Finite Group Databasebeta

← #28347907

Subgroups — #28347907

Order 13696.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#20293239C211yesmaximal, characteristic, complemented
2#20292931C212yesmaximal, complemented
3#20292927C212yesmaximal, complemented
4#20292871C212yesmaximal, complemented
5#20292854C212yesmaximal, complemented
6#20292832C212yesmaximal, complemented
7#20292824C212yesmaximal, complemented
8#20292654C211yesmaximal, characteristic, complemented
9#540×C107C211yesmaximal, characteristic
10C2×#13133806C2212yes—
11C2×#13133806C2212yes—
12C2×#13133802C2211yescharacteristic
13C22×C107⋊C8C2211yescharacteristic
14#16812109C2211yescharacteristic
15#16812102—22no—
16#16812101C2211yescharacteristic
17#16812099—22no—
18#16812088C2212yes—
19C4×Dic214C2212yes—
20#16812042C2212yes—
21#16812040C2212yes—
22C4⋊(C107⋊C8)C2212yes—
23C4⋊(C107⋊C8)C2212yes—
24#16812032C2212yes—
25#16812032C2212yes—
26#16812030C2212yes—
27#16812030C2212yes—
28#127×C107C2211yescharacteristic
29#125×C107C2211yescharacteristic
30#123×C107—22no—
31#121×C107—22no—
32#121×C107C2212yes—
33C4×D4×C107C2212yes—
34#117×C107C2211yescharacteristic
35C2×Dic214C2312yes—
36#13133806—42no—
37#13133806—42no—
38#13133802—41no—
39C2×C107⋊C8—41no—
40C2×C107⋊C8C2312yes—
41C2×C107⋊C8C2312yes—
42C4⋊Dic107—22no—
43C4⋊Dic107C2312yes—
44Dic107⋊C4—22no—
45Dic107⋊C4—22no—
46Dic107⋊C4—22no—
47C22⋊Dic107—22no—
48C4×Dic107—22no—
49C2×D4×C107—22no—
50C2×D4×C107—22no—
51C2×D4×C107C2312yes—
52C22×C4×C107—22no—
53C22×C4×C107C2311yescharacteristic
54C4⋊C4×C107—22no—
55C4⋊C4×C107D412yes—
56C4⋊C4×C107C2312yes—
57C22⋊C4×C107—22no—
58C22⋊C4×C107—22no—
59C22⋊C4×C107D412yes—
60C42×C107C2311yescharacteristic
61C42×C107C2311yescharacteristic
62C23×C107—22no—
63C23×C107C2×D411yescharacteristic
64D4×C107—22no—
65D4×C107—22no—
66D4×C107—82no—
67D4×C107—82no—
68C2×C4×C107—21no—
69C2×C4×C107—21no—
70C2×C4×C107—22no—
71C2×C4×C107—82no—
72C2×C4×C107C2×D412yes—
73C2×C4×C107C2×D412yes—
74C2×C4×C107C2×D411yescharacteristic
75C2×C4×C107C2411yescharacteristic
76C2×Dic107—22no—
77C2×Dic107—22no—
78Dic214—22no—
79Dic214—22no—
80C107⋊C8—22no—
81C107⋊C8—22no—
82C22×C107—21no—
83C22×C107—22no—
84C22×C107—42no—
85C22×C107—42no—
86C22×C107—42no—
87C22×C107C22×D411yescharacteristic
88C4×C107—21no—
89C4×C107—22no—
90C4×C107—22no—
91C4×C107—42no—
92C4×C107H(5,𝔽2)11yescharacteristic
93C4×C1072−1+411yescharacteristic
94Dic107—42no—
95Dic107—42no—
96C2×C107—41no—
97C2×C107—82no—
98C2×C107#53711yescharacteristic
99C2×C107#57712yes—
100#3260—1071nomaximal, Sylow
101C107#326011yescharacteristic, complemented, Sylow
102#541—1071no—
103#540D10711yescharacteristic
104#479—1072no—
105#478—1072no—
106#468—1072no—
107#465—1072no—
108#461—1072no—
109#460—1072no—
110#424—1071no—
111C2×SD16—1072no—
112C2×SD16—1072no—
113C2×M16—1071no—
114C22×C8—1071no—
115C4⋊Q8—1071no—
116#127C2×D10711yescharacteristic
117#126—2142no—
118#125—1071no—
119#125—2142no—
120#125C2×D10711yescharacteristic
121#123—22no—
122C22⋊Q8—1072no—
123#121—22no—
124#121C2×D10712yes—
125C4×Q8—1072no—
126C4×D4C2×D10712yes—
127#117C2×D10711yescharacteristic
128#107—1072no—
129#106—1072no—
130C4⋊C8—1072no—
131C4⋊C8—1072no—
132#103—1072no—
133#103—1072no—
134#102—1072no—
135#102—1072no—
136C2×Q8—1072no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4C22×D10712yes—
140C22×C4—22no—
141C22×C4C22×D10711yescharacteristic
142SD16—4282no—
143SD16—4282no—
144M16—4281no—
145C2×C8—1072no—
146C2×C8—1072no—
147C2×C8—4281no—
148C4⋊C4—22no—
149C4⋊C4—1072no—
150C4⋊C4—2142no—
151C4⋊C4—2142no—
152C4⋊C4—2142no—
153C4⋊C4—2142no—
154C4⋊C4C22⋊D10712yes—
155C4⋊C4C22×D10712yes—
156C22⋊C4—22no—
157C22⋊C4—22no—
158C22⋊C4—2142no—
159C22⋊C4C22⋊D10712yes—
160C42—2142no—
161C42C22×D10711yescharacteristic
162C42C22×D10711yescharacteristic
163C23—22no—
164C23C2×C22⋊D10711yescharacteristic
165Q8—2142no—
166Q8—2142no—
167D4—22no—
168D4—22no—
169D4—82no—
170D4—82no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—82no—
175C2×C4—2142no—
176C2×C4—2142no—
177C2×C4C2×C22⋊D10712yes—
178C2×C4C2×C22⋊D10712yes—
179C2×C4C2×C22⋊D10711yescharacteristic
180C2×C4C23×D10711yescharacteristic
181C8—2142no—
182C8—2142no—
183C22—22no—
184C22—21no—
185C22—42no—
186C22—42no—
187C22—42no—
188C22C22×C22⋊D10711yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—42no—
193C4—4282no—
194C4—4282no—
195C4#1681215511yescharacteristic
196C4#1681215611yescharacteristic
197C2—41no—
198C2—82no—
199C2#2029320511yescharacteristic
200C2#2029343812yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting, prime radical#540×C107C2
socleC22×C107C22×D4
hypercenter#540D107
hypocenterC107#3260
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #20293239C2semidirect
2. #20292931C2semidirect
3. #20292927C2semidirect
4. #20292871C2semidirect
5. #20292854C2semidirect
6. #20292832C2semidirect
7. #20292824C2semidirect
8. #20292654C2semidirect
101. C107#3260semidirect

Maximal subgroups

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

subgroupindex
#202932392
#202929312
#202929272
#202928712
#202928542
#202928322
#202928242
#202926542
#540×C1072
#3260107