Finite Group Databasebeta

← #27181739

Subgroups — #27181739

Order 12928.

364 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 125 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#20265734C211yesmaximal, characteristic, complemented
2#20265587C211yesmaximal, characteristic, complemented
3#20265572C211yesmaximal, characteristic, complemented
4#20265547C211yesmaximal, characteristic, complemented
5#20265536C211yesmaximal, characteristic, complemented
6#20265433C212yesmaximal, complemented
7#20265421C212yesmaximal
8C4×#13108064C212yesmaximal, complemented
9C4×#13108062C212yesmaximal, complemented
10C4×#13108058C211yesmaximal, characteristic, complemented
11#529×C101C211yesmaximal, characteristic
12C2×#13108064C2212yes—
13C2×#13108062C2212yes—
14C2×#13108058C2212yes—
15#15692255C2211yescharacteristic
16#15692255C2211yescharacteristic
17#15692247—21no—
18#15692234—21no—
19#15692234C2212yes—
20C4×Dic202—21no—
21C4×Dic202C2212yes—
22#15692214—21no—
23#15692188C2212yes—
24#15692186C2212yes—
25#15692179C2212yes—
26#15692178C2212yes—
27#15692178C2212yes—
28#15692176C2212yes—
29#15692166C2211yescharacteristic
30#15692166C2211yescharacteristic
31C4×#11717579C2212yes—
32#126×C101—21no—
33#125×C101C2211yescharacteristic
34#124×C101C2211yescharacteristic
35#123×C101—21no—
36C22⋊Q8×C101C2211yescharacteristic
37#121×C101C2211yescharacteristic
38C4×Q8×C101C2211yescharacteristic
39C4×D4×C101—21no—
40C4×D4×C101C2211yescharacteristic
41#117×C101—21no—
42C2×C42×C101C2211yescharacteristic
43C2×Dic202—21no—
44C2×Dic202C2312yes—
45#13108064—24no—
46#13108062—24no—
47#13108058—24no—
48C2×#11717579C2312yes—
49C2×#11717579C2312yes—
50C4⋊Dic101—21no—
51C4⋊Dic101—21no—
52C4⋊Dic101—21no—
53C4⋊Dic101C2312yes—
54#13108053—22no—
55#13108053—22no—
56C22⋊Dic101—22no—
57C4×Dic101—22no—
58C2×Q8×C101C2311yescharacteristic
59C2×D4×C101—21no—
60C2×D4×C101C2311yescharacteristic
61C22×C4×C101—21no—
62C22×C4×C101D411yescharacteristic
63C22×C4×C101D411yescharacteristic
64C22×C4×C101C2311yescharacteristic
65C4⋊C4×C101—21no—
66C4⋊C4×C101—21no—
67C4⋊C4×C101—21no—
68C4⋊C4×C101—21no—
69C4⋊C4×C101C2311yescharacteristic
70C4⋊C4×C101C2311yescharacteristic
71C22⋊C4×C101—21no—
72C22⋊C4×C101—21no—
73C22⋊C4×C101—21no—
74C22⋊C4×C101—21no—
75C22⋊C4×C101—21no—
76C42×C101—21no—
77C42×C101D411yescharacteristic
78C42×C101D411yescharacteristic
79C42×C101C2311yescharacteristic
80C42×C101C2311yescharacteristic
81C23×C101—21no—
82C23×C101C2×D411yescharacteristic
83Q8×C101—22no—
84D4×C101—22no—
85D4×C101—42no—
86C2×C4×C101—22no—
87C2×C4×C101—22no—
88C2×C4×C101—21no—
89C2×C4×C101—21no—
90C2×C4×C101—21no—
91C2×C4×C101—41no—
92C2×C4×C101—42no—
93C2×C4×C101C2×D411yescharacteristic
94C2×C4×C101C2×D411yescharacteristic
95C2×C4×C101C2×D411yescharacteristic
96C2×C4×C101C2×D411yescharacteristic
97C2×C4×C101C2×D411yescharacteristic
98C2×C4×C101C2411yescharacteristic
99C2×Dic101—22no—
100C2×Dic101—22no—
101Dic202—22no—
102Dic202—22no—
103Dic202—42no—
104#11717579—22no—
105#11717579#4114yes—
106C22×C101—21no—
107C22×C101—21no—
108C22×C101—21no—
109C22×C101—41no—
110C22×C101—41no—
111C22×C101—41no—
112C22×C101C22×D411yescharacteristic
113C4×C101—21no—
114C4×C101—21no—
115C4×C101—21no—
116C4×C101—21no—
117C4×C101—41no—
118C4×C101—41no—
119C4×C101—41no—
120C4×C101#13712yes—
121C4×C101C2×#4112yes—
122Dic101—42no—
123Dic101—42no—
124C2×C101—41no—
125C2×C101—81no—
126C2×C101#53211yescharacteristic
127C2×C101C2×#13711yescharacteristic
128C2×C101#57511yescharacteristic
129#3172—1011nomaximal, Sylow
130C101#317211yescharacteristic, complemented, Sylow
131#533—1011no—
132#529D10111yescharacteristic
133#501—1011no—
134#497—1011no—
135#491—1011no—
136#489—1011no—
137#470—1012no—
138#468—1012no—
139C4×Q16—1012no—
140C4×SD16—1012no—
141C4×M16—1011no—
142C2×Q16—1012no—
143C2×SD16—1012no—
144C2×M16—1012no—
145C4⋊Q8—1011no—
146C4⋊Q8—1011no—
147#126—21no—
148#125—2021no—
149#125C2×D10111yescharacteristic
150#124C2×D10111yescharacteristic
151#123—21no—
152C22⋊Q8—1012no—
153C22⋊Q8—2021no—
154C22⋊Q8C2×D10111yescharacteristic
155#121C2×D10111yescharacteristic
156C4×Q8—1012no—
157C4×Q8—2021no—
158C4×Q8C2×D10111yescharacteristic
159C4×D4—21no—
160C4×D4C2×D10111yescharacteristic
161#117—21no—
162#117—2021no—
163C2×C42C2×D10111yescharacteristic
164#107—1012no—
165#106—1012no—
166#103—1012no—
167#103—1012no—
168#103—1012no—
169#102—1012no—
170#97—1011no—
171#97—1011no—
172C4×C8—1012no—
173C2×Q8—1012no—
174C2×Q8—2021no—
175C2×Q8C22×D10111yescharacteristic
176C2×D4—21no—
177C2×D4C22×D10111yescharacteristic
178C22×C4—21no—
179C22×C4C22⋊D10111yescharacteristic
180C22×C4C22⋊D10111yescharacteristic
181C22×C4C22×D10111yescharacteristic
182Q16—2024no—
183SD16—2024no—
184M16—2024no—
185C2×C8—1012no—
186C2×C8—1012no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—1012no—
192C4⋊C4—2022no—
193C4⋊C4—2022no—
194C4⋊C4—2021no—
195C4⋊C4—2021no—
196C4⋊C4—2021no—
197C4⋊C4C22×D10111yescharacteristic
198C4⋊C4C22×D10111yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—21no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—2022no—
205C42—21no—
206C42—2022no—
207C42C22⋊D10111yescharacteristic
208C42C22⋊D10111yescharacteristic
209C42C22×D10111yescharacteristic
210C42C22×D10111yescharacteristic
211C23—21no—
212C23C2×C22⋊D10111yescharacteristic
213Q8—22no—
214Q8—4042no—
215Q8—2022no—
216Q8—2022no—
217D4—22no—
218D4—42no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—2022no—
227C2×C4—2022no—
228C2×C4C2×C22⋊D10111yescharacteristic
229C2×C4C2×C22⋊D10111yescharacteristic
230C2×C4C2×C22⋊D10111yescharacteristic
231C2×C4C2×C22⋊D10111yescharacteristic
232C2×C4C2×C22⋊D10111yescharacteristic
233C2×C4C23×D10111yescharacteristic
234C8—1014no—
235C8—2022no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22C22×C22⋊D10111yescharacteristic
243C4—21no—
244C4—21no—
245C4—21no—
246C4—21no—
247C4—41no—
248C4—41no—
249C4—41no—
250C4—4042no—
251C4—4042no—
252C4#1569228512yes—
253C4#41×D10112yes—
254C2—41no—
255C2—81no—
256C2#2026572311yescharacteristic
257C2C2×#1569228511yescharacteristic
258C2#2026597511yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#529×C101C2
socleC22×C101C22×D4
prime radical#121×C101C22
hypercenter#529D101
hypocenterC101#3172
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #20265734C2semidirect
2. #20265587C2semidirect
3. #20265572C2semidirect
4. #20265547C2semidirect
5. #20265536C2semidirect
6. #20265433C2semidirect
8. C4×#13108064C2semidirect
9. C4×#13108062C2semidirect
10. C4×#13108058C2semidirect
130. C101#3172semidirect

Maximal subgroups

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

subgroupindex
#202657342
#202655872
#202655722
#202655472
#202655362
#202654332
#202654212
C4×#131080642
C4×#131080622
C4×#131080582
#529×C1012
#3172101