Finite Group Databasebeta

← #28347910

Subgroups — #28347910

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#20292936C212yesmaximal, complemented
3#20292933C212yesmaximal, complemented
4#20292883C212yesmaximal
5#20292868C211yesmaximal, characteristic
6#20292836C212yesmaximal, complemented
7#20292835C212yesmaximal, complemented
8#20292653C211yesmaximal, characteristic
9#467×C107C211yesmaximal, characteristic
10C2×Dic428C2211yescharacteristic
11C2×#13133808C2212yes—
12C2×#13133802C2211yescharacteristic
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#16812043C2211yescharacteristic
24#16812040C2212yes—
25Dic107⋊C8C2212yes—
26Dic107⋊C8C2212yes—
27#16812034C2212yes—
28#16812033C2212yes—
29#16812032C2212yes—
30C2×Q16×C107C2211yescharacteristic
31C22×C8×C107C2211yescharacteristic
32C22⋊Q8×C107C2212yes—
33#107×C107C2211yescharacteristic
34#103×C107C2212yes—
35C2×Dic214C2312yes—
36Dic428—41no—
37#13133808—42no—
38#13133802—41no—
39C2×C107⋊C8C2312yes—
40C4⋊Dic107—22no—
41C4⋊Dic107—22no—
42C4⋊Dic107D411yescharacteristic
43C4⋊Dic107D411yescharacteristic
44C4⋊Dic107C2312yes—
45Dic107⋊C4—22no—
46Dic107⋊C4—22no—
47Dic107⋊C4—22no—
48Dic107⋊C4—22no—
49Dic107⋊C4—22no—
50C22⋊Dic107—22no—
51C22⋊Dic107D412yes—
52C4×Dic107—22no—
53C4×Dic107—22no—
54C4×Dic107C2312yes—
55C2×Q8×C107C2312yes—
56C22×C4×C107C2311yescharacteristic
57Q16×C107—41no—
58C2×C8×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—21no—
84C22×C107—21no—
85C22×C107C22×D411yescharacteristic
86C4×C107—21no—
87C4×C107—42no—
88C4×C107—42no—
89C4×C107H(5,𝔽2)11yescharacteristic
90C4×C1072−1+411yescharacteristic
91Dic107—22no—
92Dic107—22no—
93Dic107—42no—
94Dic107—42no—
95C2×C107—41no—
96C2×C107#53711yescharacteristic
97C2×C107#57811yescharacteristic
98C2×C107#57811yescharacteristic
99#3261—1071nomaximal, Sylow
100C107#326111yescharacteristic, complemented, Sylow
101#541—1072no—
102#479—1072no—
103#479—1072no—
104#470—1072no—
105#467—1071no—
106#467D10711yescharacteristic
107#462—1072no—
108#462—1072no—
109#424—1071no—
110C2×Q16—1072no—
111C2×Q16—1071no—
112C2×Q16C2×D10711yescharacteristic
113C2×M16—1071no—
114C22×C8C2×D10711yescharacteristic
115C4⋊Q8—1072no—
116#126—2142no—
117#126—2142no—
118#125—1072no—
119#125—2142no—
120#125—2142no—
121C22⋊Q8—1072no—
122C22⋊Q8C2×D10712yes—
123C4×Q8—1072no—
124C4×Q8—1072no—
125#117—1071no—
126#107—1071no—
127#107C2×D10711yescharacteristic
128#106—1072no—
129C4⋊C8—1072no—
130C4⋊C8—1072no—
131#103—1072no—
132#103—1072no—
133#103—1072no—
134#103C2×D10712yes—
135C2×Q8—1072no—
136C2×Q8C22×D10712yes—
137C22×C4C22×D10711yescharacteristic
138Q16—41no—
139Q16—4281no—
140Q16—4282no—
141M16—4281no—
142C2×C8—41no—
143C2×C8—1072no—
144C2×C8C22×D10711yescharacteristic
145C2×C8C22×D10711yescharacteristic
146C4⋊C4—22no—
147C4⋊C4—1071no—
148C4⋊C4—1071no—
149C4⋊C4—1072no—
150C4⋊C4—2142no—
151C4⋊C4—2142no—
152C4⋊C4—2142no—
153C4⋊C4—2142no—
154C4⋊C4—2142no—
155C4⋊C4—2142no—
156C4⋊C4—2142no—
157C4⋊C4C22×D10712yes—
158C22⋊C4—22no—
159C22⋊C4—1072no—
160C22⋊C4—2142no—
161C42—1072no—
162C42—2142no—
163C42—2142no—
164C23D4×D10711yescharacteristic
165Q8—22no—
166Q8—22no—
167Q8—2142no—
168Q8—2142no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—1072no—
174C2×C4—1072no—
175C2×C4—2142no—
176C2×C4—2142no—
177C2×C4D4×D10711yescharacteristic
178C2×C4C23×D10711yescharacteristic
179C8—21no—
180C8—21no—
181C8—2142no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22C2×D4×D10711yescharacteristic
186C4—21no—
187C4—42no—
188C4—42no—
189C4—4282no—
190C4—4282no—
191C4—2142no—
192C4—2142no—
193C4#1681215411yescharacteristic
194C4#1681215611yescharacteristic
195C2—41no—
196C2#2029320411yescharacteristic
197C2#2029344011yescharacteristic
198C2#2029344111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#467×C107C2
socleC22×C107C22×D4
prime radicalC23×C107C2×D4
hypercenter#467D107
hypocenterC107#3261
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
2. #20292936C2semidirect
3. #20292933C2semidirect
6. #20292836C2semidirect
7. #20292835C2semidirect
100. C107#3261semidirect

Maximal subgroups

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

subgroupindex
#202932372
#202929362
#202929332
#202928832
#202928682
#202928362
#202928352
#202926532
#467×C1072
#3261107