Finite Group Databasebeta

← #28338124

Subgroups — #28338124

Order 13696.

258 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 65 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#20293173C211yesmaximal, characteristic
2#20292601C211yesmaximal, characteristic, complemented
3#20292421C212yesmaximal, complemented
4#20292419C212yesmaximal, complemented
5#410×C107C211yesmaximal, characteristic, complemented
6C2×#13133818C411yescharacteristic, complemented
7#16812107C411yescharacteristic, complemented
8#16812105C2211yescharacteristic
9#16812098C411yescharacteristic, complemented
10#16812096C2211yescharacteristic
11#16812081—21no—
12#16812081—21no—
13C4×C22⋊D107—21no—
14C4×C22⋊D107—21no—
15#16812068—21no—
16C2×C4×Dic107C411yescharacteristic, complemented
17#16812024—21no—
18#16812024C2212yes—
19C2×#41×C107C2211yescharacteristic
20#117×C107—21no—
21#99×C107—21no—
22#99×C107C2212yes—
23#13133818—41no—
24#13133818—41no—
25C2×Dic214C2×C411yescharacteristic
26C2×D428C2×C411yescharacteristic
27C2×C22⋊D107—21no—
28C2×C22⋊D107—21no—
29C2×C22⋊D107—21no—
30C22×Dic107—21no—
31C2×C4×D107—21no—
32C4⋊Dic107—41no—
33Dic107⋊C4—21no—
34Dic107⋊C4—21no—
35C22⋊Dic107—21no—
36C22⋊Dic107—21no—
37C22⋊Dic107—22no—
38(C2×D107)⋊C4—21no—
39(C2×D107)⋊C4—21no—
40C4×Dic107—21no—
41C4×Dic107—41no—
42C4×Dic107C2×C411yescharacteristic
43C4×Dic107C2×C411yescharacteristic
44#41×C107—41no—
45#41×C107—41no—
46C2×Q8×C107C2×C411yescharacteristic
47C2×D4×C107D411yescharacteristic
48C2×D4×C107D411yescharacteristic
49C2×D4×C107C2311yescharacteristic
50C22×C4×C107C2×C411yescharacteristic
51C22×C4×C107D411yescharacteristic
52C22×C4×C107D411yescharacteristic
53C4⋊C4×C107—41no—
54C22⋊C4×C107—22no—
55C42×C107—41no—
56C23×C107C22⋊C411yescharacteristic
57C23×C107C2×D411yescharacteristic
58C23×C107C2×D411yescharacteristic
59C22×D107—21no—
60Q8×C107—41no—
61D4×C107—41no—
62D4×C107—41no—
63D4×C107—41no—
64C2×C4×C107—21no—
65C2×C4×C107—21no—
66C2×C4×C107—21no—
67C2×C4×C107—21no—
68C2×C4×C107—21no—
69C2×C4×C107—21no—
70C2×C4×C107—42no—
71C2×C4×C107C22⋊C411yescharacteristic
72C2×C4×C107C22⋊C411yescharacteristic
73C2×C4×C107C22⋊C411yescharacteristic
74C2×C4×C107C22×C411yescharacteristic
75C22⋊D107—42no—
76C22⋊D107—42no—
77C22⋊D107—42no—
78C2×Dic107—21no—
79C2×Dic107—21no—
80C2×Dic107—21no—
81C2×Dic107—42no—
82C2×Dic107—42no—
83D428—41no—
84C4×D107—41no—
85C4×D107—41no—
86Dic214—41no—
87C22×C107—21no—
88C22×C107—21no—
89C22×C107—21no—
90C22×C107—41no—
91C22×C107—41no—
92C22×C107—41no—
93C22×C107C2×C22⋊C411yescharacteristic
94C2×D107—41no—
95C2×D107—41no—
96C2×D107—41no—
97C4×C107—21no—
98C4×C107—21no—
99C4×C107—21no—
100C4×C107—82no—
101C4×C107#9911yescharacteristic
102C4×C107#9911yescharacteristic
103Dic107—22no—
104Dic107—41no—
105Dic107—41no—
106Dic107—82no—
107C2×C107—21no—
108C2×C107—41no—
109C2×C107—41no—
110C2×C107—41no—
111C2×C107C2×#9911yescharacteristic
112D107—81no—
113#2126—1071nomaximal, Sylow
114C107#212611yescharacteristic, complemented, Sylow
115#532—1071no—
116#410—1071no—
117#410D10711yescharacteristic, complemented
118#354—1072no—
119#353—1072no—
120C2×#41—1071no—
121C2×#41C2×D10711yescharacteristic
122C4⋊Q8—1071no—
123#127—1071no—
124#124—1071no—
125#124—1071no—
126#121—2141no—
127#121—2141no—
128C4×D4—2141no—
129C4×D4—2141no—
130#117—21no—
131#117—2141no—
132C2×C42—1071no—
133#99—21no—
134#99—1072no—
135#99—2141no—
136#99C2×D10712yes—
137#41—41no—
138#41—41no—
139#41—4281no—
140#41—4281no—
141C2×Q8—1071no—
142C2×Q8C4×D10711yescharacteristic
143C2×D4—1071no—
144C2×D4—2141no—
145C2×D4—2141no—
146C2×D4—2141no—
147C2×D4D42811yescharacteristic
148C2×D4C22⋊D10711yescharacteristic
149C2×D4C22×D10711yescharacteristic
150C22×C4—2141no—
151C22×C4—2141no—
152C22×C4C4×D10711yescharacteristic
153C22×C4D42811yescharacteristic
154C22×C4C22⋊D10711yescharacteristic
155C4⋊C4—41no—
156C4⋊C4—4281no—
157C4⋊C4—2141no—
158C4⋊C4—2141no—
159C22⋊C4—22no—
160C22⋊C4—2142no—
161C22⋊C4—2141no—
162C22⋊C4—2141no—
163C22⋊C4—2141no—
164C22⋊C4—2141no—
165C42—41no—
166C42—1071no—
167C42—1071no—
168C42—4281no—
169C42—2141no—
170C23—2141no—
171C23(C2×D107)⋊C411yescharacteristic
172C23C2×C22⋊D10711yescharacteristic
173C23C2×D42811yescharacteristic
174Q8—41no—
175Q8—4281no—
176D4—41no—
177D4—41no—
178D4—41no—
179D4—4281no—
180D4—4282no—
181D4—4282no—
182D4—4282no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—4282no—
191C2×C4—4282no—
192C2×C4—4281no—
193C2×C4—4281no—
194C2×C4—2141no—
195C2×C4—2141no—
196C2×C4—2141no—
197C2×C4(C2×D107)⋊C411yescharacteristic
198C2×C4(C2×D107)⋊C411yescharacteristic
199C2×C4(C2×D107)⋊C411yescharacteristic
200C2×C4C2×C4×D10711yescharacteristic
201C22—21no—
202C22—21no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—4281no—
208C22—4281no—
209C22—4281no—
210C22C2×(C2×D107)⋊C411yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—82no—
215C4—8562no—
216C4—4281no—
217C4—4281no—
218C4—2142no—
219C4#1681202311yescharacteristic
220C4#1681202311yescharacteristic
221C2—21no—
222C2—41no—
223C2—41no—
224C2—41no—
225C2—8561no—
226C2C2×#1681202311yescharacteristic
subgroupG/H
centerC4#16812023
commutatorC2×C4×C107C22×C4
FrattiniC2×D4C22×D107
Fitting#410×C107C2
socleC2×C107C2×#99
prime radical#20293173C2
cosocleC2×D4×C107C23
hypercenter#410D107
hypocenterC107#2126
LCSG ▷ C2×C4×C107 ▷ C22×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #20292601C2semidirect
3. #20292421C2semidirect
4. #20292419C2semidirect
5. #410×C107C2semidirect
6. C2×#13133818C4semidirect
7. #16812107C4semidirect
9. #16812098C4semidirect
16. C2×C4×Dic107C4semidirect
114. C107#2126semidirect
117. #410D107semidirect

Maximal subgroups

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

subgroupindex
#202931732
#202926012
#202924212
#202924192
#410×C1072
#2126107