Finite Group Databasebeta

← #28336959

Subgroups — #28336959

Order 13696.

332 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 79 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#20293425C211yesmaximal, characteristic
2#20293095C211yesmaximal, characteristic, complemented
3#20292645C211yesmaximal, characteristic, complemented
4C2×#16812036C211yesmaximal, characteristic, complemented
5#20292397C211yesmaximal, characteristic, complemented
6#20292386C211yesmaximal, characteristic, complemented
7#410×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9#16812140C412yescomplemented
10#16812139—21no—
11#16812135C412yescomplemented
12#16812134—21no—
13#16812129—22no—
14C2×#13133802C2211yescharacteristic
15C2×#13133802C2211yescharacteristic
16#16812074—21no—
17#16812071—21no—
18(C22⋊D107)⋊C4—22no—
19#16812066—21no—
20C2×C4×Dic107C2211yescharacteristic
21C42×D107—21no—
22#16812037—22no—
23#16812036—21no—
24#16812036—21no—
25C2×#41×C107C2211yescharacteristic
26#117×C107C2211yescharacteristic
27#117×C107C2211yescharacteristic
28#99×C107—22no—
29#13133818—21no—
30#13133818—21no—
31#13133818C2×C412yes—
32#13133818D412yes—
33C2×Dic214D411yescharacteristic
34C2×D428D411yescharacteristic
35C2×C22⋊D107—21no—
36C22×Dic107—21no—
37C2×C4×D107—21no—
38C2×C4×D107—22no—
39#13133808—22no—
40#13133806—22no—
41#13133805—22no—
42#13133803—22no—
43#13133802—21no—
44#13133802—21no—
45#13133802—21no—
46#13133802C2×C412yescomplemented
47C2×C107⋊C8—21no—
48C2×C107⋊C8—21no—
49Dic107⋊C4—22no—
50(C2×D107)⋊C4—22no—
51C4×Dic107—22no—
52C4×Dic107—21no—
53C4×Dic107D411yescharacteristic
54C4×Dic107D411yescharacteristic
55#41×C107—21no—
56#41×C107—21no—
57#41×C107—21no—
58#41×C107—21no—
59C2×Q8×C107C2×C411yescharacteristic, complemented
60C2×D4×C107—21no—
61C2×D4×C107C2×C411yescharacteristic, complemented
62C22×C4×C107—21no—
63C22×C4×C107C2311yescharacteristic
64C4⋊C4×C107—21no—
65C4⋊C4×C107—21no—
66C22⋊C4×C107—21no—
67C22⋊C4×C107D412yes—
68C42×C107—21no—
69C42×C107—21no—
70C23×C107—21no—
71C23×C107C2×D411yescharacteristic
72C22×D107—21no—
73Q8×C107—21no—
74Q8×C107—21no—
75D4×C107—21no—
76D4×C107—21no—
77D4×C107—41no—
78D4×C107—41no—
79C2×C4×C107—21no—
80C2×C4×C107—22no—
81C2×C4×C107—41no—
82C2×C4×C107—41no—
83C2×C4×C107—41no—
84C2×C4×C107C22×C411yescharacteristic
85C2×C4×C107C2×D411yescharacteristic
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107C2×D411yescharacteristic
88C2×C4×C107#4111yescharacteristic
89C2×C4×C107#4111yescharacteristic
90C22⋊D107—22no—
91C22⋊D107—22no—
92C2×Dic107—21no—
93C2×Dic107—21no—
94C2×Dic107—21no—
95C2×Dic107—42no—
96D428—21no—
97D428C22⋊C412yes—
98C4×D107—22no—
99C4×D107—22no—
100C4×D107—44no—
101Dic214—21no—
102Dic214C22⋊C412yes—
103C107⋊C8—22no—
104C107⋊C8—41no—
105C22×C107—41no—
106C22×C107—41no—
107C22×C107—81no—
108C22×C107C4×D411yescharacteristic
109C22×C107#12011yescharacteristic
110C22×C107#12111yescharacteristic
111C2×D107—22no—
112C2×D107—42no—
113C4×C107—42no—
114C4×C107—41no—
115C4×C107—81no—
116C4×C107C2×C22⋊C411yescharacteristic
117C4×C107C4×D411yescharacteristic
118C4×C107#12111yescharacteristic
119C4×C107#12311yescharacteristic
120Dic107—22no—
121Dic107—22no—
122Dic107—41no—
123C2×C107—21no—
124C2×C107—21no—
125C2×C107—21no—
126C2×C107—81no—
127C2×C107#38611yescharacteristic
128D107—42no—
129#1906—1071nomaximal, Sylow
130C107#190611yescharacteristic, complemented, Sylow
131#575—1071no—
132C4×#41—1071no—
133#421—1071no—
134C2×#104—1071no—
135#410D10711yescharacteristic, complemented
136#344—1071no—
137#339—1071no—
138C2×#41—1071no—
139C2×#41C2×D10711yescharacteristic
140#137—1072no—
141#137—2141no—
142AGL(1,ℤ/8)—1072no—
143AGL(1,ℤ/8)—2141no—
144#135—2142no—
145C2×M16—1071no—
146C2×M16—1071no—
147C4×Q8—2141no—
148C4×D4—2141no—
149C4×D4—2142no—
150#117—2141no—
151#117C2×D10711yescharacteristic
152#117C2×D10711yescharacteristic
153C2×C42—1071no—
154C2×C42—2141no—
155#104—2142no—
156#104—2141no—
157#104—2141no—
158#99—22no—
159#41—21no—
160#41—21no—
161#41—21no—
162#41—21no—
163#41—1072no—
164#41—1072no—
165#41—2141no—
166#41—2141no—
167C2×Q8—1071no—
168C2×Q8C4×D10711yescharacteristic, complemented
169C2×D4—21no—
170C2×D4—1071no—
171C2×D4—2141no—
172C2×D4C4×D10711yescharacteristic, complemented
173C22×C4—21no—
174C22×C4—2142no—
175C22×C4—2141no—
176C22×C4—2141no—
177C22×C4C22×D10711yescharacteristic
178Q16—2142no—
179SD16—2142no—
180SD16—2142no—
181D8—2142no—
182M16—1072no—
183M16—2141no—
184M16—2141no—
185M16—2141no—
186C2×C8—2141no—
187C2×C8—2141no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—2142no—
191C22⋊C4—21no—
192C22⋊C4—2142no—
193C22⋊C4C22⋊D10712yes—
194C42—21no—
195C42—21no—
196C42—1071no—
197C42—1071no—
198C42—2142no—
199C42—2141no—
200C23—21no—
201C23—2141no—
202C23C2×C22⋊D10711yescharacteristic
203Q8—21no—
204Q8—21no—
205Q8—1072no—
206Q8—2141no—
207D4—21no—
208D4—21no—
209D4—41no—
210D4—41no—
211D4—1072no—
212D4—2142no—
213D4—2142no—
214D4—2141no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—4284no—
221C2×C4—4282no—
222C2×C4—2141no—
223C2×C4—2141no—
224C2×C4—2141no—
225C2×C4—2142no—
226C2×C4—2142no—
227C2×C4C2×C4×D10711yescharacteristic
228C2×C4D4×D10711yescharacteristic
229C2×C4D4×D10711yescharacteristic
230C2×C4D4×D10711yescharacteristic
231C2×C4#1313381711yescharacteristic
232C2×C4#1313381811yescharacteristic
233C8—4281no—
234C8—2142no—
235C22—41no—
236C22—41no—
237C22—81no—
238C22—4282no—
239C22—2142no—
240C22C4×C22⋊D10711yescharacteristic
241C22#1681207711yescharacteristic
242C22#1681208011yescharacteristic
243C4—42no—
244C4—41no—
245C4—81no—
246C4—4281no—
247C4—2142no—
248C4—2142no—
249C4C22⋊C4×D10711yescharacteristic
250C4(C22⋊D107)⋊C411yescharacteristic
251C4#1681208211yescharacteristic
252C4#1681209111yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—81no—
257C2—4282no—
258C2#2029250311yescharacteristic
subgroupG/H
centerC4#16812082
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#410×C107C2
socleC2×C107#386
prime radical#20293425C2
cosocleC22×C4×C107C23
hypercenter#410D107
hypocenterC107#1906
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #20293095C2semidirect
3. #20292645C2semidirect
4. C2×#16812036C2semidirect
5. #20292397C2semidirect
6. #20292386C2semidirect
7. #410×C107C2semidirect
9. #16812140C4semidirect
11. #16812135C4semidirect
46. #13133802C2×C4semidirect
59. C2×Q8×C107C2×C4semidirect
61. C2×D4×C107C2×C4semidirect
130. C107#1906semidirect
135. #410D107semidirect
168. C2×Q8C4×D107semidirect
172. C2×D4C4×D107semidirect

Maximal subgroups

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

subgroupindex
#202934252
#202930952
#202926452
C2×#168120362
#202923972
#202923862
#410×C1072
#1906107