Finite Group Databasebeta

← #28336871

Subgroups — #28336871

Order 13696.

386 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16812092C211yesmaximal, characteristic
2#20293100C211yesmaximal, characteristic
3C2×#16812023C211yesmaximal, characteristic
4#20292490C211yesmaximal, characteristic, complemented
5#20292394C211yesmaximal, characteristic, complemented
6#20292393C211yesmaximal, characteristic, complemented
7#381×C107C211yesmaximal, characteristic, complemented
8C22×C22⋊D107C2211yescharacteristic
9#16812092—22no—
10#16812092C414yescomplemented
11C4×C22⋊D107—22no—
12(C22⋊D107)⋊C4—22no—
13#16812065—21no—
14C2×Dic107⋊C4—21no—
15C2×C22⋊Dic107C2211yescharacteristic
16C2×C22⋊Dic107C2211yescharacteristic
17C2×C22⋊Dic107C2211yescharacteristic
18C2×(C2×D107)⋊C4—21no—
19C22⋊C4×D107—21no—
20#16812023—22no—
21C107⋊#95—22no—
22C23×C4×C107C2211yescharacteristic
23C2×C22⋊C4×C107C2211yescharacteristic
24C2×C22⋊C4×C107C2211yescharacteristic
25#95×C107—22no—
26C23×D107D411yescharacteristic
27C2×C22⋊D107—21no—
28C2×C22⋊D107—21no—
29C2×C22⋊D107—21no—
30C2×C22⋊D107—21no—
31C2×C22⋊D107C2×C412yes—
32C2×C22⋊D107D412yes—
33C22×Dic107—21no—
34C22×Dic107—21no—
35C22×Dic107D411yescharacteristic
36C22×Dic107D412yes—
37C2×C4×D107—22no—
38Dic107⋊C4—22no—
39Dic107⋊C4—24no—
40C22⋊Dic107—21no—
41C22⋊Dic107—21no—
42C22⋊Dic107—21no—
43C22⋊Dic107—21no—
44C22⋊Dic107—21no—
45C22⋊Dic107C2×C412yescomplemented
46(C2×D107)⋊C4—22no—
47(C2×D107)⋊C4—24no—
48C4×Dic107—22no—
49C24×C107C2311yescharacteristic
50C22×C4×C107—21no—
51C22×C4×C107—21no—
52C22×C4×C107—21no—
53C22×C4×C107—21no—
54C22×C4×C107—21no—
55C22×C4×C107—21no—
56C22×C4×C107—21no—
57C22×C4×C107C2×C412yescomplemented
58C22×C4×C107D412yes—
59C22⋊C4×C107—21no—
60C22⋊C4×C107—21no—
61C22⋊C4×C107—21no—
62C22⋊C4×C107—21no—
63C23×C107—42no—
64C23×C107C22×C411yescharacteristic
65C23×C107C2×D411yescharacteristic
66C23×C107C2×D411yescharacteristic
67C23×C107C2×D411yescharacteristic
68C23×C107C2×D411yescharacteristic
69C23×C107#4111yescharacteristic
70C23×C107#4111yescharacteristic
71C22×D107—21no—
72C22×D107—21no—
73C22×D107—42no—
74C22×D107C22⋊C412yes—
75C2×C4×C107—22no—
76C2×C4×C107—22no—
77C2×C4×C107—22no—
78C2×C4×C107—22no—
79C2×C4×C107—41no—
80C2×C4×C107—41no—
81C2×C4×C107—41no—
82C2×C4×C107—41no—
83C2×C4×C107—42no—
84C2×C4×C107—42no—
85C2×C4×C107—41no—
86C22⋊D107—22no—
87C22⋊D107—22no—
88C22⋊D107—22no—
89C22⋊D107—22no—
90C2×Dic107—21no—
91C2×Dic107—21no—
92C2×Dic107—22no—
93C2×Dic107—22no—
94C2×Dic107—42no—
95C2×Dic107—42no—
96C2×Dic107—42no—
97C2×Dic107—41no—
98C2×Dic107C22⋊C412yes—
99C4×D107—82no—
100C22×C107—22no—
101C22×C107—22no—
102C22×C107—22no—
103C22×C107—41no—
104C22×C107—41no—
105C22×C107—41no—
106C22×C107—41no—
107C22×C107C2×C22⋊C411yescharacteristic
108C22×C107C4×D411yescharacteristic
109C22×C107C4×D411yescharacteristic
110C22×C107#12011yescharacteristic
111C22×C107#12111yescharacteristic
112C22×C107#12111yescharacteristic
113C22×C107#12311yescharacteristic
114C2×D107—22no—
115C2×D107—42no—
116C2×D107—42no—
117C2×D107—81no—
118C4×C107—42no—
119C4×C107—42no—
120C4×C107—81no—
121Dic107—22no—
122Dic107—42no—
123Dic107—42no—
124Dic107—81no—
125C2×C107—21no—
126C2×C107—21no—
127C2×C107—21no—
128C2×C107—21no—
129C2×C107—21no—
130C2×C107—21no—
131C2×C107#38611yescharacteristic
132C2×C107#45812yes—
133D107—42no—
134#1890—1071nomaximal, Sylow
135C107#189011yescharacteristic, complemented, Sylow
136C2×#123—1071no—
137#518—1071no—
138C2×#99—1071no—
139#381—1071no—
140#381D10711yescharacteristic, complemented
141#342—1071no—
142#342—1071no—
143C22×D4—1071no—
144C23×C4C2×D10711yescharacteristic
145#123—1074no—
146#123—2142no—
147C4×D4—2142no—
148C4×D4—2142no—
149#117—2141no—
150C2×C4⋊C4—2141no—
151C2×C22⋊C4—1071no—
152C2×C22⋊C4—1071no—
153C2×C22⋊C4—1071no—
154C2×C22⋊C4—2141no—
155C2×C22⋊C4—2141no—
156C2×C22⋊C4C2×D10711yescharacteristic
157C2×C22⋊C4C2×D10711yescharacteristic
158#99—2142no—
159#95—22no—
160#95—2142no—
161C24—1071no—
162C24C22×D10711yescharacteristic
163C2×D4—1072no—
164C2×D4—1072no—
165C2×D4—2141no—
166C2×D4—2141no—
167C2×D4—2141no—
168C2×D4—2141no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—1071no—
177C22×C4—1072no—
178C22×C4—2142no—
179C22×C4—2141no—
180C22×C4—2141no—
181C22×C4C4×D10712yescomplemented
182C22×C4C22⋊D10712yes—
183C4⋊C4—2144no—
184C4⋊C4—2142no—
185C22⋊C4—21no—
186C22⋊C4—21no—
187C22⋊C4—21no—
188C22⋊C4—21no—
189C22⋊C4—1072no—
190C22⋊C4—2144no—
191C22⋊C4—2142no—
192C22⋊C4—2141no—
193C22⋊C4—2141no—
194C22⋊C4—2141no—
195C22⋊C4—2141no—
196C22⋊C4—2141no—
197C42—2142no—
198C23—42no—
199C23—1072no—
200C23—4282no—
201C23—2141no—
202C23—2141no—
203C23C2×C4×D10711yescharacteristic
204C23D4×D10711yescharacteristic
205C23D4×D10711yescharacteristic
206C23D4×D10711yescharacteristic
207C23C2×C22⋊D10711yescharacteristic
208C23#1313381711yescharacteristic
209C23#1313381811yescharacteristic
210D4—2142no—
211D4—2142no—
212D4—2142no—
213D4—2142no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—8562no—
226C2×C4—1072no—
227C2×C4—4282no—
228C2×C4—4281no—
229C2×C4—4282no—
230C2×C4—4282no—
231C2×C4—2142no—
232C2×C4—2142no—
233C2×C4—2141no—
234C2×C4—2141no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—8561no—
243C22—4282no—
244C22—4282no—
245C22—2142no—
246C22C22⋊C4×D10711yescharacteristic
247C22(C22⋊D107)⋊C411yescharacteristic
248C22C4×C22⋊D10711yescharacteristic
249C22#1681207711yescharacteristic
250C22#1681208011yescharacteristic
251C22#1681208211yescharacteristic
252C22#1681209111yescharacteristic
253C4—42no—
254C4—42no—
255C4—81no—
256C4—8561no—
257C4—4282no—
258C4—4282no—
259C4—2142no—
260C2—21no—
261C2—21no—
262C2—21no—
263C2—21no—
264C2—21no—
265C2—21no—
266C2—4282no—
267C2#2029250311yescharacteristic
268C2#2029281612yes—
subgroupG/H
centerC22#16812077
commutatorC23×C107C22×C4
FrattiniC24C22×D107
Fitting#381×C107C2
socleC22×C107#120
prime radicalC22×C22⋊D107C22
cosocleC24×C107C23
hypercenter#381D107
hypocenterC107#1890
LCSG ▷ C23×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C24 ◁ #381

Complements

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

NH ≅ G/Ntype
4. #20292490C2semidirect
5. #20292394C2semidirect
6. #20292393C2semidirect
7. #381×C107C2semidirect
10. #16812092C4semidirect
45. C22⋊Dic107C2×C4semidirect
57. C22×C4×C107C2×C4semidirect
135. C107#1890semidirect
140. #381D107semidirect
181. C22×C4C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120922
#202931002
C2×#168120232
#202924902
#202923942
#202923932
#381×C1072
#1890107