Finite Group Databasebeta

← #28347912

Subgroups — #28347912

Order 13696.

322 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 109 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#20293237C212yesmaximal
2#20292937C212yesmaximal, complemented
3#20292932C212yesmaximal, complemented
4#20292885C211yesmaximal, characteristic
5#20292866C212yesmaximal
6#20292836C212yesmaximal, complemented
7#20292835C212yesmaximal, complemented
8#20292655C211yesmaximal, characteristic
9#470×C107C211yesmaximal, characteristic
10C2×Dic428C2211yescharacteristic
11C2×#13133808C2212yes—
12C22×C107⋊C8C2211yescharacteristic
13#16812107C2212yes—
14#16812102—22no—
15#16812102—22no—
16#16812100—22no—
17#16812100C2212yes—
18#16812099—22no—
19#16812088C2212yes—
20Q8×Dic107C2212yes—
21#16812074C2212yes—
22#16812068C2211yescharacteristic
23#16812042C2212yes—
24#16812041C2211yescharacteristic
25Dic107⋊C8C2212yes—
26Dic107⋊C8C2212yes—
27#16812034C2212yes—
28#16812033C2212yes—
29#16812032C2212yes—
30C2×Q16×C107C2211yescharacteristic
31C2×M16×C107C2211yescharacteristic
32C22⋊Q8×C107C2212yes—
33#106×C107C2211yescharacteristic
34#103×C107C2212yes—
35C2×Dic214C2312yes—
36Dic428—41no—
37#13133808—42no—
38C2×C107⋊C8—41no—
39C2×C107⋊C8C2312yes—
40C4⋊Dic107—22no—
41C4⋊Dic107—22no—
42C4⋊Dic107D412yes—
43C4⋊Dic107C2312yes—
44Dic107⋊C4—22no—
45Dic107⋊C4—22no—
46Dic107⋊C4—22no—
47Dic107⋊C4—22no—
48Dic107⋊C4—22no—
49C22⋊Dic107—22no—
50C22⋊Dic107D411yescharacteristic
51C22⋊Dic107D411yescharacteristic
52C4×Dic107—22no—
53C4×Dic107—22no—
54C4×Dic107C2312yes—
55C2×Q8×C107C2312yes—
56C22×C4×C107C2311yescharacteristic
57Q16×C107—41no—
58M16×C107—41no—
59C2×C8×C107C2311yescharacteristic
60C2×C8×C107C2311yescharacteristic
61C4⋊C4×C107—22no—
62C4⋊C4×C107C2312yes—
63C22⋊C4×C107—22no—
64C23×C107C2×D411yescharacteristic
65Q8×C107—22no—
66Q8×C107—22no—
67C2×C4×C107—21no—
68C2×C4×C107—21no—
69C2×C4×C107—22no—
70C2×C4×C107—22no—
71C2×C4×C107C2×D411yescharacteristic
72C2×C4×C107C2411yescharacteristic
73C2×Dic107—22no—
74C2×Dic107—22no—
75C2×Dic107C2×D412yes—
76C2×Dic107C2×D412yes—
77Dic214—22no—
78Dic214—22no—
79C8×C107—21no—
80C8×C107—21no—
81C107⋊C8—22no—
82C22×C107—21no—
83C22×C107—22no—
84C22×C107C22×D411yescharacteristic
85C4×C107—21no—
86C4×C107—42no—
87C4×C107—42no—
88C4×C107H(5,𝔽2)11yescharacteristic
89C4×C1072−1+411yescharacteristic
90Dic107—22no—
91Dic107—22no—
92Dic107—42no—
93Dic107—42no—
94C2×C107—41no—
95C2×C107#53711yescharacteristic
96C2×C107#57812yes—
97#3261—1071nomaximal, Sylow
98C107#326111yescharacteristic, complemented, Sylow
99#541—1072no—
100#479—1072no—
101#479—1072no—
102#470—1071no—
103#470D10711yescharacteristic
104#467—1072no—
105#462—1072no—
106#462—1072no—
107#424—1071no—
108C2×Q16—1072no—
109C2×Q16—1071no—
110C2×Q16C2×D10711yescharacteristic
111C2×M16C2×D10711yescharacteristic
112C22×C8—1071no—
113C4⋊Q8—1072no—
114#126—2142no—
115#126—2142no—
116#125—1072no—
117#125—2142no—
118#125—2142no—
119C22⋊Q8—1072no—
120C22⋊Q8C2×D10712yes—
121C4×Q8—1072no—
122C4×Q8—1072no—
123#117—1071no—
124#107—1072no—
125#106—1071no—
126#106C2×D10711yescharacteristic
127C4⋊C8—1072no—
128C4⋊C8—1072no—
129#103—1072no—
130#103—1072no—
131#103—1072no—
132#103C2×D10712yes—
133C2×Q8—1072no—
134C2×Q8C22×D10712yes—
135C22×C4C22×D10711yescharacteristic
136Q16—41no—
137Q16—4281no—
138Q16—4282no—
139M16—41no—
140C2×C8—1072no—
141C2×C8—4281no—
142C2×C8C22×D10711yescharacteristic
143C2×C8C22×D10711yescharacteristic
144C4⋊C4—22no—
145C4⋊C4—1072no—
146C4⋊C4—1072no—
147C4⋊C4—2142no—
148C4⋊C4—2142no—
149C4⋊C4—2142no—
150C4⋊C4—2142no—
151C4⋊C4—2142no—
152C4⋊C4—2142no—
153C4⋊C4—2142no—
154C4⋊C4C22×D10712yes—
155C22⋊C4—22no—
156C22⋊C4—1071no—
157C22⋊C4—1071no—
158C22⋊C4—2142no—
159C42—1072no—
160C42—2142no—
161C42—2142no—
162C23D4×D10711yescharacteristic
163Q8—22no—
164Q8—22no—
165Q8—2142no—
166Q8—2142no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—1072no—
172C2×C4—1072no—
173C2×C4—2142no—
174C2×C4—2142no—
175C2×C4D4×D10711yescharacteristic
176C2×C4C23×D10711yescharacteristic
177C8—21no—
178C8—21no—
179C8—2142no—
180C22—22no—
181C22—21no—
182C22C2×D4×D10711yescharacteristic
183C4—21no—
184C4—42no—
185C4—42no—
186C4—4282no—
187C4—4282no—
188C4—2142no—
189C4—2142no—
190C4#1681215411yescharacteristic
191C4#1681215611yescharacteristic
192C2—41no—
193C2#2029320411yescharacteristic
194C2#2029344312yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#470×C107C2
socleC22×C107C22×D4
prime radicalC23×C107C2×D4
hypercenter#470D107
hypocenterC107#3261
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
2. #20292937C2semidirect
3. #20292932C2semidirect
6. #20292836C2semidirect
7. #20292835C2semidirect
98. C107#3261semidirect

Maximal subgroups

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

subgroupindex
#202932372
#202929372
#202929322
#202928852
#202928662
#202928362
#202928352
#202926552
#470×C1072
#3261107