Finite Group Databasebeta

← #28344793

Subgroups — #28344793

Order 13696.

688 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 177 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×#16812093C212yesmaximal, complemented
2C2×#16812083C211yesmaximal, characteristic
3C22×C4×Dic107C211yesmaximal, characteristic, complemented
4#20292528C212yesmaximal, complemented
5#20292526C212yesmaximal, complemented
6#20292519C214yesmaximal, complemented
7#20292504C212yesmaximal, complemented
8#380×C107C211yesmaximal, characteristic, complemented
9C22×D428C2211yescharacteristic
10C22×C22⋊D107C2212yes—
11C23×Dic107C2211yescharacteristic
12C23×Dic107C2211yescharacteristic
13#16812093—28no—
14#16812083—24no—
15C2×C4⋊Dic107C2214yes—
16C2×C4⋊Dic107C2211yescharacteristic
17C2×(C2×D107)⋊C4C2214yes—
18C2×(C2×D107)⋊C4C2214yes—
19C2×(C2×D107)⋊C4C2212yes—
20C2×C4×Dic107—24no—
21C2×C4×Dic107C2212yes—
22C2×C4×Dic107C2212yes—
23C107⋊#95C2214yes—
24C23×C4×C107C2211yescharacteristic
25C2×C4⋊C4×C107C2212yes—
26C2×C22⋊C4×C107C2212yes—
27#95×C107C2212yes—
28C23×D107C2312yes—
29C2×D428—22no—
30C2×D428—24no—
31C2×C22⋊D107—24no—
32C2×C22⋊D107—24no—
33C2×C22⋊D107—24no—
34C22×Dic107—22no—
35C22×Dic107—22no—
36C22×Dic107—22no—
37C22×Dic107—24no—
38C22×Dic107D414yescomplemented
39C22×Dic107C2312yes—
40C22×Dic107C2312yes—
41C22×Dic107C2312yes—
42C4⋊Dic107—22no—
43C4⋊Dic107—28no—
44(C2×D107)⋊C4—24no—
45(C2×D107)⋊C4—28no—
46(C2×D107)⋊C4—28no—
47C4×Dic107—24no—
48C4×Dic107—24no—
49C24×C107C2311yescharacteristic
50C22×C4×C107—24no—
51C22×C4×C107D414yes—
52C22×C4×C107C2314yes—
53C22×C4×C107C2311yescharacteristic
54C22×C4×C107C2311yescharacteristic
55C4⋊C4×C107—24no—
56C22⋊C4×C107—24no—
57C23×C107—22no—
58C23×C107—22no—
59C23×C107C2×D412yes—
60C23×C107C2×D412yes—
61C23×C107#4112yes—
62C23×C107C2411yescharacteristic
63C22×D107—22no—
64C22×D107—22no—
65C22×D107—24no—
66C22×D107—22no—
67C22×D107—24no—
68C2×C4×C107—22no—
69C2×C4×C107—28no—
70C2×C4×C107—24no—
71C2×C4×C107—24no—
72C2×C4×C107—24no—
73C2×C4×C107C2×D412yes—
74C2×C4×C107C2×D412yes—
75C2×C4×C107#4114yes—
76C22⋊D107—48no—
77C2×Dic107—28no—
78C2×Dic107—28no—
79C2×Dic107—22no—
80C2×Dic107—22no—
81C2×Dic107—22no—
82C2×Dic107—22no—
83C2×Dic107—22no—
84C2×Dic107—22no—
85C2×Dic107—22no—
86C2×Dic107C2×D414yes—
87C2×Dic107#4114yes—
88D428—44no—
89C22×C107—22no—
90C22×C107—22no—
91C22×C107—24no—
92C22×C107—24no—
93C22×C107#12114yes—
94C22×C107C22×D411yescharacteristic
95C22×C107C22×D411yescharacteristic
96C22×C107C2×#4112yes—
97C22×C107C2×#4112yes—
98C22×C107C2×#4111yescharacteristic
99C2×D107—42no—
100C2×D107—42no—
101C2×D107—42no—
102C2×D107—44no—
103C2×D107—44no—
104C4×C107—22no—
105C4×C107—44no—
106C4×C107#12314yescomplemented
107Dic107—24no—
108Dic107—24no—
109Dic107—42no—
110C2×C107—24no—
111C2×C107C2×#12111yescharacteristic
112C2×C107C2×#12311yescharacteristic
113C2×C107#52511yescharacteristic
114C2×C107#52912yes—
115C2×C107#53212yes—
116D107—82no—
117#2854—1071nomaximal, Sylow
118C107#285411yescharacteristic, complemented, Sylow
119C2×#123—1072no—
120C2×#121—1071no—
121C22×C42—1071no—
122#390—1072no—
123#389—1072no—
124#388—1074no—
125#386—1072no—
126#380D10711yescharacteristic, complemented
127C22×D4—1072no—
128C22×D4—1071no—
129C23×C4—1071no—
130C23×C4—1071no—
131C23×C4C2×D10711yescharacteristic
132#123—2148no—
133#121—2144no—
134C2×C4⋊C4—1071no—
135C2×C4⋊C4—1074no—
136C2×C4⋊C4C2×D10712yes—
137C2×C22⋊C4—1072no—
138C2×C22⋊C4—1074no—
139C2×C22⋊C4—1074no—
140C2×C22⋊C4C2×D10712yes—
141C2×C42—1072no—
142C2×C42—1072no—
143C2×C42—2144no—
144#95—1074no—
145#95C2×D10712yes—
146C24—1072no—
147C24C22×D10711yescharacteristic
148C2×D4—2144no—
149C2×D4—2144no—
150C2×D4—2144no—
151C2×D4—2142no—
152C2×D4—2144no—
153C22×C4—24no—
154C22×C4—1074no—
155C22×C4—1072no—
156C22×C4—1072no—
157C22×C4—1072no—
158C22×C4—2144no—
159C22×C4—2142no—
160C22×C4—2142no—
161C22×C4—2142no—
162C22×C4D42814yes—
163C22×C4C22×D10714yes—
164C22×C4C22×D10711yescharacteristic
165C22×C4C22×D10711yescharacteristic
166C4⋊C4—24no—
167C4⋊C4—2148no—
168C4⋊C4—2142no—
169C22⋊C4—24no—
170C22⋊C4—2148no—
171C22⋊C4—2148no—
172C22⋊C4—2144no—
173C42—2144no—
174C42—2144no—
175C23—22no—
176C23—22no—
177C23—2142no—
178C23—2142no—
179C23—2144no—
180C23—2144no—
181C23—2142no—
182C23D4×D10712yes—
183C23C2×D42812yes—
184C23#1313381912yes—
185C23C23×D10711yescharacteristic
186D4—4288no—
187D4—4284no—
188C2×C4—22no—
189C2×C4—28no—
190C2×C4—24no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—1074no—
194C2×C4—1074no—
195C2×C4—2142no—
196C2×C4—2142no—
197C2×C4—2142no—
198C2×C4—2148no—
199C2×C4—2148no—
200C2×C4—2142no—
201C2×C4—2142no—
202C2×C4—2142no—
203C2×C4—2142no—
204C2×C4C2×D42812yes—
205C2×C4C2×D42812yes—
206C2×C4#1313381714yes—
207C22—22no—
208C22—22no—
209C22—24no—
210C22—24no—
211C22—4282no—
212C22—4282no—
213C22—4282no—
214C22—4284no—
215C22—4284no—
216C22#1681208414yes—
217C22C2×D4×D10711yescharacteristic
218C22C22×D42811yescharacteristic
219C22#41×D10712yes—
220C22C2×#1313381712yes—
221C22C2×#1313381911yescharacteristic
222C4—22no—
223C4—44no—
224C4—4282no—
225C4—2144no—
226C4—2144no—
227C4#1681209314yescomplemented
228C2—24no—
229C2—8562no—
230C2C2×#1681208411yescharacteristic
231C2C2×#1681209311yescharacteristic
232C2#2029313911yescharacteristic
233C2#2029316312yes—
234C2#2029317212yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#380×C107C2
prime radicalC2×#16812083C2
hypercenter#380D107
hypocenterC107#2854
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C2×#16812093C2semidirect
3. C22×C4×Dic107C2semidirect
4. #20292528C2semidirect
5. #20292526C2semidirect
6. #20292519C2semidirect
7. #20292504C2semidirect
8. #380×C107C2semidirect
38. C22×Dic107D4semidirect
106. C4×C107#123semidirect
118. C107#2854semidirect
126. #380D107semidirect
227. C4#16812093semidirect

Maximal subgroups

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

subgroupindex
C2×#168120932
C2×#168120832
C22×C4×Dic1072
#202925282
#202925262
#202925192
#202925042
#380×C1072
#2854107