Finite Group Databasebeta

← #28349543

Subgroups — #28349543

Order 13696.

356 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 113 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#20293346C212yesmaximal, complemented
2#20293018C212yesmaximal, complemented
3#20292846C212yesmaximal, complemented
4#20292840C212yesmaximal
5#20292835C212yesmaximal, complemented
6#20292757C211yesmaximal, characteristic, complemented
7#20292728C212yesmaximal, complemented
8C4×Dic428C211yesmaximal, characteristic, complemented
9#495×C107C211yesmaximal, characteristic
10C2×Dic428C2211yescharacteristic
11C2×#13133808C2212yes—
12C2×#13133806C2212yes—
13#16812107C2212yes—
14#16812102—22no—
15#16812102—22no—
16#16812096C2212yes—
17#16812086—22no—
18Q8×Dic107C2212yes—
19C4×Dic214C2212yes—
20C4×Dic214C2211yescharacteristic
21D4×Dic107C2212yes—
22#16812065—22no—
23#16812043C2211yescharacteristic
24C4⋊(C107⋊C8)C2211yescharacteristic
25Dic107⋊C8C2212yes—
26#16812034C2212yes—
27#16812033C2212yes—
28#16812030C2212yes—
29#16812019C2212yes—
30C2×Q16×C107C2211yescharacteristic
31C2×SD16×C107C2212yes—
32C2×D8×C107C2211yescharacteristic
33#124×C107C2212yes—
34C4×C8×C107C2211yescharacteristic
35C2×Dic214D411yescharacteristic
36C2×Dic214C2312yes—
37C22×Dic107—22no—
38Dic428—22no—
39#13133808—42no—
40#13133806—42no—
41C2×C107⋊C8C2312yes—
42C4⋊Dic107—22no—
43C4⋊Dic107D411yescharacteristic
44C4⋊Dic107C2312yes—
45Dic107⋊C4—22no—
46Dic107⋊C4—22no—
47Dic107⋊C4—22no—
48Dic107⋊C4D412yes—
49C22⋊Dic107—22no—
50C22⋊Dic107—22no—
51C22⋊Dic107—22no—
52C4×Dic107—22no—
53C4×Dic107—22no—
54C4×Dic107C2312yes—
55C2×Q8×C107C2312yes—
56C2×D4×C107C2312yes—
57Q16×C107—41no—
58SD16×C107—24no—
59D8×C107—41no—
60C2×C8×C107C2311yescharacteristic
61C2×C8×C107C2311yescharacteristic
62C22⋊C4×C107—22no—
63C22⋊C4×C107—22no—
64C42×C107C2311yescharacteristic
65C23×C107—22no—
66Q8×C107—22no—
67Q8×C107—22no—
68D4×C107—22no—
69D4×C107—22no—
70C2×C4×C107—22no—
71C2×C4×C107C2×D411yescharacteristic
72C2×C4×C107C2×D411yescharacteristic
73C2×C4×C107C2411yescharacteristic
74C2×Dic107—22no—
75C2×Dic107—22no—
76C2×Dic107—44no—
77C2×Dic107C2×D412yes—
78C2×Dic107C2×D412yes—
79Dic214—22no—
80Dic214—22no—
81Dic214—41no—
82C8×C107—21no—
83C8×C107#4112yes—
84C107⋊C8—22no—
85C22×C107—42no—
86C22×C107—42no—
87C22×C107—42no—
88C22×C107C22×D411yescharacteristic
89C4×C107—21no—
90C4×C107—21no—
91C4×C107—42no—
92C4×C107C2×#4111yescharacteristic
93C4×C107H(5,𝔽2)11yescharacteristic
94Dic107—22no—
95Dic107—22no—
96Dic107—42no—
97Dic107—42no—
98C2×C107—82no—
99C2×C107#55011yescharacteristic
100C2×C107#57511yescharacteristic
101C2×C107#57811yescharacteristic
102#3402—1071nomaximal, Sylow
103C107#340211yescharacteristic, complemented, Sylow
104#556—1072no—
105#497—1072no—
106#495D10711yescharacteristic
107#464—1072no—
108#463—1072no—
109#462—1072no—
110#446—1071no—
111#440—1072no—
112C4×Q16—1071no—
113C2×Q16—1072no—
114C2×Q16—1071no—
115C2×Q16C2×D10711yescharacteristic
116C2×SD16—1072no—
117C2×SD16C2×D10712yes—
118C2×D8C2×D10711yescharacteristic
119C4⋊Q8—1072no—
120#126—2142no—
121#126—2142no—
122#124—1072no—
123#124C2×D10712yes—
124C22⋊Q8—2142no—
125C4×Q8—1071no—
126C4×Q8—1072no—
127C4×Q8—1072no—
128C4×D4—1072no—
129#117—2142no—
130#107—1071no—
131C4⋊C8—1072no—
132C4⋊C8—1071no—
133#103—1072no—
134#103—1072no—
135#102—1072no—
136#97—1072no—
137C4×C8C2×D10711yescharacteristic
138C2×Q8—1071no—
139C2×Q8—1072no—
140C2×Q8C22×D10712yes—
141C2×D4C22×D10712yes—
142C22×C4—2142no—
143Q16—41no—
144Q16—4282no—
145Q16—2142no—
146SD16—24no—
147SD16—4282no—
148D8—41no—
149C2×C8—1072no—
150C2×C8C22×D10711yescharacteristic
151C2×C8C22×D10711yescharacteristic
152C4⋊C4—1072no—
153C4⋊C4—1071no—
154C4⋊C4—1072no—
155C4⋊C4—2142no—
156C4⋊C4—2142no—
157C4⋊C4—2142no—
158C4⋊C4—2142no—
159C22⋊C4—22no—
160C22⋊C4—22no—
161C22⋊C4—2142no—
162C22⋊C4—2142no—
163C22⋊C4—2142no—
164C42—1072no—
165C42—2142no—
166C42—2142no—
167C42C22×D10711yescharacteristic
168C23—22no—
169Q8—22no—
170Q8—22no—
171Q8—4281no—
172Q8—2142no—
173Q8—2142no—
174D4—22no—
175D4—22no—
176C2×C4—22no—
177C2×C4—1072no—
178C2×C4—1072no—
179C2×C4—4284no—
180C2×C4—2142no—
181C2×C4—2142no—
182C2×C4D4×D10711yescharacteristic
183C2×C4D4×D10711yescharacteristic
184C2×C4C23×D10711yescharacteristic
185C8—21no—
186C8—2142no—
187C8#1313381712yes—
188C22—42no—
189C22—42no—
190C22—42no—
191C22C2×D4×D10711yescharacteristic
192C4—21no—
193C4—21no—
194C4—42no—
195C4—4282no—
196C4—4282no—
197C4—2142no—
198C4—2142no—
199C4C2×#1313381711yescharacteristic
200C4#1681215411yescharacteristic
201C2—82no—
202C2#2029329711yescharacteristic
203C2#2029342411yescharacteristic
204C2#2029344111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#495×C107C2
socleC22×C107C22×D4
prime radicalC2×D8×C107C22
hypercenter#495D107
hypocenterC107#3402
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #20293346C2semidirect
2. #20293018C2semidirect
3. #20292846C2semidirect
5. #20292835C2semidirect
6. #20292757C2semidirect
7. #20292728C2semidirect
8. C4×Dic428C2semidirect
103. C107#3402semidirect

Maximal subgroups

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

subgroupindex
#202933462
#202930182
#202928462
#202928402
#202928352
#202927572
#202927282
C4×Dic4282
#495×C1072
#3402107