Finite Group Databasebeta

← #28339702

Subgroups — #28339702

Order 13696.

696 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 155 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×#16812071C212yesmaximal, complemented
2C2×#16812071C211yesmaximal, characteristic, complemented
3C2×#16812067C211yesmaximal, characteristic, complemented
4#20292553C211yesmaximal, characteristic, complemented
5#20292549C212yesmaximal, complemented
6#20292545C212yesmaximal, complemented
7#20292544C211yesmaximal, characteristic, complemented
8#20292537C211yesmaximal, characteristic
9#20292478C211yesmaximal, characteristic, complemented
10#20292474C212yesmaximal, complemented
11#398×C107C211yesmaximal, characteristic, complemented
12C22×D428C2211yescharacteristic
13C22×D428C2212yes—
14C22×C4×D107C2212yes—
15C22×C4×D107C2211yescharacteristic
16#16812071—28no—
17#16812071—24no—
18#16812067—24no—
19C2×C4⋊Dic107C2211yescharacteristic
20C2×Dic107⋊C4C2212yes—
21C2×(C2×D107)⋊C4C2212yes—
22C2×(C2×D107)⋊C4C2212yes—
23C2×(C2×D107)⋊C4C2211yescharacteristic
24C2×(C2×D107)⋊C4C2212yes—
25C2×(C2×D107)⋊C4C2211yescharacteristic
26C2×(C2×D107)⋊C4C2212yes—
27C2×(C2×D107)⋊C4C2212yes—
28C2×C4×Dic107C2211yescharacteristic
29C2×C4×Dic107C2212yes—
30C2×C4×Dic107C2211yescharacteristic
31C107⋊#95C2212yes—
32C107⋊#95C2211yescharacteristic
33C2×C4⋊C4×C107C2211yescharacteristic
34C2×C4⋊C4×C107C2212yes—
35C2×C4⋊C4×C107C2211yescharacteristic
36#95×C107C2212yes—
37#95×C107C2211yescharacteristic
38C23×D107C2311yescharacteristic
39C23×D107C2312yes—
40C2×D428—24no—
41C2×D428—24no—
42C2×D428—24no—
43C2×D428—22no—
44C2×D428—24no—
45C22×Dic107C2312yes—
46C22×Dic107C2311yescharacteristic
47C22×Dic107C2311yescharacteristic
48C22×Dic107C2311yescharacteristic
49C2×C4×D107—22no—
50C2×C4×D107—22no—
51C2×C4×D107—24no—
52C2×C4×D107—24no—
53C2×C4×D107—24no—
54C2×C4×D107D414yescomplemented
55C4⋊Dic107—22no—
56Dic107⋊C4—24no—
57(C2×D107)⋊C4—24no—
58(C2×D107)⋊C4—24no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4—24no—
61(C2×D107)⋊C4—24no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4—24no—
64C4×Dic107—22no—
65C4×Dic107—24no—
66C4×Dic107D414yescomplemented
67C22×C4×C107C2311yescharacteristic
68C22×C4×C107C2312yes—
69C22×C4×C107C2311yescharacteristic
70C22×C4×C107C2311yescharacteristic
71C22×C4×C107C2312yes—
72C4⋊C4×C107—22no—
73C4⋊C4×C107—24no—
74C4⋊C4×C107—22no—
75C23×C107C2411yescharacteristic
76C22×D107—21no—
77C22×D107—22no—
78C22×D107—22no—
79C22×D107—22no—
80C22×D107—21no—
81C22×D107—22no—
82C22×D107—22no—
83C22×D107—22no—
84C22×D107—22no—
85C22×D107—22no—
86C22×D107C2×D412yes—
87C22×D107#4114yes—
88C2×C4×C107—22no—
89C2×C4×C107—21no—
90C2×C4×C107—21no—
91C2×C4×C107—22no—
92C2×C4×C107—22no—
93C2×C4×C107—21no—
94C2×C4×C107—22no—
95C2×C4×C107—22no—
96C2×C4×C107—22no—
97C2×C4×C107—22no—
98C2×C4×C107C2×D412yes—
99C2×C4×C107C2×D412yes—
100C2×C4×C107#4114yes—
101C2×Dic107—22no—
102C2×Dic107—22no—
103C2×Dic107—21no—
104C2×Dic107—22no—
105C2×Dic107—22no—
106C2×Dic107—22no—
107C2×Dic107C2×D412yes—
108C2×Dic107C2×D412yes—
109C2×Dic107C2×D412yes—
110C2×Dic107#4112yes—
111D428—28no—
112D428—216no—
113C4×D107—28no—
114C4×D107—48no—
115C22×C107C22×D411yescharacteristic
116C22×C107C22×D411yescharacteristic
117C22×C107C2×#4111yescharacteristic
118C22×C107C2×#4112yes—
119C22×C107C2×#4112yes—
120C2×D107—22no—
121C2×D107—24no—
122C2×D107—24no—
123C2×D107—24no—
124C2×D107—24no—
125C2×D107—42no—
126C2×D107—41no—
127C2×D107—42no—
128C2×D107—42no—
129C2×D107—42no—
130C2×D107—41no—
131C2×D107—42no—
132C4×C107—22no—
133C4×C107—22no—
134C4×C107—24no—
135C4×C107—42no—
136C4×C107—41no—
137Dic107—22no—
138Dic107—42no—
139Dic107—41no—
140Dic107#12114yes—
141C2×C107C2×#12111yescharacteristic
142C2×C107#52511yescharacteristic
143C2×C107#52512yes—
144C2×C107#52912yes—
145C2×C107#53211yescharacteristic
146D107—42no—
147D107—44no—
148#2392—1071nomaximal, Sylow
149C107#239211yescharacteristic, complemented, Sylow
150C2×C4×D4—1071no—
151C2×C4×D4—1072no—
152C2×#117—1071no—
153#398D10711yescharacteristic, complemented
154#397—1071no—
155#396—1072no—
156#394—1071no—
157#394—1072no—
158#392—1071no—
159C4×C4⋊C4—1071no—
160C4×C22⋊C4—1072no—
161C22×D4—1071no—
162C22×D4—1072no—
163C23×C4—1071no—
164C23×C4—1072no—
165C4×D4—2144no—
166C4×D4—2148no—
167#117—2144no—
168C2×C4⋊C4—1071no—
169C2×C4⋊C4—1072no—
170C2×C4⋊C4C2×D10711yescharacteristic
171C2×C4⋊C4C2×D10712yes—
172C2×C4⋊C4C2×D10711yescharacteristic
173C2×C22⋊C4—1071no—
174C2×C22⋊C4—1071no—
175C2×C22⋊C4—1072no—
176C2×C22⋊C4—1072no—
177C2×C22⋊C4—1072no—
178C2×C22⋊C4—1072no—
179C2×C22⋊C4—1072no—
180C2×C42—1071no—
181C2×C42—1071no—
182C2×C42—1072no—
183#95—1072no—
184#95—1071no—
185#95C2×D10712yes—
186#95C2×D10711yescharacteristic
187C24—1071no—
188C24—1072no—
189C2×D4—2144no—
190C2×D4—2144no—
191C2×D4—2144no—
192C2×D4—2142no—
193C2×D4—2144no—
194C22×C4—1071no—
195C22×C4—1071no—
196C22×C4—1074no—
197C22×C4—1072no—
198C22×C4—1071no—
199C22×C4—2144no—
200C22×C4—2144no—
201C22×C4—2144no—
202C22×C4—2142no—
203C22×C4—2142no—
204C22×C4C22×D10712yes—
205C22×C4C22×D10711yescharacteristic
206C22×C4C22×D10711yescharacteristic
207C22×C4C22×D10711yescharacteristic
208C22×C4C22×D10712yes—
209C4⋊C4—22no—
210C4⋊C4—24no—
211C4⋊C4—22no—
212C4⋊C4—2142no—
213C4⋊C4—2144no—
214C22⋊C4—2142no—
215C22⋊C4—2144no—
216C22⋊C4—2144no—
217C22⋊C4—2144no—
218C22⋊C4—2142no—
219C22⋊C4—2144no—
220C22⋊C4—2144no—
221C42—1074no—
222C42—2142no—
223C42—2144no—
224C23—1072no—
225C23—1074no—
226C23—2141no—
227C23—2142no—
228C23—2142no—
229C23—2141no—
230C23—2142no—
231C23—2142no—
232C23—2142no—
233C23—2142no—
234C23—2142no—
235C23—2142no—
236C23C23×D10711yescharacteristic
237D4—2148no—
238D4—21416no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—1072no—
250C2×C4—1072no—
251C2×C4—1072no—
252C2×C4—1072no—
253C2×C4—4288no—
254C2×C4—2148no—
255C2×C4—2142no—
256C2×C4—2141no—
257C2×C4—2142no—
258C2×C4—2142no—
259C2×C4—2142no—
260C2×C4—2142no—
261C2×C4D4×D10712yes—
262C2×C4D4×D10712yes—
263C2×C4#1313381914yes—
264C22—4281no—
265C22—4282no—
266C22—4282no—
267C22—4282no—
268C22—4281no—
269C22—4282no—
270C22—4282no—
271C22—2144no—
272C22—2144no—
273C22—2144no—
274C22—2144no—
275C22—2142no—
276C22C2×D4×D10711yescharacteristic
277C22C2×D4×D10711yescharacteristic
278C22#41×D10712yes—
279C22#41×D10711yescharacteristic
280C22C2×#1313381912yes—
281C4—22no—
282C4—24no—
283C4—22no—
284C4—41no—
285C4—42no—
286C4—1074no—
287C4—4282no—
288C4—4281no—
289C4—2142no—
290C2—4282no—
291C2—4284no—
292C2#121×D10711yescharacteristic
293C2#2029313511yescharacteristic
294C2#2029314012yes—
295C2#2029316212yes—
296C2#2029317811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#398×C107C2
prime radical#20292537C2
hypercenter#398D107
hypocenterC107#2392
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#16812071C2semidirect
2. C2×#16812071C2semidirect
3. C2×#16812067C2semidirect
4. #20292553C2semidirect
5. #20292549C2semidirect
6. #20292545C2semidirect
7. #20292544C2semidirect
9. #20292478C2semidirect
10. #20292474C2semidirect
11. #398×C107C2semidirect
54. C2×C4×D107D4semidirect
66. C4×Dic107D4semidirect
149. C107#2392semidirect
153. #398D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120712
C2×#168120672
#202925532
#202925492
#202925452
#202925442
#202925372
#202924782
#202924742
#398×C1072
#2392107