Finite Group Databasebeta

← #28349205

Subgroups — #28349205

Order 13696.

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

normal subgroups only

This group has 338 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
1Q8×Dic214C211yesmaximal, characteristic, complemented
2#20293351C211yesmaximal, characteristic
3#20292988C212yesmaximal
4#20292977C211yesmaximal, characteristic, complemented
5#20292918C211yesmaximal, characteristic
6#20292910C212yesmaximal, complemented
7#20292851C212yesmaximal
8#20292754C211yesmaximal, characteristic, complemented
9#20292734C212yesmaximal, complemented
10C4×#13133805C211yesmaximal, characteristic
11#462×C107C211yesmaximal, characteristic, complemented
12C2×#13133805C2211yescharacteristic
13#16812109—21no—
14#16812109C2211yescharacteristic
15#16812108—22no—
16#16812108C2212yes—
17#16812104—22no—
18#16812098C2212yes—
19#16812089—21no—
20Q8×Dic107C2212yes—
21Q8×Dic107C2212yes—
22C4×Dic214—21no—
23C4×Dic214C2211yescharacteristic
24C4×D428C2211yescharacteristic
25#16812067—21no—
26#16812041C2212yes—
27#16812040C2211yescharacteristic
28Dic107⋊C8C2212yes—
29#16812033C2212yes—
30#16812033C2211yescharacteristic
31#16812033C2212yes—
32#16812031C2212yes—
33#16812029C2211yescharacteristic
34#16812019C2212yes—
35C4×C107⋊C8C2211yescharacteristic
36C2×Q16×C107C2212yes—
37C4⋊Q8×C107C2211yescharacteristic
38C4×Q8×C107C2211yescharacteristic
39C4⋊C8×C107C2211yescharacteristic
40#103×C107C2212yes—
41C2×Dic214—21no—
42C2×Dic214—22no—
43C2×Dic214D411yescharacteristic
44C2×D428C2311yescharacteristic
45C2×C4×D107—21no—
46#13133805—22no—
47C2×C107⋊C8C2311yescharacteristic
48C2×C107⋊C8C2311yescharacteristic
49C4⋊Dic107—21no—
50C4⋊Dic107—22no—
51C4⋊Dic107—22no—
52C4⋊Dic107D411yescharacteristic
53C4⋊Dic107C2312yes—
54C4⋊Dic107C2311yescharacteristic
55Dic107⋊C4—22no—
56Dic107⋊C4D412yes—
57(C2×D107)⋊C4—21no—
58(C2×D107)⋊C4—22no—
59(C2×D107)⋊C4—22no—
60C4×Dic107—22no—
61C4×Dic107—22no—
62C4×Dic107C2312yes—
63C2×Q8×C107C2312yes—
64C2×Q8×C107C2311yescharacteristic
65Q16×C107—24no—
66C2×C8×C107C2312yes—
67C4⋊C4×C107—21no—
68C4⋊C4×C107—21no—
69C4⋊C4×C107—21no—
70C4⋊C4×C107C2311yescharacteristic
71C42×C107—21no—
72C42×C107C2311yescharacteristic
73C22×D107—21no—
74Q8×C107—21no—
75Q8×C107—22no—
76Q8×C107—22no—
77Q8×C107#4112yes—
78C2×C4×C107—21no—
79C2×C4×C107—21no—
80C2×C4×C107—22no—
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107C2411yescharacteristic
84C2×Dic107—21no—
85C2×Dic107—22no—
86C2×Dic107C2×D412yes—
87C2×Dic107C2×D412yes—
88D428—21no—
89D428—21no—
90C4×D107—42no—
91Dic214—22no—
92Dic214—82no—
93Dic214—81no—
94C8×C107—22no—
95C107⋊C8—21no—
96C107⋊C8—21no—
97C22×C107C22×D411yescharacteristic
98C2×D107—41no—
99C2×D107—41no—
100C2×D107—41no—
101C4×C107—22no—
102C4×C107—21no—
103C4×C107—42no—
104C4×C107—41no—
105C4×C107#13712yes—
106C4×C107C2×#4111yescharacteristic
107C4×C1072−1+411yescharacteristic
108Dic107—22no—
109Dic107—22no—
110Dic107—41no—
111Dic107—42no—
112C2×C107#54811yescharacteristic
113C2×C107C2×#13711yescharacteristic
114C2×C107#57711yescharacteristic
115D107—81no—
116#3369—1071nomaximal, Sylow
117C107#336911yescharacteristic, complemented, Sylow
118Q82—1071no—
119#556—1071no—
120#489—1072no—
121#487—1071no—
122#476—1071no—
123#475—1072no—
124#464—1072no—
125#462D10711yescharacteristic, complemented
126#446—1071no—
127#441—1072no—
128C4×SD16—1071no—
129C2×Q16C2×D10712yes—
130C2×SD16—1071no—
131C4⋊Q8—1072no—
132C4⋊Q8—1071no—
133C4⋊Q8—2141no—
134C4⋊Q8—2142no—
135C4⋊Q8C2×D10711yescharacteristic
136#126—2142no—
137#124—1072no—
138C22⋊Q8—2141no—
139C4×Q8—1071no—
140C4×Q8—1072no—
141C4×Q8—1072no—
142C4×Q8—2141no—
143C4×Q8C2×D10711yescharacteristic
144C4×D4—1071no—
145#117—2141no—
146#106—1072no—
147#106—1071no—
148C4⋊C8—1072no—
149C4⋊C8C2×D10711yescharacteristic
150#103—1071no—
151#103—1072no—
152#103—1072no—
153#103C2×D10712yes—
154#102—1072no—
155#102—1071no—
156#97—1072no—
157C4×C8—1071no—
158C2×Q8—1071no—
159C2×Q8—2142no—
160C2×Q8—2141no—
161C2×Q8C22×D10712yes—
162C2×Q8C22×D10711yescharacteristic
163C2×D4—1071no—
164C22×C4—2141no—
165Q16—24no—
166SD16—2142no—
167C2×C8—1071no—
168C2×C8—1071no—
169C2×C8C22×D10712yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—1071no—
174C4⋊C4—1072no—
175C4⋊C4—1072no—
176C4⋊C4—1071no—
177C4⋊C4—2142no—
178C4⋊C4—2142no—
179C4⋊C4—2142no—
180C4⋊C4—2141no—
181C4⋊C4C22×D10711yescharacteristic
182C22⋊C4—2142no—
183C22⋊C4—2142no—
184C22⋊C4—2141no—
185C42—21no—
186C42—1072no—
187C42—2142no—
188C42—2142no—
189C42C22×D10711yescharacteristic
190C23—2141no—
191Q8—21no—
192Q8—22no—
193Q8—22no—
194Q8—8561no—
195Q8—8562no—
196Q8—2142no—
197Q8#1313381712yes—
198D4—2141no—
199D4—2141no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—1072no—
204C2×C4—1072no—
205C2×C4—4282no—
206C2×C4—2141no—
207C2×C4—2142no—
208C2×C4D4×D10711yescharacteristic
209C2×C4D4×D10711yescharacteristic
210C2×C4C23×D10711yescharacteristic
211C8—22no—
212C8—2141no—
213C8—2141no—
214C22—4281no—
215C22—4281no—
216C22—4281no—
217C22C2×D4×D10711yescharacteristic
218C4—21no—
219C4—22no—
220C4—41no—
221C4—42no—
222C4—4281no—
223C4—4282no—
224C4—2142no—
225C4—2142no—
226C4#1681214112yes—
227C4C2×#1313381711yescharacteristic
228C4#1681215611yescharacteristic
229C2—8561no—
230C2#2029328311yescharacteristic
231C2C2×#1681214111yescharacteristic
232C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#462×C107C2
socleC22×C107C22×D4
prime radicalC2×D428C23
hypercenter#462D107
hypocenterC107#3369
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. Q8×Dic214C2semidirect
4. #20292977C2semidirect
6. #20292910C2semidirect
8. #20292754C2semidirect
9. #20292734C2semidirect
11. #462×C107C2semidirect
117. C107#3369semidirect
125. #462D107semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic2142
#202933512
#202929882
#202929772
#202929182
#202929102
#202928512
#202927542
#202927342
C4×#131338052
#462×C1072
#3369107