Finite Group Databasebeta

← #28346292

Subgroups — #28346292

Order 13696.

398 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 149 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#20293255C211yesmaximal, characteristic, complemented
2#20292927C212yesmaximal, complemented
3#20292927C212yesmaximal, complemented
4#20292917C212yesmaximal, complemented
5#20292907C212yesmaximal, complemented
6#20292670C211yesmaximal, characteristic, complemented
7#20292663C211yesmaximal, characteristic, complemented
8#20292654C211yesmaximal, characteristic, complemented
9#20292632C211yesmaximal, characteristic
10C2×#16812030C211yesmaximal, characteristic
11#518×C107C211yesmaximal, characteristic
12C2×#13133802C2211yescharacteristic
13C22×C107⋊C8C2211yescharacteristic
14#16812109C2212yescomplemented
15#16812108—21no—
16#16812101C2212yescomplemented
17#16812100—21no—
18#16812086—22no—
19#16812068C2211yescharacteristic
20C2×C4⋊Dic107C2211yescharacteristic
21#16812042C2211yescharacteristic, complemented
22#16812042C2212yescomplemented
23#16812042C2211yescharacteristic, complemented
24#16812040C2211yescharacteristic, complemented
25#16812040C2212yescomplemented
26#16812040C2211yescharacteristic, complemented
27C4⋊(C107⋊C8)C2212yescomplemented
28C4⋊(C107⋊C8)C2212yescomplemented
29#16812030C2212yes—
30#16812030C2212yes—
31#16812030C2212yes—
32#16812030C2212yes—
33C22×D4×C107C2211yescharacteristic
34C4×D4×C107—21no—
35C4×D4×C107—21no—
36C4×D4×C107C2212yes—
37C4×D4×C107C2212yes—
38#117×C107C2211yescharacteristic
39#117×C107C2211yescharacteristic
40C2×C22⋊C4×C107—22no—
41C2×Dic214—22no—
42C22×Dic107—21no—
43#13133802—22no—
44C2×C107⋊C8—22no—
45C2×C107⋊C8C2311yescharacteristic
46C2×C107⋊C8C2312yes—
47C2×C107⋊C8C2311yescharacteristic
48C4⋊Dic107—21no—
49C4⋊Dic107—22no—
50C4⋊Dic107C2312yes—
51C4⋊Dic107C2311yescharacteristic
52C4⋊Dic107C2311yescharacteristic
53Dic107⋊C4—22no—
54Dic107⋊C4—22no—
55C22⋊Dic107—21no—
56C4×Dic107—21no—
57C24×C107—21no—
58C2×D4×C107—21no—
59C2×D4×C107—22no—
60C2×D4×C107Q814yescomplemented
61C2×D4×C107C2312yes—
62C22×C4×C107—22no—
63C22×C4×C107—22no—
64C22×C4×C107C2311yescharacteristic
65C4⋊C4×C107D411yescharacteristic
66C4⋊C4×C107D411yescharacteristic
67C4⋊C4×C107C2311yescharacteristic
68C4⋊C4×C107C2311yescharacteristic
69C22⋊C4×C107—21no—
70C22⋊C4×C107—22no—
71C22⋊C4×C107—22no—
72C22⋊C4×C107D412yes—
73C42×C107—21no—
74C42×C107C2312yes—
75C23×C107—22no—
76C23×C107—24no—
77C23×C107—81no—
78C23×C107—81no—
79C23×C107C2×D411yescharacteristic
80D4×C107—24no—
81D4×C107—22no—
82D4×C107C2×Q814yes—
83C2×C4×C107—21no—
84C2×C4×C107—21no—
85C2×C4×C107—82no—
86C2×C4×C107—82no—
87C2×C4×C107C2×D412yes—
88C2×C4×C107C2×D412yes—
89C2×C4×C107C2×D411yescharacteristic
90C2×C4×C107C2×Q812yes—
91C2×C4×C107#4112yes—
92C2×C4×C107C2411yescharacteristic
93C2×Dic107—21no—
94C2×Dic107—21no—
95C2×Dic107—22no—
96C2×Dic107—81no—
97Dic214—82no—
98C107⋊C8—21no—
99C107⋊C8—21no—
100C107⋊C8—22no—
101C22×C107—21no—
102C22×C107—21no—
103C22×C107—24no—
104C22×C107—42no—
105C22×C107—42no—
106C22×C107—84no—
107C22×C107C22⋊Q812yes—
108C22×C107C22×D411yescharacteristic
109C4×C107—22no—
110C4×C107—22no—
111C4×C107—41no—
112C4×C107—41no—
113C4×C107C22⋊Q812yes—
114C4×C107C22×Q811yescharacteristic
115C4×C107C2×#4111yescharacteristic
116Dic107—41no—
117Dic107—42no—
118Dic107—41no—
119C2×C107—22no—
120C2×C107—44no—
121C2×C107C2×C22⋊Q811yescharacteristic
122C2×C107#57611yescharacteristic
123C2×C107#57711yescharacteristic
124#3081—1071nomaximal, Sylow
125C107#308111yescharacteristic, complemented, Sylow
126#544—1071no—
127#518D10711yescharacteristic
128#478—1072no—
129#478—1072no—
130#476—1072no—
131#474—1072no—
132#428—1071no—
133#427—1071no—
134#424—1071no—
135#418—1071no—
136C2×#102—1071no—
137C22×D4C2×D10711yescharacteristic
138C2×M16—1071no—
139C22×C8—1071no—
140C4⋊Q8—1072no—
141C4⋊Q8—2141no—
142#125—1072no—
143#125—2141no—
144C22⋊Q8—2142no—
145C4×D4—21no—
146C4×D4—21no—
147C4×D4C2×D10712yes—
148C4×D4C2×D10712yes—
149#117—1071no—
150#117C2×D10711yescharacteristic
151#117C2×D10711yescharacteristic
152C2×C4⋊C4—1071no—
153C2×C22⋊C4—22no—
154#107—1071no—
155#107—1071no—
156#107—1072no—
157#106—1071no—
158#106—1071no—
159#106—1072no—
160C4⋊C8—1072no—
161C4⋊C8—1072no—
162#102—1072no—
163#102—1072no—
164#102—1072no—
165#102—1072no—
166C24—21no—
167C2×Q8—2142no—
168C2×D4—21no—
169C2×D4—22no—
170C2×D4Dic21414yescomplemented
171C2×D4C22×D10712yes—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—2141no—
175C22×C4C22×D10711yescharacteristic
176M16—2142no—
177C2×C8—1071no—
178C2×C8—1072no—
179C2×C8—1071no—
180C2×C8—2142no—
181C4⋊C4—1072no—
182C4⋊C4—1071no—
183C4⋊C4—1071no—
184C4⋊C4—2141no—
185C4⋊C4—2142no—
186C4⋊C4—2142no—
187C4⋊C4—2142no—
188C4⋊C4C22⋊D10711yescharacteristic
189C4⋊C4C22⋊D10711yescharacteristic
190C4⋊C4C22×D10711yescharacteristic
191C4⋊C4C22×D10711yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—2141no—
196C22⋊C4C22⋊D10712yes—
197C42—21no—
198C42—2141no—
199C42C22×D10712yes—
200C23—22no—
201C23—24no—
202C23—81no—
203C23—81no—
204C23C2×C22⋊D10711yescharacteristic
205Q8—8562no—
206D4—24no—
207D4—22no—
208D4C2×Dic21414yes—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—8561no—
214C2×C4—2141no—
215C2×C4—2141no—
216C2×C4—2142no—
217C2×C4C2×C22⋊D10712yes—
218C2×C4C2×C22⋊D10712yes—
219C2×C4C2×C22⋊D10711yescharacteristic
220C2×C4C2×Dic21412yes—
221C2×C4#1313381812yes—
222C2×C4C23×D10711yescharacteristic
223C8—2141no—
224C8—2141no—
225C8—2142no—
226C22—21no—
227C22—21no—
228C22—24no—
229C22—42no—
230C22—42no—
231C22—84no—
232C22#1681208812yes—
233C22C22×C22⋊D10711yescharacteristic
234C4—22no—
235C4—22no—
236C4—41no—
237C4—41no—
238C4—4282no—
239C4—4281no—
240C4—4281no—
241C4#1681208812yes—
242C4C22×Dic21411yescharacteristic
243C4C2×#1313381811yescharacteristic
244C2—22no—
245C2—44no—
246C2C2×#1681208811yescharacteristic
247C2#2029343011yescharacteristic
248C2#2029343811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#518×C107C2
socleC22×C107C22×D4
prime radicalC22×D4×C107C22
hypercenter#518D107
hypocenterC107#3081
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #20293255C2semidirect
2. #20292927C2semidirect
3. #20292927C2semidirect
4. #20292917C2semidirect
5. #20292907C2semidirect
6. #20292670C2semidirect
7. #20292663C2semidirect
8. #20292654C2semidirect
14. #16812109C22semidirect
16. #16812101C22semidirect
21. #16812042C22semidirect
22. #16812042C22semidirect
23. #16812042C22semidirect
24. #16812040C22semidirect
25. #16812040C22semidirect
26. #16812040C22semidirect
27. C4⋊(C107⋊C8)C22semidirect
28. C4⋊(C107⋊C8)C22semidirect
60. C2×D4×C107Q8semidirect
125. C107#3081semidirect
170. C2×D4Dic214semidirect

Maximal subgroups

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

subgroupindex
#202932552
#202929272
#202929172
#202929072
#202926702
#202926632
#202926542
#202926322
C2×#168120302
#518×C1072
#3081107