Finite Group Databasebeta

← #28344483

Subgroups — #28344483

Order 13696.

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

normal subgroups only

This group has 644 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×#16812083C211yesmaximal, characteristic, complemented
2#20292553C212yesmaximal, complemented
3#20292552C212yesmaximal, complemented
4#20292536C212yesmaximal, complemented
5#20292535C212yesmaximal, complemented
6#20292535C214yesmaximal, complemented
7#20292490C211yesmaximal, characteristic, complemented
8C2×#121×C107C211yesmaximal, characteristic, complemented
9C22×D428C2211yescharacteristic
10C22×C22⋊D107C2212yes—
11C22×C22⋊D107C2214yes—
12#16812083—24no—
13C2×C4⋊Dic107C2211yescharacteristic
14C2×C22⋊Dic107C2212yes—
15C2×C22⋊Dic107C2214yes—
16C2×(C2×D107)⋊C4C2212yes—
17C2×(C2×D107)⋊C4C2214yes—
18C107⋊#95C2212yes—
19C107⋊#95C2212yes—
20C107⋊#95C2214yes—
21C22×D4×C107C2212yes—
22C22×D4×C107C2211yescharacteristic
23C23×C4×C107C2211yescharacteristic
24#121×C107—24no—
25C2×C4⋊C4×C107C2211yescharacteristic
26C2×C22⋊C4×C107C2212yes—
27C23×D107C2312yes—
28C2×D428—22no—
29C2×D428—42no—
30C2×C22⋊D107—24no—
31C2×C22⋊D107—44no—
32C2×C22⋊D107—44no—
33C2×C22⋊D107—44no—
34C2×C22⋊D107—44no—
35C22×Dic107C2312yes—
36C22×Dic107C2314yes—
37C4⋊Dic107—22no—
38C22⋊Dic107—42no—
39C22⋊Dic107—44no—
40(C2×D107)⋊C4—24no—
41(C2×D107)⋊C4—44no—
42C24×C107C2312yes—
43C24×C107C2311yescharacteristic
44C2×D4×C107—24no—
45C2×D4×C107—22no—
46C2×D4×C107—42no—
47C2×D4×C107—44no—
48C22×C4×C107—42no—
49C22×C4×C107D414yescomplemented
50C22×C4×C107C2312yes—
51C22×C4×C107C2311yescharacteristic
52C22×C4×C107C2311yescharacteristic
53C4⋊C4×C107—22no—
54C22⋊C4×C107—24no—
55C23×C107—22no—
56C23×C107—22no—
57C23×C107—22no—
58C23×C107—22no—
59C23×C107—24no—
60C23×C107—24no—
61C23×C107—22no—
62C23×C107—22no—
63C23×C107C2×D412yes—
64C23×C107C2411yescharacteristic
65C22×D107—22no—
66C22×D107—22no—
67C22×D107—22no—
68C22×D107—24no—
69C22×D107—24no—
70D4×C107—44no—
71D4×C107—48no—
72C2×C4×C107—22no—
73C2×C4×C107—22no—
74C2×C4×C107—22no—
75C2×C4×C107—24no—
76C2×C4×C107—44no—
77C2×C4×C107C2×D412yes—
78C2×C4×C107C2×D412yes—
79C22⋊D107—48no—
80C22⋊D107—416no—
81C2×Dic107—22no—
82C2×Dic107—24no—
83C2×Dic107—24no—
84C2×Dic107—24no—
85C2×Dic107—24no—
86D428—44no—
87C22×C107—22no—
88C22×C107—22no—
89C22×C107—22no—
90C22×C107—44no—
91C22×C107—42no—
92C22×C107—42no—
93C22×C107—42no—
94C22×C107—44no—
95C22×C107—44no—
96C22×C107C22×D411yescharacteristic
97C22×C107H(5,𝔽2)12yes—
98C22×C107H(5,𝔽2)12yes—
99C22×C107H(5,𝔽2)12yes—
100C2×D107—42no—
101C2×D107—42no—
102C2×D107—42no—
103C2×D107—44no—
104C2×D107—44no—
105C4×C107—22no—
106C4×C107—22no—
107C4×C107—42no—
108Dic107—42no—
109Dic107—44no—
110C2×C107—42no—
111C2×C107—82no—
112C2×C107#53411yescharacteristic
113C2×C107#53511yescharacteristic
114C2×C107#53511yescharacteristic
115C2×C107#56014yes—
116D107—82no—
117#2815—1071nomaximal, Sylow
118C107#281511yescharacteristic, complemented, Sylow
119C2×#121—1071no—
120C2×#121D10711yescharacteristic, complemented
121#397—1072no—
122#397—1072no—
123#392—1072no—
124#392—1074no—
125#392—1072no—
126#381—1071no—
127C22×D4—1074no—
128C22×D4—1072no—
129C22×D4—1071no—
130C22×D4C2×D10712yes—
131C22×D4C2×D10711yescharacteristic
132C23×C4C2×D10711yescharacteristic
133#121—24no—
134#121—2144no—
135C2×C4⋊C4—1071no—
136C2×C4⋊C4C2×D10711yescharacteristic
137C2×C22⋊C4—1072no—
138C2×C22⋊C4—1074no—
139C2×C22⋊C4—1074no—
140C2×C22⋊C4—1072no—
141C2×C22⋊C4C2×D10712yes—
142#95—1072no—
143#95—1072no—
144#95—1074no—
145C24—1072no—
146C24C22×D10712yes—
147C24C22×D10711yescharacteristic
148C2×D4—24no—
149C2×D4—22no—
150C2×D4—42no—
151C2×D4—44no—
152C2×D4—4282no—
153C2×D4—4284no—
154C2×D4—4284no—
155C2×D4—4284no—
156C2×D4—4284no—
157C2×D4—2144no—
158C2×D4—2142no—
159C22×C4—42no—
160C22×C4—1074no—
161C22×C4—1072no—
162C22×C4C22⋊D10714yescomplemented
163C22×C4C22×D10712yes—
164C22×C4C22×D10711yescharacteristic
165C22×C4C22×D10711yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—2142no—
168C22⋊C4—24no—
169C22⋊C4—4284no—
170C22⋊C4—4282no—
171C22⋊C4—4284no—
172C22⋊C4—2144no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—24no—
178C23—24no—
179C23—22no—
180C23—22no—
181C23—2142no—
182C23—2142no—
183C23—2144no—
184C23—2142no—
185C23—2144no—
186C23C2×C22⋊D10712yes—
187C23C23×D10711yescharacteristic
188D4—44no—
189D4—48no—
190D4—4284no—
191D4—4288no—
192D4—42816no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—44no—
198C2×C4—2142no—
199C2×C4—2144no—
200C2×C4—2144no—
201C2×C4—2144no—
202C2×C4—2144no—
203C2×C4C2×C22⋊D10712yes—
204C2×C4C2×C22⋊D10712yes—
205C22—22no—
206C22—22no—
207C22—22no—
208C22—44no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—44no—
213C22—44no—
214C22—4282no—
215C22—4282no—
216C22—4282no—
217C22—4284no—
218C22—4284no—
219C22C22×C22⋊D10711yescharacteristic
220C22#1681215412yes—
221C22#1681215412yes—
222C22#1681215512yes—
223C4—22no—
224C4—22no—
225C4—42no—
226C4—4282no—
227C4—4284no—
228C2—42no—
229C2—82no—
230C2—8562no—
231C2#2029318611yescharacteristic
232C2#2029319111yescharacteristic
233C2#2029319411yescharacteristic
234C2#2029336514yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×#121×C107C2
hypercenterC2×#121D107
hypocenterC107#2815
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16812083C2semidirect
2. #20292553C2semidirect
3. #20292552C2semidirect
4. #20292536C2semidirect
5. #20292535C2semidirect
6. #20292535C2semidirect
7. #20292490C2semidirect
8. C2×#121×C107C2semidirect
49. C22×C4×C107D4semidirect
118. C107#2815semidirect
120. C2×#121D107semidirect
162. C22×C4C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120832
#202925532
#202925522
#202925362
#202925352
#202924902
C2×#121×C1072
#2815107