Finite Group Databasebeta

← #28337678

Subgroups — #28337678

Order 13696.

278 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 67 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#20293201C211yesmaximal, characteristic
2#20292667C211yesmaximal, characteristic, complemented
3C2×#16812037C211yesmaximal, characteristic
4#20292626C211yesmaximal, characteristic
5C2×#16812028C211yesmaximal, characteristic, complemented
6#20292387C211yesmaximal, characteristic, complemented
7#391×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9C2×#13133802C2211yescharacteristic
10C2×#13133802C2211yescharacteristic
11#16812107—21no—
12#16812098—21no—
13#16812092—22no—
14#16812085—21no—
15#16812085—21no—
16#16812068C2211yescharacteristic
17#16812042—21no—
18#16812040—21no—
19#16812037—22no—
20#16812032—21no—
21#16812029—21no—
22#16812028—22no—
23C22×Q8×C107C2211yescharacteristic
24C2×C4⋊C4×C107C2211yescharacteristic
25C2×C42×C107C2211yescharacteristic
26#95×C107—22no—
27#13133818—21no—
28#13133818—21no—
29#13133818—21no—
30#13133818—21no—
31C2×Dic214D411yescharacteristic
32C2×D428D411yescharacteristic
33C2×C22⋊D107—21no—
34C2×C4×D107—21no—
35#13133802—21no—
36#13133802—21no—
37#13133802—21no—
38#13133802D412yes—
39C2×C107⋊C8—21no—
40C2×C107⋊C8—21no—
41C4⋊Dic107D411yescharacteristic
42C4⋊Dic107D411yescharacteristic
43Dic107⋊C4—22no—
44Dic107⋊C4—22no—
45C22⋊Dic107—21no—
46(C2×D107)⋊C4—22no—
47(C2×D107)⋊C4—22no—
48C4×Dic107—21no—
49C2×Q8×C107—21no—
50C2×Q8×C107—21no—
51C2×Q8×C107—21no—
52C2×Q8×C107—21no—
53C2×Q8×C107—21no—
54C2×Q8×C107—21no—
55C22×C4×C107—21no—
56C22×C4×C107—21no—
57C22×C4×C107D412yes—
58C22×C4×C107C2311yescharacteristic
59C4⋊C4×C107—21no—
60C4⋊C4×C107—21no—
61C42×C107—21no—
62C42×C107—21no—
63C23×C107C2×D411yescharacteristic
64C22×D107—21no—
65Q8×C107—41no—
66Q8×C107—41no—
67Q8×C107—41no—
68Q8×C107—41no—
69C2×C4×C107—22no—
70C2×C4×C107—22no—
71C2×C4×C107—22no—
72C2×C4×C107—22no—
73C2×C4×C107—41no—
74C2×C4×C107—41no—
75C2×C4×C107—42no—
76C2×C4×C107—41no—
77C2×C4×C107C2×D411yescharacteristic
78C2×C4×C107C2×D411yescharacteristic
79C2×C4×C107C2×D411yescharacteristic
80C2×C4×C107#4111yescharacteristic
81C2×C4×C107#4111yescharacteristic
82C2×C4×C107#4111yescharacteristic
83C22⋊D107—41no—
84C22⋊D107—41no—
85C2×Dic107—21no—
86C2×Dic107—21no—
87C2×Dic107—21no—
88D428—21no—
89D428—21no—
90C4×D107—41no—
91C4×D107—41no—
92Dic214—21no—
93Dic214—21no—
94C107⋊C8—22no—
95C107⋊C8—41no—
96C22×C107—21no—
97C22×C107—21no—
98C22×C107#12011yescharacteristic
99C22×C107#12111yescharacteristic
100C22×C107#12311yescharacteristic
101C2×D107—41no—
102C2×D107—41no—
103C2×D107—41no—
104C4×C107—42no—
105C4×C107—42no—
106C4×C107—81no—
107C4×C107#12111yescharacteristic
108C4×C107#12111yescharacteristic
109C4×C107#12311yescharacteristic
110C4×C107#12411yescharacteristic
111Dic107—41no—
112Dic107—41no—
113Dic107—41no—
114C2×C107—21no—
115C2×C107—21no—
116C2×C107#39411yescharacteristic
117C2×C107#45411yescharacteristic
118C2×C107#45611yescharacteristic
119D107—81no—
120#2038—1071nomaximal, Sylow
121C107#203811yescharacteristic, complemented, Sylow
122#536—1071no—
123#428—1071no—
124C2×#104—1071no—
125#417—1071no—
126C2×#101—1071no—
127#391D10711yescharacteristic, complemented
128#339—1071no—
129C2×#41—1071no—
130C22×Q8C2×D10711yescharacteristic
131C2×M16—1071no—
132C2×M16—1071no—
133C4⋊Q8—2141no—
134#124—2141no—
135#123—2142no—
136C22⋊Q8—2141no—
137C22⋊Q8—2141no—
138#117—1071no—
139C2×C4⋊C4C2×D10711yescharacteristic
140C2×C42C2×D10711yescharacteristic
141#107—2141no—
142#106—2141no—
143#104—2142no—
144#103—2141no—
145#102—2141no—
146#101—2142no—
147#95—22no—
148#41—2141no—
149#41—2141no—
150#41—2141no—
151#41—2141no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—1071no—
159C2×D4—1071no—
160C2×D4—2141no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—2141no—
164C22×C4C22⋊D10712yes—
165C22×C4C22×D10711yescharacteristic
166M16—1072no—
167M16—2141no—
168M16—2141no—
169M16—2141no—
170C2×C8—2141no—
171C2×C8—2141no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—1071no—
175C4⋊C4—1071no—
176C4⋊C4—2142no—
177C4⋊C4—2142no—
178C22⋊C4—2142no—
179C22⋊C4—2142no—
180C22⋊C4—2141no—
181C42—21no—
182C42—21no—
183C42—2141no—
184C23—2141no—
185C23C2×C22⋊D10711yescharacteristic
186Q8—41no—
187Q8—41no—
188Q8—41no—
189Q8—41no—
190Q8—2141no—
191Q8—2141no—
192D4—4281no—
193D4—4281no—
194D4—2141no—
195D4—2141no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—4281no—
205C2×C4—4281no—
206C2×C4—2141no—
207C2×C4—2141no—
208C2×C4—2141no—
209C2×C4D4×D10711yescharacteristic
210C2×C4D4×D10711yescharacteristic
211C2×C4D4×D10711yescharacteristic
212C2×C4#1313381711yescharacteristic
213C2×C4#1313381811yescharacteristic
214C2×C4#1313381811yescharacteristic
215C8—4281no—
216C8—2142no—
217C22—21no—
218C22—21no—
219C22—4281no—
220C22—4281no—
221C22—4281no—
222C22#1681207711yescharacteristic
223C22#1681208111yescharacteristic
224C22#1681209211yescharacteristic
225C4—42no—
226C4—42no—
227C4—81no—
228C4—4281no—
229C4—4281no—
230C4—4281no—
231C4#1681208211yescharacteristic
232C4#1681208211yescharacteristic
233C4#1681209111yescharacteristic
234C4#1681209511yescharacteristic
235C2—21no—
236C2—21no—
237C2—8561no—
238C2#2029254211yescharacteristic
239C2#2029279911yescharacteristic
240C2#2029280911yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#391×C107C2
socleC22×C107#120
prime radicalC2×#13133818C22
hypercenter#391D107
hypocenterC107#2038
LCSG ▷ C22×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #20292667C2semidirect
5. C2×#16812028C2semidirect
6. #20292387C2semidirect
7. #391×C107C2semidirect
121. C107#2038semidirect
127. #391D107semidirect

Maximal subgroups

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

subgroupindex
#202932012
#202926672
C2×#168120372
#202926262
C2×#168120282
#202923872
#391×C1072
#2038107