Finite Group Databasebeta

← #28339262

Subgroups — #28339262

Order 13696.

428 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 177 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
1C2×#16812101C211yesmaximal, characteristic
2#20292524C212yesmaximal
3#20292514C212yesmaximal
4#20292512C212yesmaximal
5#20292510C212yesmaximal
6#20292509C212yesmaximal
7C4×C4⋊Dic107C211yesmaximal, characteristic
8#20292478C212yesmaximal
9#383×C107C211yesmaximal, characteristic
10#16812101C418yescomplemented
11C2×C4⋊Dic107C2212yes—
12C2×C4⋊Dic107C2211yescharacteristic
13C2×C4⋊Dic107C2211yescharacteristic
14C2×Dic107⋊C4C2212yes—
15C2×Dic107⋊C4C2212yes—
16C2×Dic107⋊C4C2212yes—
17C2×Dic107⋊C4C2212yes—
18C2×C4×Dic107C2212yes—
19C2×C4×Dic107C2211yescharacteristic
20C2×C4×Dic107C2212yes—
21C2×C4×Dic107C2211yescharacteristic
22C107⋊#95C2212yes—
23C107⋊#95C2212yes—
24C107⋊#95C2212yes—
25C107⋊#95C2212yes—
26C107⋊#95C2212yes—
27C2×C42×C107C2211yescharacteristic
28#95×C107C2212yes—
29#95×C107C2212yes—
30#95×C107C2211yescharacteristic
31#95×C107C2211yescharacteristic
32C22×Dic107C2312yes—
33C22×Dic107C2312yes—
34C22×Dic107C2312yes—
35C22×Dic107C2312yes—
36C4⋊Dic107—24no—
37C4⋊Dic107C2×C414yes—
38C4⋊Dic107C2×C414yes—
39Dic107⋊C4—24no—
40Dic107⋊C4—42no—
41Dic107⋊C4C2×C418yes—
42Dic107⋊C4C2×C418yes—
43C4×Dic107—24no—
44C4×Dic107—41no—
45C4×Dic107—42no—
46C4×Dic107—41no—
47C22×C4×C107C2312yes—
48C22×C4×C107C2312yes—
49C22×C4×C107C2311yescharacteristic
50C22×C4×C107C2311yescharacteristic
51C22×C4×C107C2311yescharacteristic
52C42×C107C2×C414yes—
53C23×C107C2411yescharacteristic
54C2×C4×C107—22no—
55C2×C4×C107—21no—
56C2×C4×C107—22no—
57C2×C4×C107—21no—
58C2×C4×C107—22no—
59C2×C4×C107—22no—
60C2×C4×C107—22no—
61C2×C4×C107—22no—
62C2×C4×C107—22no—
63C2×C4×C107—22no—
64C2×C4×C107C22×C412yes—
65C2×C4×C107C22×C412yes—
66C2×C4×C107C22×C412yes—
67C2×Dic107—22no—
68C2×Dic107—22no—
69C2×Dic107—22no—
70C2×Dic107—22no—
71C2×Dic107—22no—
72C2×Dic107—22no—
73C2×Dic107—22no—
74C2×Dic107—22no—
75C2×Dic107—22no—
76C2×Dic107C22×C414yes—
77C2×Dic107C22×C414yes—
78C2×Dic107#4114yes—
79C22×C107C23×C411yescharacteristic
80C22×C107C2×#4112yes—
81C22×C107H(5,𝔽2)11yescharacteristic
82C22×C1072−1+412yes—
83C22×C1072−1+411yescharacteristic
84C4×C107—22no—
85C4×C107—22no—
86C4×C107—22no—
87C4×C107—42no—
88C4×C107—42no—
89Dic107—24no—
90Dic107—24no—
91Dic107—24no—
92Dic107—42no—
93C2×C107C4×#4111yescharacteristic
94C2×C107#51911yescharacteristic
95C2×C107#52011yescharacteristic
96C2×C107#53912yes—
97C2×C107#54112yes—
98#2336—1071nomaximal, Sylow
99C107#233611yescharacteristic, complemented, Sylow
100C2×#125—1071no—
101#389—1072no—
102#387—1072no—
103#387—1072no—
104#387—1072no—
105#387—1072no—
106#383D10711yescharacteristic
107C4×C4⋊C4—1071no—
108C4×C4⋊C4—1072no—
109#125—1078no—
110C2×C4⋊C4—1072no—
111C2×C4⋊C4—1072no—
112C2×C4⋊C4—1071no—
113C2×C4⋊C4—1071no—
114C2×C4⋊C4—1072no—
115C2×C4⋊C4—1072no—
116C2×C4⋊C4—1072no—
117C2×C42—1071no—
118C2×C42—1071no—
119C2×C42—1072no—
120C2×C42—1072no—
121C2×C42C2×D10711yescharacteristic
122#95—1072no—
123#95—1072no—
124#95—1072no—
125#95—1072no—
126#95—1072no—
127#95C2×D10712yes—
128#95C2×D10711yescharacteristic
129#95C2×D10712yes—
130#95C2×D10711yescharacteristic
131C22×C4—1072no—
132C22×C4—1072no—
133C22×C4—1072no—
134C22×C4—1072no—
135C22×C4C22×D10712yes—
136C22×C4C22×D10712yes—
137C22×C4C22×D10711yescharacteristic
138C22×C4C22×D10711yescharacteristic
139C22×C4C22×D10711yescharacteristic
140C4⋊C4—1078no—
141C4⋊C4—1078no—
142C4⋊C4—1074no—
143C4⋊C4—1074no—
144C4⋊C4—4282no—
145C4⋊C4—2144no—
146C4⋊C4—2144no—
147C42—4281no—
148C42—4281no—
149C42—4282no—
150C42—2144no—
151C42C4×D10714yes—
152C23C23×D10711yescharacteristic
153C2×C4—22no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—1074no—
164C2×C4—1074no—
165C2×C4—1074no—
166C2×C4—2142no—
167C2×C4—2142no—
168C2×C4—2142no—
169C2×C4—2142no—
170C2×C4—2142no—
171C2×C4—2142no—
172C2×C4—2142no—
173C2×C4—2142no—
174C2×C4—2142no—
175C2×C4C2×C4×D10712yes—
176C2×C4C2×C4×D10712yes—
177C2×C4C2×C4×D10712yes—
178C22C22×C4×D10711yescharacteristic
179C22#41×D10712yes—
180C22#1681215511yescharacteristic
181C22#1681215612yes—
182C22#1681215611yescharacteristic
183C4—22no—
184C4—22no—
185C4—22no—
186C4—42no—
187C4—42no—
188C4—4282no—
189C4—2144no—
190C4—2144no—
191C4—2144no—
192C2#2029309511yescharacteristic
193C2#2029310411yescharacteristic
194C2#2029310711yescharacteristic
195C2#2029321812yes—
196C2#2029323412yes—
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
Fitting#383×C107C2
socle, prime radical, cosocleC23×C107C24
hypercenter#383D107
hypocenterC107#2336
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ #383

Complements

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

NH ≅ G/Ntype
10. #16812101C4semidirect
99. C107#2336semidirect

Maximal subgroups

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

subgroupindex
C2×#168121012
#202925242
#202925142
#202925122
#202925102
#202925092
C4×C4⋊Dic1072
#202924782
#383×C1072
#2336107