Finite Group Databasebeta

← #28339219

Subgroups — #28339219

Order 13696.

484 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 217 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#20292534C212yesmaximal, complemented
2#20292520C212yesmaximal
3#20292520C212yesmaximal
4#20292513C212yesmaximal, complemented
5#20292513C212yesmaximal, complemented
6#20292490C211yesmaximal, characteristic
7C4⋊C4×Dic107C212yesmaximal, complemented
8C4×C22⋊Dic107C211yesmaximal, characteristic
9C2×C22⋊Q8×C107C211yesmaximal, characteristic
10C2×C4⋊Dic107C2212yes—
11C2×C4⋊Dic107C2212yes—
12C2×C22⋊Dic107C2212yes—
13C2×C22⋊Dic107C2211yescharacteristic
14C2×C22⋊Dic107C2211yescharacteristic
15C2×C4×Dic107C2212yes—
16C2×C4×Dic107C2211yescharacteristic
17C2×C4×Dic107C2211yescharacteristic
18C2×C4×Dic107C2212yes—
19C107⋊#95C2212yes—
20C107⋊#95C2212yes—
21C107⋊#95C2212yes—
22C107⋊#95C2212yes—
23C107⋊#95C2212yes—
24C107⋊#95C2212yes—
25C107⋊#95C2212yes—
26C22×Q8×C107C2211yescharacteristic
27C23×C4×C107C2211yescharacteristic
28C22⋊Q8×C107C418yescomplemented
29C2×C4⋊C4×C107C2212yes—
30C2×C4⋊C4×C107C2211yescharacteristic
31C2×C22⋊C4×C107C2212yes—
32C22×Dic107C2312yes—
33C22×Dic107C2312yes—
34C22×Dic107C2312yes—
35C22×Dic107C2312yes—
36C4⋊Dic107—24no—
37C4⋊Dic107—42no—
38C22⋊Dic107—24no—
39C22⋊Dic107—41no—
40C22⋊Dic107—41no—
41C4×Dic107—24no—
42C4×Dic107—41no—
43C4×Dic107—42no—
44C4×Dic107—41no—
45C24×C107C2311yescharacteristic
46C2×Q8×C107—41no—
47C2×Q8×C107—41no—
48C2×Q8×C107C2×C414yes—
49C22×C4×C107—41no—
50C22×C4×C107—41no—
51C22×C4×C107C2×C414yes—
52C22×C4×C107C2311yescharacteristic
53C22×C4×C107C2312yes—
54C22×C4×C107C2312yes—
55C22×C4×C107C2311yescharacteristic
56C4⋊C4×C107C2×C414yes—
57C4⋊C4×C107C2×C418yes—
58C22⋊C4×C107C2×C418yes—
59C23×C107—22no—
60C23×C107—21no—
61C23×C107—21no—
62C23×C107—22no—
63C23×C107C22×C412yes—
64C23×C107C2411yescharacteristic
65Q8×C107—44no—
66C2×C4×C107—22no—
67C2×C4×C107—22no—
68C2×C4×C107—21no—
69C2×C4×C107—21no—
70C2×C4×C107—21no—
71C2×C4×C107—21no—
72C2×C4×C107—22no—
73C2×C4×C107—22no—
74C2×C4×C107—44no—
75C2×C4×C107C22×C414yes—
76C2×C4×C107C22×C414yes—
77C2×C4×C107C22×C412yes—
78C2×C4×C107C22×C412yes—
79C2×Dic107—22no—
80C2×Dic107—22no—
81C2×Dic107—22no—
82C2×Dic107—22no—
83C2×Dic107—22no—
84C2×Dic107—22no—
85C2×Dic107—22no—
86C2×Dic107—22no—
87C2×Dic107—22no—
88C2×Dic107—22no—
89C2×Dic107—22no—
90C2×Dic107#4114yes—
91C22×C107—22no—
92C22×C107—22no—
93C22×C107—22no—
94C22×C107—42no—
95C22×C107—42no—
96C22×C107C23×C411yescharacteristic
97C22×C107C2×#4112yes—
98C22×C107H(5,𝔽2)11yescharacteristic
99C22×C107H(5,𝔽2)11yescharacteristic
100C22×C1072−1+412yes—
101C4×C107—24no—
102C4×C107—22no—
103C4×C107—22no—
104C4×C107—24no—
105Dic107—24no—
106Dic107—42no—
107Dic107—42no—
108Dic107—42no—
109C2×C107—42no—
110C2×C107C4×#4111yescharacteristic
111C2×C107#51811yescharacteristic
112C2×C107#51911yescharacteristic
113C2×C107#53912yes—
114C2×C107#54212yes—
115#2332—1071nomaximal, Sylow
116C107#233211yescharacteristic, complemented, Sylow
117C2×C22⋊Q8D10711yescharacteristic
118#391—1072no—
119#388—1072no—
120#388—1072no—
121#387—1072no—
122#387—1072no—
123#381—1071no—
124C4×C4⋊C4—1072no—
125C4×C22⋊C4—1071no—
126C22×Q8C2×D10711yescharacteristic
127C23×C4C2×D10711yescharacteristic
128C22⋊Q8Dic10718yescomplemented
129C2×C4⋊C4—1072no—
130C2×C4⋊C4—1072no—
131C2×C4⋊C4C2×D10712yes—
132C2×C4⋊C4C2×D10711yescharacteristic
133C2×C22⋊C4—1071no—
134C2×C22⋊C4—1071no—
135C2×C22⋊C4—1072no—
136C2×C22⋊C4C2×D10712yes—
137C2×C42—1071no—
138C2×C42—1071no—
139C2×C42—1072no—
140C2×C42—1072no—
141#95—1072no—
142#95—1072no—
143#95—1072no—
144#95—1072no—
145#95—1072no—
146#95—1072no—
147#95—1072no—
148C24C22×D10711yescharacteristic
149C2×Q8—41no—
150C2×Q8—41no—
151C2×Q8C2×Dic10714yes—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—1072no—
155C22×C4—1072no—
156C22×C4—1072no—
157C22×C4—1072no—
158C22×C4C2×Dic10714yes—
159C22×C4C22×D10711yescharacteristic
160C22×C4C22×D10711yescharacteristic
161C22×C4C22×D10712yes—
162C22×C4C22×D10712yes—
163C4⋊C4—4282no—
164C4⋊C4—2144no—
165C4⋊C4C2×Dic10714yes—
166C4⋊C4C2×Dic10718yes—
167C22⋊C4—4281no—
168C22⋊C4—4281no—
169C22⋊C4—2144no—
170C22⋊C4C2×Dic10718yes—
171C42—4281no—
172C42—4281no—
173C42—4282no—
174C42—2144no—
175C23—22no—
176C23—21no—
177C23—21no—
178C23—22no—
179C23C22×Dic10712yes—
180C23C23×D10711yescharacteristic
181Q8—44no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—44no—
191C2×C4—1074no—
192C2×C4—2142no—
193C2×C4—2142no—
194C2×C4—2142no—
195C2×C4—2142no—
196C2×C4—2142no—
197C2×C4—2142no—
198C2×C4—2142no—
199C2×C4—2142no—
200C2×C4—2142no—
201C2×C4—2142no—
202C2×C4—2142no—
203C2×C4C22×Dic10714yes—
204C2×C4C22×Dic10714yes—
205C2×C4C22×Dic10712yes—
206C2×C4C22×Dic10712yes—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22C23×Dic10711yescharacteristic
213C22#41×D10712yes—
214C22#1681215411yescharacteristic
215C22#1681215411yescharacteristic
216C22#1681215712yes—
217C4—24no—
218C4—22no—
219C4—22no—
220C4—24no—
221C4—4282no—
222C4—4282no—
223C4—4282no—
224C4—2144no—
225C2—42no—
226C2#41×Dic10711yescharacteristic
227C2#2029310211yescharacteristic
228C2#2029310511yescharacteristic
229C2#2029322312yes—
230C2#2029324712yes—
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
FittingC2×C22⋊Q8×C107C2
socle, cosocleC23×C107C24
prime radicalC24×C107C23
hypercenterC2×C22⋊Q8D107
hypocenterC107#2332
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20292534C2semidirect
4. #20292513C2semidirect
5. #20292513C2semidirect
7. C4⋊C4×Dic107C2semidirect
28. C22⋊Q8×C107C4semidirect
116. C107#2332semidirect
128. C22⋊Q8Dic107semidirect

Maximal subgroups

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

subgroupindex
#202925342
#202925202
#202925132
#202924902
C4⋊C4×Dic1072
C4×C22⋊Dic1072
C2×C22⋊Q8×C1072
#2332107