Finite Group Databasebeta

← #28344404

Subgroups — #28344404

Order 13696.

480 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 137 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
1#20292569C212yesmaximal, complemented
2#20292539C212yesmaximal, complemented
3#20292539C212yesmaximal, complemented
4#20292531C212yesmaximal, complemented
5#20292530C211yesmaximal, characteristic
6#20292507C211yesmaximal, characteristic
7#20292500C212yesmaximal, complemented
8#20292499C212yesmaximal, complemented
9#386×C107C211yesmaximal, characteristic
10C22×Dic214C2212yes—
11C23×Dic107C2211yescharacteristic
12C2×C4⋊Dic107C2211yescharacteristic
13C2×Dic107⋊C4C2212yes—
14C2×Dic107⋊C4C2212yes—
15C2×C22⋊Dic107C2212yes—
16C2×C22⋊Dic107C2212yes—
17C2×C22⋊Dic107C2212yes—
18C2×C22⋊Dic107C2211yescharacteristic
19C2×C4×Dic107C2211yescharacteristic
20C107⋊#95C2212yes—
21C107⋊#95C2212yes—
22C107⋊#95C2212yes—
23C107⋊#95C2212yes—
24C107⋊#95C2212yes—
25C107⋊#95C2212yes—
26C22×D4×C107C2211yescharacteristic
27C23×C4×C107C2211yescharacteristic
28C2×C22⋊C4×C107C2212yes—
29C2×C22⋊C4×C107C2211yescharacteristic
30#95×C107C2212yes—
31C2×Dic214—24no—
32C2×Dic214—42no—
33C2×Dic214—42no—
34C22×Dic107—42no—
35C22×Dic107—41no—
36C22×Dic107C2312yes—
37C22×Dic107C2312yes—
38C22×Dic107C2312yes—
39C22×Dic107C2312yes—
40C4⋊Dic107—41no—
41Dic107⋊C4—42no—
42Dic107⋊C4—42no—
43C22⋊Dic107—24no—
44C22⋊Dic107—24no—
45C22⋊Dic107—42no—
46C22⋊Dic107—41no—
47C4×Dic107—41no—
48C24×C107C2311yescharacteristic
49C24×C107C2311yescharacteristic
50C2×D4×C107—41no—
51C2×D4×C107—41no—
52C2×D4×C107Q814yescomplemented
53C22×C4×C107—42no—
54C22×C4×C107—41no—
55C22×C4×C107C2312yes—
56C22×C4×C107C2312yes—
57C22×C4×C107C2311yescharacteristic
58C22⋊C4×C107—42no—
59C22⋊C4×C107—41no—
60C23×C107—22no—
61C23×C107—22no—
62C23×C107—21no—
63C23×C107—22no—
64C23×C107—21no—
65C23×C107—22no—
66C23×C107—21no—
67C23×C107—21no—
68C23×C107C2×Q812yes—
69C23×C107C2×Q812yes—
70C23×C107C2411yescharacteristic
71D4×C107—44no—
72C2×C4×C107—21no—
73C2×C4×C107—21no—
74C2×C4×C107—22no—
75C2×C4×C107—22no—
76C2×C4×C107—22no—
77C2×C4×C107—22no—
78C2×C4×C107—22no—
79C2×C4×C107—22no—
80C2×C4×C107—44no—
81C2×C4×C107C2×Q812yes—
82C2×Dic107—22no—
83C2×Dic107—22no—
84C2×Dic107—22no—
85C2×Dic107—22no—
86C2×Dic107—22no—
87C2×Dic107—22no—
88C2×Dic107—22no—
89C2×Dic107—22no—
90C2×Dic107—22no—
91C2×Dic107—22no—
92C2×Dic107—22no—
93C2×Dic107—44no—
94C2×Dic107#4114yes—
95Dic214—48no—
96C22×C107—22no—
97C22×C107—22no—
98C22×C107—24no—
99C22×C107—24no—
100C22×C107—42no—
101C22×C107—42no—
102C22×C107—41no—
103C22×C107—41no—
104C22×C107—41no—
105C22×C107—41no—
106C22×C107C22×Q811yescharacteristic
107C22×C107C2×#4112yes—
108C22×C107H(5,𝔽2)11yescharacteristic
109C22×C107H(5,𝔽2)12yes—
110C22×C107H(5,𝔽2)11yescharacteristic
111C4×C107—22no—
112C4×C107—42no—
113C4×C107—42no—
114Dic107—24no—
115Dic107—42no—
116Dic107—42no—
117Dic107—42no—
118C2×C107—42no—
119C2×C107—42no—
120C2×C107#53812yes—
121C2×C107#54311yescharacteristic
122C2×C107#55412yes—
123C2×C107#55911yescharacteristic
124C2×C107H(3,𝔽4)11yescharacteristic
125#2804—1071nomaximal, Sylow
126C107#280411yescharacteristic, complemented, Sylow
127#400—1072no—
128#393—1072no—
129#393—1072no—
130#391—1072no—
131#390—1071no—
132#386—1071no—
133#386D10711yescharacteristic
134#385—1072no—
135#385—1072no—
136C22×Q8—1072no—
137C22×D4C2×D10711yescharacteristic
138C23×C4—1071no—
139C23×C4C2×D10711yescharacteristic
140C2×C4⋊C4—1072no—
141C2×C4⋊C4—1072no—
142C2×C4⋊C4—1071no—
143C2×C22⋊C4—1072no—
144C2×C22⋊C4—1071no—
145C2×C22⋊C4—1072no—
146C2×C22⋊C4—1072no—
147C2×C22⋊C4C2×D10712yes—
148C2×C22⋊C4C2×D10711yescharacteristic
149C2×C42—1071no—
150#95—1072no—
151#95—1072no—
152#95—1072no—
153#95—1072no—
154#95—1072no—
155#95—1072no—
156#95C2×D10712yes—
157C24C22×D10711yescharacteristic
158C24C22×D10711yescharacteristic
159C2×Q8—4282no—
160C2×Q8—4282no—
161C2×Q8—2144no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4Dic21414yescomplemented
165C22×C4—42no—
166C22×C4—41no—
167C22×C4—1072no—
168C22×C4—1072no—
169C22×C4—1072no—
170C22×C4—1072no—
171C22×C4—4282no—
172C22×C4—4281no—
173C22×C4C22×D10712yes—
174C22×C4C22×D10712yes—
175C22×C4C22×D10711yescharacteristic
176C4⋊C4—4282no—
177C4⋊C4—4282no—
178C4⋊C4—4281no—
179C22⋊C4—42no—
180C22⋊C4—41no—
181C22⋊C4—4282no—
182C22⋊C4—4281no—
183C22⋊C4—2144no—
184C22⋊C4—2144no—
185C42—4281no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—21no—
190C23—22no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23C2×Dic21412yes—
195C23C2×Dic21412yes—
196C23C23×D10711yescharacteristic
197Q8—4288no—
198D4—44no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—44no—
208C2×C4—1074no—
209C2×C4—4284no—
210C2×C4—2142no—
211C2×C4—2142no—
212C2×C4—2142no—
213C2×C4—2142no—
214C2×C4—2142no—
215C2×C4—2142no—
216C2×C4—2142no—
217C2×C4—2142no—
218C2×C4—2142no—
219C2×C4—2142no—
220C2×C4—2142no—
221C2×C4C2×Dic21412yes—
222C22—22no—
223C22—22no—
224C22—24no—
225C22—24no—
226C22—42no—
227C22—42no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22C22×Dic21411yescharacteristic
233C22#41×D10712yes—
234C22#1681215411yescharacteristic
235C22#1681215412yes—
236C22#1681215411yescharacteristic
237C4—22no—
238C4—42no—
239C4—42no—
240C4—4282no—
241C4—4282no—
242C4—4282no—
243C4—2144no—
244C2—42no—
245C2—42no—
246C2#2029320912yes—
247C2#2029325311yescharacteristic
248C2#2029333612yes—
249C2#2029336111yescharacteristic
250C2#2029337011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#386×C107C2
prime radicalC22×D4×C107C22
hypercenter#386D107
hypocenterC107#2804
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. #20292569C2semidirect
2. #20292539C2semidirect
3. #20292539C2semidirect
4. #20292531C2semidirect
7. #20292500C2semidirect
8. #20292499C2semidirect
52. C2×D4×C107Q8semidirect
126. C107#2804semidirect
164. C2×D4Dic214semidirect

Maximal subgroups

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

subgroupindex
#202925692
#202925392
#202925312
#202925302
#202925072
#202925002
#202924992
#386×C1072
#2804107