Finite Group Databasebeta

← #28346258

Subgroups — #28346258

Order 13696.

440 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 153 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#20293184C211yesmaximal, characteristic, complemented
2#20292927C214yesmaximal, complemented
3#20292917C212yesmaximal, complemented
4#20292907C212yesmaximal, complemented
5#20292667C212yesmaximal, complemented
6#20292651C211yesmaximal, characteristic, complemented
7#20292632C212yesmaximal
8C2×C4×D4×C107C211yesmaximal, characteristic
9C2×#13133802C2212yes—
10#16812109C2211yescharacteristic, complemented
11#16812109C2211yescharacteristic, complemented
12#16812101C2212yescomplemented
13#16812088—21no—
14#16812088—21no—
15C4×Dic214—21no—
16C4×Dic214—21no—
17#16812068C2212yes—
18#16812042C2214yescomplemented
19#16812040C2214yescomplemented
20C4⋊(C107⋊C8)C2212yescomplemented
21C4⋊(C107⋊C8)C2212yescomplemented
22#16812030C2214yes—
23#16812030C2214yes—
24C22×D4×C107C2211yescharacteristic
25C23×C4×C107—21no—
26C4×D4×C107—21no—
27C4×D4×C107—21no—
28C4×D4×C107C2212yes—
29C4×D4×C107C2212yes—
30C2×C4⋊C4×C107C2211yescharacteristic
31C2×C22⋊C4×C107—21no—
32C2×C42×C107C2211yescharacteristic
33C2×Dic214—21no—
34C2×Dic214—21no—
35#13133802—24no—
36C2×C107⋊C8C2314yes—
37C4⋊Dic107—21no—
38C4⋊Dic107—21no—
39C4⋊Dic107C2312yes—
40C4⋊Dic107C2312yes—
41Dic107⋊C4—22no—
42Dic107⋊C4—22no—
43C22⋊Dic107—22no—
44C4×Dic107—22no—
45C24×C107—21no—
46C2×D4×C107—21no—
47C2×D4×C107—22no—
48C2×D4×C107Q814yescomplemented
49C2×D4×C107C2312yes—
50C22×C4×C107—21no—
51C22×C4×C107—22no—
52C22×C4×C107—22no—
53C22×C4×C107—44no—
54C22×C4×C107D411yescharacteristic
55C22×C4×C107D411yescharacteristic
56C22×C4×C107C2311yescharacteristic
57C4⋊C4×C107—21no—
58C4⋊C4×C107C2312yes—
59C22⋊C4×C107—22no—
60C22⋊C4×C107—22no—
61C42×C107D411yescharacteristic
62C42×C107D411yescharacteristic
63C42×C107C2311yescharacteristic
64C42×C107C2311yescharacteristic
65C23×C107—24no—
66C23×C107—22no—
67C23×C107—42no—
68C23×C107—42no—
69C23×C107C2×D411yescharacteristic
70D4×C107—24no—
71D4×C107—22no—
72D4×C107C2×Q814yes—
73C2×C4×C107—22no—
74C2×C4×C107—22no—
75C2×C4×C107—41no—
76C2×C4×C107—44no—
77C2×C4×C107—42no—
78C2×C4×C107—44no—
79C2×C4×C107C2×D411yescharacteristic
80C2×C4×C107C2×D411yescharacteristic
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107C2×D411yescharacteristic
84C2×C4×C107C2×Q812yes—
85C2×C4×C107#4112yes—
86C2×C4×C107C2411yescharacteristic
87C2×Dic107—22no—
88C2×Dic107—22no—
89Dic214—42no—
90Dic214—42no—
91C107⋊C8—24no—
92C22×C107—21no—
93C22×C107—21no—
94C22×C107—24no—
95C22×C107—48no—
96C22×C107—42no—
97C22×C107—42no—
98C22×C107C22⋊Q812yes—
99C22×C107C22×D411yescharacteristic
100C4×C107—21no—
101C4×C107—21no—
102C4×C107—21no—
103C4×C107—42no—
104C4×C107C22⋊Q812yes—
105C4×C107AGL(1,ℤ/8)12yes—
106C4×C107C22×Q811yescharacteristic
107C4×C107C2×#4111yescharacteristic
108Dic107—42no—
109Dic107—42no—
110C2×C107—22no—
111C2×C107—44no—
112C2×C107C2×C22⋊Q811yescharacteristic
113C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
114C2×C107#57511yescharacteristic
115#3077—1071nomaximal, Sylow
116C107#307711yescharacteristic, complemented, Sylow
117#533—1071no—
118C2×C4×D4D10711yescharacteristic
119#478—1074no—
120#476—1072no—
121#474—1072no—
122#428—1072no—
123#423—1071no—
124#418—1072no—
125C22×D4C2×D10711yescharacteristic
126C23×C4—21no—
127C2×M16—1072no—
128C4⋊Q8—1071no—
129C4⋊Q8—1071no—
130#125—1072no—
131C22⋊Q8—2141no—
132C22⋊Q8—2141no—
133C4×Q8—2141no—
134C4×Q8—2141no—
135C4×D4—21no—
136C4×D4—21no—
137C4×D4C2×D10712yes—
138C4×D4C2×D10712yes—
139#117—1072no—
140C2×C4⋊C4C2×D10711yescharacteristic
141C2×C22⋊C4—21no—
142C2×C42C2×D10711yescharacteristic
143#107—1074no—
144#106—1074no—
145C4⋊C8—1072no—
146C4⋊C8—1072no—
147#102—1074no—
148#102—1074no—
149C24—21no—
150C2×Q8—2141no—
151C2×Q8—2141no—
152C2×D4—21no—
153C2×D4—22no—
154C2×D4Dic21414yescomplemented
155C2×D4C22×D10712yes—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—44no—
160C22×C4C22⋊D10711yescharacteristic
161C22×C4C22⋊D10711yescharacteristic
162C22×C4C22×D10711yescharacteristic
163M16—2144no—
164C2×C8—1074no—
165C4⋊C4—21no—
166C4⋊C4—1072no—
167C4⋊C4—1072no—
168C4⋊C4—2142no—
169C4⋊C4—2142no—
170C4⋊C4—2141no—
171C4⋊C4—2141no—
172C4⋊C4C22×D10712yes—
173C22⋊C4—22no—
174C22⋊C4—22no—
175C22⋊C4—2142no—
176C42—2142no—
177C42C22⋊D10711yescharacteristic
178C42C22⋊D10711yescharacteristic
179C42C22×D10711yescharacteristic
180C42C22×D10711yescharacteristic
181C23—24no—
182C23—22no—
183C23—42no—
184C23—42no—
185C23C2×C22⋊D10711yescharacteristic
186Q8—4282no—
187Q8—4282no—
188D4—24no—
189D4—22no—
190D4C2×Dic21414yes—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—41no—
194C2×C4—44no—
195C2×C4—42no—
196C2×C4—44no—
197C2×C4—2142no—
198C2×C4—2142no—
199C2×C4C2×C22⋊D10711yescharacteristic
200C2×C4C2×C22⋊D10711yescharacteristic
201C2×C4C2×C22⋊D10711yescharacteristic
202C2×C4C2×C22⋊D10711yescharacteristic
203C2×C4C2×C22⋊D10711yescharacteristic
204C2×C4C2×Dic21412yes—
205C2×C4#1313381812yes—
206C2×C4C23×D10711yescharacteristic
207C8—2144no—
208C22—21no—
209C22—21no—
210C22—24no—
211C22—48no—
212C22—42no—
213C22—42no—
214C22#1681208812yes—
215C22C22×C22⋊D10711yescharacteristic
216C4—21no—
217C4—21no—
218C4—21no—
219C4—42no—
220C4—4282no—
221C4—4282no—
222C4#1681208812yes—
223C4#1681213512yes—
224C4C22×Dic21411yescharacteristic
225C4C2×#1313381811yescharacteristic
226C2—22no—
227C2—44no—
228C2C2×#1681208811yescharacteristic
229C2C2×#1681213511yescharacteristic
230C2#2029342511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×C4×D4×C107C2
socleC22×C107C22×D4
prime radicalC22×D4×C107C22
hypercenterC2×C4×D4D107
hypocenterC107#3077
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #20293184C2semidirect
2. #20292927C2semidirect
3. #20292917C2semidirect
4. #20292907C2semidirect
5. #20292667C2semidirect
6. #20292651C2semidirect
10. #16812109C22semidirect
11. #16812109C22semidirect
12. #16812101C22semidirect
18. #16812042C22semidirect
19. #16812040C22semidirect
20. C4⋊(C107⋊C8)C22semidirect
21. C4⋊(C107⋊C8)C22semidirect
48. C2×D4×C107Q8semidirect
116. C107#3077semidirect
154. C2×D4Dic214semidirect

Maximal subgroups

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

subgroupindex
#202931842
#202929272
#202929172
#202929072
#202926672
#202926512
#202926322
C2×C4×D4×C1072
#3077107