Finite Group Databasebeta

← #28344923

Subgroups — #28344923

Order 13696.

716 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 209 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×#16812156C211yesmaximal, characteristic
2#20293104C212yesmaximal, complemented
3#20292601C211yesmaximal, characteristic, complemented
4#20292600C212yesmaximal, complemented
5#20292599C218yesmaximal, complemented
6#410×C107C211yesmaximal, characteristic, complemented
7#16812156—24no—
8#16812156—24no—
9C2×#13133818C412yescomplemented
10C2×#13133818C2211yescharacteristic
11C2×#13133817C412yescomplemented
12C2×#13133817C2214yes—
13C22×Dic214C412yescomplemented
14C22×Dic214C2211yescharacteristic
15C2×Q8×D107C412yescomplemented
16C4×Dic214—24no—
17#16812074—24no—
18#16812068C2212yes—
19#16812065C2218yes—
20#16812024C2214yes—
21#16812023C2218yes—
22C2×#41×C107C2211yescharacteristic
23#117×C107C2212yes—
24#99×C107C2214yes—
25#13133818—22no—
26#13133818—22no—
27#13133818—24no—
28#13133818—24no—
29#13133817—216no—
30#13133817—28no—
31C2×Dic214—24no—
32C2×Dic214—24no—
33C2×Dic214C2×C414yes—
34C2×Dic214C2×C411yescharacteristic
35C2×Dic214C2×C412yes—
36C2×Dic214C2×C412yes—
37C2×Dic214D414yes—
38C2×Dic214D414yes—
39Q8×D107—28no—
40C2×D428C2×C411yescharacteristic
41C2×C22⋊D107C2×C414yes—
42C2×C22⋊D107C2312yes—
43C22×Dic107C2×C414yes—
44C22×Dic107C2312yes—
45C2×C4×D107C2×C412yes—
46C2×C4×D107C2×C414yes—
47C4⋊Dic107—22no—
48Dic107⋊C4—28no—
49C22⋊Dic107C2314yes—
50C4×Dic107—22no—
51C4×Dic107—28no—
52#41×C107—22no—
53#41×C107—22no—
54C2×Q8×C107C2×C411yescharacteristic
55C2×D4×C107C2×C411yescharacteristic
56C2×D4×C107C2312yes—
57C22×C4×C107C2×C412yes—
58C22×C4×C107C2311yescharacteristic
59C4⋊C4×C107—22no—
60C22⋊C4×C107C2314yes—
61C42×C107—22no—
62C23×C107C22×C412yes—
63C23×C107C2411yescharacteristic
64C22×D107C22×C412yes—
65Q8×C107—22no—
66D4×C107—24no—
67D4×C107—22no—
68C2×C4×C107—22no—
69C2×C4×C107—22no—
70C2×C4×C107—24no—
71C2×C4×C107C22×C412yes—
72C2×C4×C107C22×C411yescharacteristic
73C2×C4×C107C22×C411yescharacteristic
74C2×C4×C107C2×D412yes—
75C2×C4×C107C2×D412yes—
76C22⋊D107—24no—
77C22⋊D107—28no—
78C2×Dic107—28no—
79C2×Dic107—24no—
80C2×Dic107C22×C412yes—
81C2×Dic107C22×C414yes—
82C2×Dic107C2×D418yes—
83D428—21no—
84D428—21no—
85C4×D107—24no—
86C4×D107—28no—
87Dic214—21no—
88Dic214—21no—
89Dic214—28no—
90Dic214C22⋊C418yes—
91Dic214C22⋊C414yes—
92Dic214C22⋊C414yes—
93C22×C107—22no—
94C22×C107—41no—
95C22×C107—42no—
96C22×C107C23×C411yescharacteristic
97C22×C107C22×D412yes—
98C2×D107—22no—
99C2×D107—42no—
100C4×C107—22no—
101C4×C107—44no—
102C4×C107C2×C22⋊C411yescharacteristic
103C4×C107C2×C22⋊C411yescharacteristic
104C4×C107C2×C22⋊C412yes—
105Dic107—22no—
106Dic107—44no—
107Dic107C2×C22⋊C418yes—
108C2×C107—22no—
109C2×C107—21no—
110C2×C107—42no—
111C2×C107C22×C22⋊C411yescharacteristic
112D107—42no—
113#2884—1071nomaximal, Sylow
114C107#288411yescharacteristic, complemented, Sylow
115C2×2−1+4—1071no—
116#519—1072no—
117#410—1071no—
118#410—1072no—
119#410—1078no—
120#410D10711yescharacteristic, complemented
1212−1+4—2144no—
1222−1+4—2144no—
123C2×#41—1072no—
124C2×#41—1074no—
125C2×#41—1072no—
126C2×#41—1071no—
127C2×#41C2×D10711yescharacteristic
128C22×Q8—1072no—
129C22×Q8—1072no—
130C22×Q8—1071no—
131C4×Q8—2144no—
132C4×Q8—2144no—
133#117—1072no—
134#117—1078no—
135#117C2×D10712yes—
136#99—1074no—
137#99—1078no—
138#99C2×D10714yes—
139#41—22no—
140#41—22no—
141#41—2144no—
142#41—2144no—
143#41—21416no—
144#41—2148no—
145#41—2142no—
146#41—2142no—
147C2×Q8—1072no—
148C2×Q8—1072no—
149C2×Q8—1074no—
150C2×Q8—1071no—
151C2×Q8—1074no—
152C2×Q8—1074no—
153C2×Q8—2144no—
154C2×Q8—2144no—
155C2×Q8—2148no—
156C2×Q8C4×D10711yescharacteristic
157C2×D4—1074no—
158C2×D4—1072no—
159C2×D4—1071no—
160C2×D4C4×D10711yescharacteristic
161C2×D4C22×D10712yes—
162C22×C4—1072no—
163C22×C4—1074no—
164C22×C4—1074no—
165C22×C4—1072no—
166C22×C4C4×D10712yes—
167C22×C4C22×D10711yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—2148no—
170C4⋊C4—2142no—
171C22⋊C4—1074no—
172C22⋊C4C22×D10714yes—
173C42—22no—
174C42—2148no—
175C42—2142no—
176C23—1072no—
177C23C2×C4×D10712yes—
178C23C23×D10711yescharacteristic
179Q8—22no—
180Q8—1074no—
181Q8—1074no—
182Q8—1078no—
183Q8—2148no—
184Q8—2141no—
185Q8—2141no—
186D4—24no—
187D4—22no—
188D4—2148no—
189D4—2144no—
190D4—2141no—
191D4—2141no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—1074no—
196C2×C4—1078no—
197C2×C4—1072no—
198C2×C4—2148no—
199C2×C4—2148no—
200C2×C4—2144no—
201C2×C4—2144no—
202C2×C4C2×C4×D10712yes—
203C2×C4C2×C4×D10711yescharacteristic
204C2×C4C2×C4×D10711yescharacteristic
205C2×C4D4×D10712yes—
206C2×C4D4×D10712yes—
207C22—22no—
208C22—41no—
209C22—42no—
210C22—4282no—
211C22—2142no—
212C22C22×C4×D10711yescharacteristic
213C22C2×D4×D10712yes—
214C4—22no—
215C4—44no—
216C4—1078no—
217C4—4284no—
218C4—2142no—
219C4C22⋊C4×D10711yescharacteristic
220C4C22⋊C4×D10711yescharacteristic
221C4C22⋊C4×D10712yes—
222C2—22no—
223C2—21no—
224C2—42no—
225C2—4282no—
226C2C2×C22⋊C4×D10711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D107
commutatorC22×C107C23×C4
FrattiniC23C23×D107
Fitting#410×C107C2
socleC2×C107C22×C22⋊C4
prime radicalC2×#16812156C2
cosocleC23×C107C24
hypercenter#410D107
hypocenterC107#2884
LCSG ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #20293104C2semidirect
3. #20292601C2semidirect
4. #20292600C2semidirect
5. #20292599C2semidirect
6. #410×C107C2semidirect
9. C2×#13133818C4semidirect
11. C2×#13133817C4semidirect
13. C22×Dic214C4semidirect
15. C2×Q8×D107C4semidirect
114. C107#2884semidirect
120. #410D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121562
#202931042
#202926012
#202926002
#202925992
#410×C1072
#2884107