Finite Group Databasebeta

← #28344522

Subgroups — #28344522

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×#16812109C211yesmaximal, characteristic, complemented
2C2×#16812103C211yesmaximal, characteristic
3#20292567C212yesmaximal, complemented
4#20292564C212yesmaximal
5#20292559C212yesmaximal, complemented
6#20292554C212yesmaximal
7#20292548C212yesmaximal, complemented
8#20292540C212yesmaximal
9#383×C107C211yesmaximal, characteristic, complemented
10C22×Dic214C2212yes—
11#16812109—24no—
12#16812103—24no—
13C2×C4⋊Dic107C2212yes—
14C2×C4⋊Dic107C2212yes—
15C2×C4⋊Dic107C2214yes—
16C2×Dic107⋊C4C2211yescharacteristic
17C2×Dic107⋊C4C2211yescharacteristic
18C2×Dic107⋊C4C2211yescharacteristic
19C2×(C2×D107)⋊C4C2211yescharacteristic
20C2×(C2×D107)⋊C4C2211yescharacteristic
21C2×(C2×D107)⋊C4C2211yescharacteristic
22C2×(C2×D107)⋊C4C2214yes—
23C107⋊#95C2214yes—
24C107⋊#95C2214yes—
25C2×C42×C107C2211yescharacteristic
26#95×C107C2212yes—
27#95×C107C2212yes—
28#95×C107C2212yes—
29C23×D107C2311yescharacteristic
30C2×Dic214—24no—
31C2×Dic214—44no—
32C22×Dic107C2311yescharacteristic
33C22×Dic107C2311yescharacteristic
34C22×Dic107C2311yescharacteristic
35C22×Dic107C2314yes—
36C4⋊Dic107—24no—
37C4⋊Dic107—24no—
38C4⋊Dic107—44no—
39Dic107⋊C4—22no—
40Dic107⋊C4—22no—
41Dic107⋊C4—22no—
42(C2×D107)⋊C4—22no—
43(C2×D107)⋊C4—22no—
44(C2×D107)⋊C4—22no—
45(C2×D107)⋊C4—44no—
46C22×C4×C107C2314yes—
47C22×C4×C107C2311yescharacteristic
48C22×C4×C107C2311yescharacteristic
49C22×C4×C107C2311yescharacteristic
50C42×C107D414yes—
51C23×C107C2411yescharacteristic
52C22×D107—21no—
53C22×D107—22no—
54C22×D107—22no—
55C22×D107—22no—
56C2×C4×C107—22no—
57C2×C4×C107—22no—
58C2×C4×C107—22no—
59C2×C4×C107—24no—
60C2×C4×C107—24no—
61C2×C4×C107—24no—
62C2×C4×C107C2×D412yes—
63C2×C4×C107C2×D412yes—
64C2×C4×C107C2×D412yes—
65C2×Dic107—21no—
66C2×Dic107—21no—
67C2×Dic107—21no—
68C2×Dic107—24no—
69C2×Dic107—24no—
70C2×Dic107—22no—
71C2×Dic107—24no—
72C2×Dic107—22no—
73C2×Dic107—22no—
74Dic214—48no—
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—44no—
88Dic107—41no—
89Dic107—41no—
90Dic107—41no—
91C2×C107#53511yescharacteristic
92C2×C107#53611yescharacteristic
93C2×C107#53611yescharacteristic
94C2×C107#56314yes—
95D107—81no—
96#2821—1071nomaximal, Sylow
97C107#282111yescharacteristic, complemented, Sylow
98C2×C4⋊Q8—1071no—
99C2×#126—1071no—
100#400—1072no—
101#399—1072no—
102#398—1072no—
103#397—1072no—
104#395—1072no—
105#393—1072no—
106#383D10711yescharacteristic, complemented
107C22×Q8—1072no—
108C4⋊Q8—2144no—
109#126—2144no—
110C2×C4⋊C4—1074no—
111C2×C4⋊C4—1071no—
112C2×C4⋊C4—1071no—
113C2×C4⋊C4—1071no—
114C2×C4⋊C4—1072no—
115C2×C4⋊C4—1072no—
116C2×C22⋊C4—1074no—
117C2×C22⋊C4—1071no—
118C2×C22⋊C4—1071no—
119C2×C22⋊C4—1071no—
120C2×C42C2×D10711yescharacteristic
121#95—1074no—
122#95—1074no—
123#95C2×D10712yes—
124#95C2×D10712yes—
125#95C2×D10712yes—
126C24—1071no—
127C2×Q8—4284no—
128C2×Q8—2144no—
129C22×C4—1071no—
130C22×C4—1071no—
131C22×C4—1074no—
132C22×C4—1071no—
133C22×C4C22×D10714yes—
134C22×C4C22×D10711yescharacteristic
135C22×C4C22×D10711yescharacteristic
136C22×C4C22×D10711yescharacteristic
137C4⋊C4—4284no—
138C4⋊C4—2142no—
139C4⋊C4—2142no—
140C4⋊C4—2142no—
141C4⋊C4—2144no—
142C4⋊C4—2144no—
143C22⋊C4—4284no—
144C22⋊C4—2142no—
145C22⋊C4—2142no—
146C22⋊C4—2142no—
147C42D42814yes—
148C23—2141no—
149C23—2142no—
150C23—2142no—
151C23—2142no—
152C23C23×D10711yescharacteristic
153Q8—4288no—
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—2144no—
165C2×C4—2142no—
166C2×C4—2144no—
167C2×C4—2142no—
168C2×C4—2142no—
169C2×C4C2×D42812yes—
170C2×C4C2×D42812yes—
171C2×C4C2×D42812yes—
172C22—4284no—
173C22—4281no—
174C22—4281no—
175C22—4281no—
176C22C22×D42811yescharacteristic
177C22#1681215512yes—
178C22#1681215612yes—
179C22#1681215612yes—
180C4—22no—
181C4—22no—
182C4—22no—
183C4—44no—
184C4—4284no—
185C4—4281no—
186C4—4281no—
187C4—4281no—
188C2—8561no—
189C2#2029319611yescharacteristic
190C2#2029320311yescharacteristic
191C2#2029320311yescharacteristic
192C2#2029337914yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#383×C107C2
prime radicalC23×D107C23
hypercenter#383D107
hypocenterC107#2821
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #383

Complements

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

NH ≅ G/Ntype
1. C2×#16812109C2semidirect
3. #20292567C2semidirect
5. #20292559C2semidirect
7. #20292548C2semidirect
9. #383×C107C2semidirect
97. C107#2821semidirect
106. #383D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121092
C2×#168121032
#202925672
#202925642
#202925592
#202925542
#202925482
#202925402
#383×C1072
#2821107