Finite Group Databasebeta

← #28349125

Subgroups — #28349125

Order 13696.

362 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 117 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#20293363C211yesmaximal, characteristic, complemented
2#20293334C211yesmaximal, characteristic, complemented
3#20292980C212yesmaximal, complemented
4#20292973C211yesmaximal, characteristic, complemented
5#20292925C212yesmaximal, complemented
6#20292923C211yesmaximal, characteristic, complemented
7#20292840C212yesmaximal, complemented
8#20292751C211yesmaximal, characteristic, complemented
9C4×#13133806C211yesmaximal, characteristic, complemented
10D8×Dic107C212yesmaximal, complemented
11#459×C107C211yesmaximal, characteristic
12C2×#13133806C2211yescharacteristic
13#16812109C2211yescharacteristic
14#16812100C2212yes—
15#16812096—21no—
16#16812096C2212yes—
17#16812090—22no—
18#16812088—21no—
19#16812086—22no—
20C4×Dic214C2211yescharacteristic
21C4×Dic214C2211yescharacteristic
22D4×Dic107—21no—
23D4×Dic107C2212yes—
24D4×Dic107C2212yes—
25C2×C4⋊Dic107—21no—
26#16812043C2212yes—
27#16812040C2211yescharacteristic
28Dic107⋊C8C2212yes—
29#16812034C2212yes—
30#16812032C2211yescharacteristic
31#16812030C2212yes—
32#16812030C2211yescharacteristic
33#16812030C2212yes—
34C4×C107⋊C8C2211yescharacteristic
35C8×Dic107C2212yes—
36C2×D8×C107C2212yes—
37#127×C107C2211yescharacteristic
38C4×D4×C107C2211yescharacteristic
39C4⋊C8×C107C2211yescharacteristic
40#102×C107C2212yes—
41C2×Dic214—21no—
42C2×Dic214D411yescharacteristic
43C2×Dic214C2311yescharacteristic
44C22×Dic107—22no—
45C22×Dic107—22no—
46#13133806—22no—
47C2×C107⋊C8C2311yescharacteristic
48C2×C107⋊C8C2311yescharacteristic
49C4⋊Dic107—21no—
50C4⋊Dic107—22no—
51C4⋊Dic107D411yescharacteristic
52C4⋊Dic107C2312yes—
53C4⋊Dic107C2311yescharacteristic
54Dic107⋊C4—21no—
55Dic107⋊C4—22no—
56Dic107⋊C4D412yes—
57C22⋊Dic107—22no—
58C22⋊Dic107—22no—
59C22⋊Dic107—22no—
60C22⋊Dic107—22no—
61C4×Dic107—21no—
62C4×Dic107C2312yes—
63C2×D4×C107—21no—
64C2×D4×C107—21no—
65C2×D4×C107C2312yes—
66C2×D4×C107C2311yescharacteristic
67C22×C4×C107—21no—
68D8×C107—24no—
69C2×C8×C107C2312yes—
70C4⋊C4×C107C2311yescharacteristic
71C22⋊C4×C107—21no—
72C42×C107C2311yescharacteristic
73C23×C107—21no—
74C23×C107—22no—
75D4×C107—22no—
76D4×C107—21no—
77D4×C107—22no—
78D4×C107—42no—
79D4×C107—81no—
80D4×C107#4112yes—
81C2×C4×C107—21no—
82C2×C4×C107—42no—
83C2×C4×C107C2×D411yescharacteristic
84C2×C4×C107C2×D411yescharacteristic
85C2×C4×C107C2411yescharacteristic
86C2×Dic107—21no—
87C2×Dic107—21no—
88C2×Dic107—22no—
89C2×Dic107—82no—
90C2×Dic107—82no—
91C2×Dic107C2×D412yes—
92C2×Dic107C2×D412yes—
93Dic214—22no—
94Dic214—21no—
95Dic214—21no—
96Dic214—81no—
97C8×C107—22no—
98C107⋊C8—21no—
99C107⋊C8—21no—
100C22×C107—22no—
101C22×C107—41no—
102C22×C107—41no—
103C22×C107—42no—
104C22×C107—42no—
105C22×C107—42no—
106C22×C107C22×D411yescharacteristic
107C4×C107—21no—
108C4×C107—41no—
109C4×C107#13512yes—
110C4×C107C2×#4111yescharacteristic
111C4×C1072−1+411yescharacteristic
112Dic107—22no—
113Dic107—22no—
114Dic107—41no—
115Dic107—42no—
116Dic107—41no—
117C2×C107—42no—
118C2×C107—82no—
119C2×C107#54811yescharacteristic
120C2×C107C2×#13511yescharacteristic
121C2×C107#57611yescharacteristic
122#3361—1071nomaximal, Sylow
123C107#336111yescharacteristic, complemented, Sylow
124#559—1071no—
125#554—1071no—
126#488—1072no—
127#486—1071no—
128#478—1072no—
129#477—1071no—
130#463—1072no—
131#459D10711yescharacteristic
132C8×Q8—1071no—
133C4×SD16—1071no—
134C4×D8—1072no—
135C2×SD16—1071no—
136C2×D8C2×D10712yes—
137C4⋊Q8—1071no—
138#127C2×D10711yescharacteristic
139#125—1072no—
140#124—1072no—
141#124—2141no—
142#123—2142no—
143C22⋊Q8—2142no—
144C22⋊Q8—2141no—
145C4×Q8—1071no—
146C4×Q8—1071no—
147C4×D4—1072no—
148C4×D4—1072no—
149C4×D4—2141no—
150C4×D4C2×D10711yescharacteristic
151C2×C4⋊C4—2141no—
152#107—1072no—
153#106—1071no—
154C4⋊C8—1072no—
155C4⋊C8C2×D10711yescharacteristic
156#103—1072no—
157#103—1071no—
158#102—1071no—
159#102—1072no—
160#102—1072no—
161#102C2×D10712yes—
162C4×C8—1072no—
163C4×C8—1071no—
164C2×Q8—1071no—
165C2×Q8—1071no—
166C2×Q8—2141no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4C22×D10712yes—
170C2×D4C22×D10711yescharacteristic
171C22×C4—21no—
172C22×C4—2142no—
173C22×C4—2142no—
174SD16—2142no—
175D8—24no—
176C2×C8—1071no—
177C2×C8—1071no—
178C2×C8C22×D10712yes—
179C4⋊C4—1071no—
180C4⋊C4—1072no—
181C4⋊C4—1072no—
182C4⋊C4—1071no—
183C4⋊C4—2142no—
184C4⋊C4—2142no—
185C4⋊C4—2141no—
186C4⋊C4—2141no—
187C4⋊C4C22×D10711yescharacteristic
188C22⋊C4—21no—
189C22⋊C4—2142no—
190C22⋊C4—2142no—
191C22⋊C4—2142no—
192C22⋊C4—2142no—
193C42—1072no—
194C42—2141no—
195C42C22×D10711yescharacteristic
196C23—21no—
197C23—22no—
198Q8—8561no—
199Q8—2141no—
200Q8—2141no—
201Q8—2142no—
202D4—22no—
203D4—21no—
204D4—22no—
205D4—42no—
206D4—81no—
207D4#1313381712yes—
208C2×C4—21no—
209C2×C4—42no—
210C2×C4—8562no—
211C2×C4—8562no—
212C2×C4—1072no—
213C2×C4—1072no—
214C2×C4—2141no—
215C2×C4—2141no—
216C2×C4—2142no—
217C2×C4D4×D10711yescharacteristic
218C2×C4D4×D10711yescharacteristic
219C2×C4C23×D10711yescharacteristic
220C8—22no—
221C8—2141no—
222C8—2141no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22C2×D4×D10711yescharacteristic
230C4—21no—
231C4—41no—
232C4—4281no—
233C4—4282no—
234C4—4281no—
235C4—2142no—
236C4—2142no—
237C4#1681212812yes—
238C4C2×#1313381711yescharacteristic
239C4#1681215611yescharacteristic
240C2—42no—
241C2—82no—
242C2#2029328311yescharacteristic
243C2C2×#1681212811yescharacteristic
244C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting, prime radical#459×C107C2
socleC22×C107C22×D4
hypercenter#459D107
hypocenterC107#3361
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #20293363C2semidirect
2. #20293334C2semidirect
3. #20292980C2semidirect
4. #20292973C2semidirect
5. #20292925C2semidirect
6. #20292923C2semidirect
7. #20292840C2semidirect
8. #20292751C2semidirect
9. C4×#13133806C2semidirect
10. D8×Dic107C2semidirect
123. C107#3361semidirect

Maximal subgroups

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

subgroupindex
#202933632
#202933342
#202929802
#202929732
#202929252
#202929232
#202928402
#202927512
C4×#131338062
D8×Dic1072
#459×C1072
#3361107