Finite Group Databasebeta

← #28348201

Subgroups — #28348201

Order 13696.

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

normal subgroups only

This group has 482 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
1C2×#16812128C211yesmaximal, characteristic, complemented
2C2×Q16×D107C211yesmaximal, characteristic, complemented
3#20293285C212yesmaximal, complemented
4#20292865C211yesmaximal, characteristic, complemented
5#20292860C211yesmaximal, characteristic, complemented
6#20292846C212yesmaximal, complemented
7#20292786C212yesmaximal, complemented
8#20292764C212yesmaximal, complemented
9C4×Dic428C211yesmaximal, characteristic, complemented
10C8×D428C211yesmaximal, characteristic, complemented
11#495×C107C211yesmaximal, characteristic, complemented
12C2×#13133817C2212yes—
13C2×Q8×D107C2212yes—
14#16812128—24no—
15C2×Dic428C2211yescharacteristic
16C2×#13133808C2212yes—
17Q16×D107—24no—
18C2×#13133806C2212yes—
19C2×C8×D107C2211yescharacteristic
20C2×C8×D107C2211yescharacteristic
21#16812095—22no—
22#16812086—22no—
23#16812085C2212yes—
24#16812082—22no—
25#16812080C2212yes—
26C4×Dic214C2212yes—
27C4×D428C2211yescharacteristic
28(C22⋊D107)⋊C4—22no—
29#16812043C2211yescharacteristic
30C4⋊(C107⋊C8)C2211yescharacteristic
31#16812034C2212yes—
32#16812033C2212yes—
33#16812030C2212yes—
34(C2×D107)⋊C8C2211yescharacteristic
35(C2×D107)⋊C8C2211yescharacteristic
36C2×Q16×C107C2211yescharacteristic
37C2×SD16×C107C2212yes—
38C2×D8×C107C2211yescharacteristic
39#124×C107C2212yes—
40C4×C8×C107C2211yescharacteristic
41#13133817—24no—
42#13133817—42no—
43C2×Dic214—22no—
44C2×Dic214C2312yes—
45Q8×D107—24no—
46Q8×D107—42no—
47C2×D428D411yescharacteristic
48C2×C22⋊D107—22no—
49C2×C22⋊D107—22no—
50C22×Dic107—22no—
51C2×C4×D107—22no—
52C2×C4×D107C2311yescharacteristic
53C2×C4×D107C2311yescharacteristic
54Dic428D414yescomplemented
55#13133808—24no—
56#13133806—24no—
57C2×C107⋊C8C2311yescharacteristic
58C2×C107⋊C8C2311yescharacteristic
59C8×D107—22no—
60C8×D107—22no—
61C4⋊Dic107D411yescharacteristic
62C4⋊Dic107C2312yes—
63Dic107⋊C4—22no—
64Dic107⋊C4—22no—
65C22⋊Dic107—22no—
66(C2×D107)⋊C4—22no—
67(C2×D107)⋊C4D411yescharacteristic
68(C2×D107)⋊C4D411yescharacteristic
69C4×Dic107—22no—
70C2×Q8×C107C2312yes—
71C2×D4×C107C2312yes—
72Q16×C107—22no—
73SD16×C107—42no—
74D8×C107—22no—
75C2×C8×C107C2311yescharacteristic
76C2×C8×C107C2311yescharacteristic
77C22⋊C4×C107—22no—
78C22⋊C4×C107—22no—
79C42×C107C2311yescharacteristic
80C23×C107—22no—
81C22×D107C2×D411yescharacteristic
82C22×D107C2×D411yescharacteristic
83Q8×C107—22no—
84Q8×C107—22no—
85D4×C107—22no—
86D4×C107—22no—
87C2×C4×C107—22no—
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107C2×D411yescharacteristic
90C2×C4×C107C2411yescharacteristic
91C22⋊D107—44no—
92C22⋊D107—44no—
93C2×Dic107—22no—
94C2×Dic107—22no—
95C2×Dic107—44no—
96C2×Dic107C2×D411yescharacteristic
97C2×Dic107C2×D411yescharacteristic
98D428—22no—
99C4×D107—21no—
100C4×D107—21no—
101C4×D107—21no—
102C4×D107—21no—
103C4×D107—44no—
104Dic214—22no—
105Dic214—44no—
106Dic214C2×D414yes—
107C8×C107—21no—
108C8×C107C2×D412yes—
109C107⋊C8—21no—
110C107⋊C8—21no—
111C22×C107—42no—
112C22×C107—42no—
113C22×C107—42no—
114C22×C107C22×D411yescharacteristic
115C2×D107—21no—
116C2×D107—21no—
117C2×D107—21no—
118C2×D107—21no—
119C2×D107—21no—
120C2×D107#13512yescomplemented
121C4×C107—21no—
122C4×C107—21no—
123C4×C107—42no—
124C4×C107C22×D411yescharacteristic
125C4×C107H(5,𝔽2)11yescharacteristic
126Dic107—24no—
127Dic107—21no—
128Dic107—21no—
129Dic107—42no—
130C2×C107—82no—
131C2×C107D4211yescharacteristic
132C2×C107C2×#13511yescharacteristic
133C2×C107#57811yescharacteristic
134D107—22no—
135D107—41no—
136#3285—1071nomaximal, Sylow
137C107#328511yescharacteristic, complemented, Sylow
138C2×#135—1071no—
139C22×Q16—1071no—
140#548—1072no—
141#495D10711yescharacteristic, complemented
142#467—1071no—
143#466—1071no—
144#464—1072no—
145#452—1072no—
146#448—1072no—
147C4×Q16—1071no—
148C8×D4—1071no—
149C2×#41—1072no—
150C22×Q8—1072no—
151#135—2144no—
152C2×Q16—1072no—
153C2×Q16—1071no—
154C2×Q16—2144no—
155C2×Q16C2×D10711yescharacteristic
156C2×SD16—1072no—
157C2×SD16C2×D10712yes—
158C2×D8C2×D10711yescharacteristic
159C22×C8—1071no—
160C22×C8—1071no—
161#124—2142no—
162#124C2×D10712yes—
163C22⋊Q8—1072no—
164C22⋊Q8—2142no—
165#121—1072no—
166#121—2142no—
167C4×Q8—1072no—
168C4×D4—1071no—
169C4×D4—2142no—
170#107—1071no—
171C4⋊C8—1071no—
172#103—1072no—
173#103—1072no—
174#102—1072no—
175C22⋊C8—1071no—
176C22⋊C8—1071no—
177C4×C8C2×D10711yescharacteristic
178#41—4282no—
179#41—2144no—
180C2×Q8—1072no—
181C2×Q8—4282no—
182C2×Q8—2142no—
183C2×Q8—2144no—
184C2×Q8C22×D10712yes—
185C2×D4—1071no—
186C2×D4—2142no—
187C2×D4—2142no—
188C2×D4C22×D10712yes—
189C22×C4—1071no—
190C22×C4—1071no—
191C22×C4—2142no—
192C22×C4—2142no—
193Q16—22no—
194Q16—1074no—
195Q16—2144no—
196SD16—42no—
197SD16—2144no—
198D8—22no—
199C2×C8—1071no—
200C2×C8—1071no—
201C2×C8—2142no—
202C2×C8—2142no—
203C2×C8C22×D10711yescharacteristic
204C2×C8C22×D10711yescharacteristic
205C4⋊C4—1071no—
206C4⋊C4—1072no—
207C4⋊C4—2142no—
208C4⋊C4—2142no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—1071no—
212C22⋊C4—1071no—
213C22⋊C4—2142no—
214C22⋊C4—2142no—
215C42—2142no—
216C42C22×D10711yescharacteristic
217C23—22no—
218C23—1071no—
219C23—1071no—
220Q8—22no—
221Q8—22no—
222Q8—1074no—
223Q8—4284no—
224Q8—2142no—
225D4—22no—
226D4—22no—
227D4—4284no—
228D4—4284no—
229D4—2142no—
230C2×C4—22no—
231C2×C4—1071no—
232C2×C4—1071no—
233C2×C4—4284no—
234C2×C4—4284no—
235C2×C4—2142no—
236C2×C4—2142no—
237C2×C4—2141no—
238C2×C4—2141no—
239C2×C4—2141no—
240C2×C4—2141no—
241C2×C4D4×D10711yescharacteristic
242C2×C4D4×D10711yescharacteristic
243C2×C4C23×D10711yescharacteristic
244C8—21no—
245C8—2141no—
246C8—2141no—
247C8D4×D10712yes—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—1072no—
252C22—2141no—
253C22—2141no—
254C22—2141no—
255C22—2141no—
256C22—2141no—
257C22C2×D4×D10711yescharacteristic
258C4—21no—
259C4—21no—
260C4—42no—
261C4—4282no—
262C4—2141no—
263C4—2141no—
264C4—2144no—
265C4C2×D4×D10711yescharacteristic
266C4#1681215411yescharacteristic
267C2—82no—
268C2—4281no—
269C2—2142no—
270C2#2029326211yescharacteristic
271C2#135×D10711yescharacteristic
272C2#2029344111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#495×C107C2
socleC22×C107C22×D4
hypercenter#495D107
hypocenterC107#3285
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#16812128C2semidirect
2. C2×Q16×D107C2semidirect
3. #20293285C2semidirect
4. #20292865C2semidirect
5. #20292860C2semidirect
6. #20292846C2semidirect
7. #20292786C2semidirect
8. #20292764C2semidirect
9. C4×Dic428C2semidirect
10. C8×D428C2semidirect
11. #495×C107C2semidirect
54. Dic428D4semidirect
120. C2×D107#135semidirect
137. C107#3285semidirect
141. #495D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121282
C2×Q16×D1072
#202932852
#202928652
#202928602
#202928462
#202927862
#202927642
C4×Dic4282
C8×D4282
#495×C1072
#3285107