Finite Group Databasebeta

← #28349090

Subgroups — #28349090

Order 13696.

354 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 115 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#20293342C211yesmaximal, characteristic
2#20293338C211yesmaximal, characteristic, complemented
3#20293030C211yesmaximal, characteristic, complemented
4#20292971C212yesmaximal, complemented
5#20292939C212yesmaximal, complemented
6#20292923C211yesmaximal, characteristic, complemented
7#20292872C212yesmaximal, complemented
8C4×#13133806C211yesmaximal, characteristic, complemented
9#20292704C211yesmaximal, characteristic, complemented
10#20292666C212yesmaximal, complemented
11#476×C107C211yesmaximal, characteristic, complemented
12C2×#13133806C2211yescharacteristic
13C2×#13133801C2212yes—
14#16812109C2211yescharacteristic
15#16812100—21no—
16#16812095—22no—
17#16812089C2212yes—
18#16812085—22no—
19#16812083—21no—
20#16812080C2212yes—
21C4×Dic214C2211yescharacteristic
22C4×D428C2211yescharacteristic
23#16812071—21no—
24#16812068—21no—
25#16812067C2212yes—
26C4⋊C4×D107C2212yes—
27#16812043C2212yes—
28#16812042C2212yes—
29#16812042C2212yes—
30#16812040C2211yescharacteristic
31#16812034C2212yes—
32#16812032C2211yescharacteristic
33#16812030C2211yescharacteristic
34(C2×D107)⋊C8C2212yes—
35C4×C107⋊C8C2211yescharacteristic
36C4⋊Q8×C107C2211yescharacteristic
37C4×D4×C107C2211yescharacteristic
38#106×C107C2212yes—
39C4⋊C8×C107C2211yescharacteristic
40#102×C107C2212yes—
41C2×Dic214—21no—
42C2×Dic214C2311yescharacteristic
43C2×D428D411yescharacteristic
44C2×C22⋊D107—22no—
45C2×C4×D107—22no—
46C2×C4×D107C2312yes—
47#13133806—22no—
48#13133801—24no—
49C2×C107⋊C8C2311yescharacteristic
50C2×C107⋊C8C2311yescharacteristic
51C4⋊Dic107—21no—
52C4⋊Dic107—22no—
53C4⋊Dic107D411yescharacteristic
54C4⋊Dic107C2312yes—
55C4⋊Dic107C2311yescharacteristic
56Dic107⋊C4—21no—
57Dic107⋊C4—22no—
58C22⋊Dic107—22no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4—22no—
61(C2×D107)⋊C4D412yes—
62C4×Dic107—21no—
63C4×Dic107—22no—
64C2×Q8×C107—21no—
65C2×D4×C107C2311yescharacteristic
66C22×C4×C107—21no—
67C2×C8×C107C2312yes—
68C4⋊C4×C107—21no—
69C4⋊C4×C107C2311yescharacteristic
70C4⋊C4×C107C2312yes—
71C22⋊C4×C107—21no—
72C42×C107C2311yescharacteristic
73C23×C107—21no—
74C22×D107C2×D412yes—
75Q8×C107—81no—
76D4×C107—21no—
77D4×C107—21no—
78C2×C4×C107—21no—
79C2×C4×C107—22no—
80C2×C4×C107—42no—
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107C2411yescharacteristic
84C22⋊D107—82no—
85C2×Dic107—21no—
86C2×Dic107—22no—
87C2×Dic107—21no—
88C2×Dic107C2×D412yes—
89D428—22no—
90C4×D107—22no—
91C4×D107—22no—
92C4×D107—82no—
93Dic214—21no—
94Dic214—21no—
95Dic214—81no—
96C8×C107—22no—
97C107⋊C8—21no—
98C107⋊C8#4112yes—
99C22×C107—41no—
100C22×C107—41no—
101C22×C107—41no—
102C22×C107C22×D411yescharacteristic
103C2×D107—22no—
104C2×D107—22no—
105C2×D107—22no—
106C4×C107—21no—
107C4×C107—41no—
108C4×C107—42no—
109C4×C107#13712yes—
110C4×C107C2×#4111yescharacteristic
111C4×C1072−1+411yescharacteristic
112Dic107—22no—
113Dic107—41no—
114Dic107—42no—
115Dic107—41no—
116C2×C107—81no—
117C2×C107#54711yescharacteristic
118C2×C107C2×#13711yescharacteristic
119C2×C107#57611yescharacteristic
120D107—42no—
121#3357—1071nomaximal, Sylow
122C107#335711yescharacteristic, complemented, Sylow
123#555—1071no—
124#554—1071no—
125#500—1071no—
126#485—1072no—
127#480—1072no—
128#477—1071no—
129#476D10711yescharacteristic, complemented
130#468—1072no—
131C4×SD16—1071no—
132#436—1071no—
133#428—1072no—
134C2×SD16—1071no—
135C2×M16—1072no—
136C4⋊Q8—1071no—
137C4⋊Q8C2×D10711yescharacteristic
138#125—2141no—
139#124—2142no—
140C22⋊Q8—1072no—
141C22⋊Q8—2142no—
142#121—1072no—
143#121—2141no—
144C4×Q8—1071no—
145C4×D4—1071no—
146C4×D4—2141no—
147C4×D4C2×D10711yescharacteristic
148#117—1072no—
149#117—2141no—
150C2×C4⋊C4—1072no—
151#107—1072no—
152#107—1072no—
153#107—1072no—
154#106—1071no—
155#106C2×D10712yes—
156C4⋊C8C2×D10711yescharacteristic
157#103—1072no—
158#103—1071no—
159#102—1071no—
160#102C2×D10712yes—
161C22⋊C8—1072no—
162C4×C8—1071no—
163C2×Q8—21no—
164C2×Q8—1071no—
165C2×Q8—2141no—
166C2×D4—1071no—
167C2×D4—2142no—
168C2×D4C22×D10711yescharacteristic
169C22×C4—21no—
170C22×C4—1072no—
171C22×C4—2142no—
172SD16—2142no—
173M16—2144no—
174C2×C8—1071no—
175C2×C8—1071no—
176C2×C8C22×D10712yes—
177C4⋊C4—21no—
178C4⋊C4—1071no—
179C4⋊C4—1071no—
180C4⋊C4—1072no—
181C4⋊C4—2142no—
182C4⋊C4—2142no—
183C4⋊C4—2141no—
184C4⋊C4—2141no—
185C4⋊C4C22×D10712yes—
186C4⋊C4C22×D10711yescharacteristic
187C22⋊C4—21no—
188C22⋊C4—1072no—
189C22⋊C4—2142no—
190C22⋊C4—2142no—
191C22⋊C4—2142no—
192C42—2142no—
193C42—2141no—
194C42C22×D10711yescharacteristic
195C23—21no—
196C23—1072no—
197Q8—81no—
198Q8—8561no—
199Q8—2141no—
200Q8—2141no—
201D4—21no—
202D4—21no—
203D4—8562no—
204D4—2142no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4—8562no—
209C2×C4—1072no—
210C2×C4—2142no—
211C2×C4—2141no—
212C2×C4—2141no—
213C2×C4—2142no—
214C2×C4—2142no—
215C2×C4D4×D10711yescharacteristic
216C2×C4D4×D10711yescharacteristic
217C2×C4C23×D10711yescharacteristic
218C8—22no—
219C8—1072no—
220C8—2141no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—2142no—
225C22—2142no—
226C22—2142no—
227C22C2×D4×D10711yescharacteristic
228C4—21no—
229C4—41no—
230C4—42no—
231C4—4281no—
232C4—4282no—
233C4—4281no—
234C4—2142no—
235C4#1681213812yes—
236C4#41×D10711yescharacteristic
237C4#1681215711yescharacteristic
238C2—81no—
239C2—4282no—
240C2#2029327911yescharacteristic
241C2C2×#1681213811yescharacteristic
242C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#476×C107C2
socleC22×C107C22×D4
prime radical#20293342C2
hypercenter#476D107
hypocenterC107#3357
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #20293338C2semidirect
3. #20293030C2semidirect
4. #20292971C2semidirect
5. #20292939C2semidirect
6. #20292923C2semidirect
7. #20292872C2semidirect
8. C4×#13133806C2semidirect
9. #20292704C2semidirect
10. #20292666C2semidirect
11. #476×C107C2semidirect
122. C107#3357semidirect
129. #476D107semidirect

Maximal subgroups

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

subgroupindex
#202933422
#202933382
#202930302
#202929712
#202929392
#202929232
#202928722
C4×#131338062
#202927042
#202926662
#476×C1072
#3357107