Finite Group Databasebeta

← #28335017

Subgroups — #28335017

Order 13696.

284 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 81 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×C22⋊(C107⋊C8)C211yesmaximal, characteristic, complemented
2#20292490C211yesmaximal, characteristic, complemented
3#20292364C212yesmaximal, complemented
4#20292361C212yesmaximal, complemented
5#525×C107C211yesmaximal, characteristic
6C22×C107⋊C8—21no—
7C2×C22⋊Dic107—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—22no—
24C2×C107⋊C8—22no—
25C2×C107⋊C8—42no—
26C22⋊Dic107—42no—
27C24×C107D411yescharacteristic
28#41×C107—42no—
29C2×Q8×C107C2×C411yescharacteristic
30C2×D4×C107—21no—
31C2×D4×C107—21no—
32C2×D4×C107—21no—
33C2×D4×C107C2×C411yescharacteristic
34C22×C4×C107—21no—
35C22×C4×C107—42no—
36C22×C4×C107C2×C411yescharacteristic
37C22×C4×C107C2×C411yescharacteristic
38C22×C4×C107D411yescharacteristic
39C22×C4×C107D411yescharacteristic
40C22×C4×C107D411yescharacteristic
41C22×C4×C107C2311yescharacteristic
42C4⋊C4×C107—21no—
43C4⋊C4×C107—21no—
44C4⋊C4×C107C2×C411yescharacteristic
45C4⋊C4×C107C2×C411yescharacteristic
46C22⋊C4×C107—21no—
47C22⋊C4×C107—21no—
48C22⋊C4×C107—21no—
49C22⋊C4×C107—21no—
50C22⋊C4×C107—21no—
51C42×C107—21no—
52C23×C107—22no—
53C23×C107—22no—
54C23×C107—21no—
55C23×C107—41no—
56C23×C107C22⋊C411yescharacteristic
57C23×C107C22⋊C411yescharacteristic
58C23×C107C2×D411yescharacteristic
59Q8×C107—41no—
60D4×C107—41no—
61D4×C107—42no—
62D4×C107—42no—
63D4×C107—42no—
64C2×C4×C107—22no—
65C2×C4×C107—22no—
66C2×C4×C107—21no—
67C2×C4×C107—21no—
68C2×C4×C107—21no—
69C2×C4×C107—41no—
70C2×C4×C107—41no—
71C2×C4×C107—41no—
72C2×C4×C107—41no—
73C2×C4×C107—42no—
74C2×C4×C107C22⋊C411yescharacteristic
75C2×C4×C107C22⋊C411yescharacteristic
76C2×C4×C107C22×C411yescharacteristic
77C2×C4×C107C2×D411yescharacteristic
78C2×Dic107—42no—
79C2×Dic107—42no—
80C2×Dic107—42no—
81C107⋊C8—42no—
82C22×C107—21no—
83C22×C107—21no—
84C22×C107—21no—
85C22×C107—21no—
86C22×C107—21no—
87C22×C107—21no—
88C22×C107—21no—
89C22×C107—44no—
90C22×C107—41no—
91C22×C107—41no—
92C22×C107—41no—
93C22×C107#9912yes—
94C22×C107C2×C22⋊C411yescharacteristic
95C22×C107#13512yes—
96C4×C107—21no—
97C4×C107—21no—
98C4×C107—21no—
99C4×C107—21no—
100C4×C107—41no—
101C4×C107—41no—
102C4×C107—41no—
103Dic107—82no—
104C2×C107—22no—
105C2×C107—22no—
106C2×C107—41no—
107C2×C107—81no—
108C2×C107C2×#9911yescharacteristic
109C2×C107#41611yescharacteristic
110C2×C107#42111yescharacteristic
111#1516—1071nomaximal, Sylow
112C107#151611yescharacteristic, complemented, Sylow
113#525D10711yescharacteristic
114C2×C22⋊C8—1071no—
115#381—1071no—
116#328—1072no—
117#327—1072no—
118C2×#41Dic10711yescharacteristic, complemented
119C23×C4C2×D10711yescharacteristic
120C22×C8—2141no—
121#123—21no—
122C22⋊Q8Dic10711yescharacteristic, complemented
123C22⋊Q8C2×D10711yescharacteristic
124#121Dic10711yescharacteristic, complemented
125#121C2×D10711yescharacteristic
126#120—21no—
127C4×D4—21no—
128C4×D4—21no—
129#117Dic10711yescharacteristic, complemented
130C2×C22⋊C4—2141no—
131C22⋊C8—1072no—
132C22⋊C8—2141no—
133#95—1072no—
134#95—2141no—
135C24C22⋊D10711yescharacteristic
136#41—42no—
137C2×Q8C2×Dic10711yescharacteristic
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4C2×Dic10711yescharacteristic
142C22×C4—21no—
143C22×C4—42no—
144C22×C4—2142no—
145C22×C4C2×Dic10711yescharacteristic
146C22×C4C2×Dic10711yescharacteristic
147C22×C4C22⋊D10711yescharacteristic
148C22×C4C22⋊D10711yescharacteristic
149C22×C4C22⋊D10711yescharacteristic
150C22×C4C22×D10711yescharacteristic
151C2×C8—4282no—
152C2×C8—2142no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4C2×Dic10711yescharacteristic
156C4⋊C4C2×Dic10711yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—4282no—
163C42—21no—
164C23—22no—
165C23—22no—
166C23—21no—
167C23—41no—
168C23C22⋊Dic10711yescharacteristic
169C23C22⋊Dic10711yescharacteristic
170C23C2×C22⋊D10711yescharacteristic
171Q8—41no—
172D4—41no—
173D4—42no—
174D4—42no—
175D4—42no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—4282no—
187C2×C4—4282no—
188C2×C4—4282no—
189C2×C4C22⋊Dic10711yescharacteristic
190C2×C4C22⋊Dic10711yescharacteristic
191C2×C4C22×Dic10711yescharacteristic
192C2×C4C2×C22⋊D10711yescharacteristic
193C8—4282no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22#1681202412yes—
206C22C2×C22⋊Dic10711yescharacteristic
207C22#1681212912yes—
208C4—21no—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—41no—
213C4—41no—
214C4—41no—
215C4—8562no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—81no—
220C2C2×#1681202411yescharacteristic
221C2#2029262311yescharacteristic
222C2#2029264411yescharacteristic
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#1516
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. C2×C22⋊(C107⋊C8)C2semidirect
2. #20292490C2semidirect
3. #20292364C2semidirect
4. #20292361C2semidirect
12. C2×#41×C107C4semidirect
15. C22⋊Q8×C107C4semidirect
17. #121×C107C4semidirect
22. #117×C107C4semidirect
112. C107#1516semidirect
118. C2×#41Dic107semidirect
122. C22⋊Q8Dic107semidirect
124. #121Dic107semidirect
129. #117Dic107semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊(C107⋊C8)2
#202924902
#202923642
#202923612
#525×C1072
#1516107