Finite Group Databasebeta

← #28344466

Subgroups — #28344466

Order 13696.

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

normal subgroups only

This group has 578 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×#16812094C211yesmaximal, characteristic, complemented
2C2×#16812084C211yesmaximal, characteristic, complemented
3#20292562C211yesmaximal, characteristic, complemented
4#20292551C212yesmaximal, complemented
5#20292545C212yesmaximal, complemented
6#20292545C212yesmaximal, complemented
7#20292544C211yesmaximal, characteristic, complemented
8#20292537C211yesmaximal, characteristic, complemented
9#20292537C212yesmaximal, complemented
10#20292486C211yesmaximal, characteristic, complemented
11#400×C107C211yesmaximal, characteristic, complemented
12C22×D428C2211yescharacteristic
13C22×D428C2212yes—
14C22×D428C2211yescharacteristic
15C22×D428C2212yes—
16C22×C4×D107C2211yescharacteristic
17#16812094—24no—
18#16812084—24no—
19C2×Dic107⋊C4C2211yescharacteristic
20C2×Dic107⋊C4C2212yes—
21C2×Dic107⋊C4C2212yes—
22C2×(C2×D107)⋊C4C2212yes—
23C2×(C2×D107)⋊C4C2211yescharacteristic
24C2×(C2×D107)⋊C4C2211yescharacteristic
25C2×(C2×D107)⋊C4C2211yescharacteristic
26C2×(C2×D107)⋊C4C2212yes—
27C2×(C2×D107)⋊C4C2212yes—
28C2×(C2×D107)⋊C4C2212yes—
29C2×(C2×D107)⋊C4C2212yes—
30C2×(C2×D107)⋊C4C2212yes—
31C107⋊#95C2211yescharacteristic
32C2×C4⋊C4×C107C2211yescharacteristic
33C2×C4⋊C4×C107C2211yescharacteristic
34#95×C107C2211yescharacteristic
35#95×C107C2212yes—
36#95×C107C2212yes—
37C23×D107C2311yescharacteristic
38C23×D107C2312yes—
39C23×D107C2311yescharacteristic
40C2×D428—24no—
41C2×D428—22no—
42C2×D428—22no—
43C2×D428—42no—
44C2×D428—42no—
45C2×D428—42no—
46C2×D428—42no—
47C2×D428—42no—
48C2×D428—42no—
49C2×D428—42no—
50C22×Dic107C2311yescharacteristic
51C22×Dic107C2312yes—
52C22×Dic107C2311yescharacteristic
53C2×C4×D107—42no—
54C2×C4×D107D414yescomplemented
55Dic107⋊C4—22no—
56Dic107⋊C4—42no—
57Dic107⋊C4—42no—
58(C2×D107)⋊C4—22no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4—24no—
61(C2×D107)⋊C4—22no—
62(C2×D107)⋊C4—42no—
63(C2×D107)⋊C4—42no—
64(C2×D107)⋊C4—42no—
65(C2×D107)⋊C4—42no—
66(C2×D107)⋊C4—42no—
67C22×C4×C107C2311yescharacteristic
68C22×C4×C107C2312yes—
69C22×C4×C107C2312yes—
70C22×C4×C107C2311yescharacteristic
71C22×C4×C107C2311yescharacteristic
72C4⋊C4×C107—22no—
73C4⋊C4×C107—22no—
74C23×C107C2411yescharacteristic
75C22×D107—21no—
76C22×D107—22no—
77C22×D107—22no—
78C22×D107—21no—
79C22×D107—21no—
80C22×D107—22no—
81C22×D107—22no—
82C22×D107—22no—
83C22×D107—22no—
84C22×D107—22no—
85C22×D107—22no—
86C22×D107—22no—
87C22×D107—22no—
88C22×D107—21no—
89C22×D107—21no—
90C22×D107—22no—
91C22×D107C2×D412yes—
92C2×C4×C107—22no—
93C2×C4×C107—21no—
94C2×C4×C107—22no—
95C2×C4×C107—21no—
96C2×C4×C107—21no—
97C2×C4×C107—22no—
98C2×C4×C107—22no—
99C2×C4×C107—22no—
100C2×C4×C107—22no—
101C2×C4×C107—22no—
102C2×C4×C107—22no—
103C2×C4×C107—21no—
104C2×C4×C107C2×D412yes—
105C2×Dic107—22no—
106C2×Dic107—21no—
107C2×Dic107—22no—
108C2×Dic107—22no—
109C2×Dic107—22no—
110C2×Dic107—22no—
111C2×Dic107C2×D412yes—
112D428—44no—
113D428—44no—
114D428—48no—
115D428—48no—
116C4×D107—44no—
117C22×C107C22×D411yescharacteristic
118C22×C107H(5,𝔽2)12yes—
119C22×C107H(5,𝔽2)12yes—
120C22×C107H(5,𝔽2)11yescharacteristic
121C22×C1072−1+411yescharacteristic
122C2×D107—22no—
123C2×D107—22no—
124C2×D107—22no—
125C2×D107—42no—
126C2×D107—42no—
127C2×D107—42no—
128C2×D107—41no—
129C2×D107—42no—
130C2×D107—41no—
131C2×D107—41no—
132C2×D107—42no—
133C2×D107—42no—
134C2×D107—42no—
135C2×D107—42no—
136C2×D107—42no—
137C2×D107—42no—
138C2×D107—42no—
139C4×C107—22no—
140C4×C107—41no—
141C4×C107—42no—
142C4×C107—41no—
143C4×C107—42no—
144Dic107—22no—
145Dic107—41no—
146Dic107—42no—
147C2×C107#53411yescharacteristic
148C2×C107#53511yescharacteristic
149C2×C107#53711yescharacteristic
150C2×C107#56012yes—
151C2×C107#56212yes—
152D107—42no—
153D107—82no—
154D107—81no—
155#2813—1071nomaximal, Sylow
156C107#281311yescharacteristic, complemented, Sylow
157C2×#123—1071no—
158C2×#121—1071no—
159#400D10711yescharacteristic, complemented
160#399—1071no—
161#396—1072no—
162#394—1071no—
163#394—1072no—
164#394—1072no—
165#392—1071no—
166#392—1072no—
167#380—1071no—
168C22×D4—1072no—
169C22×D4—1072no—
170C22×D4—1071no—
171C22×D4—1071no—
172C23×C4—1071no—
173#123—2144no—
174#121—2144no—
175C2×C4⋊C4—1072no—
176C2×C4⋊C4—1072no—
177C2×C4⋊C4—1071no—
178C2×C4⋊C4C2×D10711yescharacteristic
179C2×C4⋊C4C2×D10711yescharacteristic
180C2×C22⋊C4—1072no—
181C2×C22⋊C4—1072no—
182C2×C22⋊C4—1072no—
183C2×C22⋊C4—1072no—
184C2×C22⋊C4—1072no—
185C2×C22⋊C4—1072no—
186C2×C22⋊C4—1071no—
187C2×C22⋊C4—1071no—
188C2×C22⋊C4—1071no—
189#95—1071no—
190#95C2×D10711yescharacteristic
191#95C2×D10712yes—
192#95C2×D10712yes—
193C24—1071no—
194C24—1071no—
195C24—1072no—
196C2×D4—4282no—
197C2×D4—4282no—
198C2×D4—4282no—
199C2×D4—4282no—
200C2×D4—4282no—
201C2×D4—4282no—
202C2×D4—4282no—
203C2×D4—2144no—
204C2×D4—2142no—
205C2×D4—2142no—
206C22×C4—1071no—
207C22×C4—1074no—
208C22×C4—1072no—
209C22×C4—1071no—
210C22×C4—4282no—
211C22×C4C22×D10712yes—
212C22×C4C22×D10711yescharacteristic
213C22×C4C22×D10712yes—
214C22×C4C22×D10711yescharacteristic
215C22×C4C22×D10711yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—4282no—
219C4⋊C4—4282no—
220C4⋊C4—2142no—
221C22⋊C4—4282no—
222C22⋊C4—4282no—
223C22⋊C4—4282no—
224C22⋊C4—4282no—
225C22⋊C4—4282no—
226C22⋊C4—2144no—
227C22⋊C4—2142no—
228C22⋊C4—2142no—
229C22⋊C4—2142no—
230C23—1072no—
231C23—2141no—
232C23—2142no—
233C23—2142no—
234C23—2142no—
235C23—2142no—
236C23—2142no—
237C23—2142no—
238C23—2142no—
239C23—2142no—
240C23—2141no—
241C23—2141no—
242C23—2142no—
243C23—2141no—
244C23—2142no—
245C23—2142no—
246C23—2141no—
247C23C23×D10711yescharacteristic
248D4—4284no—
249D4—4288no—
250D4—4284no—
251D4—4288no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—1072no—
265C2×C4—4284no—
266C2×C4—2141no—
267C2×C4—2142no—
268C2×C4—2142no—
269C2×C4—2142no—
270C2×C4—2142no—
271C2×C4—2142no—
272C2×C4D4×D10712yes—
273C22—4282no—
274C22—4281no—
275C22—4282no—
276C22—4281no—
277C22—4281no—
278C22—4282no—
279C22—4282no—
280C22—4282no—
281C22—4282no—
282C22—4282no—
283C22—4282no—
284C22—4282no—
285C22—4282no—
286C22—4282no—
287C22—2142no—
288C22—2142no—
289C22—2142no—
290C22C2×D4×D10711yescharacteristic
291C22#1681215512yes—
292C22#1681215512yes—
293C22#1681215511yescharacteristic
294C22#1681215711yescharacteristic
295C4—22no—
296C4—41no—
297C4—42no—
298C4—41no—
299C4—42no—
300C4—4281no—
301C4—4282no—
302C4—2142no—
303C2—8562no—
304C2—8561no—
305C2—4282no—
306C2#2029318811yescharacteristic
307C2#2029319511yescharacteristic
308C2#2029320711yescharacteristic
309C2#2029336812yes—
310C2#2029337512yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#400×C107C2
hypercenter#400D107
hypocenterC107#2813
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#16812094C2semidirect
2. C2×#16812084C2semidirect
3. #20292562C2semidirect
4. #20292551C2semidirect
5. #20292545C2semidirect
6. #20292545C2semidirect
7. #20292544C2semidirect
8. #20292537C2semidirect
9. #20292537C2semidirect
10. #20292486C2semidirect
11. #400×C107C2semidirect
54. C2×C4×D107D4semidirect
156. C107#2813semidirect
159. #400D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120942
C2×#168120842
#202925622
#202925512
#202925452
#202925442
#202925372
#202924862
#400×C1072
#2813107