Finite Group Databasebeta

← #28334965

Subgroups — #28334965

Order 13696.

334 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 85 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#20292591C211yesmaximal, characteristic, complemented
2C2×C107⋊#95C211yesmaximal, characteristic, complemented
3#20292364C212yesmaximal, complemented
4#20292361C212yesmaximal, complemented
5#525×C107C211yesmaximal, characteristic
6C23×Dic107—21no—
7C2×#13133802—21no—
8C22⋊(C107⋊C8)—21no—
9C22⋊(C107⋊C8)C2212yes—
10C107⋊#95—21no—
11C107⋊#95C2212yes—
12C2×#41×C107C411yescharacteristic, complemented
13C23×C4×C107C2211yescharacteristic
14#123×C107—21no—
15C22⋊Q8×C107C411yescharacteristic, complemented
16C22⋊Q8×C107C2211yescharacteristic
17#121×C107C411yescharacteristic, complemented
18#121×C107C2211yescharacteristic
19#120×C107—21no—
20C4×D4×C107—21no—
21C4×D4×C107—21no—
22#117×C107C411yescharacteristic, complemented
23C22×Dic107—24no—
24C22×Dic107—22no—
25C22×Dic107—42no—
26C22×Dic107—42no—
27#13133802—42no—
28C2×C107⋊C8—22no—
29C24×C107D411yescharacteristic
30#41×C107—42no—
31C2×Q8×C107C2×C411yescharacteristic, complemented
32C2×D4×C107—21no—
33C2×D4×C107—21no—
34C2×D4×C107—21no—
35C2×D4×C107C2×C411yescharacteristic, complemented
36C22×C4×C107—21no—
37C22×C4×C107—42no—
38C22×C4×C107C2×C411yescharacteristic
39C22×C4×C107C2×C411yescharacteristic
40C22×C4×C107D411yescharacteristic
41C22×C4×C107D411yescharacteristic
42C22×C4×C107D411yescharacteristic
43C22×C4×C107C2311yescharacteristic
44C4⋊C4×C107—21no—
45C4⋊C4×C107—21no—
46C4⋊C4×C107C2×C411yescharacteristic, complemented
47C4⋊C4×C107C2×C411yescharacteristic, complemented
48C22⋊C4×C107—21no—
49C22⋊C4×C107—21no—
50C22⋊C4×C107—21no—
51C22⋊C4×C107—21no—
52C22⋊C4×C107—21no—
53C42×C107—21no—
54C23×C107—22no—
55C23×C107—22no—
56C23×C107—21no—
57C23×C107—21no—
58C23×C107—21no—
59C23×C107C22⋊C411yescharacteristic
60C23×C107C22⋊C411yescharacteristic
61C23×C107C2×D411yescharacteristic
62Q8×C107—41no—
63D4×C107—41no—
64D4×C107—42no—
65D4×C107—42no—
66D4×C107—41no—
67D4×C107—41no—
68C2×C4×C107—22no—
69C2×C4×C107—22no—
70C2×C4×C107—21no—
71C2×C4×C107—21no—
72C2×C4×C107—21no—
73C2×C4×C107—41no—
74C2×C4×C107—41no—
75C2×C4×C107—41no—
76C2×C4×C107—41no—
77C2×C4×C107—42no—
78C2×C4×C107C22⋊C411yescharacteristic
79C2×C4×C107C22⋊C411yescharacteristic
80C2×C4×C107C22×C411yescharacteristic
81C2×C4×C107C2×D411yescharacteristic
82C2×Dic107—24no—
83C2×Dic107—48no—
84C2×Dic107—42no—
85C2×Dic107—42no—
86C107⋊C8—42no—
87C22×C107—24no—
88C22×C107—24no—
89C22×C107—21no—
90C22×C107—21no—
91C22×C107—21no—
92C22×C107—21no—
93C22×C107—21no—
94C22×C107—21no—
95C22×C107—41no—
96C22×C107—41no—
97C22×C107—41no—
98C22×C107#9912yes—
99C22×C107#10412yes—
100C22×C107C2×C22⋊C411yescharacteristic
101C22×C107AGL(1,ℤ/8)11yescharacteristic
102C22×C107#13711yescharacteristic
103C4×C107—21no—
104C4×C107—21no—
105C4×C107—21no—
106C4×C107—21no—
107C4×C107—41no—
108C4×C107—41no—
109C4×C107—41no—
110Dic107—44no—
111C2×C107—21no—
112C2×C107—22no—
113C2×C107—22no—
114C2×C107—21no—
115C2×C107—81no—
116C2×C107C2×#9911yescharacteristic
117C2×C107#41711yescharacteristic
118C2×C107C2×#10411yescharacteristic
119#1504—1071nomaximal, Sylow
120C107#150411yescharacteristic, complemented, Sylow
121#525D10711yescharacteristic
122#407—1071no—
123C2×#95—1071no—
124#328—1072no—
125#327—1072no—
126C2×#41Dic10711yescharacteristic, complemented
127C23×C4—2141no—
128C23×C4C2×D10711yescharacteristic
129C2×M16—2141no—
130#123—21no—
131C22⋊Q8Dic10711yescharacteristic, complemented
132C22⋊Q8C2×D10711yescharacteristic
133#121Dic10711yescharacteristic, complemented
134#121C2×D10711yescharacteristic
135#120—21no—
136C4×D4—21no—
137C4×D4—21no—
138#117Dic10711yescharacteristic, complemented
139C22⋊C8—1072no—
140C22⋊C8—2141no—
141#95—1072no—
142#95—2141no—
143C24C22⋊D10711yescharacteristic
144#41—42no—
145C2×Q8C2×Dic10711yescharacteristic, complemented
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4C2×Dic10711yescharacteristic, complemented
150C22×C4—21no—
151C22×C4—42no—
152C22×C4—4282no—
153C22×C4—4282no—
154C22×C4—2144no—
155C22×C4—2142no—
156C22×C4C2×Dic10711yescharacteristic
157C22×C4C2×Dic10711yescharacteristic
158C22×C4C22⋊D10711yescharacteristic
159C22×C4C22⋊D10711yescharacteristic
160C22×C4C22⋊D10711yescharacteristic
161C22×C4C22×D10711yescharacteristic
162M16—4282no—
163C2×C8—2142no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4C2×Dic10711yescharacteristic, complemented
167C4⋊C4C2×Dic10711yescharacteristic, complemented
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—21no—
173C42—21no—
174C23—22no—
175C23—21no—
176C23—21no—
177C23—22no—
178C23—21no—
179C23C22⋊Dic10711yescharacteristic
180C23C22⋊Dic10711yescharacteristic
181C23C2×C22⋊D10711yescharacteristic
182Q8—41no—
183D4—41no—
184D4—42no—
185D4—41no—
186D4—41no—
187D4—42no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—4288no—
199C2×C4—4282no—
200C2×C4—4282no—
201C2×C4—2144no—
202C2×C4C22⋊Dic10711yescharacteristic
203C2×C4C22⋊Dic10711yescharacteristic
204C2×C4C22×Dic10711yescharacteristic
205C2×C4C2×C22⋊D10711yescharacteristic
206C8—4282no—
207C22—24no—
208C22—24no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22#1681202412yes—
219C22#1681203612yes—
220C22C2×C22⋊Dic10711yescharacteristic
221C22#1681213411yescharacteristic
222C22#1681213911yescharacteristic
223C4—21no—
224C4—21no—
225C4—21no—
226C4—21no—
227C4—41no—
228C4—41no—
229C4—41no—
230C4—4284no—
231C2—21no—
232C2—22no—
233C2—21no—
234C2—22no—
235C2—81no—
236C2C2×#1681202411yescharacteristic
237C2#2029262811yescharacteristic
238C2C2×#1681203611yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting, prime radical#525×C107C2
socleC22×C107C2×C22⋊C4
cosocleC22×C4×C107C23
hypercenter#525D107
hypocenterC107#1504
LCSG ▷ C2×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #20292591C2semidirect
2. C2×C107⋊#95C2semidirect
3. #20292364C2semidirect
4. #20292361C2semidirect
12. C2×#41×C107C4semidirect
15. C22⋊Q8×C107C4semidirect
17. #121×C107C4semidirect
22. #117×C107C4semidirect
31. C2×Q8×C107C2×C4semidirect
35. C2×D4×C107C2×C4semidirect
46. C4⋊C4×C107C2×C4semidirect
47. C4⋊C4×C107C2×C4semidirect
120. C107#1504semidirect
126. C2×#41Dic107semidirect
131. C22⋊Q8Dic107semidirect
133. #121Dic107semidirect
138. #117Dic107semidirect
145. C2×Q8C2×Dic107semidirect
149. C2×D4C2×Dic107semidirect
166. C4⋊C4C2×Dic107semidirect
167. C4⋊C4C2×Dic107semidirect

Maximal subgroups

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

subgroupindex
#202925912
C2×C107⋊#952
#202923642
#202923612
#525×C1072
#1504107