Finite Group Databasebeta

← #28344513

Subgroups — #28344513

Order 13696.

464 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 133 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, complemented
2C2×#16812097C211yesmaximal, characteristic
3#20292564C212yesmaximal, complemented
4#20292554C212yesmaximal, complemented
5#20292549C214yesmaximal, complemented
6#20292548C212yesmaximal, complemented
7#20292545C212yesmaximal
8#382×C107C211yesmaximal, characteristic, complemented
9C22×Dic214C2211yescharacteristic
10C22×D428C2211yescharacteristic
11#16812101—24no—
12#16812097—24no—
13C2×C4⋊Dic107C2211yescharacteristic
14C2×C4⋊Dic107C2211yescharacteristic
15C2×C4⋊Dic107C2214yes—
16C2×Dic107⋊C4C2212yes—
17C2×Dic107⋊C4C2212yes—
18C2×(C2×D107)⋊C4C2212yes—
19C2×(C2×D107)⋊C4C2212yes—
20C2×(C2×D107)⋊C4C2214yes—
21C2×(C2×D107)⋊C4C2214yes—
22C107⋊#95C2214yes—
23C2×C4⋊C4×C107C2212yes—
24C2×C42×C107C2211yescharacteristic
25#95×C107C2212yes—
26#95×C107C2212yes—
27C23×D107C2312yes—
28C2×Dic214—22no—
29C2×Dic214—42no—
30C2×D428—22no—
31C2×D428—42no—
32C22×Dic107C2312yes—
33C22×Dic107C2312yes—
34C22×Dic107C2312yes—
35C4⋊Dic107—22no—
36C4⋊Dic107—22no—
37C4⋊Dic107—44no—
38Dic107⋊C4—24no—
39Dic107⋊C4—24no—
40(C2×D107)⋊C4—24no—
41(C2×D107)⋊C4—24no—
42(C2×D107)⋊C4—44no—
43(C2×D107)⋊C4—44no—
44C22×C4×C107C2314yes—
45C22×C4×C107C2311yescharacteristic
46C22×C4×C107C2311yescharacteristic
47C22×C4×C107C2311yescharacteristic
48C4⋊C4×C107—42no—
49C42×C107D414yes—
50C23×C107C2411yescharacteristic
51C22×D107—22no—
52C22×D107—22no—
53C22×D107—24no—
54C22×D107—24no—
55C22×D107—22no—
56C2×C4×C107—21no—
57C2×C4×C107—22no—
58C2×C4×C107—22no—
59C2×C4×C107—21no—
60C2×C4×C107—24no—
61C2×C4×C107—24no—
62C2×C4×C107—24no—
63C2×C4×C107C2×D412yes—
64C2×C4×C107C2×D412yes—
65C2×C4×C107C2×D412yes—
66C2×Dic107—22no—
67C2×Dic107—22no—
68C2×Dic107—22no—
69C2×Dic107—22no—
70C2×Dic107—24no—
71C2×Dic107—24no—
72C2×Dic107—22no—
73D428—44no—
74Dic214—44no—
75C22×C107C22×D411yescharacteristic
76C22×C107H(5,𝔽2)11yescharacteristic
77C22×C107H(5,𝔽2)12yes—
78C22×C1072−1+412yes—
79C22×C1072−1+411yescharacteristic
80C2×D107—42no—
81C2×D107—42no—
82C2×D107—42no—
83C2×D107—44no—
84C2×D107—44no—
85C4×C107—22no—
86C4×C107—22no—
87C4×C107—22no—
88C4×C107—44no—
89Dic107—42no—
90Dic107—42no—
91Dic107—42no—
92C2×C107#53511yescharacteristic
93C2×C107#53611yescharacteristic
94C2×C107#53711yescharacteristic
95C2×C107#56212yes—
96C2×C107#56312yes—
97D107—82no—
98#2820—1071nomaximal, Sylow
99C107#282011yescharacteristic, complemented, Sylow
100C2×#125—1071no—
101C2×#124—1071no—
102#399—1072no—
103#397—1072no—
104#396—1074no—
105#395—1072no—
106#394—1072no—
107#382D10711yescharacteristic, complemented
108C22×Q8—1071no—
109C22×D4—1071no—
110#125—2144no—
111#124—2144no—
112C2×C4⋊C4—1074no—
113C2×C4⋊C4—1072no—
114C2×C4⋊C4—1072no—
115C2×C4⋊C4—1071no—
116C2×C4⋊C4—1071no—
117C2×C4⋊C4C2×D10712yes—
118C2×C22⋊C4—1074no—
119C2×C22⋊C4—1074no—
120C2×C22⋊C4—1072no—
121C2×C22⋊C4—1072no—
122C2×C42C2×D10711yescharacteristic
123#95—1074no—
124#95C2×D10712yes—
125#95C2×D10712yes—
126C24—1072no—
127C2×Q8—4282no—
128C2×Q8—2142no—
129C2×D4—4282no—
130C2×D4—2142no—
131C22×C4—1072no—
132C22×C4—1072no—
133C22×C4—1072no—
134C22×C4C22×D10714yes—
135C22×C4C22×D10711yescharacteristic
136C22×C4C22×D10711yescharacteristic
137C22×C4C22×D10711yescharacteristic
138C4⋊C4—42no—
139C4⋊C4—4284no—
140C4⋊C4—2144no—
141C4⋊C4—2144no—
142C4⋊C4—2142no—
143C4⋊C4—2142no—
144C22⋊C4—4284no—
145C22⋊C4—4284no—
146C22⋊C4—2144no—
147C22⋊C4—2144no—
148C42D42814yes—
149C23—2142no—
150C23—2142no—
151C23—2144no—
152C23—2144no—
153C23—2142no—
154C23C23×D10711yescharacteristic
155Q8—4284no—
156D4—4284no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—24no—
164C2×C4—2142no—
165C2×C4—2142no—
166C2×C4—2142no—
167C2×C4—2142no—
168C2×C4—2144no—
169C2×C4—2144no—
170C2×C4—2142no—
171C2×C4C2×D42812yes—
172C2×C4C2×D42812yes—
173C2×C4C2×D42812yes—
174C22—4284no—
175C22—4282no—
176C22—4282no—
177C22—4282no—
178C22—4284no—
179C22C22×D42811yescharacteristic
180C22#1681215411yescharacteristic
181C22#1681215512yes—
182C22#1681215612yes—
183C22#1681215711yescharacteristic
184C4—22no—
185C4—22no—
186C4—22no—
187C4—44no—
188C4—4282no—
189C4—4282no—
190C4—4282no—
191C2—8562no—
192C2#2029319611yescharacteristic
193C2#2029320311yescharacteristic
194C2#2029320811yescharacteristic
195C2#2029337112yes—
196C2#2029338112yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#382×C107C2
prime radicalC22×D428C22
hypercenter#382D107
hypocenterC107#2820
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
1. C2×#16812101C2semidirect
3. #20292564C2semidirect
4. #20292554C2semidirect
5. #20292549C2semidirect
6. #20292548C2semidirect
8. #382×C107C2semidirect
99. C107#2820semidirect
107. #382D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121012
C2×#168120972
#202925642
#202925542
#202925492
#202925482
#202925452
#382×C1072
#2820107