Finite Group Databasebeta

← #28335015

Subgroups — #28335015

Order 13696.

284 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 73 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#20293138C211yesmaximal, characteristic
2C2×C22⋊(C107⋊C8)C211yesmaximal, characteristic, complemented
3#20292363C212yesmaximal, complemented
4#20292360C212yesmaximal, complemented
5#381×C107C211yesmaximal, characteristic, complemented
6C2×#13133818C411yescharacteristic, complemented
7C22×C107⋊C8—21no—
8#16812092—21no—
9#16812088C411yescharacteristic, complemented
10#16812088C2211yescharacteristic
11#16812083C411yescharacteristic, complemented
12#16812083C2211yescharacteristic
13#16812078—21no—
14C4×C22⋊D107—21no—
15C4×C22⋊D107—21no—
16#16812068C411yescharacteristic, complemented
17C22⋊(C107⋊C8)—21no—
18C22⋊(C107⋊C8)C2212yes—
19C23×C4×C107C2211yescharacteristic
20C2×C22⋊C4×C107—21no—
21#95×C107—21no—
22#95×C107C2212yes—
23#13133818—42no—
24C2×Dic214C2×C411yescharacteristic
25C2×D428C2×C411yescharacteristic
26C2×C22⋊D107—21no—
27C2×C22⋊D107—21no—
28C2×C22⋊D107—21no—
29C2×C4×D107—21no—
30C2×C107⋊C8—22no—
31C2×C107⋊C8—42no—
32C4⋊Dic107C2×C411yescharacteristic
33C4⋊Dic107C2×C411yescharacteristic
34Dic107⋊C4—21no—
35Dic107⋊C4—21no—
36C22⋊Dic107—21no—
37C22⋊Dic107—21no—
38C22⋊Dic107—21no—
39(C2×D107)⋊C4—21no—
40(C2×D107)⋊C4—21no—
41C4×Dic107—21no—
42C24×C107D411yescharacteristic
43C22×C4×C107—22no—
44C22×C4×C107—42no—
45C22×C4×C107C2×C411yescharacteristic
46C22×C4×C107C2×C411yescharacteristic
47C22×C4×C107D411yescharacteristic
48C22×C4×C107D411yescharacteristic
49C22×C4×C107D411yescharacteristic
50C22×C4×C107C2311yescharacteristic
51C22⋊C4×C107—42no—
52C23×C107—22no—
53C23×C107—22no—
54C23×C107—41no—
55C23×C107C22⋊C411yescharacteristic
56C23×C107C22⋊C411yescharacteristic
57C23×C107C2×D411yescharacteristic
58C22×D107—21no—
59C2×C4×C107—22no—
60C2×C4×C107—22no—
61C2×C4×C107—41no—
62C2×C4×C107—41no—
63C2×C4×C107—41no—
64C2×C4×C107—41no—
65C2×C4×C107—42no—
66C2×C4×C107—42no—
67C2×C4×C107—42no—
68C2×C4×C107C22⋊C411yescharacteristic
69C2×C4×C107C22⋊C411yescharacteristic
70C2×C4×C107C22×C411yescharacteristic
71C2×C4×C107C2×D411yescharacteristic
72C22⋊D107—42no—
73C22⋊D107—42no—
74C22⋊D107—42no—
75C2×Dic107—21no—
76C2×Dic107—21no—
77C2×Dic107—21no—
78D428—41no—
79C4×D107—42no—
80Dic214—41no—
81C107⋊C8—42no—
82C22×C107—21no—
83C22×C107—21no—
84C22×C107—21no—
85C22×C107—21no—
86C22×C107—21no—
87C22×C107—21no—
88C22×C107—21no—
89C22×C107—44no—
90C22×C107#9912yes—
91C22×C107C2×C22⋊C411yescharacteristic
92C22×C107#13512yes—
93C2×D107—41no—
94C2×D107—41no—
95C2×D107—41no—
96C4×C107—21no—
97C4×C107—21no—
98C4×C107—21no—
99C4×C107—21no—
100C4×C107—82no—
101Dic107—41no—
102Dic107—41no—
103Dic107—41no—
104C2×C107—22no—
105C2×C107—22no—
106C2×C107—41no—
107C2×C107C2×#9911yescharacteristic
108C2×C107#41611yescharacteristic
109C2×C107#42111yescharacteristic
110D107—81no—
111#1516—1071nomaximal, Sylow
112C107#151611yescharacteristic, complemented, Sylow
113#525—1071no—
114C2×C22⋊C8—1071no—
115#381D10711yescharacteristic, complemented
116#328—1072no—
117#327—1072no—
118C2×#41—1071no—
119C23×C4C2×D10711yescharacteristic
120C22×C8—2141no—
121#123—2141no—
122C22⋊Q8—1071no—
123C22⋊Q8—1071no—
124#121—1071no—
125#121—1071no—
126#120—2141no—
127C4×D4—2141no—
128C4×D4—2141no—
129#117—1071no—
130C2×C22⋊C4—21no—
131C22⋊C8—1072no—
132C22⋊C8—2141no—
133#95—21no—
134#95C2×D10712yes—
135C24C22⋊D10711yescharacteristic
136#41—4282no—
137C2×Q8—1071no—
138C2×D4—1071no—
139C2×D4—2141no—
140C2×D4—2141no—
141C2×D4—2141no—
142C22×C4—22no—
143C22×C4—42no—
144C22×C4—2141no—
145C22×C4C4×D10711yescharacteristic
146C22×C4C4×D10711yescharacteristic
147C22×C4D42811yescharacteristic
148C22×C4D42811yescharacteristic
149C22×C4C22⋊D10711yescharacteristic
150C22×C4C22×D10711yescharacteristic
151C2×C8—4282no—
152C2×C8—2142no—
153C4⋊C4—1071no—
154C4⋊C4—1071no—
155C4⋊C4—2141no—
156C4⋊C4—2141no—
157C22⋊C4—42no—
158C22⋊C4—2141no—
159C22⋊C4—2141no—
160C22⋊C4—2141no—
161C22⋊C4—2141no—
162C22⋊C4—2141no—
163C42—2141no—
164C23—22no—
165C23—22no—
166C23—41no—
167C23—2141no—
168C23(C2×D107)⋊C411yescharacteristic
169C23(C2×D107)⋊C411yescharacteristic
170C23C2×C22⋊D10711yescharacteristic
171Q8—4281no—
172D4—4281no—
173D4—4282no—
174D4—4282no—
175D4—4282no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—4282no—
186C2×C4—2141no—
187C2×C4—2141no—
188C2×C4—2141no—
189C2×C4(C2×D107)⋊C411yescharacteristic
190C2×C4(C2×D107)⋊C411yescharacteristic
191C2×C4C2×C4×D10711yescharacteristic
192C2×C4C2×D42811yescharacteristic
193C8—4282no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—4281no—
203C22—4281no—
204C22—4281no—
205C22#1681202312yes—
206C22C2×(C2×D107)⋊C411yescharacteristic
207C22#1681212912yes—
208C4—21no—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—82no—
213C4—4281no—
214C4—4281no—
215C4—4281no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—8561no—
220C2C2×#1681202311yescharacteristic
221C2#2029262111yescharacteristic
222C2#2029264511yescharacteristic
subgroupG/H
centerC22C2×(C2×D107)⋊C4
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#381×C107C2
socleC22×C107C2×C22⋊C4
prime radical#20293138C2
cosocleC22×C4×C107C23
hypercenter#381D107
hypocenterC107#1516
LCSG ▷ C2×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊(C107⋊C8)C2semidirect
3. #20292363C2semidirect
4. #20292360C2semidirect
5. #381×C107C2semidirect
6. C2×#13133818C4semidirect
9. #16812088C4semidirect
11. #16812083C4semidirect
16. #16812068C4semidirect
112. C107#1516semidirect
115. #381D107semidirect

Maximal subgroups

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

subgroupindex
#202931382
C2×C22⋊(C107⋊C8)2
#202923632
#202923602
#381×C1072
#1516107