Finite Group Databasebeta

← #28349488

Subgroups — #28349488

Order 13696.

350 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 123 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#20293350C211yesmaximal, characteristic
2#20293019C212yesmaximal
3#20293006C211yesmaximal, characteristic, complemented
4#20292849C212yesmaximal
5#20292837C211yesmaximal, characteristic, complemented
6#20292834C212yesmaximal, complemented
7#20292829C211yesmaximal, characteristic
8#20292758C211yesmaximal, characteristic, complemented
9#20292732C212yesmaximal, complemented
10C4×#13133805C211yesmaximal, characteristic
11#557×C107C211yesmaximal, characteristic, complemented
12C2×#13133808C2212yes—
13C2×#13133808C2212yes—
14C2×#13133808C2212yes—
15C2×#13133805C2211yescharacteristic
16C2×#13133805C2212yes—
17#16812109C2211yescharacteristic
18#16812104—22no—
19#16812097C2212yes—
20#16812085—21no—
21C4×Dic214C2212yes—
22C4×D428C2211yescharacteristic
23#16812067—21no—
24#16812040C2211yescharacteristic
25C4⋊(C107⋊C8)C2211yescharacteristic
26C4⋊(C107⋊C8)C2212yes—
27#16812033C2211yescharacteristic
28#16812033C2212yes—
29#16812032C2212yes—
30#16812029C2211yescharacteristic
31#16812020C2212yes—
32C4×C107⋊C8C2211yescharacteristic
33C4⋊Q8×C107C2211yescharacteristic
34C4⋊Q8×C107C2212yes—
35#125×C107—21no—
36#125×C107—22no—
37C4×Q8×C107—21no—
38C4×Q8×C107C2211yescharacteristic
39C4×Q8×C107C2212yes—
40C4×Q8×C107C2211yescharacteristic
41C2×Dic214C2312yes—
42C2×D428C2311yescharacteristic
43C2×C4×D107—21no—
44#13133808—24no—
45#13133808—42no—
46#13133808—42no—
47#13133805—22no—
48#13133805—42no—
49C2×C107⋊C8C2312yes—
50C2×C107⋊C8C2311yescharacteristic
51C2×C107⋊C8C2311yescharacteristic
52C4⋊Dic107—21no—
53C4⋊Dic107—21no—
54C4⋊Dic107C2311yescharacteristic
55Dic107⋊C4—22no—
56(C2×D107)⋊C4—22no—
57(C2×D107)⋊C4—21no—
58(C2×D107)⋊C4—22no—
59C4×Dic107—22no—
60C2×Q8×C107D411yescharacteristic
61C2×Q8×C107C2311yescharacteristic
62C2×Q8×C107C2312yes—
63C4⋊C4×C107—22no—
64C4⋊C4×C107—21no—
65C4⋊C4×C107—22no—
66C4⋊C4×C107—21no—
67C4⋊C4×C107—22no—
68C4⋊C4×C107—21no—
69C4⋊C4×C107D411yescharacteristic
70C4⋊C4×C107D412yes—
71C4⋊C4×C107C2311yescharacteristic
72C42×C107—22no—
73C42×C107—21no—
74C42×C107C2311yescharacteristic
75C42×C107C2312yes—
76C22×D107—21no—
77Q8×C107—22no—
78Q8×C107—21no—
79Q8×C107—22no—
80Q8×C107—22no—
81Q8×C107—21no—
82C2×C4×C107—21no—
83C2×C4×C107—21no—
84C2×C4×C107—22no—
85C2×C4×C107C2×D412yes—
86C2×C4×C107C2×D412yes—
87C2×C4×C107C2×D411yescharacteristic
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107C2411yescharacteristic
90C2×Dic107—21no—
91C2×Dic107—22no—
92D428—21no—
93D428—21no—
94C4×D107—42no—
95Dic214—22no—
96Dic214—22no—
97C107⋊C8—22no—
98C107⋊C8—21no—
99C107⋊C8#4112yes—
100C22×C107C22×D411yescharacteristic
101C2×D107—41no—
102C2×D107—41no—
103C2×D107—41no—
104C4×C107—22no—
105C4×C107—22no—
106C4×C107—21no—
107C4×C107—41no—
108C4×C107—42no—
109C4×C107—41no—
110C4×C107#13712yes—
111C4×C107C2×#4111yescharacteristic
112C4×C107H(5,𝔽2)11yescharacteristic
113Dic107—42no—
114Dic107—41no—
115C2×C107#55011yescharacteristic
116C2×C107C2×#13711yescharacteristic
117C2×C107#57811yescharacteristic
118D107—81no—
119#3398—1071nomaximal, Sylow
120C107#339811yescharacteristic, complemented, Sylow
121#557D10711yescharacteristic, complemented
122#556—1071no—
123#497—1072no—
124#494—1071no—
125#464—1072no—
126#462—1072no—
127#462—1071no—
128#461—1071no—
129#446—1071no—
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⋊Q8C2×D10711yescharacteristic
139C4⋊Q8C2×D10712yes—
140#126—2142no—
141#125—21no—
142#125—22no—
143#124—1072no—
144C22⋊Q8—2141no—
145C4×Q8—21no—
146C4×Q8—1072no—
147C4×Q8C2×D10711yescharacteristic
148C4×Q8C2×D10712yes—
149C4×Q8C2×D10711yescharacteristic
150C4×D4—1071no—
151#117—2141no—
152#106—1071no—
153C4⋊C8—1071no—
154C4⋊C8—1072no—
155#103—1071no—
156#103—1072no—
157#103—1072no—
158#102—1071no—
159#97—1072no—
160C4×C8—1071no—
161C2×Q8—1072no—
162C2×Q8C22⋊D10711yescharacteristic
163C2×Q8C22×D10711yescharacteristic
164C2×Q8C22×D10712yes—
165C2×D4—1071no—
166C22×C4—2141no—
167Q16—4282no—
168Q16—4282no—
169Q16—2144no—
170SD16—4282no—
171SD16—2142no—
172C2×C8—1071no—
173C2×C8—1071no—
174C2×C8—1072no—
175C4⋊C4—22no—
176C4⋊C4—21no—
177C4⋊C4—22no—
178C4⋊C4—21no—
179C4⋊C4—22no—
180C4⋊C4—21no—
181C4⋊C4—1071no—
182C4⋊C4—2141no—
183C4⋊C4—2142no—
184C4⋊C4—2141no—
185C4⋊C4C22⋊D10711yescharacteristic
186C4⋊C4C22⋊D10712yes—
187C4⋊C4C22×D10711yescharacteristic
188C22⋊C4—2142no—
189C22⋊C4—2141no—
190C22⋊C4—2142no—
191C42—22no—
192C42—21no—
193C42—2142no—
194C42C22×D10711yescharacteristic
195C42C22×D10712yes—
196C23—2141no—
197Q8—22no—
198Q8—21no—
199Q8—22no—
200Q8—22no—
201Q8—21no—
202Q8—2142no—
203Q8—2142no—
204D4—2141no—
205D4—2141no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—4282no—
210C2×C4—2141no—
211C2×C4—2142no—
212C2×C4C2×C22⋊D10712yes—
213C2×C4C2×C22⋊D10712yes—
214C2×C4C2×C22⋊D10711yescharacteristic
215C2×C4C2×C22⋊D10711yescharacteristic
216C2×C4C23×D10711yescharacteristic
217C8—1072no—
218C8—2142no—
219C8—2141no—
220C22—4281no—
221C22—4281no—
222C22—4281no—
223C22C22×C22⋊D10711yescharacteristic
224C4—22no—
225C4—22no—
226C4—21no—
227C4—41no—
228C4—41no—
229C4—42no—
230C4—4281no—
231C4—4282no—
232C4#1681214012yes—
233C4#41×D10711yescharacteristic
234C4#1681215511yescharacteristic
235C2—8561no—
236C2#2029329611yescharacteristic
237C2C2×#1681214011yescharacteristic
238C2#2029344211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#557×C107C2
socleC22×C107C22×D4
prime radicalC2×D428C23
hypercenter#557D107
hypocenterC107#3398
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
3. #20293006C2semidirect
5. #20292837C2semidirect
6. #20292834C2semidirect
8. #20292758C2semidirect
9. #20292732C2semidirect
11. #557×C107C2semidirect
120. C107#3398semidirect
121. #557D107semidirect

Maximal subgroups

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

subgroupindex
#202933502
#202930192
#202930062
#202928492
#202928372
#202928342
#202928292
#202927582
#202927322
C4×#131338052
#557×C1072
#3398107