Finite Group Databasebeta

← #28344499

Subgroups — #28344499

Order 13696.

444 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×#16812086C212yesmaximal
2#20292569C212yesmaximal, complemented
3#20292568C212yesmaximal, complemented
4#20292555C212yesmaximal
5#20292552C211yesmaximal, characteristic
6#20292548C212yesmaximal, complemented
7#20292547C212yesmaximal, complemented
8#20292488C211yesmaximal, characteristic
9#397×C107C211yesmaximal, characteristic
10C22×Dic214C2212yes—
11C23×Dic107C2211yescharacteristic
12#16812086—28no—
13C2×C4⋊Dic107C2212yes—
14C2×C4⋊Dic107C2211yescharacteristic
15C2×Dic107⋊C4C2212yes—
16C2×Dic107⋊C4C2212yes—
17C2×Dic107⋊C4C2212yes—
18C2×C22⋊Dic107C2212yes—
19C2×C22⋊Dic107C2211yescharacteristic
20C107⋊#95C2212yes—
21C107⋊#95C2212yes—
22C107⋊#95C2212yes—
23C107⋊#95C2212yes—
24C107⋊#95C2212yes—
25C107⋊#95C2211yescharacteristic
26C107⋊#95C2212yes—
27C2×C4⋊C4×C107C2211yescharacteristic
28C2×C22⋊C4×C107C2212yes—
29C2×C22⋊C4×C107C2211yescharacteristic
30#95×C107C2212yes—
31#95×C107C2211yescharacteristic
32C2×Dic214—24no—
33C2×Dic214—42no—
34C2×Dic214—42no—
35C22×Dic107—41no—
36C22×Dic107—41no—
37C22×Dic107D414yes—
38C22×Dic107C2312yes—
39C22×Dic107C2312yes—
40C22×Dic107C2312yes—
41C22×Dic107C2312yes—
42C4⋊Dic107—24no—
43C4⋊Dic107—41no—
44Dic107⋊C4—24no—
45Dic107⋊C4—24no—
46Dic107⋊C4—42no—
47C22⋊Dic107—24no—
48C22⋊Dic107—41no—
49C24×C107C2311yescharacteristic
50C22×C4×C107C2312yes—
51C22×C4×C107C2312yes—
52C22×C4×C107C2311yescharacteristic
53C22×C4×C107C2311yescharacteristic
54C4⋊C4×C107—41no—
55C22⋊C4×C107—24no—
56C22⋊C4×C107—41no—
57C23×C107—21no—
58C23×C107—21no—
59C23×C107—22no—
60C23×C107—22no—
61C23×C107C2×D412yes—
62C23×C107C2411yescharacteristic
63C2×C4×C107—22no—
64C2×C4×C107—22no—
65C2×C4×C107—21no—
66C2×C4×C107—21no—
67C2×C4×C107—22no—
68C2×C4×C107—22no—
69C2×C4×C107—22no—
70C2×C4×C107—22no—
71C2×C4×C107—22no—
72C2×C4×C107—22no—
73C2×Dic107—22no—
74C2×Dic107—22no—
75C2×Dic107—22no—
76C2×Dic107—22no—
77C2×Dic107—22no—
78C2×Dic107—22no—
79C2×Dic107—22no—
80C2×Dic107—22no—
81C2×Dic107—22no—
82C2×Dic107—22no—
83C2×Dic107—22no—
84C2×Dic107—44no—
85C2×Dic107C2×D414yes—
86Dic214—48no—
87C22×C107—22no—
88C22×C107—24no—
89C22×C107—41no—
90C22×C107—42no—
91C22×C107—41no—
92C22×C107C22×D411yescharacteristic
93C22×C107H(5,𝔽2)11yescharacteristic
94C22×C107H(5,𝔽2)11yescharacteristic
95C22×C1072−1+412yes—
96C22×C1072−1+412yes—
97C4×C107—42no—
98C4×C107—42no—
99C4×C107—41no—
100C4×C107—41no—
101Dic107—24no—
102Dic107—42no—
103Dic107—42no—
104Dic107—42no—
105C2×C107—42no—
106C2×C107#53411yescharacteristic
107C2×C107#53612yes—
108C2×C107#56311yescharacteristic
109C2×C107#56312yes—
110C2×C107#56311yescharacteristic
111#2817—1071nomaximal, Sylow
112C107#281711yescharacteristic, complemented, Sylow
113C2×C22⋊Q8—1072no—
114#400—1072no—
115#400—1072no—
116#397—1071no—
117#397—1072no—
118#397D10711yescharacteristic
119#395—1072no—
120#395—1072no—
121#381—1071no—
122C22×Q8—1072no—
123C23×C4—1071no—
124C22⋊Q8—2148no—
125C2×C4⋊C4—1072no—
126C2×C4⋊C4—1071no—
127C2×C4⋊C4—1072no—
128C2×C4⋊C4—1072no—
129C2×C4⋊C4—1072no—
130C2×C4⋊C4C2×D10711yescharacteristic
131C2×C22⋊C4—1071no—
132C2×C22⋊C4—1072no—
133C2×C22⋊C4C2×D10712yes—
134C2×C22⋊C4C2×D10711yescharacteristic
135#95—1072no—
136#95—1072no—
137#95—1072no—
138#95—1072no—
139#95—1072no—
140#95—1072no—
141#95—1071no—
142#95C2×D10712yes—
143#95C2×D10711yescharacteristic
144C24C22×D10711yescharacteristic
145C2×Q8—4282no—
146C2×Q8—4282no—
147C2×Q8—2144no—
148C22×C4—1074no—
149C22×C4—1072no—
150C22×C4—1072no—
151C22×C4—1072no—
152C22×C4—1072no—
153C22×C4—4281no—
154C22×C4—4281no—
155C22×C4C22×D10711yescharacteristic
156C22×C4C22×D10711yescharacteristic
157C22×C4C22×D10712yes—
158C22×C4C22×D10712yes—
159C4⋊C4—41no—
160C4⋊C4—4282no—
161C4⋊C4—4281no—
162C4⋊C4—2144no—
163C4⋊C4—2144no—
164C4⋊C4—2144no—
165C22⋊C4—24no—
166C22⋊C4—41no—
167C22⋊C4—4281no—
168C22⋊C4—2144no—
169C23—21no—
170C23—21no—
171C23—22no—
172C23—22no—
173C23D4×D10712yes—
174C23C23×D10711yescharacteristic
175Q8—4288no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—1074no—
187C2×C4—4284no—
188C2×C4—2142no—
189C2×C4—2142no—
190C2×C4—2142no—
191C2×C4—2142no—
192C2×C4—2142no—
193C2×C4—2142no—
194C2×C4—2142no—
195C2×C4—2142no—
196C2×C4—2142no—
197C2×C4—2142no—
198C2×C4—2142no—
199C22—22no—
200C22—24no—
201C22—41no—
202C22—42no—
203C22—41no—
204C22C2×D4×D10711yescharacteristic
205C22#1681215411yescharacteristic
206C22#1681215511yescharacteristic
207C22#1681215612yes—
208C22#1681215612yes—
209C4—41no—
210C4—41no—
211C4—42no—
212C4—42no—
213C4—4282no—
214C4—4282no—
215C4—4282no—
216C4—2144no—
217C2—42no—
218C2#2029318511yescharacteristic
219C2#2029319912yes—
220C2#2029337711yescharacteristic
221C2#2029337912yes—
222C2#2029338011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#397×C107C2
prime radicalC24×C107C23
hypercenter#397D107
hypocenterC107#2817
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #20292569C2semidirect
3. #20292568C2semidirect
6. #20292548C2semidirect
7. #20292547C2semidirect
112. C107#2817semidirect

Maximal subgroups

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

subgroupindex
C2×#168120862
#202925692
#202925682
#202925552
#202925522
#202925482
#202925472
#202924882
#397×C1072
#2817107