Finite Group Databasebeta

← #28349494

Subgroups — #28349494

Order 13696.

350 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 125 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#20293356C211yesmaximal, characteristic, complemented
2#20293350C211yesmaximal, characteristic
3#20293021C212yesmaximal
4#20293007C211yesmaximal, characteristic, complemented
5#20292850C212yesmaximal
6#20292838C212yesmaximal, complemented
7#20292838C211yesmaximal, characteristic, complemented
8#20292833C211yesmaximal, characteristic
9#20292735C212yesmaximal, complemented
10C4×#13133807C211yesmaximal, characteristic
11#446×C107C211yesmaximal, characteristic, complemented
12C2×Dic428C2212yes—
13C2×Dic428C2212yes—
14C2×Dic428C2212yes—
15C2×#13133807C2211yescharacteristic
16C2×#13133807C2212yes—
17#16812109C2211yescharacteristic
18#16812109C2211yescharacteristic
19#16812109C2212yes—
20#16812104—22no—
21#16812100—21no—
22#16812100—22no—
23#16812097C2212yes—
24#16812085—21no—
25Q8×Dic107—21no—
26C4×Dic214C2211yescharacteristic
27C4×Dic214C2212yes—
28C4×Dic214C2212yes—
29C4×D428C2211yescharacteristic
30#16812067—21no—
31#16812041C2211yescharacteristic
32#16812034C2211yescharacteristic
33#16812034C2212yes—
34#16812034C2212yes—
35#16812031C2211yescharacteristic
36C4×Q8×C107C2211yescharacteristic
37C4⋊C8×C107C2211yescharacteristic
38C4⋊C8×C107C2212yes—
39#97×C107C2212yes—
40C4×C8×C107C2211yescharacteristic
41C2×Dic214C2312yes—
42C2×Dic214C2311yescharacteristic
43C2×Dic214C2312yes—
44C2×D428C2311yescharacteristic
45C2×C4×D107—21no—
46Dic428—24no—
47Dic428—42no—
48Dic428—42no—
49#13133807—22no—
50#13133807—42no—
51C4⋊Dic107—21no—
52C4⋊Dic107—22no—
53C4⋊Dic107—21no—
54C4⋊Dic107—22no—
55C4⋊Dic107—21no—
56C4⋊Dic107—21no—
57C4⋊Dic107C2311yescharacteristic
58C4⋊Dic107C2311yescharacteristic
59Dic107⋊C4—22no—
60Dic107⋊C4—22no—
61Dic107⋊C4—21no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4—21no—
64(C2×D107)⋊C4—22no—
65C4×Dic107—22no—
66C4×Dic107—22no—
67C4×Dic107—21no—
68C2×Q8×C107D411yescharacteristic
69C2×C8×C107C2312yes—
70C2×C8×C107C2311yescharacteristic
71C2×C8×C107C2311yescharacteristic
72C4⋊C4×C107D411yescharacteristic
73C4⋊C4×C107D412yes—
74C42×C107C2311yescharacteristic
75C42×C107C2312yes—
76C22×D107—21no—
77Q8×C107—22no—
78C2×C4×C107C2×D412yes—
79C2×C4×C107C2×D411yescharacteristic
80C2×C4×C107C2×D412yes—
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2411yescharacteristic
83C2×Dic107—21no—
84C2×Dic107—21no—
85C2×Dic107—22no—
86C2×Dic107—21no—
87C2×Dic107—22no—
88D428—21no—
89D428—21no—
90C4×D107—42no—
91Dic214—21no—
92Dic214—22no—
93Dic214—22no—
94Dic214—22no—
95Dic214—22no—
96Dic214—21no—
97C8×C107—22no—
98C8×C107—21no—
99C8×C107#4112yes—
100C22×C107C22×D411yescharacteristic
101C2×D107—41no—
102C2×D107—41no—
103C2×D107—41no—
104C4×C107—22no—
105C4×C107—22no—
106C4×C107—21no—
107C4×C107#13712yes—
108C4×C107C2×#4111yescharacteristic
109C4×C107H(5,𝔽2)11yescharacteristic
110Dic107—41no—
111Dic107—42no—
112Dic107—42no—
113Dic107—41no—
114Dic107—41no—
115C2×C107#55011yescharacteristic
116C2×C107C2×#13711yescharacteristic
117C2×C107#57811yescharacteristic
118D107—81no—
119#3398—1071nomaximal, Sylow
120C107#339811yescharacteristic, complemented, Sylow
121#557—1071no—
122#556—1071no—
123#497—1072no—
124#494—1071no—
125#464—1072no—
126#462—1072no—
127#462—1071no—
128#461—1071no—
129#446D10711yescharacteristic, complemented
130#441—1072no—
131C4×SD16—1071no—
132C2×Q16—1072no—
133C2×Q16—1072no—
134C2×Q16—1072no—
135C2×SD16—1072no—
136C2×SD16—1071no—
137C4⋊Q8—1071no—
138C4⋊Q8—1071no—
139C4⋊Q8—1072no—
140#126—2142no—
141#125—2141no—
142#125—2142no—
143#124—1072no—
144C22⋊Q8—2141no—
145C4×Q8—1072no—
146C4×Q8—1071no—
147C4×Q8—1072no—
148C4×Q8—2141no—
149C4×Q8C2×D10711yescharacteristic
150C4×D4—1071no—
151#117—2141no—
152#106—1071no—
153C4⋊C8C2×D10711yescharacteristic
154C4⋊C8C2×D10712yes—
155#103—1071no—
156#103—1072no—
157#103—1072no—
158#102—1071no—
159#97C2×D10712yes—
160C4×C8C2×D10711yescharacteristic
161C2×Q8—1072no—
162C2×Q8—1071no—
163C2×Q8—1072no—
164C2×Q8D42811yescharacteristic
165C2×D4—1071no—
166C22×C4—2141no—
167Q16—4282no—
168Q16—4282no—
169Q16—2144no—
170SD16—4282no—
171SD16—2142no—
172C2×C8C22×D10711yescharacteristic
173C2×C8C22×D10712yes—
174C2×C8C22×D10711yescharacteristic
175C4⋊C4—1071no—
176C4⋊C4—1071no—
177C4⋊C4—2141no—
178C4⋊C4—2142no—
179C4⋊C4—2142no—
180C4⋊C4—2141no—
181C4⋊C4—2142no—
182C4⋊C4—2141no—
183C4⋊C4—2142no—
184C4⋊C4—2141no—
185C4⋊C4—2141no—
186C4⋊C4D42811yescharacteristic
187C4⋊C4D42812yes—
188C22⋊C4—2142no—
189C22⋊C4—2141no—
190C22⋊C4—2142no—
191C42—2142no—
192C42—2142no—
193C42—2141no—
194C42C22×D10711yescharacteristic
195C42C22×D10712yes—
196C23—2141no—
197Q8—22no—
198Q8—2141no—
199Q8—2142no—
200Q8—2142no—
201Q8—2142no—
202Q8—2142no—
203Q8—2141no—
204D4—2141no—
205D4—2141no—
206C2×C4—4282no—
207C2×C4—2141no—
208C2×C4—2141no—
209C2×C4—2142no—
210C2×C4—2141no—
211C2×C4—2142no—
212C2×C4C2×D42812yes—
213C2×C4C2×D42812yes—
214C2×C4C2×D42811yescharacteristic
215C2×C4C2×D42811yescharacteristic
216C2×C4C23×D10711yescharacteristic
217C8—22no—
218C8—21no—
219C8#1313381912yes—
220C22—4281no—
221C22—4281no—
222C22—4281no—
223C22C22×D42811yescharacteristic
224C4—22no—
225C4—22no—
226C4—21no—
227C4—4281no—
228C4—4281no—
229C4—4282no—
230C4—4281no—
231C4—4282no—
232C4#1681214212yes—
233C4C2×#1313381911yescharacteristic
234C4#1681215511yescharacteristic
235C2—8561no—
236C2#2029329911yescharacteristic
237C2C2×#1681214211yescharacteristic
238C2#2029344411yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#446×C107C2
socleC22×C107C22×D4
prime radicalC2×D428C23
hypercenter#446D107
hypocenterC107#3398
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #446

Complements

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

NH ≅ G/Ntype
1. #20293356C2semidirect
4. #20293007C2semidirect
6. #20292838C2semidirect
7. #20292838C2semidirect
9. #20292735C2semidirect
11. #446×C107C2semidirect
120. C107#3398semidirect
129. #446D107semidirect

Maximal subgroups

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

subgroupindex
#202933562
#202933502
#202930212
#202930072
#202928502
#202928382
#202928332
#202927352
C4×#131338072
#446×C1072
#3398107