Finite Group Databasebeta

← #24915406

Subgroups — #24915406

Order 10624.

480 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 137 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#19127643C212yesmaximal, complemented
2#19127613C212yesmaximal, complemented
3#19127613C212yesmaximal, complemented
4#19127605C212yesmaximal, complemented
5#19127604C211yesmaximal, characteristic
6#19127581C211yesmaximal, characteristic
7#19127574C212yesmaximal, complemented
8#19127573C212yesmaximal, complemented
9#386×C83C211yesmaximal, characteristic
10C22×Dic166C2212yes—
11C23×Dic83C2211yescharacteristic
12C2×C4⋊Dic83C2211yescharacteristic
13C2×Dic83⋊C4C2212yes—
14C2×Dic83⋊C4C2212yes—
15C2×C22⋊Dic83C2212yes—
16C2×C22⋊Dic83C2212yes—
17C2×C22⋊Dic83C2212yes—
18C2×C22⋊Dic83C2211yescharacteristic
19C2×C4×Dic83C2211yescharacteristic
20C83⋊#95C2212yes—
21C83⋊#95C2212yes—
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2212yes—
26C22×D4×C83C2211yescharacteristic
27C23×C4×C83C2211yescharacteristic
28C2×C22⋊C4×C83C2212yes—
29C2×C22⋊C4×C83C2211yescharacteristic
30#95×C83C2212yes—
31C2×Dic166—24no—
32C2×Dic166—42no—
33C2×Dic166—42no—
34C22×Dic83—42no—
35C22×Dic83—41no—
36C22×Dic83C2312yes—
37C22×Dic83C2312yes—
38C22×Dic83C2312yes—
39C22×Dic83C2312yes—
40C4⋊Dic83—41no—
41Dic83⋊C4—42no—
42Dic83⋊C4—42no—
43C22⋊Dic83—24no—
44C22⋊Dic83—24no—
45C22⋊Dic83—42no—
46C22⋊Dic83—41no—
47C4×Dic83—41no—
48C24×C83C2311yescharacteristic
49C24×C83C2311yescharacteristic
50C2×D4×C83—41no—
51C2×D4×C83—41no—
52C2×D4×C83Q814yescomplemented
53C22×C4×C83—42no—
54C22×C4×C83—41no—
55C22×C4×C83C2312yes—
56C22×C4×C83C2312yes—
57C22×C4×C83C2311yescharacteristic
58C22⋊C4×C83—42no—
59C22⋊C4×C83—41no—
60C23×C83—22no—
61C23×C83—22no—
62C23×C83—21no—
63C23×C83—22no—
64C23×C83—21no—
65C23×C83—22no—
66C23×C83—21no—
67C23×C83—21no—
68C23×C83C2×Q812yes—
69C23×C83C2×Q812yes—
70C23×C83C2411yescharacteristic
71D4×C83—44no—
72C2×C4×C83—21no—
73C2×C4×C83—21no—
74C2×C4×C83—22no—
75C2×C4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83—22no—
79C2×C4×C83—22no—
80C2×C4×C83—44no—
81C2×C4×C83C2×Q812yes—
82C2×Dic83—22no—
83C2×Dic83—22no—
84C2×Dic83—22no—
85C2×Dic83—22no—
86C2×Dic83—22no—
87C2×Dic83—22no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—44no—
94C2×Dic83#4114yes—
95Dic166—48no—
96C22×C83—22no—
97C22×C83—22no—
98C22×C83—24no—
99C22×C83—24no—
100C22×C83—42no—
101C22×C83—42no—
102C22×C83—41no—
103C22×C83—41no—
104C22×C83—41no—
105C22×C83—41no—
106C22×C83C22×Q811yescharacteristic
107C22×C83C2×#4112yes—
108C22×C83H(5,𝔽2)11yescharacteristic
109C22×C83H(5,𝔽2)12yes—
110C22×C83H(5,𝔽2)11yescharacteristic
111C4×C83—22no—
112C4×C83—42no—
113C4×C83—42no—
114Dic83—24no—
115Dic83—42no—
116Dic83—42no—
117Dic83—42no—
118C2×C83—42no—
119C2×C83—42no—
120C2×C83#53812yes—
121C2×C83#54311yescharacteristic
122C2×C83#55412yes—
123C2×C83#55911yescharacteristic
124C2×C83H(3,𝔽4)11yescharacteristic
125#2804—831nomaximal, Sylow
126C83#280411yescharacteristic, complemented, Sylow
127#400—832no—
128#393—832no—
129#393—832no—
130#391—832no—
131#390—831no—
132#386—831no—
133#386D8311yescharacteristic
134#385—832no—
135#385—832no—
136C22×Q8—832no—
137C22×D4C2×D8311yescharacteristic
138C23×C4—831no—
139C23×C4C2×D8311yescharacteristic
140C2×C4⋊C4—832no—
141C2×C4⋊C4—832no—
142C2×C4⋊C4—831no—
143C2×C22⋊C4—832no—
144C2×C22⋊C4—831no—
145C2×C22⋊C4—832no—
146C2×C22⋊C4—832no—
147C2×C22⋊C4C2×D8312yes—
148C2×C22⋊C4C2×D8311yescharacteristic
149C2×C42—831no—
150#95—832no—
151#95—832no—
152#95—832no—
153#95—832no—
154#95—832no—
155#95—832no—
156#95C2×D8312yes—
157C24C22×D8311yescharacteristic
158C24C22×D8311yescharacteristic
159C2×Q8—3322no—
160C2×Q8—3322no—
161C2×Q8—1664no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4Dic16614yescomplemented
165C22×C4—42no—
166C22×C4—41no—
167C22×C4—3322no—
168C22×C4—3321no—
169C22×C4—832no—
170C22×C4—832no—
171C22×C4—832no—
172C22×C4—832no—
173C22×C4C22×D8312yes—
174C22×C4C22×D8312yes—
175C22×C4C22×D8311yescharacteristic
176C4⋊C4—3322no—
177C4⋊C4—3322no—
178C4⋊C4—3321no—
179C22⋊C4—42no—
180C22⋊C4—41no—
181C22⋊C4—3322no—
182C22⋊C4—3321no—
183C22⋊C4—1664no—
184C22⋊C4—1664no—
185C42—3321no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—21no—
190C23—22no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23C2×Dic16612yes—
195C23C2×Dic16612yes—
196C23C23×D8311yescharacteristic
197Q8—3328no—
198D4—44no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—44no—
208C2×C4—3324no—
209C2×C4—834no—
210C2×C4—1662no—
211C2×C4—1662no—
212C2×C4—1662no—
213C2×C4—1662no—
214C2×C4—1662no—
215C2×C4—1662no—
216C2×C4—1662no—
217C2×C4—1662no—
218C2×C4—1662no—
219C2×C4—1662no—
220C2×C4—1662no—
221C2×C4C2×Dic16612yes—
222C22—22no—
223C22—22no—
224C22—24no—
225C22—24no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—42no—
232C22C22×Dic16611yescharacteristic
233C22#41×D8312yes—
234C22#1458183711yescharacteristic
235C22#1458183711yescharacteristic
236C22#1458183712yes—
237C4—22no—
238C4—42no—
239C4—42no—
240C4—3322no—
241C4—3322no—
242C4—3322no—
243C4—1664no—
244C2—42no—
245C2—42no—
246C2#1912828312yes—
247C2#1912832711yescharacteristic
248C2#1912841012yes—
249C2#1912843511yescharacteristic
250C2#1912844411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#386×C83C2
prime radicalC22×D4×C83C22
hypercenter#386D83
hypocenterC83#2804
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. #19127643C2semidirect
2. #19127613C2semidirect
3. #19127613C2semidirect
4. #19127605C2semidirect
7. #19127574C2semidirect
8. #19127573C2semidirect
52. C2×D4×C83Q8semidirect
126. C83#2804semidirect
164. C2×D4Dic166semidirect

Maximal subgroups

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

subgroupindex
#191276432
#191276132
#191276052
#191276042
#191275812
#191275742
#191275732
#386×C832
#280483