Finite Group Databasebeta

← #28346838

Subgroups — #28346838

Order 13696.

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

normal subgroups only

This group has 380 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#20293182C211yesmaximal, characteristic
2C4⋊Q8×D107C211yesmaximal, characteristic
3#20292994C214yesmaximal, complemented
4#20292979C211yesmaximal, characteristic, complemented
5#20292978C211yesmaximal, characteristic, complemented
6#20292977C211yesmaximal, characteristic, complemented
7#20292634C214yesmaximal
8C4×#13133801C211yesmaximal, characteristic
9#487×C107C211yesmaximal, characteristic, complemented
10C2×Q8×D107C2212yes—
11C2×#13133801C2212yes—
12#16812109C2211yescharacteristic
13#16812109C2211yescharacteristic
14#16812108—21no—
15#16812108—21no—
16#16812107C2212yes—
17#16812100C2212yes—
18#16812089—21no—
19#16812085—21no—
20Q8×Dic107—21no—
21#16812074—21no—
22#16812067C2212yes—
23C4⋊C4×D107—21no—
24C4⋊C4×D107—21no—
25C42×D107C2211yescharacteristic
26#16812034C2214yes—
27#16812033C2214yes—
28#16812032C2214yes—
29#16812019C2212yes—
30C4×C107⋊C8C2211yescharacteristic
31C4⋊Q8×C107C2211yescharacteristic
32C4⋊Q8×C107C2211yescharacteristic
33#103×C107C2214yes—
34C4×C8×C107C2211yescharacteristic
35C2×Dic214—21no—
36C2×Dic214—22no—
37C2×Dic214C2312yes—
38Q8×D107—28no—
39C2×C4×D107—22no—
40C2×C4×D107D411yescharacteristic
41C2×C4×D107D411yescharacteristic
42C2×C4×D107C2311yescharacteristic
43#13133801—24no—
44C2×C107⋊C8C2312yes—
45C4⋊Dic107—21no—
46C4⋊Dic107—21no—
47C4⋊Dic107—21no—
48C4⋊Dic107—22no—
49C4⋊Dic107C2312yes—
50Dic107⋊C4—22no—
51Dic107⋊C4—22no—
52Dic107⋊C4—22no—
53(C2×D107)⋊C4—22no—
54C4×Dic107—22no—
55C4×Dic107D411yescharacteristic
56C4×Dic107D411yescharacteristic
57C4×Dic107C2311yescharacteristic
58C2×Q8×C107—21no—
59C2×Q8×C107C2312yes—
60C2×C8×C107C2312yes—
61C4⋊C4×C107—21no—
62C4⋊C4×C107—21no—
63C4⋊C4×C107—21no—
64C4⋊C4×C107C2312yes—
65C42×C107C2311yescharacteristic
66C22×D107C2×D411yescharacteristic
67Q8×C107—24no—
68Q8×C107—42no—
69C2×C4×C107—22no—
70C2×C4×C107—22no—
71C2×C4×C107C2×D411yescharacteristic
72C2×C4×C107C2×D411yescharacteristic
73C2×C4×C107C2411yescharacteristic
74C2×Dic107—22no—
75C2×Dic107—22no—
76C2×Dic107C2×D411yescharacteristic
77C2×Dic107C2×D411yescharacteristic
78C2×Dic107C2×D411yescharacteristic
79C4×D107—22no—
80C4×D107—22no—
81C4×D107—44no—
82C4×D107#4114yes—
83Dic214—24no—
84Dic214—42no—
85Dic214—44no—
86C8×C107—22no—
87C107⋊C8—22no—
88C22×C107C22×D411yescharacteristic
89C2×D107—21no—
90C2×D107—21no—
91C2×D107#12412yes—
92C4×C107—42no—
93C4×C107—42no—
94C4×C107#13712yes—
95C4×C107#13712yes—
96C4×C107C2×#4112yes—
97Dic107—21no—
98Dic107—21no—
99Dic107—42no—
100Dic107—42no—
101Dic107#12412yes—
102C2×C107C2×#12411yescharacteristic
103C2×C107C2×#13711yescharacteristic
104C2×C107C2×#13711yescharacteristic
105D107—22no—
106#3139—1071nomaximal, Sylow
107C107#313911yescharacteristic, complemented, Sylow
108#533—1071no—
109C2×C4⋊Q8—1071no—
110#491—1074no—
111#487—1071no—
112#487—1071no—
113#487—1071no—
114#487D10711yescharacteristic, complemented
115#419—1074no—
116C4×M16—1071no—
117C22×Q8—1072no—
118C2×M16—1072no—
119C4⋊Q8—1072no—
120C4⋊Q8—1071no—
121C4⋊Q8—1071no—
122C4⋊Q8—2141no—
123C4⋊Q8—2141no—
124C4⋊Q8C2×D10711yescharacteristic
125C4⋊Q8C2×D10711yescharacteristic
126#125—1072no—
127C22⋊Q8—2141no—
128C22⋊Q8—2141no—
129C4×Q8—2141no—
130C4×Q8—2141no—
131#117—1072no—
132C2×C4⋊C4—2141no—
133C2×C4⋊C4—2141no—
134C2×C42—1071no—
135#103—1074no—
136#103—1074no—
137#103—1074no—
138#103C2×D10714yes—
139#97—1072no—
140C4×C8—1071no—
141C4×C8C2×D10711yescharacteristic
142C2×Q8—21no—
143C2×Q8—1072no—
144C2×Q8—2142no—
145C2×Q8—2148no—
146C2×Q8—2141no—
147C2×Q8C22×D10712yes—
148C22×C4—1071no—
149C22×C4—1071no—
150C22×C4—1071no—
151C22×C4—2142no—
152M16—2144no—
153C2×C8—1072no—
154C2×C8C22×D10712yes—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—1072no—
159C4⋊C4—2142no—
160C4⋊C4—2142no—
161C4⋊C4—2142no—
162C4⋊C4—2142no—
163C4⋊C4—2141no—
164C4⋊C4—2141no—
165C4⋊C4—2141no—
166C4⋊C4C22×D10712yes—
167C22⋊C4—2142no—
168C42—1071no—
169C42—1071no—
170C42—1071no—
171C42—2142no—
172C42C22×D10711yescharacteristic
173C23—1071no—
174Q8—24no—
175Q8—42no—
176Q8—4284no—
177Q8—4282no—
178Q8—2144no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—1071no—
182C2×C4—1074no—
183C2×C4—1071no—
184C2×C4—1071no—
185C2×C4—4284no—
186C2×C4—2142no—
187C2×C4—2142no—
188C2×C4—2142no—
189C2×C4—2142no—
190C2×C4D4×D10711yescharacteristic
191C2×C4D4×D10711yescharacteristic
192C2×C4C23×D10711yescharacteristic
193C8—22no—
194C8—2142no—
195C22—1072no—
196C22—2141no—
197C22—2141no—
198C22C2×D4×D10711yescharacteristic
199C4—42no—
200C4—42no—
201C4—1072no—
202C4—4282no—
203C4—4282no—
204C4—2141no—
205C4—2141no—
206C4#1681213812yes—
207C4#1681214112yes—
208C4#41×D10712yes—
209C2—2142no—
210C2#124×D10711yescharacteristic
211C2C2×#1681213811yescharacteristic
212C2C2×#1681214111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#487×C107C2
socleC22×C107C22×D4
prime radicalC22×D107C2×D4
hypercenter#487D107
hypocenterC107#3139
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. #20292994C2semidirect
4. #20292979C2semidirect
5. #20292978C2semidirect
6. #20292977C2semidirect
9. #487×C107C2semidirect
107. C107#3139semidirect
114. #487D107semidirect

Maximal subgroups

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

subgroupindex
#202931822
C4⋊Q8×D1072
#202929942
#202929792
#202929782
#202929772
#202926342
C4×#131338012
#487×C1072
#3139107