Finite Group Databasebeta

← #27174414

Subgroups — #27174414

Order 12928.

568 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 147 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×Q8×Dic101C211yesmaximal, characteristic, complemented
2C2×#15692215C211yesmaximal, characteristic
3#20265119C212yesmaximal, complemented
4#20265104C211yesmaximal, characteristic, complemented
5#20265102C211yesmaximal, characteristic, complemented
6#20265093C212yesmaximal
7#20265088C211yesmaximal, characteristic, complemented
8C22⋊C4×Dic101C212yesmaximal, complemented
9#20265024C212yesmaximal, complemented
10#20265022C211yesmaximal, characteristic, complemented
11#393×C101C211yesmaximal, characteristic, complemented
12C22×C22⋊D101C2211yescharacteristic
13C23×Dic101C2211yescharacteristic
14C22×C4×D101C2211yescharacteristic
15Q8×Dic101—24no—
16#15692215—24no—
17C2×C4⋊Dic101C2212yes—
18C2×C4⋊Dic101C2211yescharacteristic
19C2×#13108053C2211yescharacteristic
20C2×C22⋊Dic101C2212yes—
21C2×#13108051C2212yes—
22C2×#13108051C2212yes—
23C2×#13108051C2211yescharacteristic
24C2×C4×Dic101C2212yes—
25C2×C4×Dic101C2212yes—
26C2×C4×Dic101C2211yescharacteristic
27#15692162C2211yescharacteristic
28#15692162C2212yes—
29#15692162C2212yes—
30#15692162C2212yes—
31#15692162C2212yes—
32C22×Q8×C101C2211yescharacteristic
33C2×C22⋊C4×C101C2212yes—
34C2×C22⋊C4×C101C2211yescharacteristic
35#95×C101C2211yescharacteristic
36#95×C101C2212yes—
37C23×D101C2311yescharacteristic
38C2×C22⋊D101—22no—
39C2×C22⋊D101—22no—
40C2×C22⋊D101—22no—
41C22×Dic101—22no—
42C22×Dic101—24no—
43C22×Dic101C2311yescharacteristic
44C22×Dic101C2312yes—
45C22×Dic101C2312yes—
46C22×Dic101C2311yescharacteristic
47C22×Dic101C2311yescharacteristic
48C2×C4×D101—22no—
49C2×C4×D101—24no—
50C4⋊Dic101—22no—
51C4⋊Dic101—24no—
52#13108053—22no—
53C22⋊Dic101—24no—
54#13108051—24no—
55#13108051—22no—
56#13108051—24no—
57C4×Dic101—24no—
58C4×Dic101—24no—
59C4×Dic101D414yes—
60C24×C101C2311yescharacteristic
61C2×Q8×C101—22no—
62C2×Q8×C101—24no—
63C22×C4×C101C2311yescharacteristic
64C22×C4×C101C2312yes—
65C22×C4×C101C2311yescharacteristic
66C22×C4×C101C2312yes—
67C22⋊C4×C101—22no—
68C22⋊C4×C101—24no—
69C23×C101—21no—
70C23×C101—22no—
71C23×C101—22no—
72C23×C101—21no—
73C23×C101#4112yes—
74C23×C101C2411yescharacteristic
75C22×D101—21no—
76C22×D101—22no—
77C22×D101—22no—
78C22×D101—21no—
79C22×D101#4112yes—
80Q8×C101—28no—
81C2×C4×C101—22no—
82C2×C4×C101—22no—
83C2×C4×C101—22no—
84C2×C4×C101—21no—
85C2×C4×C101—22no—
86C2×C4×C101—22no—
87C2×C4×C101—22no—
88C2×C4×C101—22no—
89C2×C4×C101C2×D412yes—
90C2×C4×C101#4114yes—
91C22⋊D101—28no—
92C2×Dic101—21no—
93C2×Dic101—21no—
94C2×Dic101—22no—
95C2×Dic101—21no—
96C2×Dic101—22no—
97C2×Dic101—22no—
98C2×Dic101—22no—
99C2×Dic101—22no—
100C2×Dic101—22no—
101C2×Dic101—22no—
102C2×Dic101—44no—
103C2×Dic101C2×D412yes—
104C2×Dic101C2×D412yes—
105C2×Dic101#4114yes—
106C4×D101—44no—
107C22×C101—22no—
108C22×C101—24no—
109C22×C101—41no—
110C22×C101—41no—
111C22×C101—42no—
112C22×C101C22×D411yescharacteristic
113C22×C101C2×#4111yescharacteristic
114C22×C101C2×#4112yes—
115C22×C101C2×#4112yes—
116C22×C101C2×#4111yescharacteristic
117C2×D101—22no—
118C2×D101—24no—
119C2×D101—41no—
120C2×D101—41no—
121C2×D101—42no—
122C4×C101—22no—
123C4×C101—24no—
124C4×C101—41no—
125C4×C101—42no—
126Dic101—22no—
127Dic101—42no—
128Dic101—41no—
129Dic101—42no—
130Dic101#12414yes—
131C2×C101—42no—
132C2×C101#52511yescharacteristic
133C2×C101#52511yescharacteristic
134C2×C101C2×#12411yescharacteristic
135C2×C101#52912yes—
136C2×C101#52912yes—
137D101—42no—
138#2398—1011nomaximal, Sylow
139C101#239811yescharacteristic, complemented, Sylow
140C2×C4×Q8—1011no—
141C2×C4×D4—1011no—
142#400—1012no—
143#397—1011no—
144#397—1011no—
145#394—1012no—
146#393—1011no—
147#393D10111yescharacteristic, complemented
148C4×C22⋊C4—1012no—
149C4×C22⋊C4—1012no—
150#376—1011no—
151C22×Q8C2×D10111yescharacteristic
152C22×D4—1011no—
153C23×C4—1011no—
154C23×C4—1011no—
155C4×Q8—2024no—
156C4×D4—2024no—
157C2×C4⋊C4—1011no—
158C2×C4⋊C4—1012no—
159C2×C4⋊C4—1011no—
160C2×C22⋊C4—1012no—
161C2×C22⋊C4—1011no—
162C2×C22⋊C4—1012no—
163C2×C22⋊C4—1012no—
164C2×C22⋊C4C2×D10111yescharacteristic
165C2×C22⋊C4C2×D10112yes—
166C2×C42—1011no—
167C2×C42—1012no—
168C2×C42—1012no—
169#95—1011no—
170#95—1012no—
171#95—1012no—
172#95—1012no—
173#95—1012no—
174#95C2×D10111yescharacteristic
175#95C2×D10112yes—
176C24—1011no—
177C24C22×D10111yescharacteristic
178C2×Q8—22no—
179C2×Q8—24no—
180C2×D4—2022no—
181C2×D4—2022no—
182C2×D4—2022no—
183C22×C4—1011no—
184C22×C4—1011no—
185C22×C4—1012no—
186C22×C4—1011no—
187C22×C4—1012no—
188C22×C4—2022no—
189C22×C4—2024no—
190C22×C4—2022no—
191C22×C4—2024no—
192C22×C4C22×D10112yes—
193C22×C4C22×D10111yescharacteristic
194C22×C4C22×D10111yescharacteristic
195C22×C4C22×D10112yes—
196C4⋊C4—2022no—
197C4⋊C4—2024no—
198C4⋊C4—2022no—
199C22⋊C4—22no—
200C22⋊C4—24no—
201C22⋊C4—2024no—
202C22⋊C4—2024no—
203C22⋊C4—2022no—
204C22⋊C4—2024no—
205C42—1014no—
206C42—2024no—
207C42—2024no—
208C23—21no—
209C23—22no—
210C23—21no—
211C23—22no—
212C23—1012no—
213C23—2021no—
214C23—2022no—
215C23—2022no—
216C23—2021no—
217C23#1310807512yes—
218C23C23×D10111yescharacteristic
219Q8—28no—
220D4—2028no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—1012no—
230C2×C4—1014no—
231C2×C4—1012no—
232C2×C4—4044no—
233C2×C4—4044no—
234C2×C4—2022no—
235C2×C4—2021no—
236C2×C4—2022no—
237C2×C4—2022no—
238C2×C4—2022no—
239C2×C4—2022no—
240C2×C4—2022no—
241C2×C4—2022no—
242C2×C4—2021no—
243C2×C4—2021no—
244C2×C4D4×D10112yes—
245C2×C4#1310807314yes—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—41no—
250C22—42no—
251C22—4041no—
252C22—4041no—
253C22—4042no—
254C22—2022no—
255C22—2024no—
256C22C2×D4×D10111yescharacteristic
257C22#41×D10111yescharacteristic
258C22#41×D10112yes—
259C22C2×#1310807312yes—
260C22C2×#1310807511yescharacteristic
261C4—22no—
262C4—24no—
263C4—41no—
264C4—42no—
265C4—1014no—
266C4—4042no—
267C4—4041no—
268C4—4042no—
269C4—2022no—
270C2—42no—
271C2—4042no—
272C2#2026568311yescharacteristic
273C2#2026569011yescharacteristic
274C2#124×D10111yescharacteristic
275C2#2026570612yes—
276C2#2026571312yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#393×C101C2
prime radicalC22×C22⋊D101C22
hypercenter#393D101
hypocenterC101#2398
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic101C2semidirect
3. #20265119C2semidirect
4. #20265104C2semidirect
5. #20265102C2semidirect
7. #20265088C2semidirect
8. C22⋊C4×Dic101C2semidirect
9. #20265024C2semidirect
10. #20265022C2semidirect
11. #393×C101C2semidirect
139. C101#2398semidirect
147. #393D101semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1012
C2×#156922152
#202651192
#202651042
#202651022
#202650932
#202650882
C22⋊C4×Dic1012
#202650242
#202650222
#393×C1012
#2398101