Finite Group Databasebeta

← #28337639

Subgroups — #28337639

Order 13696.

278 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 63 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#20293253C211yesmaximal, characteristic, complemented
2C2×#16812023C211yesmaximal, characteristic
3C2×#16812023C211yesmaximal, characteristic
4C2×#16812023C211yesmaximal, characteristic
5#20292394C211yesmaximal, characteristic, complemented
6#20292393C211yesmaximal, characteristic, complemented
7#342×C107C211yesmaximal, characteristic, complemented
8C22×C22⋊D107C2211yescharacteristic
9#16812088—22no—
10#16812086—22no—
11#16812086—22no—
12C2×C22⋊Dic107C2211yescharacteristic
13C2×C22⋊Dic107C2211yescharacteristic
14C2×C22⋊Dic107C2211yescharacteristic
15#16812023—22no—
16#16812023—22no—
17#16812023—22no—
18C2×C22⋊C4×C107C2211yescharacteristic
19C2×C22⋊C4×C107C2211yescharacteristic
20C2×C22⋊C4×C107C2211yescharacteristic
21C23×D107Q811yescharacteristic, complemented
22C2×Dic214—22no—
23C2×C22⋊D107—21no—
24C2×C22⋊D107—21no—
25C2×C22⋊D107—21no—
26C2×C22⋊D107—21no—
27C2×C22⋊D107—21no—
28C2×C22⋊D107—21no—
29C22×Dic107—21no—
30C22×Dic107—21no—
31C22×Dic107—21no—
32C22×Dic107Q811yescharacteristic, complemented
33C4⋊Dic107—22no—
34Dic107⋊C4—24no—
35C22⋊Dic107—21no—
36C22⋊Dic107—21no—
37C22⋊Dic107—21no—
38C22⋊Dic107—21no—
39C22⋊Dic107D412yes—
40C22⋊Dic107D412yes—
41C24×C107C2311yescharacteristic
42C22×C4×C107—21no—
43C22×C4×C107—21no—
44C22×C4×C107—21no—
45C22⋊C4×C107—21no—
46C22⋊C4×C107—21no—
47C22⋊C4×C107—21no—
48C22⋊C4×C107—21no—
49C22⋊C4×C107—21no—
50C22⋊C4×C107D412yes—
51C23×C107—81no—
52C23×C107C2×D411yescharacteristic
53C23×C107C2×D411yescharacteristic
54C23×C107C2×D411yescharacteristic
55C23×C107C2×Q811yescharacteristic
56C23×C107#4111yescharacteristic
57C23×C107#4111yescharacteristic
58C23×C107#4111yescharacteristic
59C22×D107—21no—
60C22×D107—21no—
61C22×D107—21no—
62C22×D107—81no—
63C2×C4×C107—22no—
64C2×C4×C107—41no—
65C2×C4×C107—41no—
66C2×C4×C107—81no—
67C2×C4×C107—81no—
68C2×C4×C107—81no—
69C22⋊D107—41no—
70C22⋊D107—41no—
71C22⋊D107—41no—
72C22⋊D107—41no—
73C2×Dic107—22no—
74C2×Dic107—22no—
75C2×Dic107—21no—
76C2×Dic107—21no—
77C2×Dic107—21no—
78C2×Dic107—41no—
79C2×Dic107—81no—
80C2×Dic107—81no—
81C2×Dic107—81no—
82Dic214—82no—
83C22×C107—41no—
84C22×C107—41no—
85C22×C107—41no—
86C22×C107—41no—
87C22×C107—41no—
88C22×C107—41no—
89C22×C107—41no—
90C22×C107#12011yescharacteristic
91C22×C107C22⋊Q811yescharacteristic
92C22×C107C22⋊Q811yescharacteristic
93C22×C107C22⋊Q811yescharacteristic
94C22×C107#12411yescharacteristic
95C22×C107#12411yescharacteristic
96C22×C107#12411yescharacteristic
97C2×D107—41no—
98C2×D107—81no—
99C2×D107—81no—
100C2×D107—81no—
101C4×C107—42no—
102C4×C107—81no—
103C4×C107—81no—
104Dic107—42no—
105Dic107—42no—
106Dic107—41no—
107Dic107—81no—
108C2×C107—21no—
109C2×C107—21no—
110C2×C107—21no—
111C2×C107—21no—
112C2×C107—21no—
113C2×C107—21no—
114C2×C107—21no—
115C2×C107#39311yescharacteristic
116D107—81no—
117#2032—1071nomaximal, Sylow
118C107#203211yescharacteristic, complemented, Sylow
119#543—1071no—
120C2×#99—1071no—
121C2×#99—1071no—
122C2×#99—1071no—
123#342—1071no—
124#342—1071no—
125#342D10711yescharacteristic, complemented
126C22×D4—1071no—
127C22⋊Q8—2142no—
128C22⋊Q8—2142no—
129C22⋊Q8—2142no—
130C2×C22⋊C4—1071no—
131C2×C22⋊C4—1071no—
132C2×C22⋊C4—1071no—
133C2×C22⋊C4C2×D10711yescharacteristic
134C2×C22⋊C4C2×D10711yescharacteristic
135C2×C22⋊C4C2×D10711yescharacteristic
136#99—2142no—
137#99—2142no—
138#99—2142no—
139C24—1071no—
140C24C22×D10711yescharacteristic
141C2×Q8—2142no—
142C2×D4—2141no—
143C2×D4—2141no—
144C2×D4—2141no—
145C2×D4—2141no—
146C2×D4—2141no—
147C2×D4—2141no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—1071no—
152C22×C4—2141no—
153C22×C4—2141no—
154C22×C4—2141no—
155C4⋊C4—2144no—
156C4⋊C4—2142no—
157C22⋊C4—21no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—1072no—
163C22⋊C4—1072no—
164C22⋊C4—2141no—
165C22⋊C4—2141no—
166C22⋊C4—2141no—
167C22⋊C4—2141no—
168C22⋊C4D42812yes—
169C23—81no—
170C23—8561no—
171C23—2141no—
172C23—2141no—
173C23—2141no—
174C23D4×D10711yescharacteristic
175C23D4×D10711yescharacteristic
176C23C2×D42811yescharacteristic
177C23Q8×D10711yescharacteristic
178C23#1313381711yescharacteristic
179C23#1313381811yescharacteristic
180C23#1313381811yescharacteristic
181Q8—8562no—
182D4—4281no—
183D4—4281no—
184D4—4281no—
185D4—4281no—
186C2×C4—22no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—8561no—
193C2×C4—8561no—
194C2×C4—8561no—
195C2×C4—4281no—
196C2×C4—2142no—
197C2×C4—2142no—
198C2×C4—2141no—
199C2×C4—2141no—
200C2×C4—2141no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—8561no—
209C22—8561no—
210C22—8561no—
211C22—4281no—
212C22#1681207911yescharacteristic
213C22#1681208711yescharacteristic
214C22#1681208711yescharacteristic
215C22#1681208911yescharacteristic
216C22#1681209511yescharacteristic
217C22#1681209511yescharacteristic
218C22#1681209711yescharacteristic
219C4—42no—
220C4—81no—
221C4—81no—
222C4—8561no—
223C4—4282no—
224C4—4282no—
225C4—4281no—
226C2—21no—
227C2—21no—
228C2—21no—
229C2—21no—
230C2—21no—
231C2—21no—
232C2—21no—
233C2—8561no—
234C2#2029254011yescharacteristic
subgroupG/H
centerC2#20292540
commutator, cosocleC24×C107C23
FrattiniC24C22×D107
Fitting#342×C107C2
socleC2×C107#393
prime radicalC22×C22⋊D107C22
hypercenter#342D107
hypocenterC107#2032
LCSG ▷ C24×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #20293253C2semidirect
5. #20292394C2semidirect
6. #20292393C2semidirect
7. #342×C107C2semidirect
21. C23×D107Q8semidirect
32. C22×Dic107Q8semidirect
118. C107#2032semidirect
125. #342D107semidirect

Maximal subgroups

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

subgroupindex
#202932532
C2×#168120232
#202923942
#202923932
#342×C1072
#2032107