Finite Group Databasebeta

← #28346426

Subgroups — #28346426

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#20293201C211yesmaximal, characteristic
2#20293108C211yesmaximal, characteristic
3#20292967C212yesmaximal, complemented
4#20292966C212yesmaximal, complemented
5#20292961C212yesmaximal, complemented
6#20292956C212yesmaximal, complemented
7#20292667C211yesmaximal, characteristic, complemented
8#20292665C211yesmaximal, characteristic, complemented
9#20292636C211yesmaximal, characteristic, complemented
10#20292593C211yesmaximal, characteristic
11C2×#103×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C2×#13133802C2211yescharacteristic
14#16812107—21no—
15#16812098—21no—
16#16812092—22no—
17#16812085C2212yes—
18#16812085C2212yes—
19#16812074—21no—
20#16812071—21no—
21C4×C22⋊D107—22no—
22#16812068C2211yescharacteristic
23#16812068C2211yescharacteristic
24#16812067C2212yes—
25C4⋊C4×D107C2212yes—
26#16812043C2211yescharacteristic
27#16812043C2211yescharacteristic
28#16812042C2212yes—
29#16812041C2211yescharacteristic
30#16812041C2211yescharacteristic
31#16812040C2212yes—
32#16812033C2212yes—
33#16812033C2212yes—
34(C2×D107)⋊C8C2212yes—
35(C2×D107)⋊C8C2212yes—
36C22×Q8×C107C2211yescharacteristic
37C22×C8×C107C2211yescharacteristic
38C2×C4⋊C4×C107C2211yescharacteristic
39#103×C107C2212yes—
40#103×C107C2212yes—
41#13133818—22no—
42#13133818—22no—
43C2×Dic214D411yescharacteristic
44C2×D428D411yescharacteristic
45C2×C22⋊D107D412yes—
46C2×C4×D107—22no—
47C2×C4×D107C2312yes—
48#13133802—22no—
49C2×C107⋊C8C2312yes—
50C4⋊Dic107C2311yescharacteristic
51C4⋊Dic107C2311yescharacteristic
52C4⋊Dic107C2311yescharacteristic
53C4⋊Dic107C2311yescharacteristic
54Dic107⋊C4—22no—
55Dic107⋊C4—22no—
56Dic107⋊C4—22no—
57C22⋊Dic107—21no—
58C22⋊Dic107—21no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4—22no—
61(C2×D107)⋊C4—22no—
62C4×Dic107—21no—
63C4×Dic107—21no—
64C4×Dic107—22no—
65C2×Q8×C107—21no—
66C2×Q8×C107—22no—
67C2×Q8×C107—22no—
68C2×Q8×C107C2312yes—
69C22×C4×C107—21no—
70C22×C4×C107—21no—
71C22×C4×C107C2311yescharacteristic
72C2×C8×C107—22no—
73C2×C8×C107C2311yescharacteristic
74C2×C8×C107C2311yescharacteristic
75C4⋊C4×C107—21no—
76C4⋊C4×C107C2312yes—
77C23×C107C2×D411yescharacteristic
78C22×D107C2×D412yes—
79Q8×C107—22no—
80Q8×C107—22no—
81Q8×C107—42no—
82C2×C4×C107—22no—
83C2×C4×C107—22no—
84C2×C4×C107—42no—
85C2×C4×C107—81no—
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107#4112yes—
88C2×C4×C107#4112yes—
89C2×C4×C107C2411yescharacteristic
90C22⋊D107—24no—
91C2×Dic107—21no—
92C2×Dic107—21no—
93C2×Dic107—21no—
94C2×Dic107—21no—
95C2×Dic107C2×D412yes—
96D428—22no—
97C4×D107—22no—
98C4×D107—22no—
99C4×D107—82no—
100Dic214—22no—
101C8×C107—21no—
102C8×C107—21no—
103C107⋊C8—22no—
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—42no—
113C4×C107#12312yes—
114C4×C107C2×#4111yescharacteristic
115C4×C107C2×#4111yescharacteristic
116Dic107—22no—
117Dic107—41no—
118Dic107—41no—
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#428—1071no—
135#427—1071no—
136#419—1071no—
137C2×#103D10711yescharacteristic, complemented
138#408—1071no—
139C2×#41—1071no—
140C22×Q8C2×D10711yescharacteristic
141C2×M16—1071no—
142C22×C8C2×D10711yescharacteristic
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—1071no—
153#117—1072no—
154C2×C4⋊C4—1072no—
155C2×C4⋊C4C2×D10711yescharacteristic
156#107—1071no—
157#107—1071no—
158#107—1072no—
159#106—1071no—
160#106—1071no—
161#106—1072no—
162#103—1072no—
163#103—1072no—
164#103C2×D10712yes—
165#103C2×D10712yes—
166C22⋊C8—1072no—
167C22⋊C8—1072no—
168#41—2142no—
169#41—2142no—
170C2×Q8—21no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8—1071no—
174C2×Q8C22×D10712yes—
175C2×D4—1072no—
176C2×D4—1071no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—1072no—
180C22×C4—2142no—
181C22×C4C22×D10711yescharacteristic
182M16—2142no—
183C2×C8—22no—
184C2×C8—1072no—
185C2×C8C22×D10711yescharacteristic
186C2×C8C22×D10711yescharacteristic
187C4⋊C4—21no—
188C4⋊C4—1071no—
189C4⋊C4—1071no—
190C4⋊C4—1071no—
191C4⋊C4—1071no—
192C4⋊C4—2142no—
193C4⋊C4—2142no—
194C4⋊C4—2142no—
195C4⋊C4C22×D10712yes—
196C22⋊C4—2142no—
197C22⋊C4—2142no—
198C22⋊C4—2142no—
199C22⋊C4—2141no—
200C22⋊C4—2141no—
201C42—2142no—
202C42—2141no—
203C42—2141no—
204C23—1072no—
205C23D4×D10711yescharacteristic
206Q8—22no—
207Q8—22no—
208Q8—42no—
209Q8—2142no—
210D4—2144no—
211D4—2142no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—42no—
215C2×C4—81no—
216C2×C4—8562no—
217C2×C4—1072no—
218C2×C4—2141no—
219C2×C4—2141no—
220C2×C4—2141no—
221C2×C4—2141no—
222C2×C4—2142no—
223C2×C4—2142no—
224C2×C4D4×D10711yescharacteristic
225C2×C4#1313381712yes—
226C2×C4#1313381812yes—
227C2×C4C23×D10711yescharacteristic
228C8—21no—
229C8—21no—
230C8—2142no—
231C22—21no—
232C22—21no—
233C22—2142no—
234C22—2142no—
235C22—2142no—
236C22#1681209112yes—
237C22C2×D4×D10711yescharacteristic
238C4—42no—
239C4—42no—
240C4—4281no—
241C4—4281no—
242C4—4281no—
243C4—4281no—
244C4—2142no—
245C4#1681209112yes—
246C4C2×#1313381711yescharacteristic
247C4C2×#1313381811yescharacteristic
248C2—22no—
249C2—4282no—
250C2C2×#1681209111yescharacteristic
251C2#2029343311yescharacteristic
252C2#2029344011yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#103×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133818C22
hypercenterC2×#103D107
hypocenterC107#3097
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. #20292967C2semidirect
4. #20292966C2semidirect
5. #20292961C2semidirect
6. #20292956C2semidirect
7. #20292667C2semidirect
8. #20292665C2semidirect
9. #20292636C2semidirect
11. C2×#103×C107C2semidirect
127. C107#3097semidirect
137. C2×#103D107semidirect

Maximal subgroups

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

subgroupindex
#202932012
#202931082
#202929672
#202929662
#202929612
#202929562
#202926672
#202926652
#202926362
#202925932
C2×#103×C1072
#3097107