Finite Group Databasebeta

← #28344506

Subgroups — #28344506

Order 13696.

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

normal subgroups only

This group has 580 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×#16812097C211yesmaximal, characteristic, complemented
2#20292543C214yesmaximal, complemented
3#20292539C212yesmaximal, complemented
4#20292538C212yesmaximal, complemented
5#20292536C212yesmaximal, complemented
6#20292535C212yesmaximal, complemented
7#20292493C211yesmaximal, characteristic, complemented
8C2×#124×C107C211yesmaximal, characteristic, complemented
9C22×Dic214C2211yescharacteristic
10C22×D428C2211yescharacteristic
11C22×C22⋊D107C2214yes—
12#16812097—24no—
13C2×C4⋊Dic107C2212yes—
14C2×C22⋊Dic107C2214yes—
15C2×(C2×D107)⋊C4C2212yes—
16C2×(C2×D107)⋊C4C2212yes—
17C2×(C2×D107)⋊C4C2214yes—
18C107⋊#95C2212yes—
19C107⋊#95C2212yes—
20C107⋊#95C2214yes—
21C22×Q8×C107C2211yescharacteristic
22C22×D4×C107C2211yescharacteristic
23#124×C107—24no—
24C2×C22⋊C4×C107C2212yes—
25C2×C22⋊C4×C107C2212yes—
26C2×C42×C107C2211yescharacteristic
27C23×D107C2312yes—
28C2×Dic214—22no—
29C2×Dic214—42no—
30C2×D428—22no—
31C2×D428—42no—
32C2×C22⋊D107—44no—
33C2×C22⋊D107—44no—
34C2×C22⋊D107—44no—
35C22×Dic107C2312yes—
36C22×Dic107C2314yes—
37C4⋊Dic107—42no—
38C22⋊Dic107—44no—
39(C2×D107)⋊C4—24no—
40(C2×D107)⋊C4—24no—
41(C2×D107)⋊C4—44no—
42C24×C107C2312yes—
43C2×Q8×C107—22no—
44C2×Q8×C107—42no—
45C2×D4×C107—22no—
46C2×D4×C107—42no—
47C22×C4×C107C2312yes—
48C22×C4×C107C2311yescharacteristic
49C22×C4×C107C2311yescharacteristic
50C22×C4×C107C2311yescharacteristic
51C22⋊C4×C107—24no—
52C22⋊C4×C107—24no—
53C42×C107D414yescomplemented
54C23×C107—22no—
55C23×C107—24no—
56C23×C107—24no—
57C23×C107—22no—
58C23×C107—22no—
59C23×C107C2411yescharacteristic
60C22×D107—22no—
61C22×D107—24no—
62C22×D107—22no—
63C22×D107—24no—
64C22×D107—22no—
65Q8×C107—44no—
66D4×C107—44no—
67C2×C4×C107—22no—
68C2×C4×C107—21no—
69C2×C4×C107—22no—
70C2×C4×C107—21no—
71C2×C4×C107—22no—
72C2×C4×C107—24no—
73C2×C4×C107C2×D412yes—
74C2×C4×C107C2×D412yes—
75C2×C4×C107C2×D412yes—
76C22⋊D107—416no—
77C2×Dic107—22no—
78C2×Dic107—24no—
79C2×Dic107—24no—
80C2×Dic107—24no—
81C2×Dic107—24no—
82D428—44no—
83Dic214—44no—
84C22×C107—42no—
85C22×C107—42no—
86C22×C107—42no—
87C22×C107—44no—
88C22×C107—44no—
89C22×C107C22×D411yescharacteristic
90C22×C107H(5,𝔽2)12yes—
91C22×C107H(5,𝔽2)12yes—
92C22×C107H(5,𝔽2)11yescharacteristic
93C22×C1072−1+411yescharacteristic
94C2×D107—42no—
95C2×D107—42no—
96C2×D107—42no—
97C2×D107—44no—
98C2×D107—44no—
99C4×C107—22no—
100C4×C107—22no—
101C4×C107—22no—
102C4×C107—42no—
103Dic107—42no—
104Dic107—44no—
105C2×C107—82no—
106C2×C107#53511yescharacteristic
107C2×C107#53511yescharacteristic
108C2×C107#53711yescharacteristic
109C2×C107H(3,𝔽4)12yes—
110C2×C107#56212yes—
111D107—82no—
112#2819—1071nomaximal, Sylow
113C107#281911yescharacteristic, complemented, Sylow
114C2×#124—1071no—
115C2×#124D10711yescharacteristic, complemented
116#394—1074no—
117#393—1072no—
118#393—1072no—
119#392—1072no—
120#392—1072no—
121#382—1071no—
122C22×Q8—1071no—
123C22×Q8C2×D10711yescharacteristic
124C22×D4—1074no—
125C22×D4—1071no—
126C22×D4C2×D10711yescharacteristic
127#124—24no—
128#124—2144no—
129C2×C4⋊C4—1072no—
130C2×C22⋊C4—1074no—
131C2×C22⋊C4—1074no—
132C2×C22⋊C4—1072no—
133C2×C22⋊C4—1072no—
134C2×C22⋊C4C2×D10712yes—
135C2×C22⋊C4C2×D10712yes—
136C2×C42C2×D10711yescharacteristic
137#95—1072no—
138#95—1072no—
139#95—1074no—
140C24—1072no—
141C24C22×D10712yes—
142C2×Q8—22no—
143C2×Q8—42no—
144C2×Q8—4282no—
145C2×Q8—2142no—
146C2×D4—22no—
147C2×D4—42no—
148C2×D4—4282no—
149C2×D4—4284no—
150C2×D4—4284no—
151C2×D4—4284no—
152C2×D4—2142no—
153C22×C4—1074no—
154C22×C4—1072no—
155C22×C4C22×D10712yes—
156C22×C4C22×D10711yescharacteristic
157C22×C4C22×D10711yescharacteristic
158C22×C4C22×D10711yescharacteristic
159C4⋊C4—4282no—
160C22⋊C4—24no—
161C22⋊C4—24no—
162C22⋊C4—4284no—
163C22⋊C4—4284no—
164C22⋊C4—2144no—
165C22⋊C4—2144no—
166C42C22⋊D10714yescomplemented
167C23—22no—
168C23—24no—
169C23—24no—
170C23—22no—
171C23—22no—
172C23—2142no—
173C23—2144no—
174C23—2144no—
175C23—2142no—
176C23—2142no—
177C23C23×D10711yescharacteristic
178Q8—44no—
179Q8—4284no—
180D4—44no—
181D4—4284no—
182D4—42816no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—2142no—
190C2×C4—2144no—
191C2×C4—2144no—
192C2×C4—2144no—
193C2×C4—2144no—
194C2×C4C2×C22⋊D10712yes—
195C2×C4C2×C22⋊D10712yes—
196C2×C4C2×C22⋊D10712yes—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—44no—
201C22—44no—
202C22—4282no—
203C22—4282no—
204C22—4282no—
205C22—4284no—
206C22—4284no—
207C22C22×C22⋊D10711yescharacteristic
208C22#1681215412yes—
209C22#1681215512yes—
210C22#1681215511yescharacteristic
211C22#1681215611yescharacteristic
212C4—22no—
213C4—22no—
214C4—22no—
215C4—42no—
216C4—4282no—
217C4—4284no—
218C2—82no—
219C2—8562no—
220C2#2029319111yescharacteristic
221C2#2029319411yescharacteristic
222C2#2029320511yescharacteristic
223C2#2029336912yes—
224C2#2029337112yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC2×#124×C107C2
hypercenterC2×#124D107
hypocenterC107#2819
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#16812097C2semidirect
2. #20292543C2semidirect
3. #20292539C2semidirect
4. #20292538C2semidirect
5. #20292536C2semidirect
6. #20292535C2semidirect
7. #20292493C2semidirect
8. C2×#124×C107C2semidirect
53. C42×C107D4semidirect
113. C107#2819semidirect
115. C2×#124D107semidirect
166. C42C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120972
#202925432
#202925392
#202925382
#202925362
#202925352
#202924932
C2×#124×C1072
#2819107