Finite Group Databasebeta

← #28344482

Subgroups — #28344482

Order 13696.

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

normal subgroups only

This group has 644 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×#16812082C211yesmaximal, characteristic, complemented
2C2×#16812080C211yesmaximal, characteristic, complemented
3#20292555C212yesmaximal, complemented
4#20292554C211yesmaximal, characteristic, complemented
5#20292552C211yesmaximal, characteristic, complemented
6#20292536C212yesmaximal, complemented
7#20292535C211yesmaximal, characteristic, complemented
8#20292535C212yesmaximal, complemented
9#20292535C212yesmaximal, complemented
10#20292489C211yesmaximal, characteristic, complemented
11#392×C107C211yesmaximal, characteristic, complemented
12C22×D428C2211yescharacteristic
13C22×C22⋊D107C2211yescharacteristic
14C22×C22⋊D107C2211yescharacteristic
15C22×C22⋊D107C2212yes—
16C22×C22⋊D107C2212yes—
17C22×C4×D107C2211yescharacteristic
18#16812082—24no—
19#16812080—24no—
20C2×C4⋊Dic107C2211yescharacteristic
21C2×Dic107⋊C4C2211yescharacteristic
22C2×C22⋊Dic107C2212yes—
23C2×C22⋊Dic107C2212yes—
24C2×C22⋊Dic107C2212yes—
25C2×(C2×D107)⋊C4C2211yescharacteristic
26C2×(C2×D107)⋊C4C2212yes—
27C2×(C2×D107)⋊C4C2212yes—
28C107⋊#95C2211yescharacteristic
29C107⋊#95C2212yes—
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C22×D4×C107C2211yescharacteristic
33C22×D4×C107C2212yes—
34C2×C22⋊C4×C107C2211yescharacteristic
35C2×C22⋊C4×C107C2212yes—
36#95×C107C2211yescharacteristic
37C23×D107C2311yescharacteristic
38C23×D107C2311yescharacteristic
39C2×D428—22no—
40C2×D428—42no—
41C2×C22⋊D107—24no—
42C2×C22⋊D107—22no—
43C2×C22⋊D107—22no—
44C2×C22⋊D107—42no—
45C2×C22⋊D107—42no—
46C2×C22⋊D107—42no—
47C2×C22⋊D107—42no—
48C2×C22⋊D107—42no—
49C2×C22⋊D107—42no—
50C2×C22⋊D107—42no—
51C22×Dic107C2311yescharacteristic
52C22×Dic107C2312yes—
53C22×Dic107C2312yes—
54C22×Dic107C2311yescharacteristic
55C2×C4×D107—42no—
56C2×C4×D107D414yescomplemented
57C4⋊Dic107—22no—
58Dic107⋊C4—22no—
59C22⋊Dic107—24no—
60C22⋊Dic107—42no—
61C22⋊Dic107—42no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4—42no—
64(C2×D107)⋊C4—42no—
65C24×C107C2311yescharacteristic
66C24×C107C2312yes—
67C2×D4×C107—22no—
68C2×D4×C107—42no—
69C2×D4×C107—42no—
70C2×D4×C107—42no—
71C2×D4×C107—42no—
72C22×C4×C107C2312yes—
73C22×C4×C107C2311yescharacteristic
74C22×C4×C107C2311yescharacteristic
75C22⋊C4×C107—22no—
76C22⋊C4×C107—42no—
77C23×C107—21no—
78C23×C107—22no—
79C23×C107—22no—
80C23×C107—22no—
81C23×C107—21no—
82C23×C107—22no—
83C23×C107—22no—
84C23×C107—22no—
85C23×C107—22no—
86C23×C107—22no—
87C23×C107—21no—
88C23×C107—22no—
89C23×C107C2411yescharacteristic
90C22×D107—22no—
91C22×D107—21no—
92C22×D107—21no—
93C22×D107—22no—
94C22×D107—21no—
95C22×D107—21no—
96C22×D107—22no—
97C22×D107—21no—
98C22×D107—22no—
99C22×D107C2×D412yes—
100D4×C107—44no—
101D4×C107—48no—
102C2×C4×C107—22no—
103C2×C4×C107—21no—
104C2×C4×C107—22no—
105C2×C4×C107—22no—
106C2×C4×C107—22no—
107C2×C4×C107—22no—
108C2×C4×C107C2×D412yes—
109C22⋊D107—44no—
110C22⋊D107—44no—
111C22⋊D107—48no—
112C22⋊D107—48no—
113C2×Dic107—22no—
114C2×Dic107—21no—
115C2×Dic107—22no—
116C2×Dic107—22no—
117C2×Dic107—22no—
118C2×Dic107—22no—
119C2×Dic107—22no—
120C2×Dic107—22no—
121C2×Dic107—22no—
122C2×Dic107C2×D412yes—
123D428—44no—
124C4×D107—44no—
125C22×C107—41no—
126C22×C107—42no—
127C22×C107—41no—
128C22×C107—42no—
129C22×C107—42no—
130C22×C107—41no—
131C22×C107—42no—
132C22×C107—42no—
133C22×C107—42no—
134C22×C107—42no—
135C22×C107—42no—
136C22×C107—42no—
137C22×C107C22×D411yescharacteristic
138C22×C107H(5,𝔽2)12yes—
139C22×C107H(5,𝔽2)12yes—
140C22×C107H(5,𝔽2)11yescharacteristic
141C22×C107H(5,𝔽2)11yescharacteristic
142C2×D107—22no—
143C2×D107—22no—
144C2×D107—22no—
145C2×D107—42no—
146C2×D107—42no—
147C2×D107—41no—
148C2×D107—41no—
149C2×D107—41no—
150C2×D107—42no—
151C2×D107—42no—
152C4×C107—22no—
153C4×C107—41no—
154C4×C107—42no—
155Dic107—22no—
156Dic107—41no—
157Dic107—42no—
158Dic107—42no—
159C2×C107—81no—
160C2×C107—82no—
161C2×C107#53411yescharacteristic
162C2×C107#53511yescharacteristic
163C2×C107#53511yescharacteristic
164C2×C107#56012yes—
165C2×C107#56012yes—
166D107—42no—
167D107—81no—
168#2815—1071nomaximal, Sylow
169C107#281511yescharacteristic, complemented, Sylow
170C2×#121—1071no—
171C2×#121—1071no—
172#397—1071no—
173#397—1072no—
174#397—1071no—
175#392—1071no—
176#392—1072no—
177#392—1072no—
178#392—1072no—
179#392D10711yescharacteristic, complemented
180#381—1071no—
181C22×D4—1072no—
182C22×D4—1071no—
183C22×D4—1072no—
184C22×D4—1071no—
185C22×D4—1071no—
186C22×D4C2×D10711yescharacteristic
187C22×D4C2×D10712yes—
188C23×C4—1071no—
189#121—2144no—
190#121—2144no—
191C2×C4⋊C4—1071no—
192C2×C4⋊C4—1071no—
193C2×C22⋊C4—1072no—
194C2×C22⋊C4—1072no—
195C2×C22⋊C4—1072no—
196C2×C22⋊C4—1072no—
197C2×C22⋊C4—1072no—
198C2×C22⋊C4—1071no—
199C2×C22⋊C4C2×D10711yescharacteristic
200C2×C22⋊C4C2×D10712yes—
201#95—1072no—
202#95—1071no—
203#95—1072no—
204#95—1072no—
205#95C2×D10711yescharacteristic
206C24—1071no—
207C24—1071no—
208C24C22×D10711yescharacteristic
209C24C22×D10712yes—
210C2×D4—22no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—4282no—
216C2×D4—4282no—
217C2×D4—4282no—
218C2×D4—4282no—
219C2×D4—4282no—
220C2×D4—4282no—
221C2×D4—4282no—
222C2×D4—4282no—
223C2×D4—2142no—
224C2×D4—2142no—
225C2×D4—2142no—
226C2×D4—2144no—
227C22×C4—1071no—
228C22×C4—1074no—
229C22×C4—1072no—
230C22×C4—1071no—
231C22×C4—1072no—
232C22×C4—4282no—
233C22×C4C22×D10712yes—
234C22×C4C22×D10711yescharacteristic
235C22×C4C22×D10711yescharacteristic
236C4⋊C4—2142no—
237C4⋊C4—2142no—
238C22⋊C4—22no—
239C22⋊C4—42no—
240C22⋊C4—4282no—
241C22⋊C4—4282no—
242C22⋊C4—4282no—
243C22⋊C4—4282no—
244C22⋊C4—2144no—
245C22⋊C4—2142no—
246C23—21no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—21no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—21no—
257C23—22no—
258C23—1072no—
259C23—2141no—
260C23—2141no—
261C23—2142no—
262C23—2142no—
263C23—2141no—
264C23—2142no—
265C23—2142no—
266C23—2141no—
267C23—2141no—
268C23C23×D10711yescharacteristic
269D4—44no—
270D4—48no—
271D4—4284no—
272D4—4288no—
273D4—4284no—
274D4—4288no—
275D4—4284no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—1072no—
283C2×C4—4284no—
284C2×C4—2141no—
285C2×C4—2142no—
286C2×C4—2142no—
287C2×C4—2142no—
288C2×C4—2142no—
289C2×C4—2142no—
290C2×C4—2142no—
291C2×C4—2142no—
292C2×C4—2142no—
293C2×C4D4×D10712yes—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—4281no—
307C22—4281no—
308C22—4281no—
309C22—4282no—
310C22—4282no—
311C22—4282no—
312C22—4282no—
313C22—2142no—
314C22—2142no—
315C22—2142no—
316C22C2×D4×D10711yescharacteristic
317C22#1681215412yes—
318C22#1681215412yes—
319C22#1681215411yescharacteristic
320C22#1681215511yescharacteristic
321C4—22no—
322C4—41no—
323C4—42no—
324C4—4281no—
325C4—4282no—
326C4—4282no—
327C4—2142no—
328C2—82no—
329C2—81no—
330C2—8561no—
331C2—4282no—
332C2#2029318711yescharacteristic
333C2#2029319011yescharacteristic
334C2#2029319311yescharacteristic
335C2#2029336512yes—
336C2#2029336612yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#392×C107C2
hypercenter#392D107
hypocenterC107#2815
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#16812082C2semidirect
2. C2×#16812080C2semidirect
3. #20292555C2semidirect
4. #20292554C2semidirect
5. #20292552C2semidirect
6. #20292536C2semidirect
7. #20292535C2semidirect
8. #20292535C2semidirect
9. #20292535C2semidirect
10. #20292489C2semidirect
11. #392×C107C2semidirect
56. C2×C4×D107D4semidirect
169. C107#2815semidirect
179. #392D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120822
C2×#168120802
#202925552
#202925542
#202925522
#202925362
#202925352
#202924892
#392×C1072
#2815107