Finite Group Databasebeta

← #27174405

Subgroups — #27174405

Order 12928.

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

normal subgroups only

This group has 528 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×#15692220C211yesmaximal, characteristic, complemented
2C2×#15692213C211yesmaximal, characteristic
3#20265112C211yesmaximal, characteristic
4#20265111C212yesmaximal, complemented
5#20265101C212yesmaximal
6#20265097C211yesmaximal, characteristic, complemented
7#20265090C211yesmaximal, characteristic
8#20265028C211yesmaximal, characteristic, complemented
9#20265028C212yesmaximal, complemented
10#20265024C212yesmaximal
11#400×C101C211yesmaximal, characteristic, complemented
12C22×Dic202C2211yescharacteristic
13C22×C4×D101C2211yescharacteristic
14#15692220—24no—
15#15692213—24no—
16C2×C4⋊Dic101C2211yescharacteristic
17C2×#13108053C2212yes—
18C2×#13108053C2212yes—
19C2×#13108053C2212yes—
20C2×#13108053C2212yes—
21C2×#13108051C2212yes—
22C2×#13108051C2212yes—
23C2×#13108051C2211yescharacteristic
24C2×#13108051C2211yescharacteristic
25C2×C4×Dic101C2212yes—
26C2×C4×Dic101C2212yes—
27C2×C4×Dic101C2211yescharacteristic
28C2×C4×Dic101C2211yescharacteristic
29#15692162C2212yes—
30#15692162C2212yes—
31#15692162C2211yescharacteristic
32C2×C4⋊C4×C101C2211yescharacteristic
33C2×C4⋊C4×C101C2211yescharacteristic
34#95×C101C2212yes—
35#95×C101C2211yescharacteristic
36#95×C101C2212yes—
37C23×D101C2311yescharacteristic
38C2×Dic202—22no—
39C2×Dic202—24no—
40C22×Dic101C2311yescharacteristic
41C22×Dic101C2311yescharacteristic
42C22×Dic101C2312yes—
43C22×Dic101C2312yes—
44C22×Dic101C2311yescharacteristic
45C2×C4×D101—22no—
46C2×C4×D101—22no—
47C2×C4×D101Q814yescomplemented
48C4⋊Dic101—22no—
49#13108053—24no—
50#13108053—24no—
51#13108053—24no—
52#13108053—24no—
53#13108051—22no—
54#13108051—22no—
55#13108051—24no—
56#13108051—24no—
57C4×Dic101—22no—
58C4×Dic101—24no—
59C4×Dic101—24no—
60C4×Dic101D414yes—
61C22×C4×C101C2312yes—
62C22×C4×C101C2311yescharacteristic
63C22×C4×C101C2312yes—
64C22×C4×C101C2311yescharacteristic
65C22×C4×C101C2311yescharacteristic
66C4⋊C4×C101—22no—
67C4⋊C4×C101—22no—
68C23×C101C2411yescharacteristic
69C22×D101—21no—
70C22×D101—22no—
71C22×D101—22no—
72C22×D101—21no—
73C22×D101C2×Q812yes—
74C2×C4×C101—21no—
75C2×C4×C101—21no—
76C2×C4×C101—22no—
77C2×C4×C101—21no—
78C2×C4×C101—22no—
79C2×C4×C101—22no—
80C2×C4×C101—22no—
81C2×C4×C101—22no—
82C2×C4×C101—22no—
83C2×C4×C101—22no—
84C2×C4×C101—22no—
85C2×C4×C101C2×D412yes—
86C2×C4×C101C2×Q812yes—
87C2×Dic101—22no—
88C2×Dic101—22no—
89C2×Dic101—22no—
90C2×Dic101—22no—
91C2×Dic101—22no—
92C2×Dic101—21no—
93C2×Dic101—22no—
94C2×Dic101C2×D412yes—
95C2×Dic101C2×D412yes—
96C2×Dic101C2×Q812yes—
97C2×Dic101#4114yes—
98C2×Dic101#4114yes—
99C2×Dic101#4112yes—
100C4×D101—28no—
101Dic202—28no—
102C22×C101C22×D411yescharacteristic
103C22×C101C22×Q811yescharacteristic
104C22×C101C2×#4111yescharacteristic
105C22×C101C2×#4112yes—
106C22×C101C2×#4112yes—
107C2×D101—22no—
108C2×D101—24no—
109C2×D101—41no—
110C2×D101—42no—
111C2×D101—41no—
112C4×C101—22no—
113C4×C101—22no—
114C4×C101—41no—
115C4×C101—42no—
116C4×C101—42no—
117Dic101—24no—
118Dic101—24no—
119Dic101—22no—
120Dic101—41no—
121Dic101C22⋊Q814yes—
122C2×C101C2×C22⋊Q811yescharacteristic
123C2×C101#52511yescharacteristic
124C2×C101#52912yes—
125C2×C101#53211yescharacteristic
126C2×C101#53312yes—
127D101—42no—
128#2396—1011nomaximal, Sylow
129C101#239611yescharacteristic, complemented, Sylow
130C2×C4×Q8—1011no—
131C2×#117—1011no—
132#400D10111yescharacteristic, complemented
133#399—1011no—
134#398—1012no—
135#396—1012no—
136#395—1011no—
137#393—1011no—
138C4×C4⋊C4—1012no—
139C4×C4⋊C4—1011no—
140C4×C22⋊C4—1012no—
141C22×Q8—1011no—
142C23×C4—1011no—
143C4×Q8—2024no—
144#117—2024no—
145C2×C4⋊C4—1011no—
146C2×C4⋊C4—1012no—
147C2×C4⋊C4—1012no—
148C2×C4⋊C4—1012no—
149C2×C4⋊C4—1012no—
150C2×C4⋊C4C2×D10111yescharacteristic
151C2×C4⋊C4C2×D10111yescharacteristic
152C2×C22⋊C4—1011no—
153C2×C22⋊C4—1011no—
154C2×C22⋊C4—1012no—
155C2×C22⋊C4—1012no—
156C2×C42—1011no—
157C2×C42—1011no—
158C2×C42—1012no—
159C2×C42—1012no—
160#95—1012no—
161#95—1012no—
162#95—1011no—
163#95C2×D10112yes—
164#95C2×D10111yescharacteristic
165#95C2×D10112yes—
166C24—1011no—
167C2×Q8—2024no—
168C2×Q8—2022no—
169C22×C4—1011no—
170C22×C4—1012no—
171C22×C4—1012no—
172C22×C4—1011no—
173C22×C4—1014no—
174C22×C4—1011no—
175C22×C4—2022no—
176C22×C4—2022no—
177C22×C4C22×D10112yes—
178C22×C4C22×D10111yescharacteristic
179C22×C4C22×D10112yes—
180C22×C4C22×D10111yescharacteristic
181C22×C4C22×D10111yescharacteristic
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—2024no—
185C4⋊C4—2024no—
186C4⋊C4—2024no—
187C4⋊C4—2024no—
188C4⋊C4—2022no—
189C22⋊C4—2024no—
190C22⋊C4—2024no—
191C22⋊C4—2022no—
192C22⋊C4—2022no—
193C42—1014no—
194C42—2024no—
195C42—2022no—
196C42—2024no—
197C23—1012no—
198C23—2021no—
199C23—2022no—
200C23—2022no—
201C23—2021no—
202C23C23×D10111yescharacteristic
203Q8—2028no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—1012no—
216C2×C4—1012no—
217C2×C4—1012no—
218C2×C4—1012no—
219C2×C4—1014no—
220C2×C4—1014no—
221C2×C4—2028no—
222C2×C4—2021no—
223C2×C4—2022no—
224C2×C4—2022no—
225C2×C4—2022no—
226C2×C4—2022no—
227C2×C4—2022no—
228C2×C4—2022no—
229C2×C4D4×D10112yes—
230C2×C4Q8×D10112yes—
231C22—4041no—
232C22—4042no—
233C22—4041no—
234C22—2024no—
235C22—2022no—
236C22C2×D4×D10111yescharacteristic
237C22C2×Q8×D10111yescharacteristic
238C22#41×D10112yes—
239C22#41×D10112yes—
240C22#41×D10111yescharacteristic
241C4—22no—
242C4—22no—
243C4—41no—
244C4—42no—
245C4—42no—
246C4—1014no—
247C4—4041no—
248C4—2024no—
249C4—2024no—
250C4—2022no—
251C2—4042no—
252C2C22⋊Q8×D10111yescharacteristic
253C2#2026568511yescharacteristic
254C2#2026571012yes—
255C2#2026572611yescharacteristic
256C2#2026573312yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#400×C101C2
prime radicalC23×D101C23
hypercenter#400D101
hypocenterC101#2396
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#15692220C2semidirect
4. #20265111C2semidirect
6. #20265097C2semidirect
8. #20265028C2semidirect
9. #20265028C2semidirect
11. #400×C101C2semidirect
47. C2×C4×D101Q8semidirect
129. C101#2396semidirect
132. #400D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922202
C2×#156922132
#202651122
#202651112
#202651012
#202650972
#202650902
#202650282
#202650242
#400×C1012
#2396101