Finite Group Databasebeta

← #27142039

Subgroups — #27142039

Order 12416.

376 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#13095495C211yesmaximal, characteristic, complemented
2C2×#15690737C212yesmaximal, complemented
3C4×C22⋊Dic97C211yesmaximal, characteristic, complemented
4#20260494C212yesmaximal, complemented
5#525×C97C211yesmaximal, characteristic
6C2×#13095495—24no—
7C2×#13095495—21no—
8C2×#13095495C2212yes—
9C22×#11715895—21no—
10C2×C22⋊Dic97—21no—
11C2×C4×Dic97C2212yes—
12#15690737—24no—
13#15690717—21no—
14C2×#41×C97C2211yescharacteristic
15C23×C4×C97C2211yescharacteristic
16#123×C97—21no—
17C22⋊Q8×C97C411yescharacteristic, complemented
18C22⋊Q8×C97C411yescharacteristic, complemented
19#121×C97C411yescharacteristic, complemented
20#121×C97C411yescharacteristic, complemented
21#120×C97—21no—
22C4×D4×C97—21no—
23C4×D4×C97—21no—
24#117×C97C2211yescharacteristic
25C22×Dic97—22no—
26#13095495—24no—
27#13095495—22no—
28#13095495D414yescomplemented
29C2×#11715895—24no—
30C2×#11715895—22no—
31C22⋊Dic97—24no—
32C4×Dic97—22no—
33C4×Dic97—22no—
34C24×C97D411yescharacteristic
35#41×C97—22no—
36#41×C97—22no—
37C2×Q8×C97C2×C411yescharacteristic
38C2×D4×C97—21no—
39C2×D4×C97—21no—
40C2×D4×C97—21no—
41C2×D4×C97C2×C411yescharacteristic
42C22×C4×C97—22no—
43C22×C4×C97—22no—
44C22×C4×C97—21no—
45C22×C4×C97C2×C411yescharacteristic
46C22×C4×C97C2×C411yescharacteristic
47C22×C4×C97D411yescharacteristic
48C22×C4×C97D411yescharacteristic
49C22×C4×C97D411yescharacteristic
50C22×C4×C97C2311yescharacteristic
51C4⋊C4×C97—21no—
52C4⋊C4×C97—21no—
53C4⋊C4×C97C2×C411yescharacteristic
54C4⋊C4×C97C2×C411yescharacteristic
55C22⋊C4×C97—21no—
56C22⋊C4×C97—21no—
57C22⋊C4×C97—21no—
58C22⋊C4×C97—21no—
59C22⋊C4×C97—21no—
60C42×C97—21no—
61C23×C97—22no—
62C23×C97—22no—
63C23×C97—21no—
64C23×C97—41no—
65C23×C97C22⋊C411yescharacteristic
66C23×C97C22⋊C411yescharacteristic
67C23×C97C2×D411yescharacteristic
68Q8×C97—22no—
69D4×C97—22no—
70D4×C97—42no—
71D4×C97—42no—
72D4×C97—42no—
73C2×C4×C97—22no—
74C2×C4×C97—22no—
75C2×C4×C97—22no—
76C2×C4×C97—21no—
77C2×C4×C97—21no—
78C2×C4×C97—21no—
79C2×C4×C97—21no—
80C2×C4×C97—21no—
81C2×C4×C97—21no—
82C2×C4×C97—21no—
83C2×C4×C97—41no—
84C2×C4×C97—41no—
85C2×C4×C97C22⋊C411yescharacteristic
86C2×C4×C97C22⋊C411yescharacteristic
87C2×C4×C97C22×C411yescharacteristic
88C2×C4×C97C2×D412yes—
89C2×C4×C97C2×D411yescharacteristic
90C2×C4×C97#4112yes—
91C2×Dic97—24no—
92C2×Dic97—42no—
93C2×Dic97—42no—
94#11715895—24no—
95C22×C97—21no—
96C22×C97—21no—
97C22×C97—21no—
98C22×C97—21no—
99C22×C97—21no—
100C22×C97—21no—
101C22×C97—21no—
102C22×C97—44no—
103C22×C97—41no—
104C22×C97—41no—
105C22×C97—41no—
106C22×C97#10412yescomplemented
107C22×C97C2×C22⋊C411yescharacteristic
108C22×C97#12112yes—
109C4×C97—21no—
110C4×C97—21no—
111C4×C97—41no—
112C4×C97—41no—
113C4×C97—41no—
114C4×C97C4×D412yes—
115C4×C97#12011yescharacteristic
116C4×C97#12311yescharacteristic
117Dic97—44no—
118C2×C97—22no—
119C2×C97—22no—
120C2×C97—41no—
121C2×C97—81no—
122C2×C97#38611yescharacteristic
123C2×C97C2×#10411yescharacteristic
124C2×C97#42111yescharacteristic
125#1871—971nomaximal, Sylow
126C97#187111yescharacteristic, complemented, Sylow
127C22×M16—971no—
128#525D9711yescharacteristic
129C2×#104—972no—
130C4×C22⋊C4—971no—
131#339—972no—
132C2×#41C2×D9711yescharacteristic
133C23×C4C2×D9711yescharacteristic
134C2×M16—972no—
135C2×M16—1944no—
136C2×M16—1941no—
137C22×C8—1941no—
138#123—21no—
139C22⋊Q8Dic9711yescharacteristic, complemented
140C22⋊Q8Dic9711yescharacteristic, complemented
141#121Dic9711yescharacteristic, complemented
142#121Dic9711yescharacteristic, complemented
143#120—21no—
144C4×D4—21no—
145C4×D4—21no—
146#117C2×D9711yescharacteristic
147C2×C22⋊C4—1941no—
148C2×C42—972no—
149#104—1944no—
150#95—1941no—
151C24C22⋊D9711yescharacteristic
152#41—22no—
153#41—22no—
154C2×Q8C2×Dic9711yescharacteristic
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4C2×Dic9711yescharacteristic
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—1942no—
163C22×C4C2×Dic9711yescharacteristic
164C22×C4C2×Dic9711yescharacteristic
165C22×C4C22⋊D9711yescharacteristic
166C22×C4C22⋊D9711yescharacteristic
167C22×C4C22⋊D9711yescharacteristic
168C22×C4C22×D9711yescharacteristic
169M16—974no—
170M16—1944no—
171M16—1942no—
172C2×C8—1944no—
173C2×C8—1942no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4C2×Dic9711yescharacteristic
177C4⋊C4C2×Dic9711yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—21no—
181C22⋊C4—21no—
182C22⋊C4—21no—
183C22⋊C4—1944no—
184C42—21no—
185C42—1942no—
186C42—1942no—
187C23—22no—
188C23—22no—
189C23—21no—
190C23—41no—
191C23C22⋊Dic9711yescharacteristic
192C23C22⋊Dic9711yescharacteristic
193C23C2×C22⋊D9711yescharacteristic
194Q8—22no—
195D4—22no—
196D4—42no—
197D4—42no—
198D4—42no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—3882no—
212C2×C4—3882no—
213C2×C4—1944no—
214C2×C4C22⋊Dic9711yescharacteristic
215C2×C4C22⋊Dic9711yescharacteristic
216C2×C4C22×Dic9711yescharacteristic
217C2×C4D4×D9712yes—
218C2×C4C2×C22⋊D9711yescharacteristic
219C2×C4#1309551012yes—
220C8—1944no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—44no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22#1569073712yescomplemented
233C22C2×C22⋊Dic9711yescharacteristic
234C22#1569078212yes—
235C4—21no—
236C4—21no—
237C4—41no—
238C4—41no—
239C4—41no—
240C4—3884no—
241C4D4×Dic9712yes—
242C4#1569077811yescharacteristic
243C4#1569079111yescharacteristic
244C2—22no—
245C2—22no—
246C2—41no—
247C2—81no—
248C2#2026061511yescharacteristic
249C2C2×#1569073711yescharacteristic
250C2#2026075211yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting, prime radical#525×C97C2
socleC22×C97C2×C22⋊C4
cosocleC22×C4×C97C23
hypercenter#525D97
hypocenterC97#1871
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#13095495C2semidirect
2. C2×#15690737C2semidirect
3. C4×C22⋊Dic97C2semidirect
4. #20260494C2semidirect
17. C22⋊Q8×C97C4semidirect
18. C22⋊Q8×C97C4semidirect
19. #121×C97C4semidirect
20. #121×C97C4semidirect
28. #13095495D4semidirect
106. C22×C97#104semidirect
126. C97#1871semidirect
139. C22⋊Q8Dic97semidirect
140. C22⋊Q8Dic97semidirect
141. #121Dic97semidirect
142. #121Dic97semidirect
232. C22#15690737semidirect

Maximal subgroups

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

subgroupindex
C22×#130954952
C2×#156907372
C4×C22⋊Dic972
#202604942
#525×C972
#187197