Finite Group Databasebeta

← #28337334

Subgroups — #28337334

Order 13696.

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

normal subgroups only

This group has 376 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
1C22×#13133803C211yesmaximal, characteristic
2C2×#16812040C211yesmaximal, characteristic, complemented
3C2×#16812031C211yesmaximal, characteristic, complemented
4C2×#16812019C211yesmaximal, characteristic, complemented
5#20292530C211yesmaximal, characteristic, complemented
6#20292528C211yesmaximal, characteristic, complemented
7C2×#102×C107C211yesmaximal, characteristic, complemented
8C22×D428C2211yescharacteristic
9C2×#13133803—22no—
10C2×#13133803—22no—
11C2×#13133803C414yescomplemented
12C22×C107⋊C8C2211yescharacteristic
13C2×C4⋊Dic107C2211yescharacteristic
14C2×C22⋊Dic107—22no—
15C2×(C2×D107)⋊C4—22no—
16C2×C4×Dic107C2211yescharacteristic
17#16812040—22no—
18#16812031—22no—
19#16812019—22no—
20C22×D4×C107C2211yescharacteristic
21C22×C8×C107C2211yescharacteristic
22C2×C4⋊C4×C107C2211yescharacteristic
23#102×C107—22no—
24C23×D107—21no—
25C2×D428—21no—
26C2×D428—21no—
27C2×D428—21no—
28C2×D428—21no—
29C2×D428—21no—
30C2×D428C2×C412yes—
31C22×Dic107—21no—
32C22×Dic107D412yes—
33#13133803—28no—
34C2×C107⋊C8C2×C412yes—
35C2×C107⋊C8D412yes—
36C2×C107⋊C8D412yes—
37C4⋊Dic107—21no—
38C4⋊Dic107—21no—
39C22⋊Dic107—44no—
40(C2×D107)⋊C4—44no—
41C4×Dic107—21no—
42C4×Dic107—21no—
43C24×C107—21no—
44C2×D4×C107—21no—
45C2×D4×C107—21no—
46C2×D4×C107—21no—
47C2×D4×C107—21no—
48C2×D4×C107—21no—
49C2×D4×C107C2×C412yes—
50C22×C4×C107—21no—
51C22×C4×C107C2311yescharacteristic
52C2×C8×C107—21no—
53C2×C8×C107—21no—
54C2×C8×C107D412yes—
55C4⋊C4×C107—21no—
56C4⋊C4×C107—21no—
57C23×C107—22no—
58C23×C107—41no—
59C23×C107—41no—
60C23×C107—42no—
61C23×C107—42no—
62C23×C107C2×D411yescharacteristic
63C22×D107—22no—
64C22×D107—41no—
65C22×D107—41no—
66C22×D107—42no—
67C22×D107—42no—
68D4×C107—22no—
69D4×C107—22no—
70D4×C107—41no—
71D4×C107—41no—
72C2×C4×C107—41no—
73C2×C4×C107—42no—
74C2×C4×C107C22×C411yescharacteristic
75C2×C4×C107C2×D411yescharacteristic
76C2×C4×C107C2×D411yescharacteristic
77C2×C4×C107C2×D411yescharacteristic
78C2×C4×C107#4111yescharacteristic
79C2×C4×C107#4111yescharacteristic
80C2×Dic107—22no—
81C2×Dic107—22no—
82C2×Dic107—22no—
83C2×Dic107—41no—
84C2×Dic107—42no—
85D428—22no—
86D428—22no—
87D428—41no—
88D428—41no—
89C8×C107—22no—
90C107⋊C8C22⋊C414yescomplemented
91C22×C107—42no—
92C22×C107—44no—
93C22×C107—82no—
94C22×C107—81no—
95C22×C107—81no—
96C22×C107C4×D411yescharacteristic
97C22×C107#12111yescharacteristic
98C22×C107#12711yescharacteristic
99C22×C107AGL(1,ℤ/8)12yes—
100C22×C107AGL(1,ℤ/8)12yes—
101C2×D107—42no—
102C2×D107—44no—
103C2×D107—82no—
104C2×D107—81no—
105C2×D107—81no—
106C4×C107—81no—
107C4×C107C2×C22⋊C411yescharacteristic
108C4×C107C4×D411yescharacteristic
109C4×C107#12111yescharacteristic
110C4×C107#12411yescharacteristic
111Dic107—42no—
112Dic107—81no—
113C2×C107—82no—
114C2×C107#39011yescharacteristic
115C2×C107#44212yes—
116C2×C107#46912yes—
117C2×C107#49611yescharacteristic
118C2×C107#49611yescharacteristic
119D107—82no—
120#1972—1071nomaximal, Sylow
121C107#197211yescharacteristic, complemented, Sylow
122C22×D8—1071no—
123C2×#106—1071no—
124C2×#102—1071no—
125C2×#102D10711yescharacteristic, complemented
126C2×#97—1071no—
127#390—1071no—
128#390—1071no—
129C22×D4—1071no—
130C22×D4C2×D10711yescharacteristic
131C2×D8—1074no—
132C2×D8—2142no—
133C2×D8—2142no—
134C22×C8—1071no—
135C22×C8C2×D10711yescharacteristic
136C2×C4⋊C4—1071no—
137C2×C4⋊C4C2×D10711yescharacteristic
138C2×C22⋊C4—2142no—
139C2×C22⋊C4—2142no—
140C2×C42—1071no—
141#106—2142no—
142#102—22no—
143#102—2142no—
144#97—2142no—
145C24—21no—
146C24—2141no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—1072no—
153C2×D4—2141no—
154C2×D4—2141no—
155C2×D4—2141no—
156C2×D4—2141no—
157C2×D4—2141no—
158C2×D4C4×D10712yes—
159C22×C4—21no—
160C22×C4—1072no—
161C22×C4—2141no—
162C22×C4C22×D10711yescharacteristic
163D8—2148no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—1072no—
167C2×C8—1072no—
168C2×C8—1072no—
169C2×C8C22⋊D10712yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—2141no—
173C4⋊C4—2141no—
174C22⋊C4—4284no—
175C22⋊C4—4284no—
176C42—2141no—
177C42—2141no—
178C23—22no—
179C23—41no—
180C23—41no—
181C23—42no—
182C23—42no—
183C23—4281no—
184C23—4281no—
185C23—4282no—
186C23—4282no—
187C23—2142no—
188C23D4×D10711yescharacteristic
189D4—22no—
190D4—22no—
191D4—41no—
192D4—41no—
193D4—4281no—
194D4—4281no—
195D4—2142no—
196D4—2142no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—4281no—
200C2×C4—4282no—
201C2×C4—2142no—
202C2×C4—2142no—
203C2×C4—2142no—
204C2×C4C2×C4×D10711yescharacteristic
205C2×C4D4×D10711yescharacteristic
206C2×C4D4×D10711yescharacteristic
207C2×C4C2×C22⋊D10711yescharacteristic
208C2×C4#1313381711yescharacteristic
209C2×C4#1313381811yescharacteristic
210C8—22no—
211C8—1074no—
212C22—44no—
213C22—42no—
214C22—82no—
215C22—81no—
216C22—81no—
217C22—8562no—
218C22—8561no—
219C22—8561no—
220C22—4284no—
221C22—4282no—
222C22(C22⋊D107)⋊C411yescharacteristic
223C22#1681208211yescharacteristic
224C22#1681210511yescharacteristic
225C22#1681213312yes—
226C22#1681213612yes—
227C4—81no—
228C4—8561no—
229C4—4282no—
230C4C22⋊C4×D10711yescharacteristic
231C4C4×C22⋊D10711yescharacteristic
232C4#1681208111yescharacteristic
233C4#1681209511yescharacteristic
234C2—82no—
235C2—8562no—
236C2#2029252711yescharacteristic
237C2#2029273612yes—
238C2#2029287912yes—
239C2#2029301311yescharacteristic
240C2#2029301511yescharacteristic
subgroupG/H
centerC23D4×D107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
FittingC2×#102×C107C2
socleC23×C107C2×D4
prime radicalC22×#13133803C2
cosocleC22×C4×C107C23
hypercenterC2×#102D107
hypocenterC107#1972
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×#16812040C2semidirect
3. C2×#16812031C2semidirect
4. C2×#16812019C2semidirect
5. #20292530C2semidirect
6. #20292528C2semidirect
7. C2×#102×C107C2semidirect
11. C2×#13133803C4semidirect
90. C107⋊C8C22⋊C4semidirect
121. C107#1972semidirect
125. C2×#102D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338032
C2×#168120402
C2×#168120312
C2×#168120192
#202925302
#202925282
C2×#102×C1072
#1972107