Finite Group Databasebeta

← #28335446

Subgroups — #28335446

Order 13696.

280 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 53 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#20293189C211yesmaximal, characteristic
2#20292940C212yesmaximal, complemented
3#20292360C212yesmaximal, complemented
4#20292351C211yesmaximal, characteristic, complemented
5#397×C107C211yesmaximal, characteristic, complemented
6C22×C22⋊D107—21no—
7#16812093—22no—
8#16812083—21no—
9#16812083C2212yes—
10#16812079—22no—
11#16812042—22no—
12#16812029—22no—
13C22⋊(C107⋊C8)C2212yescomplemented
14C2×C4⋊C4×C107C2211yescharacteristic, complemented
15C2×C22⋊C4×C107—21no—
16C2×C22⋊C4×C107C2211yescharacteristic
17#95×C107—21no—
18#95×C107C2211yescharacteristic, complemented
19C23×D107—21no—
20C2×D428—21no—
21C2×D428D412yes—
22C2×C22⋊D107—22no—
23C2×C22⋊D107—22no—
24C2×C22⋊D107—42no—
25C2×C22⋊D107—42no—
26C22×Dic107—21no—
27C2×C107⋊C8—22no—
28C4⋊Dic107—21no—
29C4⋊Dic107D412yes—
30(C2×D107)⋊C4—22no—
31(C2×D107)⋊C4—22no—
32C24×C107D411yescharacteristic
33C22×C4×C107—21no—
34C22×C4×C107—21no—
35C22×C4×C107D411yescharacteristic
36C22×C4×C107C2311yescharacteristic
37C4⋊C4×C107—22no—
38C22⋊C4×C107—22no—
39C22⋊C4×C107—42no—
40C23×C107—21no—
41C23×C107—21no—
42C23×C107—41no—
43C23×C107—41no—
44C23×C107D812yes—
45C23×C107C2×D411yescharacteristic
46C22×D107—22no—
47C22×D107—42no—
48C22×D107—81no—
49C22×D107—81no—
50C2×C4×C107—21no—
51C2×C4×C107—41no—
52C2×C4×C107—41no—
53C2×C4×C107—41no—
54C2×C4×C107—41no—
55C2×C4×C107—41no—
56C2×C4×C107—41no—
57C2×C4×C107—41no—
58C2×C4×C107—41no—
59C2×C4×C107C2×D412yes—
60C22⋊D107—44no—
61C22⋊D107—42no—
62C22⋊D107—42no—
63C2×Dic107—22no—
64C2×Dic107—42no—
65D428—42no—
66D428—81no—
67C107⋊C8—42no—
68C22×C107—21no—
69C22×C107—21no—
70C22×C107—22no—
71C22×C107—42no—
72C22×C107—41no—
73C22×C107—41no—
74C22×C107—81no—
75C22×C107#12011yescharacteristic
76C22×C107C2×D811yescharacteristic
77C22×C107AGL(1,ℤ/8)11yescharacteristic
78C2×D107—42no—
79C2×D107—42no—
80C2×D107—42no—
81C2×D107—84no—
82C4×C107—22no—
83C4×C107—41no—
84C4×C107—81no—
85C4×C107—81no—
86Dic107—42no—
87C2×C107—21no—
88C2×C107—21no—
89C2×C107—42no—
90C2×C107#44711yescharacteristic
91C2×C107#45511yescharacteristic
92C2×C107#45811yescharacteristic
93D107—82no—
94#1607—1071nomaximal, Sylow
95C107#160711yescharacteristic, complemented, Sylow
96#534—1071no—
97#480—1072no—
98#397D10711yescharacteristic, complemented
99#327—1072no—
100#323—1071no—
101C22×D4—2141no—
102#123—2142no—
103#121—1072no—
104#121—2141no—
105#120—2142no—
106C2×C4⋊C4C2×D10711yescharacteristic, complemented
107C2×C22⋊C4—21no—
108C2×C22⋊C4C2×D10711yescharacteristic
109#107—2142no—
110#102—2142no—
111C22⋊C8—1072no—
112#95—21no—
113#95C2×D10711yescharacteristic, complemented
114C24—2141no—
115C24C22⋊D10711yescharacteristic
116C2×D4—1072no—
117C2×D4—4282no—
118C2×D4—4282no—
119C2×D4—2141no—
120C2×D4—2142no—
121C2×D4—2142no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—2141no—
125C22×C4C22⋊D10711yescharacteristic
126C22×C4C22×D10711yescharacteristic
127C2×C8—2142no—
128C4⋊C4—22no—
129C4⋊C4—1072no—
130C4⋊C4—2141no—
131C22⋊C4—22no—
132C22⋊C4—42no—
133C22⋊C4—2142no—
134C22⋊C4—2142no—
135C23—21no—
136C23—21no—
137C23—41no—
138C23—41no—
139C23—8561no—
140C23—8561no—
141C23—4282no—
142C23—2142no—
143C23#1313380312yes—
144C23C2×C22⋊D10711yescharacteristic
145D4—8561no—
146D4—4282no—
147D4—4284no—
148D4—4282no—
149D4—4282no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—4282no—
160C2×C4—2142no—
161C2×C4D4×D10712yes—
162C8—4282no—
163C22—21no—
164C22—21no—
165C22—22no—
166C22—42no—
167C22—41no—
168C22—41no—
169C22—81no—
170C22—8564no—
171C22—4282no—
172C22—4282no—
173C22—4282no—
174C22#1681207711yescharacteristic
175C22C2×#1313380311yescharacteristic
176C22#1681213411yescharacteristic
177C4—22no—
178C4—41no—
179C4—81no—
180C4—81no—
181C4—4282no—
182C2—21no—
183C2—21no—
184C2—42no—
185C2—8562no—
186C2#2029276011yescharacteristic
187C2#2029280411yescharacteristic
188C2#2029281611yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#397×C107C2
socleC22×C107#120
prime radical#20293189C2
hypercenter#397D107
hypocenterC107#1607
LCSG ▷ C22×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #20292940C2semidirect
3. #20292360C2semidirect
4. #20292351C2semidirect
5. #397×C107C2semidirect
13. C22⋊(C107⋊C8)C22semidirect
14. C2×C4⋊C4×C107C22semidirect
18. #95×C107C22semidirect
95. C107#1607semidirect
98. #397D107semidirect
106. C2×C4⋊C4C2×D107semidirect
113. #95C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931892
#202929402
#202923602
#202923512
#397×C1072
#1607107