Finite Group Databasebeta

← #28346432

Subgroups — #28346432

Order 13696.

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

normal subgroups only

This group has 386 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#20293203C211yesmaximal, characteristic
2#20293108C211yesmaximal, characteristic
3#20292971C212yesmaximal, complemented
4#20292970C212yesmaximal, complemented
5#20292964C212yesmaximal, complemented
6#20292959C212yesmaximal, complemented
7#20292663C211yesmaximal, characteristic, complemented
8#20292637C211yesmaximal, characteristic, complemented
9C2×#16812032C211yesmaximal, characteristic, complemented
10#20292595C211yesmaximal, characteristic
11#428×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×Dic214C2211yescharacteristic
14C22×C107⋊C8C2211yescharacteristic
15#16812109—21no—
16#16812097—21no—
17#16812093—22no—
18#16812089C2212yes—
19#16812089C2212yes—
20#16812074—21no—
21#16812071—21no—
22C4×C22⋊D107—22no—
23#16812068C2211yescharacteristic
24#16812067C2212yes—
25C4⋊C4×D107C2212yes—
26#16812042C2211yescharacteristic
27#16812042C2211yescharacteristic
28#16812040C2211yescharacteristic
29#16812040C2211yescharacteristic
30#16812034C2212yes—
31#16812034C2212yes—
32#16812032C2212yes—
33#16812032C2212yes—
34(C2×D107)⋊C8C2212yes—
35(C2×D107)⋊C8C2212yes—
36C2×M16×C107C2211yescharacteristic
37#117×C107C2211yescharacteristic
38C2×C4⋊C4×C107C2211yescharacteristic
39#107×C107C2212yes—
40#106×C107C2212yes—
41#13133818—22no—
42#13133818—22no—
43C2×Dic214—22no—
44C2×Dic214—22no—
45C2×Dic214—21no—
46C2×Dic214D411yescharacteristic
47C2×Dic214C2312yes—
48C2×D428D411yescharacteristic
49C2×C22⋊D107D412yes—
50C22×Dic107—21no—
51C2×C4×D107—22no—
52C2×C4×D107C2312yes—
53C2×C107⋊C8—22no—
54C2×C107⋊C8C2311yescharacteristic
55C2×C107⋊C8C2311yescharacteristic
56C4⋊Dic107—22no—
57C4⋊Dic107—22no—
58C4⋊Dic107C2311yescharacteristic
59C4⋊Dic107C2311yescharacteristic
60Dic107⋊C4—22no—
61C22⋊Dic107—21no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4—22no—
64(C2×D107)⋊C4—22no—
65C4×Dic107—21no—
66C4×Dic107—22no—
67C22×C4×C107—21no—
68C22×C4×C107C2311yescharacteristic
69M16×C107—22no—
70C2×C8×C107C2312yes—
71C4⋊C4×C107—21no—
72C4⋊C4×C107C2312yes—
73C4⋊C4×C107C2311yescharacteristic
74C4⋊C4×C107C2311yescharacteristic
75C22⋊C4×C107—21no—
76C42×C107—21no—
77C23×C107C2×D411yescharacteristic
78C22×D107C2×D412yes—
79C2×C4×C107—21no—
80C2×C4×C107—21no—
81C2×C4×C107—22no—
82C2×C4×C107—81no—
83C2×C4×C107C2×D411yescharacteristic
84C2×C4×C107#4112yes—
85C2×C4×C107#4112yes—
86C2×C4×C107C2411yescharacteristic
87C22⋊D107—24no—
88C2×Dic107—21no—
89C2×Dic107—21no—
90C2×Dic107—22no—
91C2×Dic107—42no—
92C2×Dic107C2×D412yes—
93D428—22no—
94C4×D107—22no—
95C4×D107—22no—
96C4×D107—82no—
97Dic214—22no—
98Dic214—22no—
99Dic214—22no—
100Dic214—42no—
101C8×C107—22no—
102C107⋊C8—21no—
103C107⋊C8—21no—
104C22×C107—21no—
105C22×C107—21no—
106C22×C107#12312yes—
107C22×C107C22×D411yescharacteristic
108C2×D107—22no—
109C2×D107—22no—
110C2×D107—22no—
111C4×C107—42no—
112C4×C107—41no—
113C4×C107—41no—
114C4×C107#12312yes—
115C4×C107C2×#4111yescharacteristic
116C4×C107C2×#4111yescharacteristic
117Dic107—22no—
118Dic107—42no—
119Dic107—41no—
120Dic107—41no—
121C2×C107—22no—
122C2×C107C2×#12311yescharacteristic
123C2×C107#57711yescharacteristic
124C2×C107#57811yescharacteristic
125D107—42no—
126#3097—1071nomaximal, Sylow
127C107#309711yescharacteristic, complemented, Sylow
128#536—1071no—
129#520—1071no—
130#485—1072no—
131#485—1072no—
132#484—1072no—
133#483—1072no—
134#428D10711yescharacteristic, complemented
135#427—1071no—
136#419—1071no—
137C2×#103—1071no—
138#408—1071no—
139C2×#41—1071no—
140C22×Q8—1071no—
141C2×M16C2×D10711yescharacteristic
142C22×C8—1071no—
143C4⋊Q8—2141no—
144#124—2141no—
145#123—2142no—
146C22⋊Q8—1072no—
147C22⋊Q8—1072no—
148C4×Q8—2141no—
149C4×D4—2141no—
150C4×D4—2142no—
151#117—1071no—
152#117—1072no—
153#117C2×D10711yescharacteristic
154C2×C4⋊C4—1072no—
155C2×C4⋊C4C2×D10711yescharacteristic
156#107—1071no—
157#107—1071no—
158#107C2×D10712yes—
159#106—1071no—
160#106—1071no—
161#106C2×D10712yes—
162#103—1072no—
163#103—1072no—
164#103—1072no—
165#103—1072no—
166C22⋊C8—1072no—
167C22⋊C8—1072no—
168#41—2142no—
169#41—2142no—
170C2×Q8—1071no—
171C2×Q8—1072no—
172C2×Q8—2142no—
173C2×Q8—2142no—
174C2×Q8—2141no—
175C2×D4—1072no—
176C2×D4—1071no—
177C22×C4—21no—
178C22×C4—1072no—
179C22×C4—2142no—
180C22×C4—2141no—
181C22×C4C22×D10711yescharacteristic
182M16—22no—
183C2×C8—1071no—
184C2×C8—1071no—
185C2×C8—2142no—
186C2×C8C22×D10712yes—
187C4⋊C4—21no—
188C4⋊C4—1071no—
189C4⋊C4—1071no—
190C4⋊C4—2142no—
191C4⋊C4—2142no—
192C4⋊C4—2142no—
193C4⋊C4C22×D10712yes—
194C4⋊C4C22×D10711yescharacteristic
195C4⋊C4C22×D10711yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—2142no—
198C22⋊C4—2142no—
199C22⋊C4—2142no—
200C22⋊C4—2141no—
201C42—21no—
202C42—2142no—
203C42—2141no—
204C23—1072no—
205C23D4×D10711yescharacteristic
206Q8—4282no—
207Q8—2142no—
208Q8—2142no—
209Q8—2142no—
210D4—2144no—
211D4—2142no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—81no—
216C2×C4—8562no—
217C2×C4—1072no—
218C2×C4—4282no—
219C2×C4—2141no—
220C2×C4—2141no—
221C2×C4—2142no—
222C2×C4—2142no—
223C2×C4—2142no—
224C2×C4D4×D10711yescharacteristic
225C2×C4#1313381812yes—
226C2×C4#1313381912yes—
227C2×C4C23×D10711yescharacteristic
228C8—22no—
229C8—2141no—
230C8—2141no—
231C22—21no—
232C22—21no—
233C22—2142no—
234C22—2142no—
235C22—2142no—
236C22#1681209412yes—
237C22C2×D4×D10711yescharacteristic
238C4—42no—
239C4—41no—
240C4—41no—
241C4—4282no—
242C4—4281no—
243C4—4281no—
244C4—2142no—
245C4#1681209412yes—
246C4C2×#1313381811yescharacteristic
247C4C2×#1313381911yescharacteristic
248C2—22no—
249C2—4282no—
250C2C2×#1681209411yescharacteristic
251C2#2029343711yescharacteristic
252C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#428×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133818C22
hypercenter#428D107
hypocenterC107#3097
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #20292971C2semidirect
4. #20292970C2semidirect
5. #20292964C2semidirect
6. #20292959C2semidirect
7. #20292663C2semidirect
8. #20292637C2semidirect
9. C2×#16812032C2semidirect
11. #428×C107C2semidirect
127. C107#3097semidirect
134. #428D107semidirect

Maximal subgroups

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

subgroupindex
#202932032
#202931082
#202929712
#202929702
#202929642
#202929592
#202926632
#202926372
C2×#168120322
#202925952
#428×C1072
#3097107