Finite Group Databasebeta

← #28339796

Subgroups — #28339796

Order 13696.

864 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C22×#13133818C211yesmaximal, characteristic
2C2×#16812092C212yesmaximal, complemented
3#20292547C212yesmaximal, complemented
4#20292544C212yesmaximal, complemented
5#20292506C212yesmaximal, complemented
6#20292498C214yesmaximal, complemented
7#20292490C211yesmaximal, characteristic, complemented
8C22×C4⋊C4×C107C211yesmaximal, characteristic, complemented
9C2×#13133818—28no—
10C2×#13133818—24no—
11C22×Dic214C2211yescharacteristic
12C22×D428C2211yescharacteristic
13C22×C22⋊D107C2212yes—
14C22×C4×D107C2212yes—
15#16812092—28no—
16C2×Dic107⋊C4C2214yes—
17C2×C22⋊Dic107C2212yes—
18C2×(C2×D107)⋊C4C2214yes—
19C2×(C2×D107)⋊C4C2214yes—
20C107⋊#95C2212yes—
21C107⋊#95C2212yes—
22C107⋊#95C2214yes—
23C23×C4×C107C2211yescharacteristic
24C23×C4×C107C2211yescharacteristic
25C23×C4×C107C2211yescharacteristic
26C2×C4⋊C4×C107—24no—
27C2×C4⋊C4×C107C2212yes—
28C2×C4⋊C4×C107C2212yes—
29C23×D107C2312yes—
30#13133818—232no—
31C2×Dic214—22no—
32C2×Dic214—24no—
33C2×D428—24no—
34C2×D428—22no—
35C2×C22⋊D107—24no—
36C2×C22⋊D107—24no—
37C2×C22⋊D107D418yes—
38C22×Dic107C2314yes—
39C22×Dic107C2312yes—
40C2×C4×D107—24no—
41C2×C4×D107—24no—
42C2×C4×D107—24no—
43Dic107⋊C4—28no—
44C22⋊Dic107—24no—
45(C2×D107)⋊C4—28no—
46(C2×D107)⋊C4—28no—
47C24×C107C2311yescharacteristic
48C22×C4×C107—24no—
49C22×C4×C107—22no—
50C22×C4×C107—24no—
51C22×C4×C107—24no—
52C22×C4×C107—22no—
53C22×C4×C107D414yes—
54C22×C4×C107C2312yes—
55C22×C4×C107C2311yescharacteristic
56C22×C4×C107C2311yescharacteristic
57C22×C4×C107C2312yes—
58C4⋊C4×C107—24no—
59C4⋊C4×C107—24no—
60C23×C107—22no—
61C23×C107—22no—
62C23×C107C2×D414yes—
63C23×C107C2×D412yes—
64C23×C107C2411yescharacteristic
65C22×D107—22no—
66C22×D107—22no—
67C22×D107—24no—
68C22×D107—24no—
69C22×D107C2×D414yes—
70C2×C4×C107—28no—
71C2×C4×C107—28no—
72C2×C4×C107—22no—
73C2×C4×C107—22no—
74C2×C4×C107—22no—
75C2×C4×C107—22no—
76C2×C4×C107—22no—
77C2×C4×C107—24no—
78C2×C4×C107—44no—
79C2×C4×C107C2×D412yes—
80C2×C4×C107C2×D412yes—
81C2×C4×C107#4114yescomplemented
82C22⋊D107—216no—
83C2×Dic107—22no—
84C2×Dic107—22no—
85C2×Dic107—24no—
86C2×Dic107—24no—
87C2×Dic107—24no—
88C2×Dic107C2×D414yes—
89D428—28no—
90C4×D107—216no—
91Dic214—28no—
92C22×C107—22no—
93C22×C107—22no—
94C22×C107—28no—
95C22×C107#12014yes—
96C22×C107C22×D412yes—
97C22×C107C22×D411yescharacteristic
98C22×C107C2×#4112yes—
99C22×C1072−1+412yes—
100C2×D107—24no—
101C2×D107—28no—
102C2×D107—42no—
103C2×D107—42no—
104C2×D107—44no—
105C4×C107—22no—
106C4×C107—22no—
107C4×C107—24no—
108C4×C107—42no—
109Dic107—24no—
110Dic107—44no—
111C2×C107—24no—
112C2×C107C2×#12011yescharacteristic
113C2×C107#52511yescharacteristic
114C2×C107#53611yescharacteristic
115C2×C107#54714yes—
116D107—44no—
117#2406—1071nomaximal, Sylow
118C107#240611yescharacteristic, complemented, Sylow
119C22×#41—1071no—
120C2×#123—1072no—
121C22×C4⋊C4D10711yescharacteristic, complemented
122#395—1072no—
123#394—1072no—
124#386—1072no—
125#385—1074no—
126#381—1071no—
127C2×#41—2148no—
128C2×#41—2144no—
129C22×Q8—1071no—
130C22×D4—1072no—
131C22×D4—1071no—
132C23×C4—1072no—
133C23×C4C2×D10711yescharacteristic
134C23×C4C2×D10711yescharacteristic
135C23×C4C2×D10711yescharacteristic
136#123—2148no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—1074no—
139C2×C4⋊C4C2×D10712yes—
140C2×C4⋊C4C2×D10712yes—
141C2×C22⋊C4—1072no—
142C2×C22⋊C4—1074no—
143C2×C22⋊C4—1074no—
144#95—1072no—
145#95—1072no—
146#95—1074no—
147C24—1072no—
148C24C22×D10711yescharacteristic
149#41—21432no—
150C2×Q8—2144no—
151C2×Q8—2142no—
152C2×D4—1078no—
153C2×D4—2144no—
154C2×D4—2144no—
155C2×D4—2144no—
156C2×D4—2142no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—1072no—
163C22×C4—1074no—
164C22×C4—2144no—
165C22×C4—2144no—
166C22×C4—2144no—
167C22×C4C22⋊D10714yes—
168C22×C4C22×D10712yes—
169C22×C4C22×D10711yescharacteristic
170C22×C4C22×D10711yescharacteristic
171C22×C4C22×D10712yes—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—2148no—
175C22⋊C4—2148no—
176C22⋊C4—2144no—
177C22⋊C4—2148no—
178C23—22no—
179C23—22no—
180C23—1074no—
181C23—2142no—
182C23—2142no—
183C23—2144no—
184C23—2144no—
185C23D4×D10714yes—
186C23C2×C22⋊D10712yes—
187C23C23×D10711yescharacteristic
188Q8—2148no—
189D4—2148no—
190D4—21416no—
191C2×C4—28no—
192C2×C4—28no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—44no—
200C2×C4—1074no—
201C2×C4—21416no—
202C2×C4—2144no—
203C2×C4—2144no—
204C2×C4—2144no—
205C2×C4—2142no—
206C2×C4—2142no—
207C2×C4C2×C22⋊D10712yes—
208C2×C4C2×C22⋊D10712yes—
209C2×C4#1313381814yescomplemented
210C22—22no—
211C22—22no—
212C22—28no—
213C22—4282no—
214C22—4282no—
215C22—4284no—
216C22—2148no—
217C22—2144no—
218C22#1681207714yes—
219C22C2×D4×D10712yes—
220C22C22×C22⋊D10711yescharacteristic
221C22C2×#1313381812yes—
222C22#1681215712yes—
223C4—24no—
224C4—22no—
225C4—22no—
226C4—42no—
227C4—4284no—
228C4—2144no—
229C2—24no—
230C2—4284no—
231C2C2×#1681207711yescharacteristic
232C2#2029313811yescharacteristic
233C2#2029320111yescharacteristic
234C2#2029327814yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
FittingC22×C4⋊C4×C107C2
prime radicalC22×#13133818C2
hypercenterC22×C4⋊C4D107
hypocenterC107#2406
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16812092C2semidirect
3. #20292547C2semidirect
4. #20292544C2semidirect
5. #20292506C2semidirect
6. #20292498C2semidirect
7. #20292490C2semidirect
8. C22×C4⋊C4×C107C2semidirect
81. C2×C4×C107#41semidirect
118. C107#2406semidirect
121. C22×C4⋊C4D107semidirect
209. C2×C4#13133818semidirect

Maximal subgroups

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

subgroupindex
C22×#131338182
C2×#168120922
#202925472
#202925442
#202925062
#202924982
#202924902
C22×C4⋊C4×C1072
#2406107