Finite Group Databasebeta

← #28339646

Subgroups — #28339646

Order 13696.

408 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 173 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#20292522C212yesmaximal
4#20292514C212yesmaximal
5#20292512C212yesmaximal
6#20292510C212yesmaximal
7#20292508C212yesmaximal
8#20292493C211yesmaximal, characteristic
9#382×C107C211yesmaximal, characteristic
10#16812101C418yescomplemented
11C2×C4⋊Dic107C2211yescharacteristic
12C2×C4⋊Dic107C2211yescharacteristic
13C2×C4⋊Dic107C2211yescharacteristic
14C2×C4⋊Dic107C2211yescharacteristic
15C2×Dic107⋊C4C2212yes—
16C2×Dic107⋊C4C2212yes—
17C2×Dic107⋊C4C2212yes—
18C2×Dic107⋊C4C2212yes—
19C2×C4×Dic107C2212yes—
20C2×C4×Dic107C2212yes—
21C107⋊#95C2212yes—
22C107⋊#95C2212yes—
23C107⋊#95C2212yes—
24C107⋊#95C2212yes—
25C107⋊#95C2212yes—
26C107⋊#95C2212yes—
27C2×C4⋊C4×C107C2211yescharacteristic
28C2×C4⋊C4×C107C2211yescharacteristic
29C2×C42×C107C2211yescharacteristic
30#95×C107C2212yes—
31#95×C107C2212yes—
32C22×Dic107C2312yes—
33C22×Dic107C2312yes—
34C22×Dic107C2312yes—
35C22×Dic107C2312yes—
36C4⋊Dic107—41no—
37C4⋊Dic107—41no—
38C4⋊Dic107C2×C414yes—
39C4⋊Dic107C2×C414yes—
40Dic107⋊C4—42no—
41Dic107⋊C4—42no—
42Dic107⋊C4C2×C418yes—
43Dic107⋊C4C2×C418yes—
44C4×Dic107—42no—
45C4×Dic107—42no—
46C22×C4×C107C2312yes—
47C22×C4×C107C2312yes—
48C22×C4×C107C2311yescharacteristic
49C22×C4×C107C2311yescharacteristic
50C22×C4×C107C2311yescharacteristic
51C4⋊C4×C107—41no—
52C4⋊C4×C107—41no—
53C42×C107C2×C414yes—
54C23×C107C2411yescharacteristic
55C2×C4×C107—21no—
56C2×C4×C107—21no—
57C2×C4×C107—21no—
58C2×C4×C107—21no—
59C2×C4×C107—21no—
60C2×C4×C107—21no—
61C2×C4×C107—22no—
62C2×C4×C107—22no—
63C2×C4×C107—22no—
64C2×C4×C107—22no—
65C2×C4×C107—22no—
66C2×C4×C107—22no—
67C2×C4×C107C22×C412yes—
68C2×C4×C107C22×C412yes—
69C2×C4×C107C22×C412yes—
70C2×Dic107—22no—
71C2×Dic107—22no—
72C2×Dic107—22no—
73C2×Dic107—22no—
74C2×Dic107—22no—
75C2×Dic107—22no—
76C2×Dic107—22no—
77C2×Dic107—22no—
78C2×Dic107—22no—
79C2×Dic107—22no—
80C2×Dic107C22×C414yes—
81C2×Dic107C22×C414yes—
82C22×C107C23×C411yescharacteristic
83C22×C107H(5,𝔽2)11yescharacteristic
84C22×C107H(5,𝔽2)11yescharacteristic
85C22×C1072−1+411yescharacteristic
86C22×C1072−1+411yescharacteristic
87C22×C1072−1+411yescharacteristic
88C22×C1072−1+411yescharacteristic
89C4×C107—22no—
90C4×C107—22no—
91C4×C107—22no—
92C4×C107—42no—
93C4×C107—42no—
94Dic107—24no—
95Dic107—24no—
96Dic107—42no—
97Dic107—42no—
98C2×C107#51911yescharacteristic
99C2×C107#52011yescharacteristic
100C2×C107#52011yescharacteristic
101C2×C107#56211yescharacteristic
102C2×C107#56311yescharacteristic
103C2×C107#56311yescharacteristic
104C2×C107#56411yescharacteristic
105#2382—1071nomaximal, Sylow
106C107#238211yescharacteristic, complemented, Sylow
107C2×#125—1071no—
108#389—1072no—
109#389—1072no—
110#387—1072no—
111#387—1072no—
112#387—1072no—
113#387—1072no—
114#382—1071no—
115#382D10711yescharacteristic
116#125—1078no—
117C2×C4⋊C4—1072no—
118C2×C4⋊C4—1072no—
119C2×C4⋊C4—1071no—
120C2×C4⋊C4—1071no—
121C2×C4⋊C4—1072no—
122C2×C4⋊C4—1072no—
123C2×C4⋊C4—1071no—
124C2×C4⋊C4—1071no—
125C2×C4⋊C4C2×D10711yescharacteristic
126C2×C4⋊C4C2×D10711yescharacteristic
127C2×C42—1072no—
128C2×C42—1072no—
129C2×C42C2×D10711yescharacteristic
130#95—1072no—
131#95—1072no—
132#95—1072no—
133#95—1072no—
134#95—1072no—
135#95—1072no—
136#95C2×D10712yes—
137#95C2×D10712yes—
138C22×C4—1072no—
139C22×C4—1072no—
140C22×C4—1072no—
141C22×C4—1072no—
142C22×C4C22×D10712yes—
143C22×C4C22×D10712yes—
144C22×C4C22×D10711yescharacteristic
145C22×C4C22×D10711yescharacteristic
146C22×C4C22×D10711yescharacteristic
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—1078no—
150C4⋊C4—1078no—
151C4⋊C4—1074no—
152C4⋊C4—1074no—
153C4⋊C4—4282no—
154C4⋊C4—4282no—
155C4⋊C4—4281no—
156C4⋊C4—4281no—
157C42—4282no—
158C42—4282no—
159C42C4×D10714yes—
160C23C23×D10711yescharacteristic
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—1074no—
174C2×C4—1074no—
175C2×C4—2142no—
176C2×C4—2142no—
177C2×C4—2142no—
178C2×C4—2142no—
179C2×C4—2142no—
180C2×C4—2142no—
181C2×C4—2142no—
182C2×C4—2142no—
183C2×C4—2142no—
184C2×C4—2142no—
185C2×C4C2×C4×D10712yes—
186C2×C4C2×C4×D10712yes—
187C2×C4C2×C4×D10712yes—
188C22C22×C4×D10711yescharacteristic
189C22#1681215411yescharacteristic
190C22#1681215511yescharacteristic
191C22#1681215611yescharacteristic
192C22#1681215611yescharacteristic
193C22#1681215611yescharacteristic
194C22#1681215711yescharacteristic
195C4—22no—
196C4—22no—
197C4—22no—
198C4—42no—
199C4—42no—
200C4—4282no—
201C4—4282no—
202C4—2144no—
203C4—2144no—
204C2#2029310411yescharacteristic
205C2#2029310711yescharacteristic
206C2#2029310811yescharacteristic
207C2#2029337111yescharacteristic
208C2#2029337711yescharacteristic
209C2#2029338111yescharacteristic
210C2C107⋊#56411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutatorC22×C107C23×C4
Fitting#382×C107C2
socle, prime radical, cosocleC23×C107C24
hypercenter#382D107
hypocenterC107#2382
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
10. #16812101C4semidirect
106. C107#2382semidirect

Maximal subgroups

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

subgroupindex
C2×#168121012
#202925242
#202925222
#202925142
#202925122
#202925102
#202925082
#202924932
#382×C1072
#2382107