Finite Group Databasebeta

← #28344797

Subgroups — #28344797

Order 13696.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16812086C211yesmaximal, characteristic
2C22×C4×Dic107C211yesmaximal, characteristic
3#20292531C212yesmaximal, complemented
4#20292522C212yesmaximal, complemented
5#20292513C212yesmaximal, complemented
6#20292509C212yesmaximal, complemented
7#20292500C211yesmaximal, characteristic
8#20292488C211yesmaximal, characteristic
9#20292488C211yesmaximal, characteristic
10#20292485C211yesmaximal, characteristic
11#385×C107C211yesmaximal, characteristic
12C22×Dic214C2211yescharacteristic
13C23×Dic107C2211yescharacteristic
14C23×Dic107C2211yescharacteristic
15#16812086—24no—
16C2×C4⋊Dic107C2211yescharacteristic
17C2×Dic107⋊C4C2211yescharacteristic
18C2×Dic107⋊C4C2211yescharacteristic
19C2×Dic107⋊C4C2212yes—
20C2×Dic107⋊C4C2212yes—
21C2×C22⋊Dic107C2211yescharacteristic
22C2×C22⋊Dic107C2211yescharacteristic
23C2×C4×Dic107—24no—
24C2×C4×Dic107C2212yes—
25C2×C4×Dic107C2212yes—
26C107⋊#95C2212yes—
27C107⋊#95C2212yes—
28C107⋊#95C2212yes—
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C23×C4×C107C2211yescharacteristic
33C2×C4⋊C4×C107C2212yes—
34C2×C22⋊C4×C107C2211yescharacteristic
35C2×C22⋊C4×C107C2211yescharacteristic
36#95×C107C2212yes—
37C2×Dic214—22no—
38C2×Dic214—24no—
39C22×Dic107—22no—
40C22×Dic107—22no—
41C22×Dic107—24no—
42C22×Dic107D414yes—
43C22×Dic107Q814yescomplemented
44C22×Dic107C2311yescharacteristic
45C22×Dic107C2312yes—
46C22×Dic107C2311yescharacteristic
47C22×Dic107C2311yescharacteristic
48C22×Dic107C2312yes—
49C22×Dic107C2311yescharacteristic
50C4⋊Dic107—22no—
51Dic107⋊C4—24no—
52Dic107⋊C4—22no—
53Dic107⋊C4—22no—
54Dic107⋊C4—24no—
55C22⋊Dic107—22no—
56C22⋊Dic107—22no—
57C4×Dic107—24no—
58C4×Dic107—24no—
59C24×C107C2311yescharacteristic
60C22×C4×C107—22no—
61C22×C4×C107—24no—
62C22×C4×C107C2311yescharacteristic
63C22×C4×C107C2311yescharacteristic
64C22×C4×C107C2312yes—
65C22×C4×C107C2312yes—
66C4⋊C4×C107—24no—
67C22⋊C4×C107—22no—
68C22⋊C4×C107—22no—
69C23×C107—22no—
70C23×C107—22no—
71C23×C107C2×D412yes—
72C23×C107C2×Q812yes—
73C23×C107#4112yes—
74C23×C107C2411yescharacteristic
75C2×C4×C107—28no—
76C2×C4×C107—22no—
77C2×C4×C107—22no—
78C2×C4×C107—21no—
79C2×C4×C107—21no—
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107—22no—
84C2×C4×C107—22no—
85C2×C4×C107#4114yes—
86C2×Dic107—28no—
87C2×Dic107—22no—
88C2×Dic107—22no—
89C2×Dic107—22no—
90C2×Dic107—28no—
91C2×Dic107—21no—
92C2×Dic107—21no—
93C2×Dic107—22no—
94C2×Dic107—22no—
95C2×Dic107—22no—
96C2×Dic107—22no—
97C2×Dic107—22no—
98C2×Dic107C2×D412yes—
99C2×Dic107C2×D412yes—
100C2×Dic107C2×Q814yes—
101C2×Dic107#4114yes—
102Dic214—44no—
103C22×C107—22no—
104C22×C107—22no—
105C22×C107—24no—
106C22×C107—24no—
107C22×C107C22⋊Q814yes—
108C22×C107C22×D411yescharacteristic
109C22×C107C22×Q811yescharacteristic
110C22×C107C2×#4112yes—
111C22×C107C2×#4112yes—
112C22×C107C2×#4111yescharacteristic
113C4×C107—24no—
114C4×C107—41no—
115C4×C107—41no—
116C4×C107—42no—
117Dic107—24no—
118Dic107—22no—
119Dic107—41no—
120Dic107—41no—
121Dic107—42no—
122Dic107#12314yes—
123C2×C107—24no—
124C2×C107C2×C22⋊Q811yescharacteristic
125C2×C107C2×#12311yescharacteristic
126C2×C107#52511yescharacteristic
127C2×C107#52912yes—
128C2×C107#53312yes—
129#2855—1071nomaximal, Sylow
130C107#285511yescharacteristic, complemented, Sylow
131C2×C22⋊Q8—1071no—
132C22×C42—1071no—
133#391—1072no—
134#389—1072no—
135#387—1072no—
136#387—1072no—
137#385—1071no—
138#385D10711yescharacteristic
139#381—1071no—
140#381—1071no—
141#380—1071no—
142C22×Q8—1071no—
143C23×C4—1071no—
144C23×C4—1071no—
145C23×C4C2×D10711yescharacteristic
146C22⋊Q8—2144no—
147C2×C4⋊C4—1071no—
148C2×C4⋊C4—1071no—
149C2×C4⋊C4—1071no—
150C2×C4⋊C4—1072no—
151C2×C4⋊C4—1072no—
152C2×C4⋊C4C2×D10712yes—
153C2×C22⋊C4—1071no—
154C2×C22⋊C4—1071no—
155C2×C22⋊C4C2×D10711yescharacteristic
156C2×C22⋊C4C2×D10711yescharacteristic
157C2×C42—1072no—
158C2×C42—1072no—
159C2×C42—2144no—
160#95—1072no—
161#95—1072no—
162#95—1072no—
163#95—1072no—
164#95—1072no—
165#95—1072no—
166#95C2×D10712yes—
167C24C22×D10711yescharacteristic
168C2×Q8—2142no—
169C2×Q8—2144no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—1071no—
173C22×C4—1074no—
174C22×C4—1074no—
175C22×C4—1072no—
176C22×C4—1071no—
177C22×C4—1072no—
178C22×C4—1071no—
179C22×C4—1071no—
180C22×C4—2144no—
181C22×C4—2142no—
182C22×C4—2142no—
183C22×C4C22×D10712yes—
184C22×C4C22×D10711yescharacteristic
185C22×C4C22×D10711yescharacteristic
186C22×C4C22×D10712yes—
187C4⋊C4—24no—
188C4⋊C4—2142no—
189C4⋊C4—2142no—
190C4⋊C4—2142no—
191C4⋊C4—2144no—
192C4⋊C4—2144no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—2142no—
196C22⋊C4—2142no—
197C42—2144no—
198C42—2144no—
199C23—22no—
200C23—22no—
201C23D4×D10712yes—
202C23Q8×D10712yes—
203C23#1313381812yes—
204C23C23×D10711yescharacteristic
205Q8—4284no—
206C2×C4—28no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—1072no—
217C2×C4—1074no—
218C2×C4—1072no—
219C2×C4—1074no—
220C2×C4—2148no—
221C2×C4—2141no—
222C2×C4—2141no—
223C2×C4—2142no—
224C2×C4—2142no—
225C2×C4—2142no—
226C2×C4—2142no—
227C2×C4—2142no—
228C2×C4—2148no—
229C2×C4—2142no—
230C2×C4—2142no—
231C2×C4—2142no—
232C2×C4#1313381714yes—
233C22—22no—
234C22—22no—
235C22—24no—
236C22—24no—
237C22#1681208714yes—
238C22C2×D4×D10711yescharacteristic
239C22C2×Q8×D10711yescharacteristic
240C22#41×D10712yes—
241C22C2×#1313381712yes—
242C22C2×#1313381811yescharacteristic
243C4—24no—
244C4—41no—
245C4—41no—
246C4—42no—
247C4—1074no—
248C4—4281no—
249C4—4281no—
250C4—4282no—
251C4—2144no—
252C4—2142no—
253C2—24no—
254C2C2×#1681208711yescharacteristic
255C2#123×D10711yescharacteristic
256C2#2029313311yescharacteristic
257C2#2029315712yes—
258C2#2029317912yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#385×C107C2
prime radicalC24×C107C23
hypercenter#385D107
hypocenterC107#2855
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. #20292531C2semidirect
4. #20292522C2semidirect
5. #20292513C2semidirect
6. #20292509C2semidirect
43. C22×Dic107Q8semidirect
130. C107#2855semidirect

Maximal subgroups

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

subgroupindex
C2×#168120862
C22×C4×Dic1072
#202925312
#202925222
#202925132
#202925092
#202925002
#202924882
#202924852
#385×C1072
#2855107