Finite Group Databasebeta

← #28345326

Subgroups — #28345326

Order 13696.

456 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 175 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
1C2×Q16×D107C211yesmaximal, characteristic
2#20293103C212yesmaximal
3#20292735C211yesmaximal, characteristic, complemented
4#20292734C211yesmaximal, characteristic, complemented
5#20292733C211yesmaximal, characteristic, complemented
6C4×#13133808C212yesmaximal, complemented
7#20292719C212yesmaximal, complemented
8#20292662C211yesmaximal, characteristic
9#20292634C212yesmaximal
10#20292584C211yesmaximal, characteristic
11#441×C107C211yesmaximal, characteristic, complemented
12C2×Q8×D107C2212yes—
13C2×Dic428C2211yescharacteristic
14C2×#13133808C2212yes—
15Q16×D107C418yescomplemented
16C2×#13133801C2211yescharacteristic
17C2×C8×D107C2211yescharacteristic
18Q8×Dic107C2212yes—
19C4×Dic214—22no—
20C4×Dic214C2212yes—
21#16812074—22no—
22#16812074C2212yes—
23#16812067—22no—
24#16812067C2212yes—
25#16812066—22no—
26#16812066C2211yescharacteristic
27#16812042C2212yes—
28#16812041C2211yescharacteristic
29#16812034C2212yes—
30#16812033C2212yes—
31#16812032C2212yes—
32#16812019C2211yescharacteristic
33C4×C107⋊C8C2211yescharacteristic
34C8×Dic107C2211yescharacteristic
35C2×Q16×C107C2211yescharacteristic
36C4×Q8×C107C2212yes—
37#106×C107C2211yescharacteristic
38#103×C107C2212yes—
39#97×C107C2211yescharacteristic
40C2×Dic214—22no—
41C2×Dic214C2312yes—
42Q8×D107—24no—
43Q8×D107C2×C418yes—
44C2×C4×D107—22no—
45C2×C4×D107C2311yescharacteristic
46Dic428C2×C414yes—
47#13133808C2×C418yes—
48#13133801—22no—
49C2×C107⋊C8C2311yescharacteristic
50C2×C107⋊C8C2311yescharacteristic
51C8×D107C2×C414yes—
52C4⋊Dic107—22no—
53C4⋊Dic107C2312yes—
54Dic107⋊C4—22no—
55Dic107⋊C4—22no—
56Dic107⋊C4D412yes—
57(C2×D107)⋊C4—22no—
58(C2×D107)⋊C4D412yes—
59C4×Dic107—22no—
60C4×Dic107—22no—
61C4×Dic107—22no—
62C4×Dic107—22no—
63C4×Dic107C2311yescharacteristic
64C2×Q8×C107C2312yes—
65Q16×C107C2×C414yes—
66C2×C8×C107C2311yescharacteristic
67C2×C8×C107C2311yescharacteristic
68C4⋊C4×C107—22no—
69C4⋊C4×C107C2312yes—
70C42×C107—22no—
71C42×C107C2311yescharacteristic
72C22×D107C2×D411yescharacteristic
73Q8×C107—22no—
74Q8×C107C22×C414yes—
75C2×C4×C107—22no—
76C2×C4×C107—22no—
77C2×C4×C107C2×D412yes—
78C2×C4×C107C2411yescharacteristic
79C2×Dic107—22no—
80C2×Dic107—22no—
81C2×Dic107C2×D412yes—
82C2×Dic107C2×D411yescharacteristic
83C4×D107—28no—
84C4×D107C22×C412yes—
85C4×D107#4112yes—
86Dic214—22no—
87Dic214—28no—
88Dic214C22×C414yes—
89C8×C107—21no—
90C8×C107C22×C412yes—
91C107⋊C8—21no—
92C107⋊C8C22×C412yes—
93C22×C107C22×D411yescharacteristic
94C2×D107—22no—
95C2×D107C4×D412yes—
96C4×C107—24no—
97C4×C107—22no—
98C4×C107—42no—
99C4×C107C23×C411yescharacteristic
100C4×C107C2×#4111yescharacteristic
101Dic107—24no—
102Dic107—22no—
103Dic107—42no—
104Dic107C4×D412yes—
105C2×C107C2×C4×D411yescharacteristic
106C2×C107#57812yes—
107D107—22no—
108#2950—1071nomaximal, Sylow
109C107#295011yescharacteristic, complemented, Sylow
110C22×Q16—1071no—
111#519—1072no—
112#441—1071no—
113#441—1071no—
114#441—1071no—
115#441D10711yescharacteristic, complemented
116C4×Q16—1072no—
117C4×Q16—1072no—
118#427—1071no—
119#419—1072no—
120#405—1071no—
121C22×Q8—1072no—
122C2×Q16—1072no—
123C2×Q16—1078no—
124C2×Q16—1071no—
125C2×Q16C2×D10711yescharacteristic
126C2×M16—1071no—
127C22×C8—1071no—
128C4×Q8—1072no—
129C4×Q8—1072no—
130C4×Q8—1072no—
131C4×Q8—2142no—
132C4×Q8—2142no—
133C4×Q8C2×D10712yes—
134#117—1071no—
135#117—1072no—
136#117—2142no—
137#117—2142no—
138#107—1072no—
139#106—1071no—
140#106C2×D10711yescharacteristic
141#103—1072no—
142#103—1072no—
143#103—1072no—
144#103C2×D10712yes—
145#97—1071no—
146#97C2×D10711yescharacteristic
147C4×C8—1071no—
148C4×C8—1071no—
149C2×Q8—1078no—
150C2×Q8—1072no—
151C2×Q8—2142no—
152C2×Q8—2144no—
153C2×Q8C22×D10712yes—
154C22×C4—1071no—
155C22×C4—2142no—
156Q16—1078no—
157Q16—1074no—
158Q16C4×D10714yes—
159M16—2142no—
160C2×C8—1071no—
161C2×C8—1074no—
162C2×C8—1071no—
163C2×C8C22×D10711yescharacteristic
164C2×C8C22×D10711yescharacteristic
165C4⋊C4—22no—
166C4⋊C4—1072no—
167C4⋊C4—1072no—
168C4⋊C4—2142no—
169C4⋊C4—2142no—
170C4⋊C4—2142no—
171C4⋊C4C22×D10712yes—
172C22⋊C4—1072no—
173C22⋊C4—2142no—
174C42—22no—
175C42—1071no—
176C42—2142no—
177C42—2142no—
178C42—2142no—
179C42—2142no—
180C42C22×D10711yescharacteristic
181C23—1071no—
182Q8—22no—
183Q8—1074no—
184Q8—2148no—
185Q8—2142no—
186Q8C2×C4×D10714yes—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—1071no—
190C2×C4—1072no—
191C2×C4—1072no—
192C2×C4—1072no—
193C2×C4—2148no—
194C2×C4—2142no—
195C2×C4—2142no—
196C2×C4D4×D10712yes—
197C2×C4C23×D10711yescharacteristic
198C8—21no—
199C8—1072no—
200C8—2141no—
201C8C2×C4×D10712yes—
202C22—1072no—
203C22—2142no—
204C22C2×D4×D10711yescharacteristic
205C4—22no—
206C4—24no—
207C4—42no—
208C4—1072no—
209C4—4282no—
210C4—2142no—
211C4—2144no—
212C4C22×C4×D10711yescharacteristic
213C4#41×D10711yescharacteristic
214C2—2142no—
215C2C4×D4×D10711yescharacteristic
216C2#2029344312yes—
subgroupG/H
centerC22C2×D4×D107
commutatorC4×C107C23×C4
FrattiniC2×C4C23×D107
Fitting#441×C107C2
socleC22×C107C22×D4
prime radicalC22×D107C2×D4
cosocleC2×C4×C107C24
hypercenter#441D107
hypocenterC107#2950
LCSG ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
3. #20292735C2semidirect
4. #20292734C2semidirect
5. #20292733C2semidirect
6. C4×#13133808C2semidirect
7. #20292719C2semidirect
11. #441×C107C2semidirect
15. Q16×D107C4semidirect
109. C107#2950semidirect
115. #441D107semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1072
#202931032
#202927352
#202927342
#202927332
C4×#131338082
#202927192
#202926622
#202926342
#202925842
#441×C1072
#2950107