Finite Group Databasebeta

← #28337593

Subgroups — #28337593

Order 13696.

430 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 75 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
2C2×#16812093C212yesmaximal, complemented
3C2×#16812023C212yesmaximal, complemented
4#20292488C211yesmaximal, characteristic, complemented
5#342×C107C211yesmaximal, characteristic, complemented
6C22×C22⋊D107C2212yes—
7C23×Dic107C2211yescharacteristic
8#16812093—22no—
9#16812093—24no—
10#16812093—24no—
11#16812083—22no—
12#16812079—22no—
13C2×C4⋊Dic107—22no—
14C2×C22⋊Dic107C2211yescharacteristic
15C2×(C2×D107)⋊C4—22no—
16#16812023—24no—
17C107⋊#95—22no—
18C2×C22⋊C4×C107C2211yescharacteristic
19C2×C22⋊C4×C107C2212yes—
20C23×D107D412yes—
21C2×D428—22no—
22C2×C22⋊D107—22no—
23C2×C22⋊D107—22no—
24C2×C22⋊D107—22no—
25C2×C22⋊D107—22no—
26C2×C22⋊D107—22no—
27C2×C22⋊D107—22no—
28C22×Dic107—22no—
29C22×Dic107—22no—
30C22×Dic107—21no—
31C22×Dic107—21no—
32C22×Dic107—21no—
33C22×Dic107D412yes—
34C22×Dic107D412yescomplemented
35C4⋊Dic107—28no—
36C4⋊Dic107—22no—
37C22⋊Dic107—22no—
38(C2×D107)⋊C4—28no—
39(C2×D107)⋊C4—24no—
40C24×C107C2311yescharacteristic
41C22×C4×C107—22no—
42C22×C4×C107D412yes—
43C22⋊C4×C107—22no—
44C22⋊C4×C107—22no—
45C22⋊C4×C107D414yescomplemented
46C23×C107—42no—
47C23×C107C2×D412yes—
48C23×C107C2×D411yescharacteristic
49C23×C107C2×D411yescharacteristic
50C23×C107C2×D412yes—
51C23×C107#4111yescharacteristic
52C22×D107—22no—
53C22×D107—22no—
54C22×D107—22no—
55C22×D107—44no—
56C2×C4×C107—22no—
57C2×C4×C107—24no—
58C2×C4×C107—44no—
59C2×C4×C107—42no—
60C22⋊D107—42no—
61C22⋊D107—42no—
62C22⋊D107—42no—
63C22⋊D107—42no—
64C2×Dic107—24no—
65C2×Dic107—22no—
66C2×Dic107—22no—
67C2×Dic107—22no—
68C2×Dic107—22no—
69C2×Dic107—41no—
70C2×Dic107—42no—
71C2×Dic107—41no—
72C2×Dic107—42no—
73C2×Dic107—41no—
74D428—82no—
75C22×C107—22no—
76C22×C107—24no—
77C22×C107—41no—
78C22×C107—42no—
79C22×C107—41no—
80C22×C107#12012yes—
81C22×C107#12011yescharacteristic
82C22×C107#12111yescharacteristic
83C22×C107#12112yes—
84C22×C107#12711yescharacteristic
85C2×D107—44no—
86C2×D107—42no—
87C2×D107—44no—
88C2×D107—82no—
89C4×C107—44no—
90C4×C107—42no—
91Dic107—42no—
92Dic107—42no—
93Dic107—81no—
94C2×C107—21no—
95C2×C107—22no—
96C2×C107—22no—
97C2×C107—21no—
98C2×C107#39211yescharacteristic
99C2×C107#45812yes—
100D107—82no—
101#2022—1071nomaximal, Sylow
102C107#202211yescharacteristic, complemented, Sylow
103#534—1071no—
104C2×#123—1072no—
105C2×#99—1072no—
106#381—1071no—
107#342D10711yescharacteristic, complemented
108C22×D4—1072no—
109C23×C4—1071no—
110#123—2144no—
111#123—2142no—
112#123—2144no—
113#121—2142no—
114#120—2142no—
115C2×C4⋊C4—2142no—
116C2×C22⋊C4—1071no—
117C2×C22⋊C4—2142no—
118C2×C22⋊C4C2×D10711yescharacteristic
119C2×C22⋊C4C2×D10712yes—
120#99—2144no—
121#95—2142no—
122C24—1072no—
123C24C22×D10711yescharacteristic
124C2×D4—2142no—
125C2×D4—2142no—
126C2×D4—2142no—
127C2×D4—2142no—
128C2×D4—2142no—
129C2×D4—2142no—
130C2×D4—2142no—
131C22×C4—22no—
132C22×C4—1072no—
133C22×C4—1072no—
134C22×C4—2141no—
135C22×C4—2142no—
136C22×C4—2141no—
137C22×C4—2142no—
138C22×C4—2141no—
139C22×C4D42812yes—
140C4⋊C4—2148no—
141C4⋊C4—2142no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—2148no—
145C22⋊C4—2144no—
146C22⋊C4—2142no—
147C22⋊C4D42814yescomplemented
148C23—42no—
149C23—4284no—
150C23—2142no—
151C23—2142no—
152C23—2142no—
153C23D4×D10712yes—
154C23D4×D10711yescharacteristic
155C23C2×D42811yescharacteristic
156C23C2×D42812yes—
157C23#1313381911yescharacteristic
158D4—8562no—
159D4—4282no—
160D4—4282no—
161D4—4282no—
162D4—4282no—
163C2×C4—24no—
164C2×C4—22no—
165C2×C4—44no—
166C2×C4—42no—
167C2×C4—4282no—
168C2×C4—4281no—
169C2×C4—4281no—
170C2×C4—4282no—
171C2×C4—4281no—
172C2×C4—2144no—
173C2×C4—2142no—
174C2×C4—2142no—
175C2×C4—2142no—
176C2×C4—2142no—
177C22—22no—
178C22—24no—
179C22—41no—
180C22—42no—
181C22—41no—
182C22—8562no—
183C22—4284no—
184C22—4282no—
185C22—4284no—
186C22#1681207912yes—
187C22#1681207911yescharacteristic
188C22#1681208412yes—
189C22#1681208411yescharacteristic
190C22#1681210611yescharacteristic
191C4—44no—
192C4—42no—
193C4—8561no—
194C4—4282no—
195C4—4282no—
196C2—22no—
197C2—21no—
198C2—22no—
199C2—21no—
200C2—8562no—
201C2#2029253711yescharacteristic
202C2#2029281812yes—
subgroupG/H
centerC22#16812079
commutator, cosocleC24×C107C23
FrattiniC24C22×D107
Fitting#342×C107C2
socleC22×C107#120
prime radical#20293189C2
hypercenter#342D107
hypocenterC107#2022
LCSG ▷ C24×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. C2×#16812093C2semidirect
3. C2×#16812023C2semidirect
4. #20292488C2semidirect
5. #342×C107C2semidirect
34. C22×Dic107D4semidirect
45. C22⋊C4×C107D4semidirect
102. C107#2022semidirect
107. #342D107semidirect
147. C22⋊C4D428semidirect

Maximal subgroups

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

subgroupindex
#202931892
C2×#168120932
C2×#168120232
#202924882
#342×C1072
#2022107