Finite Group Databasebeta

← #28344441

Subgroups — #28344441

Order 13696.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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#20292552C212yesmaximal, complemented
2#20292541C212yesmaximal
3#20292539C212yesmaximal, complemented
4#20292533C211yesmaximal, characteristic, complemented
5#20292525C211yesmaximal, characteristic, complemented
6#20292520C212yesmaximal, complemented
7#20292520C212yesmaximal, complemented
8#20292508C212yesmaximal, complemented
9#390×C107C211yesmaximal, characteristic
10C22×Dic214C2211yescharacteristic
11C2×C4⋊Dic107C2211yescharacteristic
12C2×C4⋊Dic107C2211yescharacteristic
13C2×C4⋊Dic107C2211yescharacteristic
14C2×Dic107⋊C4C2212yes—
15C2×C22⋊Dic107C2212yes—
16C2×C22⋊Dic107C2212yes—
17C2×C22⋊Dic107C2212yes—
18C2×C22⋊Dic107C2212yes—
19C2×C4×Dic107C2212yes—
20C107⋊#95C2212yes—
21C107⋊#95C2212yes—
22C107⋊#95C2212yes—
23C107⋊#95C2212yes—
24C107⋊#95C2212yes—
25C107⋊#95C2212yes—
26C22×D4×C107C2211yescharacteristic
27C2×C4⋊C4×C107C2211yescharacteristic
28C2×C22⋊C4×C107C2212yes—
29C2×C22⋊C4×C107C2212yes—
30C2×C42×C107C2211yescharacteristic
31C2×Dic214—22no—
32C2×Dic214—41no—
33C2×Dic214—41no—
34C22×Dic107C2312yes—
35C22×Dic107C2312yes—
36C22×Dic107C2312yes—
37C22×Dic107C2312yes—
38C4⋊Dic107—22no—
39C4⋊Dic107—41no—
40C4⋊Dic107—41no—
41Dic107⋊C4—24no—
42C22⋊Dic107—42no—
43C22⋊Dic107—42no—
44C22⋊Dic107—42no—
45C22⋊Dic107—42no—
46C4×Dic107—42no—
47C24×C107C2312yes—
48C2×D4×C107—22no—
49C2×D4×C107—41no—
50C2×D4×C107—41no—
51C22×C4×C107C2312yes—
52C22×C4×C107C2311yescharacteristic
53C22×C4×C107C2311yescharacteristic
54C22×C4×C107C2311yescharacteristic
55C4⋊C4×C107—41no—
56C22⋊C4×C107—24no—
57C22⋊C4×C107—24no—
58C42×C107—41no—
59C23×C107—22no—
60C23×C107—22no—
61C23×C107—22no—
62C23×C107—22no—
63C23×C107—22no—
64C23×C107—22no—
65C23×C107—22no—
66C23×C107C2411yescharacteristic
67D4×C107—44no—
68C2×C4×C107—21no—
69C2×C4×C107—21no—
70C2×C4×C107—21no—
71C2×C4×C107—21no—
72C2×C4×C107—21no—
73C2×C4×C107—21no—
74C2×C4×C107—22no—
75C2×C4×C107—22no—
76C2×C4×C107—22no—
77C2×C4×C107#4112yes—
78C2×C4×C107#4112yes—
79C2×C4×C107#4112yes—
80C2×Dic107—22no—
81C2×Dic107—22no—
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—
92Dic214—44no—
93C22×C107—42no—
94C22×C107—42no—
95C22×C107—42no—
96C22×C107—42no—
97C22×C107—42no—
98C22×C107—42no—
99C22×C107—42no—
100C22×C107C2×#4111yescharacteristic
101C22×C107C2×#4111yescharacteristic
102C22×C107C2×#4111yescharacteristic
103C22×C107H(5,𝔽2)11yescharacteristic
104C22×C107H(5,𝔽2)11yescharacteristic
105C22×C107H(5,𝔽2)11yescharacteristic
106C22×C1072−1+411yescharacteristic
107C4×C107—22no—
108C4×C107—22no—
109C4×C107—22no—
110C4×C107—42no—
111Dic107—42no—
112Dic107—42no—
113Dic107—42no—
114Dic107—42no—
115C2×C107—82no—
116C2×C107#53811yescharacteristic
117C2×C107#53811yescharacteristic
118C2×C107#53911yescharacteristic
119C2×C107#55511yescharacteristic
120C2×C107#55611yescharacteristic
121C2×C107#55911yescharacteristic
122C2×C107#56211yescharacteristic
123#2808—1071nomaximal, Sylow
124C107#280811yescharacteristic, complemented, Sylow
125#397—1072no—
126#394—1072no—
127#393—1072no—
128#391—1071no—
129#390D10711yescharacteristic
130#389—1071no—
131#388—1072no—
132#388—1072no—
133#387—1072no—
134C22×Q8—1071no—
135C22×D4C2×D10711yescharacteristic
136C2×C4⋊C4—1071no—
137C2×C4⋊C4—1071no—
138C2×C4⋊C4—1072no—
139C2×C4⋊C4—1071no—
140C2×C4⋊C4C2×D10711yescharacteristic
141C2×C22⋊C4—1072no—
142C2×C22⋊C4—1072no—
143C2×C22⋊C4—1072no—
144C2×C22⋊C4—1072no—
145C2×C22⋊C4C2×D10712yes—
146C2×C22⋊C4C2×D10712yes—
147C2×C42—1072no—
148C2×C42C2×D10711yescharacteristic
149#95—1072no—
150#95—1072no—
151#95—1072no—
152#95—1072no—
153#95—1072no—
154#95—1072no—
155C24C22×D10712yes—
156C2×Q8—4281no—
157C2×Q8—4281no—
158C2×Q8—2142no—
159C2×D4—22no—
160C2×D4—41no—
161C2×D4—41no—
162C22×C4—1072no—
163C22×C4—1072no—
164C22×C4—1072no—
165C22×C4—1072no—
166C22×C4C22×D10712yes—
167C22×C4C22×D10711yescharacteristic
168C22×C4C22×D10711yescharacteristic
169C22×C4C22×D10711yescharacteristic
170C4⋊C4—41no—
171C4⋊C4—4281no—
172C4⋊C4—4281no—
173C4⋊C4—2142no—
174C4⋊C4—2144no—
175C22⋊C4—24no—
176C22⋊C4—24no—
177C22⋊C4—4282no—
178C22⋊C4—4282no—
179C22⋊C4—4282no—
180C22⋊C4—4282no—
181C42—41no—
182C42—4282no—
183C23—22no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—22no—
190C23C23×D10711yescharacteristic
191Q8—4284no—
192D4—44no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—2142no—
203C2×C4—2142no—
204C2×C4—2142no—
205C2×C4—2142no—
206C2×C4—2142no—
207C2×C4—2142no—
208C2×C4—2142no—
209C2×C4—2142no—
210C2×C4—2142no—
211C2×C4—2142no—
212C2×C4—2142no—
213C2×C4—2142no—
214C2×C4#1313381712yes—
215C2×C4#1313381712yes—
216C2×C4#1313381812yes—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22C2×#1313381711yescharacteristic
225C22C2×#1313381711yescharacteristic
226C22C2×#1313381811yescharacteristic
227C22#1681215411yescharacteristic
228C22#1681215411yescharacteristic
229C22#1681215511yescharacteristic
230C22#1681215611yescharacteristic
231C4—22no—
232C4—22no—
233C4—22no—
234C4—42no—
235C4—4282no—
236C4—4282no—
237C4—4282no—
238C4—4282no—
239C2—82no—
240C2#2029321411yescharacteristic
241C2#2029321511yescharacteristic
242C2#2029322211yescharacteristic
243C2#2029334211yescharacteristic
244C2#2029334711yescharacteristic
245C2#2029336311yescharacteristic
246C2#2029337111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#390×C107C2
prime radicalC22×D4×C107C22
hypercenter#390D107
hypocenterC107#2808
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #20292552C2semidirect
3. #20292539C2semidirect
4. #20292533C2semidirect
5. #20292525C2semidirect
6. #20292520C2semidirect
7. #20292520C2semidirect
8. #20292508C2semidirect
124. C107#2808semidirect

Maximal subgroups

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

subgroupindex
#202925522
#202925412
#202925392
#202925332
#202925252
#202925202
#202925082
#390×C1072
#2808107