23×C4—832no—
153C22⋊Q8—1668no—
154C4×Q8—1664no—
155C4×D4—1664no—
156C2×C4⋊C4—831no—
157C2×C4⋊C4—831no—
158C2×C4⋊C4—831no—
159C2×C4⋊C4—832no—
160C2×C4⋊C4—832no—
161C2×C4⋊C4—832no—
162C2×C4⋊C4C2×D8311yescharacteristic
163C2×C22⋊C4—832no—
164C2×C22⋊C4—832no—
165C2×C22⋊C4—832no—
166C2×C22⋊C4C2×D8312yes—
167C2×C22⋊C4C2×D8312yes—
168C2×C42—831no—
169C2×C42—831no—
170C2×C42—831no—
171#95—832no—
172#95—832no—
173#95—832no—
174#95—831no—
175#95C2×D8311yescharacteristic
176C24C22×D8312yes—
177C2×Q8—834no—
178C2×Q8—1662no—
179C2×Q8—1662no—
180C2×D4—24no—
181C2×D4—22no—
182C22×C4—831no—
183C22×C4—831no—
184C22×C4—831no—
185C22×C4—832no—
186C22×C4—831no—
187C22×C4—832no—
188C22×C4—1664no—
189C22×C4—1664no—
190C22×C4—1664no—
191C22×C4C22×D8312yes—
192C22×C4C22×D8311yescharacteristic
193C22×C4C22×D8311yescharacteristic
194C22×C4C22×D8311yescharacteristic
195C4⋊C4—22no—
196C4⋊C4—1662no—
197C4⋊C4—1664no—
198C4⋊C4—1664no—
199C4⋊C4—1662no—
200C4⋊C4—1662no—
201C4⋊C4—1664no—
202C22⋊C4—24no—
203C22⋊C4—24no—
204C22⋊C4—1664no—
205C22⋊C4—1664no—
206C22⋊C4—1664no—
207C42—834no—
208C42—1662no—
209C42—1662no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23#1305058914yes—
217C23C23×D8311yescharacteristic
218Q8—1668no—
219D4—28no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—3328no—
229C2×C4—832no—
230C2×C4—832no—
231C2×C4—832no—
232C2×C4—832no—
233C2×C4—832no—
234C2×C4—1661no—
235C2×C4—1662no—
236C2×C4—1662no—
237C2×C4—1662no—
238C2×C4—1662no—
239C2×C4—1662no—
240C2×C4—1662no—
241C2×C4—1662no—
242C2×C4—1661no—
243C2×C4—1661no—
244C2×C4—1662no—
245C2×C4D4×D8312yes—
246C2×C4D4×D8312yes—
247C22—24no—
248C22—24no—
249C22—24no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22C2×D4×D8311yescharacteristic
255C22C2×D4×D8311yescharacteristic
256C22#41×D8311yescharacteristic
257C22C2×#1305058912yes—
258C22#1458183912yes—
259C4—22no—
260C4—22no—
261C4—42no—
262C4—41no—
263C4—3322no—
264C4—3322no—
265C4—3321no—
266C4—834no—
267C4—1662no—
268C4—1662no—
269C2—44no—
270C2#121×D8311yescharacteristic
271C2#1912820711yescharacteristic
272C2#1912827111yescharacteristic
273C2#1912835712yes—
274C2#1912838812yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#394×C83C2
prime radicalC22×D4×C83C22
hypercenter#394D83
hypocenterC83#2536
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#14581769C2semidirect
2. C2×#14581757C2semidirect
4. #19127633C2semidirect
5. #19127631C2semidirect
7. #19127595C2semidirect
8. #19127594C2semidirect
9. #19127565C2semidirect
10. #19127559C2semidirect
40. C2×Dic166D4semidirect
138. C83#2536semidirect

Maximal subgroups

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

subgroupindex
C2×#145817692
C2×#145817572
C2×D4×Dic832
#191276332
#191276312
#191276152
#191275952
#191275942
#191275652
#191275592
#394×C832
#253683