Finite Group Databasebeta

← #28336663

Subgroups — #28336663

Order 13696.

316 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 99 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×#16812067C211yesmaximal, characteristic
2C2×#16812043C211yesmaximal, characteristic, complemented
3C2×#16812040C211yesmaximal, characteristic, complemented
4C2×(C2×D107)⋊C8C211yesmaximal, characteristic
5#20292484C211yesmaximal, characteristic
6#20292389C211yesmaximal, characteristic, complemented
7#340×C107C211yesmaximal, characteristic, complemented
8C22×C4×D107C2211yescharacteristic
9C22×C107⋊C8C2211yescharacteristic
10#16812067—22no—
11#16812067—22no—
12C2×C4⋊Dic107C2211yescharacteristic
13C2×C4⋊Dic107C2211yescharacteristic
14C2×(C2×D107)⋊C4—21no—
15C2×(C2×D107)⋊C4—21no—
16C2×C4×Dic107—21no—
17#16812043—22no—
18#16812040—22no—
19(C2×D107)⋊C8C414yescomplemented
20C107⋊#95—21no—
21C22×C8×C107C2211yescharacteristic
22C2×C4⋊C4×C107C2211yescharacteristic
23C2×C4⋊C4×C107C2211yescharacteristic
24C23×D107D411yescharacteristic
25C22×Dic107—21no—
26C22×Dic107—21no—
27C22×Dic107D411yescharacteristic
28C2×C4×D107—21no—
29C2×C4×D107—21no—
30C2×C4×D107—21no—
31C2×C4×D107—21no—
32C2×C4×D107C2×C412yes—
33C2×C4×D107Q812yes—
34C2×C107⋊C8—21no—
35C2×C107⋊C8—21no—
36C2×C107⋊C8C2×C412yes—
37C4⋊Dic107—21no—
38C4⋊Dic107—21no—
39C4⋊Dic107—21no—
40C4⋊Dic107D412yes—
41(C2×D107)⋊C4—24no—
42(C2×D107)⋊C4—42no—
43C4×Dic107—24no—
44C22×C4×C107—21no—
45C22×C4×C107—21no—
46C22×C4×C107C2311yescharacteristic
47C2×C8×C107—21no—
48C2×C8×C107—21no—
49C2×C8×C107C2×C412yes—
50C4⋊C4×C107—21no—
51C4⋊C4×C107—21no—
52C4⋊C4×C107—21no—
53C4⋊C4×C107D412yes—
54C23×C107C2×D411yescharacteristic
55C22×D107—21no—
56C22×D107—21no—
57C22×D107—21no—
58C22×D107—21no—
59C22×D107—21no—
60C22×D107—21no—
61C22×D107C4⋊C412yes—
62C2×C4×C107—22no—
63C2×C4×C107—41no—
64C2×C4×C107—41no—
65C2×C4×C107—41no—
66C2×C4×C107—41no—
67C2×C4×C107—41no—
68C2×C4×C107C22×C411yescharacteristic
69C2×C4×C107C2×D411yescharacteristic
70C2×C4×C107C2×D411yescharacteristic
71C2×C4×C107C2×Q811yescharacteristic
72C2×C4×C107#4111yescharacteristic
73C2×C4×C107#4111yescharacteristic
74C2×Dic107—21no—
75C2×Dic107—21no—
76C2×Dic107—22no—
77C2×Dic107—41no—
78C2×Dic107—41no—
79C2×Dic107—41no—
80C2×Dic107—41no—
81C2×Dic107—41no—
82C2×Dic107C4⋊C412yes—
83C4×D107—22no—
84C4×D107—22no—
85C4×D107—22no—
86C4×D107—22no—
87C8×C107—22no—
88C107⋊C8—22no—
89C22×C107C2×C4⋊C411yescharacteristic
90C22×C107#12011yescharacteristic
91C22×C107#12311yescharacteristic
92C22×C107#13511yescharacteristic
93C22×C107#13511yescharacteristic
94C22×C107AGL(1,ℤ/8)11yescharacteristic
95C22×C107#13711yescharacteristic
96C2×D107—22no—
97C2×D107—22no—
98C2×D107—22no—
99C2×D107—41no—
100C2×D107—41no—
101C2×D107—41no—
102C2×D107—41no—
103C4×C107—42no—
104C4×C107—81no—
105C4×C107C4×D411yescharacteristic
106C4×C107C4×D411yescharacteristic
107C4×C107C22⋊Q811yescharacteristic
108C4×C107C22⋊Q811yescharacteristic
109Dic107—22no—
110Dic107—42no—
111Dic107—81no—
112C2×C107#38511yescharacteristic
113C2×C107#42711yescharacteristic
114C2×C107#42811yescharacteristic
115C2×C107#44911yescharacteristic
116C2×C107#45211yescharacteristic
117C2×C107#48211yescharacteristic
118C2×C107#48511yescharacteristic
119D107—42no—
120#1854—1071nomaximal, Sylow
121C107#185411yescharacteristic, complemented, Sylow
122C2×#117—1071no—
123C2×#107—1071no—
124C2×#106—1071no—
125C2×C22⋊C8—1071no—
126#380—1071no—
127#340—1071no—
128#340D10711yescharacteristic, complemented
129C23×C4—1071no—
130C22×C8—1071no—
131C22×C8C2×D10711yescharacteristic
132#117—2142no—
133#117—2142no—
134C2×C4⋊C4—1071no—
135C2×C4⋊C4—1071no—
136C2×C4⋊C4C2×D10711yescharacteristic
137C2×C4⋊C4C2×D10711yescharacteristic
138C2×C22⋊C4—2141no—
139C2×C22⋊C4—2141no—
140C2×C42—2141no—
141#107—2142no—
142#106—2142no—
143C22⋊C8—1074no—
144#95—2141no—
145C24—1071no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—1071no—
149C22×C4—1072no—
150C22×C4—1072no—
151C22×C4—2141no—
152C22×C4—2141no—
153C22×C4—2141no—
154C22×C4—2141no—
155C22×C4—2141no—
156C22×C4—2141no—
157C22×C4C22×D10711yescharacteristic
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—1072no—
161C2×C8—2141no—
162C2×C8—2141no—
163C2×C8C4×D10712yes—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—1072no—
168C4⋊C4—2141no—
169C4⋊C4—2141no—
170C4⋊C4—2141no—
171C4⋊C4D42812yes—
172C22⋊C4—4282no—
173C22⋊C4—2144no—
174C42—2144no—
175C23—1072no—
176C23—2141no—
177C23—2141no—
178C23—2141no—
179C23—2141no—
180C23—2141no—
181C23—2141no—
182C23D4×D10711yescharacteristic
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—1072no—
190C2×C4—4281no—
191C2×C4—4281no—
192C2×C4—4281no—
193C2×C4—4281no—
194C2×C4—4281no—
195C2×C4—2142no—
196C2×C4—2141no—
197C2×C4—2141no—
198C2×C4—2142no—
199C2×C4—2142no—
200C2×C4—2142no—
201C2×C4—2142no—
202C2×C4C2×C4×D10711yescharacteristic
203C2×C4D4×D10711yescharacteristic
204C2×C4C2×D42811yescharacteristic
205C2×C4Q8×D10711yescharacteristic
206C2×C4#1313381711yescharacteristic
207C2×C4#1313381811yescharacteristic
208C8—22no—
209C8—2142no—
210C22—4281no—
211C22—4281no—
212C22—4281no—
213C22—4281no—
214C22—2142no—
215C22—2142no—
216C22—2142no—
217C22C4⋊C4×D10711yescharacteristic
218C22#1681207911yescharacteristic
219C22#1681209111yescharacteristic
220C22#1681212811yescharacteristic
221C22#1681213211yescharacteristic
222C22#1681213611yescharacteristic
223C22#1681213811yescharacteristic
224C4—42no—
225C4—81no—
226C4—8561no—
227C4—4282no—
228C4—2142no—
229C4(C22⋊D107)⋊C411yescharacteristic
230C4C4×D42811yescharacteristic
231C4#1681208711yescharacteristic
232C4#1681208911yescharacteristic
233C2—4282no—
234C2#2029250111yescharacteristic
235C2#2029266411yescharacteristic
236C2#2029266611yescharacteristic
237C2#2029277511yescharacteristic
238C2#2029279011yescharacteristic
239C2#2029294911yescharacteristic
240C2#2029296711yescharacteristic
subgroupG/H
centerC23D4×D107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#340×C107C2
socleC23×C107C2×D4
prime radicalC23×D107D4
cosocleC22×C4×C107C23
hypercenter#340D107
hypocenterC107#1854
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#16812043C2semidirect
3. C2×#16812040C2semidirect
6. #20292389C2semidirect
7. #340×C107C2semidirect
19. (C2×D107)⋊C8C4semidirect
121. C107#1854semidirect
128. #340D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120672
C2×#168120432
C2×#168120402
C2×(C2×D107)⋊C82
#202924842
#202923892
#340×C1072
#1854107