Finite Group Databasebeta

← #28344503

Subgroups — #28344503

Order 13696.

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

normal subgroups only

This group has 630 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×#16812105C211yesmaximal, characteristic, complemented
2C2×#16812104C211yesmaximal, characteristic, complemented
3#20292544C211yesmaximal, characteristic, complemented
4#20292543C212yesmaximal, complemented
5#20292542C212yesmaximal, complemented
6#20292537C211yesmaximal, characteristic, complemented
7#20292536C212yesmaximal, complemented
8#20292536C212yesmaximal, complemented
9#20292535C211yesmaximal, characteristic, complemented
10#20292494C211yesmaximal, characteristic, complemented
11#394×C107C211yesmaximal, characteristic, complemented
12C22×D428C2211yescharacteristic
13C22×D428C2212yes—
14C22×C22⋊D107C2212yes—
15C22×C22⋊D107C2212yes—
16C22×C22⋊D107C2212yes—
17#16812105—24no—
18#16812104—24no—
19C2×C4⋊Dic107C2211yescharacteristic
20C2×Dic107⋊C4C2211yescharacteristic
21C2×C22⋊Dic107C2212yes—
22C2×(C2×D107)⋊C4C2211yescharacteristic
23C2×(C2×D107)⋊C4C2211yescharacteristic
24C2×(C2×D107)⋊C4C2211yescharacteristic
25C2×(C2×D107)⋊C4C2212yes—
26C2×(C2×D107)⋊C4C2212yes—
27C2×(C2×D107)⋊C4C2212yes—
28C2×C4×Dic107C2211yescharacteristic
29C107⋊#95C2211yescharacteristic
30C107⋊#95C2212yes—
31C107⋊#95C2212yes—
32C22×D4×C107C2211yescharacteristic
33C2×C4⋊C4×C107C2211yescharacteristic
34C2×C22⋊C4×C107C2212yes—
35C2×C22⋊C4×C107C2212yes—
36#95×C107C2211yescharacteristic
37C23×D107C2311yescharacteristic
38C23×D107C2312yes—
39C2×D428—22no—
40C2×D428—42no—
41C2×D428—42no—
42C2×D428—42no—
43C2×D428—42no—
44C2×C22⋊D107—24no—
45C2×C22⋊D107—24no—
46C2×C22⋊D107—42no—
47C2×C22⋊D107—42no—
48C2×C22⋊D107—42no—
49C2×C22⋊D107—42no—
50C2×C22⋊D107—42no—
51C2×C22⋊D107—42no—
52C2×C22⋊D107—42no—
53C22×Dic107C2311yescharacteristic
54C22×Dic107C2312yes—
55C22×Dic107C2311yescharacteristic
56C22×Dic107C2311yescharacteristic
57C4⋊Dic107—22no—
58Dic107⋊C4—22no—
59C22⋊Dic107—42no—
60(C2×D107)⋊C4—22no—
61(C2×D107)⋊C4—22no—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4—42no—
64(C2×D107)⋊C4—42no—
65(C2×D107)⋊C4—42no—
66C4×Dic107D414yescomplemented
67C24×C107C2312yes—
68C2×D4×C107—22no—
69C2×D4×C107—42no—
70C22×C4×C107C2311yescharacteristic
71C22×C4×C107C2311yescharacteristic
72C22×C4×C107C2312yes—
73C22×C4×C107C2311yescharacteristic
74C4⋊C4×C107—22no—
75C22⋊C4×C107—42no—
76C22⋊C4×C107—42no—
77C23×C107—22no—
78C23×C107—22no—
79C23×C107—22no—
80C23×C107—22no—
81C23×C107—22no—
82C23×C107—22no—
83C23×C107—22no—
84C23×C107C2411yescharacteristic
85C22×D107—21no—
86C22×D107—22no—
87C22×D107—22no—
88C22×D107—22no—
89C22×D107—22no—
90C22×D107—21no—
91C22×D107—22no—
92C22×D107—22no—
93C22×D107—22no—
94C22×D107—22no—
95C22×D107—22no—
96C22×D107—21no—
97D4×C107—44no—
98C2×C4×C107—22no—
99C2×C4×C107—21no—
100C2×C4×C107—21no—
101C2×C4×C107—21no—
102C2×C4×C107—22no—
103C2×C4×C107—22no—
104C2×C4×C107—22no—
105C2×C4×C107—21no—
106C2×C4×C107—22no—
107C2×C4×C107C2×D412yes—
108C22⋊D107—48no—
109C22⋊D107—48no—
110C22⋊D107—48no—
111C2×Dic107—21no—
112C2×Dic107—21no—
113C2×Dic107—22no—
114C2×Dic107—21no—
115C2×Dic107—22no—
116C2×Dic107—22no—
117C2×Dic107—22no—
118C2×Dic107—22no—
119C2×Dic107C2×D412yes—
120C2×Dic107C2×D412yes—
121D428—44no—
122D428—48no—
123C22×C107—42no—
124C22×C107—42no—
125C22×C107—42no—
126C22×C107—42no—
127C22×C107—42no—
128C22×C107—42no—
129C22×C107—42no—
130C22×C107C22×D411yescharacteristic
131C22×C107H(5,𝔽2)12yes—
132C22×C107H(5,𝔽2)11yescharacteristic
133C22×C107H(5,𝔽2)11yescharacteristic
134C22×C107H(5,𝔽2)12yes—
135C2×D107—42no—
136C2×D107—42no—
137C2×D107—42no—
138C2×D107—41no—
139C2×D107—41no—
140C2×D107—41no—
141C2×D107—42no—
142C2×D107—42no—
143C2×D107—42no—
144C2×D107—42no—
145C2×D107—42no—
146C2×D107—42no—
147C4×C107—22no—
148C4×C107—41no—
149C4×C107—41no—
150C4×C107—42no—
151Dic107—22no—
152Dic107—22no—
153Dic107—41no—
154Dic107—42no—
155C2×C107—82no—
156C2×C107#53511yescharacteristic
157C2×C107#53511yescharacteristic
158C2×C107#53511yescharacteristic
159C2×C107#56012yes—
160C2×C107#56012yes—
161D107—82no—
162D107—81no—
163#2818—1071nomaximal, Sylow
164C107#281811yescharacteristic, complemented, Sylow
165C2×#127—1071no—
166C2×#126—1071no—
167#394—1071no—
168#394—1072no—
169#394—1072no—
170#394D10711yescharacteristic, complemented
171#392—1071no—
172#392—1072no—
173#392—1072no—
174#392—1071no—
175#383—1071no—
176C22×D4—1072no—
177C22×D4—1072no—
178C22×D4—1072no—
179C22×D4—1072no—
180C22×D4—1071no—
181C22×D4C2×D10711yescharacteristic
182#127—2144no—
183#126—2144no—
184C2×C4⋊C4—1071no—
185C2×C4⋊C4—1071no—
186C2×C4⋊C4C2×D10711yescharacteristic
187C2×C22⋊C4—1072no—
188C2×C22⋊C4—1072no—
189C2×C22⋊C4—1072no—
190C2×C22⋊C4—1072no—
191C2×C22⋊C4—1071no—
192C2×C22⋊C4—1071no—
193C2×C22⋊C4—1071no—
194C2×C22⋊C4C2×D10712yes—
195C2×C22⋊C4C2×D10712yes—
196C2×C42—1071no—
197#95—1072no—
198#95—1072no—
199#95—1071no—
200#95C2×D10711yescharacteristic
201C24—1071no—
202C24—1072no—
203C24C22×D10712yes—
204C2×D4—22no—
205C2×D4—42no—
206C2×D4—4282no—
207C2×D4—4282no—
208C2×D4—4282no—
209C2×D4—4282no—
210C2×D4—4282no—
211C2×D4—4282no—
212C2×D4—4282no—
213C2×D4—4282no—
214C2×D4—4282no—
215C2×D4—4282no—
216C2×D4—4282no—
217C2×D4—2144no—
218C2×D4—2144no—
219C2×D4—2142no—
220C22×C4—1071no—
221C22×C4—1071no—
222C22×C4—1072no—
223C22×C4—1071no—
224C22×C4C22×D10712yes—
225C22×C4C22×D10711yescharacteristic
226C22×C4C22×D10711yescharacteristic
227C22×C4C22×D10711yescharacteristic
228C4⋊C4—22no—
229C4⋊C4—2142no—
230C4⋊C4—2142no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—4282no—
234C22⋊C4—4282no—
235C22⋊C4—4282no—
236C22⋊C4—4282no—
237C22⋊C4—2142no—
238C22⋊C4—2142no—
239C22⋊C4—2142no—
240C42—1074no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—2141no—
249C23—2142no—
250C23—2142no—
251C23—2142no—
252C23—2142no—
253C23—2142no—
254C23—2141no—
255C23—2142no—
256C23—2142no—
257C23—2141no—
258C23—2142no—
259C23—2142no—
260C23C23×D10711yescharacteristic
261D4—44no—
262D4—4284no—
263D4—4288no—
264D4—4288no—
265D4—4288no—
266D4—4288no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—1072no—
277C2×C4—1072no—
278C2×C4—2141no—
279C2×C4—2142no—
280C2×C4—2142no—
281C2×C4—2142no—
282C2×C4—2142no—
283C2×C4—2141no—
284C2×C4—2141no—
285C2×C4—2142no—
286C2×C4D4×D10712yes—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—4282no—
295C22—4281no—
296C22—4282no—
297C22—4281no—
298C22—4282no—
299C22—4281no—
300C22—4282no—
301C22—4282no—
302C22—4282no—
303C22—4282no—
304C22—4282no—
305C22—4282no—
306C22C2×D4×D10711yescharacteristic
307C22#1681215412yes—
308C22#1681215411yescharacteristic
309C22#1681215511yescharacteristic
310C22#1681215512yes—
311C4—22no—
312C4—41no—
313C4—41no—
314C4—42no—
315C4—4281no—
316C4—4282no—
317C4—2142no—
318C4—2142no—
319C2—82no—
320C2—8562no—
321C2—8561no—
322C2#2029319211yescharacteristic
323C2#2029319311yescharacteristic
324C2#2029319511yescharacteristic
325C2#2029336512yes—
326C2#2029336712yes—
subgroupG/H
center, FrattiniC23C23×D107
commutator, socle, cosocleC23×C107C24
Fitting#394×C107C2
hypercenter#394D107
hypocenterC107#2818
LCSG ▷ C23×C107 ▷ C107
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#16812105C2semidirect
2. C2×#16812104C2semidirect
3. #20292544C2semidirect
4. #20292543C2semidirect
5. #20292542C2semidirect
6. #20292537C2semidirect
7. #20292536C2semidirect
8. #20292536C2semidirect
9. #20292535C2semidirect
10. #20292494C2semidirect
11. #394×C107C2semidirect
66. C4×Dic107D4semidirect
164. C107#2818semidirect
170. #394D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121052
C2×#168121042
#202925442
#202925432
#202925422
#202925372
#202925362
#202925352
#202924942
#394×C1072
#2818107