Finite Group Databasebeta

← #28344520

Subgroups — #28344520

Order 13696.

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

normal subgroups only

This group has 430 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×#16812103C211yesmaximal, characteristic
2#20292565C212yesmaximal, complemented
3#20292562C212yesmaximal
4#20292558C212yesmaximal, complemented
5#20292553C212yesmaximal
6#20292546C212yesmaximal, complemented
7#20292538C212yesmaximal
8#20292495C211yesmaximal, characteristic, complemented
9C2×C4⋊Q8×C107C211yesmaximal, characteristic, complemented
10#16812103—24no—
11C2×Dic107⋊C4C2211yescharacteristic
12C2×Dic107⋊C4C2211yescharacteristic
13C2×Dic107⋊C4C2211yescharacteristic
14C2×Dic107⋊C4C2214yes—
15C2×(C2×D107)⋊C4C2211yescharacteristic
16C2×(C2×D107)⋊C4C2211yescharacteristic
17C2×(C2×D107)⋊C4C2211yescharacteristic
18C2×(C2×D107)⋊C4C2214yes—
19C107⋊#95C2212yes—
20C107⋊#95C2212yes—
21C107⋊#95C2212yes—
22C107⋊#95C2214yes—
23C107⋊#95C2214yes—
24C22×Q8×C107C2212yes—
25C4⋊Q8×C107—24no—
26C2×C4⋊C4×C107C2212yes—
27C2×C4⋊C4×C107C2212yes—
28C2×C42×C107C2211yescharacteristic
29C23×D107C2311yescharacteristic
30C22×Dic107C2311yescharacteristic
31C22×Dic107C2311yescharacteristic
32C22×Dic107C2311yescharacteristic
33C22×Dic107C2314yes—
34Dic107⋊C4—22no—
35Dic107⋊C4—22no—
36Dic107⋊C4—22no—
37Dic107⋊C4—44no—
38(C2×D107)⋊C4—22no—
39(C2×D107)⋊C4—22no—
40(C2×D107)⋊C4—22no—
41(C2×D107)⋊C4—44no—
42C2×Q8×C107—24no—
43C2×Q8×C107—44no—
44C22×C4×C107C2314yes—
45C22×C4×C107C2311yescharacteristic
46C22×C4×C107C2311yescharacteristic
47C22×C4×C107C2311yescharacteristic
48C4⋊C4×C107—24no—
49C4⋊C4×C107—24no—
50C42×C107D414yes—
51C23×C107C2411yescharacteristic
52C22×D107—21no—
53C22×D107—22no—
54C22×D107—22no—
55C22×D107—22no—
56Q8×C107—48no—
57C2×C4×C107—22no—
58C2×C4×C107—22no—
59C2×C4×C107—22no—
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—21no—
67C2×Dic107—21no—
68C2×Dic107—21no—
69C2×Dic107—24no—
70C2×Dic107—22no—
71C2×Dic107—22no—
72C2×Dic107—22no—
73C2×Dic107—24no—
74C2×Dic107—24no—
75C22×C107C22×D411yescharacteristic
76C22×C107H(5,𝔽2)12yes—
77C22×C1072−1+412yes—
78C22×C1072−1+412yes—
79C2×D107—41no—
80C2×D107—41no—
81C2×D107—41no—
82C2×D107—44no—
83C4×C107—22no—
84C4×C107—22no—
85C4×C107—22no—
86C4×C107—44no—
87Dic107—41no—
88Dic107—41no—
89Dic107—41no—
90Dic107—44no—
91C2×C107#53511yescharacteristic
92C2×C107#53611yescharacteristic
93C2×C107#53611yescharacteristic
94C2×C107#56314yes—
95D107—81no—
96#2821—1071nomaximal, Sylow
97C107#282111yescharacteristic, complemented, Sylow
98C2×C4⋊Q8D10711yescharacteristic, complemented
99C2×#126—1071no—
100#400—1072no—
101#399—1072no—
102#398—1072no—
103#397—1072no—
104#395—1072no—
105#393—1072no—
106#383—1071no—
107C22×Q8C2×D10712yes—
108C4⋊Q8—24no—
109#126—2144no—
110C2×C4⋊C4—1074no—
111C2×C4⋊C4—1071no—
112C2×C4⋊C4—1071no—
113C2×C4⋊C4—1071no—
114C2×C4⋊C4C2×D10712yes—
115C2×C4⋊C4C2×D10712yes—
116C2×C22⋊C4—1074no—
117C2×C22⋊C4—1071no—
118C2×C22⋊C4—1071no—
119C2×C22⋊C4—1071no—
120C2×C42C2×D10711yescharacteristic
121#95—1072no—
122#95—1072no—
123#95—1072no—
124#95—1074no—
125#95—1074no—
126C24—1071no—
127C2×Q8—24no—
128C2×Q8—44no—
129C22×C4—1074no—
130C22×C4—1071no—
131C22×C4—1071no—
132C22×C4—1071no—
133C22×C4C22×D10714yes—
134C22×C4C22×D10711yescharacteristic
135C22×C4C22×D10711yescharacteristic
136C22×C4C22×D10711yescharacteristic
137C4⋊C4—24no—
138C4⋊C4—24no—
139C4⋊C4—4284no—
140C4⋊C4—2142no—
141C4⋊C4—2142no—
142C4⋊C4—2142no—
143C22⋊C4—4284no—
144C22⋊C4—2142no—
145C22⋊C4—2142no—
146C22⋊C4—2142no—
147C42C22⋊D10714yes—
148C23—2141no—
149C23—2142no—
150C23—2142no—
151C23—2142no—
152C23C23×D10711yescharacteristic
153Q8—48no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—2141no—
161C2×C4—2141no—
162C2×C4—2141no—
163C2×C4—2144no—
164C2×C4—2142no—
165C2×C4—2144no—
166C2×C4—2144no—
167C2×C4—2142no—
168C2×C4—2142no—
169C2×C4C2×C22⋊D10712yes—
170C2×C4C2×C22⋊D10712yes—
171C2×C4C2×C22⋊D10712yes—
172C22—4284no—
173C22—4281no—
174C22—4281no—
175C22—4281no—
176C22C22×C22⋊D10711yescharacteristic
177C22#1681215412yes—
178C22#1681215712yes—
179C22#1681215712yes—
180C4—22no—
181C4—22no—
182C4—22no—
183C4—44no—
184C4—4284no—
185C4—4281no—
186C4—4281no—
187C4—4281no—
188C2—8561no—
189C2#2029319411yescharacteristic
190C2#2029320111yescharacteristic
191C2#2029320111yescharacteristic
192C2#2029337614yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×C4⋊Q8×C107C2
prime radicalC23×D107C23
hypercenterC2×C4⋊Q8D107
hypocenterC107#2821
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #20292565C2semidirect
4. #20292558C2semidirect
6. #20292546C2semidirect
8. #20292495C2semidirect
9. C2×C4⋊Q8×C107C2semidirect
97. C107#2821semidirect
98. C2×C4⋊Q8D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121032
#202925652
#202925622
#202925582
#202925532
#202925462
#202925382
#202924952
C2×C4⋊Q8×C1072
#2821107