Finite Group Databasebeta

← #28344772

Subgroups — #28344772

Order 13696.

556 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 169 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×C42×D107C211yesmaximal, characteristic
3#20292526C212yesmaximal, complemented
4#20292512C214yesmaximal, complemented
5#20292501C214yesmaximal
6#20292484C212yesmaximal
7#382×C107C211yesmaximal, characteristic, complemented
8C22×C4×D107C2212yes—
9C22×C4×D107C2211yescharacteristic
10#16812101—24no—
11C2×C4⋊Dic107C2212yes—
12C2×C4⋊Dic107C2214yes—
13C2×Dic107⋊C4C2214yes—
14C2×(C2×D107)⋊C4C2214yes—
15C2×C4×Dic107C2212yes—
16C2×C4×Dic107C2211yescharacteristic
17C42×D107—24no—
18C107⋊#95C2214yes—
19C107⋊#95C2214yes—
20C2×C4⋊C4×C107C2212yes—
21C2×C42×C107C2211yescharacteristic
22#95×C107C2214yes—
23C23×D107C2311yescharacteristic
24C22×Dic107C2314yes—
25C22×Dic107C2311yescharacteristic
26C22×Dic107C2312yes—
27C2×C4×D107—24no—
28C2×C4×D107—24no—
29C2×C4×D107—22no—
30C2×C4×D107—22no—
31C2×C4×D107—24no—
32C2×C4×D107Q814yes—
33C4⋊Dic107—24no—
34C4⋊Dic107—28no—
35Dic107⋊C4—28no—
36(C2×D107)⋊C4—28no—
37C4×Dic107—24no—
38C4×Dic107—22no—
39C22×C4×C107C2314yes—
40C22×C4×C107C2312yes—
41C22×C4×C107C2311yescharacteristic
42C4⋊C4×C107—24no—
43C42×C107D414yes—
44C23×C107C2411yescharacteristic
45C22×D107—21no—
46C22×D107—22no—
47C22×D107—21no—
48C22×D107C2×Q812yes—
49C22×D107#4114yes—
50C2×C4×C107—22no—
51C2×C4×C107—22no—
52C2×C4×C107—24no—
53C2×C4×C107—24no—
54C2×C4×C107—24no—
55C2×C4×C107C2×D414yes—
56C2×C4×C107C2×D412yes—
57C2×C4×C107C2×Q812yes—
58C2×C4×C107#4112yes—
59C2×Dic107—21no—
60C2×Dic107—22no—
61C2×Dic107—22no—
62C2×Dic107—21no—
63C2×Dic107—24no—
64C2×Dic107—24no—
65C2×Dic107—24no—
66C2×Dic107C2×Q812yes—
67C2×Dic107#4114yes—
68C4×D107—216no—
69C4×D107—28no—
70C22×C107C22×D411yescharacteristic
71C22×C107C22×Q811yescharacteristic
72C22×C107C2×#4112yes—
73C22×C107C2×#4112yes—
74C22×C107C2×#4111yescharacteristic
75C2×D107—22no—
76C2×D107—24no—
77C2×D107—24no—
78C2×D107—22no—
79C2×D107#12514yescomplemented
80C4×C107—24no—
81C4×C107—44no—
82C4×C107C22⋊Q814yes—
83Dic107—22no—
84Dic107—24no—
85Dic107—44no—
86C2×C107C2×C22⋊Q811yescharacteristic
87C2×C107#52512yes—
88C2×C107C2×#12511yescharacteristic
89C2×C107#52912yes—
90C2×C107#53311yescharacteristic
91D107—24no—
92#2851—1071nomaximal, Sylow
93C107#285111yescharacteristic, complemented, Sylow
94C2×#125—1071no—
95C22×C42—1071no—
96#389—1072no—
97#387—1074no—
98#385—1074no—
99#382D10711yescharacteristic, complemented
100#380—1072no—
101C23×C4—1071no—
102C23×C4—1072no—
103#125—2144no—
104C2×C4⋊C4—1074no—
105C2×C4⋊C4—1072no—
106C2×C4⋊C4—1074no—
107C2×C4⋊C4C2×D10712yes—
108C2×C22⋊C4—1074no—
109C2×C42—1072no—
110C2×C42—1071no—
111C2×C42—2144no—
112C2×C42C2×D10711yescharacteristic
113#95—1074no—
114#95—1074no—
115#95C2×D10714yes—
116C24—1071no—
117C22×C4—1071no—
118C22×C4—1072no—
119C22×C4—1074no—
120C22×C4—1074no—
121C22×C4—2144no—
122C22×C4—2144no—
123C22×C4—2144no—
124C22×C4—2142no—
125C22×C4—2142no—
126C22×C4C22×D10714yes—
127C22×C4C22×D10712yes—
128C22×C4C22×D10711yescharacteristic
129C4⋊C4—24no—
130C4⋊C4—2148no—
131C4⋊C4—2148no—
132C4⋊C4—2144no—
133C22⋊C4—2148no—
134C42—2144no—
135C42—2142no—
136C42D42814yes—
137C23—1072no—
138C23—1074no—
139C23—2141no—
140C23—2142no—
141C23—2141no—
142C23C23×D10711yescharacteristic
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—24no—
146C2×C4—24no—
147C2×C4—24no—
148C2×C4—1072no—
149C2×C4—1074no—
150C2×C4—21416no—
151C2×C4—2144no—
152C2×C4—2144no—
153C2×C4—2144no—
154C2×C4—2141no—
155C2×C4—2142no—
156C2×C4—2142no—
157C2×C4—2141no—
158C2×C4—2148no—
159C2×C4C2×D42814yes—
160C2×C4C2×D42812yes—
161C2×C4Q8×D10712yes—
162C2×C4#1313381712yes—
163C22—1074no—
164C22—2142no—
165C22—2144no—
166C22—2144no—
167C22—2142no—
168C22C22×D42811yescharacteristic
169C22C2×Q8×D10711yescharacteristic
170C22#41×D10712yes—
171C22#41×D10712yes—
172C22C2×#1313381711yescharacteristic
173C4—24no—
174C4—44no—
175C4—4284no—
176C4—2142no—
177C4—2144no—
178C4#1681208914yes—
179C2—2144no—
180C2C2×#1681208911yescharacteristic
181C2#2029313912yes—
182C2#125×D10711yescharacteristic
183C2#2029315612yes—
184C2#2029318311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#382×C107C2
prime radicalC23×D107C23
hypercenter#382D107
hypocenterC107#2851
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. #20292526C2semidirect
4. #20292512C2semidirect
7. #382×C107C2semidirect
79. C2×D107#125semidirect
93. C107#2851semidirect
99. #382D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121012
C2×C42×D1072
#202925262
#202925122
#202925012
#202924842
#382×C1072
#2851107