Finite Group Databasebeta

← #31784077

Subgroups — #31784077

Order 17536.

568 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 147 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×#17984053C211yesmaximal, characteristic, complemented
2C2×D4×Dic137C211yesmaximal, characteristic
3#22577343C211yesmaximal, characteristic, complemented
4#22577342C211yesmaximal, characteristic, complemented
5#22577331C212yesmaximal, complemented
6#22577317C211yesmaximal, characteristic
7#22577315C212yesmaximal, complemented
8C22⋊C4×Dic137C212yesmaximal, complemented
9C22⋊C4×Dic137C212yesmaximal, complemented
10#22577248C211yesmaximal, characteristic, complemented
11#394×C137C211yesmaximal, characteristic
12C22×Dic274C2211yescharacteristic
13C23×Dic137C2212yes—
14#17984053—24no—
15D4×Dic137—24no—
16C2×C4⋊Dic137C2211yescharacteristic
17C2×#14555489C2211yescharacteristic
18C2×#14555489C2211yescharacteristic
19C2×C22⋊Dic137C2212yes—
20C2×C22⋊Dic137C2212yes—
21C2×C22⋊Dic137C2212yes—
22C2×C4×Dic137C2212yes—
23C2×C4×Dic137C2211yescharacteristic
24C2×C4×Dic137C2211yescharacteristic
25C2×C4×Dic137C2211yescharacteristic
26#17983995C2212yes—
27#17983995C2211yescharacteristic
28#17983995C2212yes—
29#17983995C2212yes—
30#17983995C2212yes—
31#17983995C2212yes—
32C22×D4×C137C2211yescharacteristic
33C2×C4⋊C4×C137C2211yescharacteristic
34C2×C22⋊C4×C137C2212yes—
35C2×C22⋊C4×C137C2212yes—
36#95×C137C2211yescharacteristic
37C2×Dic274—22no—
38C2×Dic274—22no—
39C2×Dic274—22no—
40C22×Dic137—24no—
41C22×Dic137—24no—
42C22×Dic137—24no—
43C22×Dic137C2311yescharacteristic
44C22×Dic137C2312yes—
45C22×Dic137C2312yes—
46C22×Dic137C2311yescharacteristic
47C22×Dic137C2311yescharacteristic
48C22×Dic137C2311yescharacteristic
49C4⋊Dic137—22no—
50#14555489—22no—
51#14555489—22no—
52C22⋊Dic137—24no—
53C22⋊Dic137—24no—
54C22⋊Dic137—24no—
55C4×Dic137—22no—
56C4×Dic137—22no—
57C4×Dic137—24no—
58C4×Dic137D414yes—
59C24×C137C2312yes—
60C2×D4×C137—24no—
61C2×D4×C137—22no—
62C22×C4×C137C2311yescharacteristic
63C22×C4×C137C2312yes—
64C22×C4×C137C2311yescharacteristic
65C22×C4×C137C2311yescharacteristic
66C4⋊C4×C137—22no—
67C22⋊C4×C137—24no—
68C22⋊C4×C137—24no—
69Dic274—28no—
70C2×Dic137—21no—
71C2×Dic137—21no—
72C2×Dic137—21no—
73C2×Dic137—21no—
74C2×Dic137—21no—
75C2×Dic137—22no—
76C2×Dic137—22no—
77C2×Dic137—22no—
78C2×Dic137—22no—
79C2×Dic137—22no—
80C2×Dic137—22no—
81C2×Dic137—22no—
82C2×Dic137—48no—
83C2×Dic137C2×D412yes—
84C2×Dic137C2×D412yes—
85C2×Dic137#4112yes—
86C2×Dic137#4114yes—
87C23×C137—22no—
88C23×C137—22no—
89C23×C137—22no—
90C23×C137—22no—
91C23×C137—22no—
92C23×C137—22no—
93C23×C137#4114yes—
94C23×C137C2411yescharacteristic
95D4×C137—28no—
96C2×C4×C137—21no—
97C2×C4×C137—22no—
98C2×C4×C137—21no—
99C2×C4×C137—21no—
100C2×C4×C137—22no—
101C2×C4×C137—22no—
102C2×C4×C137—22no—
103C2×C4×C137—22no—
104C2×C4×C137C2×D412yes—
105C2×C4×C137#4112yes—
106C22×C137—24no—
107C22×C137—24no—
108C22×C137—24no—
109C22×C137—42no—
110C22×C137—42no—
111C22×C137—42no—
112C22×C137—42no—
113C22×C137C22×D411yescharacteristic
114C22×C137C2×#4111yescharacteristic
115C22×C137C2×#4112yes—
116C22×C137C2×#4112yes—
117C22×C137C2×#4111yescharacteristic
118C4×C137—22no—
119C4×C137—22no—
120C4×C137—41no—
121C4×C137—42no—
122Dic137—22no—
123Dic137—22no—
124Dic137—41no—
125Dic137—42no—
126Dic137—42no—
127Dic137#12414yes—
128C2×C137—44no—
129C2×C137#52511yescharacteristic
130C2×C137#52511yescharacteristic
131C2×C137C2×#12411yescharacteristic
132C2×C137#52912yes—
133C2×C137#52912yes—
134C137#239811yescharacteristic, complemented, Sylow
135#2398—1371nomaximal, Sylow
136C2×C4×Q8—1371no—
137C2×C4×D4—1371no—
138#400—1371no—
139#400—1371no—
140#397—1372no—
141#394—1371no—
142#394D13711yescharacteristic
143#393—1372no—
144C4×C22⋊C4—1372no—
145C4×C22⋊C4—1372no—
146#376—1371no—
147C22×Q8—1371no—
148C22×D4C2×D13711yescharacteristic
149C23×C4—1372no—
150C4×Q8—2744no—
151C4×D4—2744no—
152C2×C4⋊C4—1371no—
153C2×C4⋊C4—1371no—
154C2×C4⋊C4—1371no—
155C2×C4⋊C4C2×D13711yescharacteristic
156C2×C22⋊C4—1372no—
157C2×C22⋊C4—1372no—
158C2×C22⋊C4—1372no—
159C2×C22⋊C4C2×D13712yes—
160C2×C22⋊C4C2×D13712yes—
161C2×C42—1371no—
162C2×C42—1371no—
163C2×C42—1372no—
164C2×C42—1371no—
165#95—1372no—
166#95—1371no—
167#95—1372no—
168#95—1372no—
169#95—1372no—
170#95—1372no—
171#95C2×D13711yescharacteristic
172C24C22×D13712yes—
173C2×Q8—2742no—
174C2×Q8—2742no—
175C2×Q8—2742no—
176C2×D4—24no—
177C2×D4—22no—
178C22×C4—2744no—
179C22×C4—2744no—
180C22×C4—2744no—
181C22×C4—1372no—
182C22×C4—1372no—
183C22×C4—1371no—
184C22×C4—1371no—
185C22×C4—1371no—
186C22×C4—1371no—
187C22×C4C22×D13712yes—
188C22×C4C22×D13711yescharacteristic
189C22×C4C22×D13711yescharacteristic
190C22×C4C22×D13711yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—2742no—
193C4⋊C4—2742no—
194C4⋊C4—2742no—
195C22⋊C4—24no—
196C22⋊C4—24no—
197C22⋊C4—2744no—
198C22⋊C4—2744no—
199C22⋊C4—2744no—
200C42—2744no—
201C42—2742no—
202C42—2742no—
203C42—1374no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23#1455550914yes—
211C23C23×D13711yescharacteristic
212Q8—2748no—
213D4—28no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—2741no—
223C2×C4—2741no—
224C2×C4—2741no—
225C2×C4—2741no—
226C2×C4—2741no—
227C2×C4—2742no—
228C2×C4—2742no—
229C2×C4—2742no—
230C2×C4—2742no—
231C2×C4—2742no—
232C2×C4—2742no—
233C2×C4—2742no—
234C2×C4—5488no—
235C2×C4—1372no—
236C2×C4—1372no—
237C2×C4—1372no—
238C2×C4—1374no—
239C2×C4D4×D13712yes—
240C2×C4#1455550912yes—
241C22—24no—
242C22—24no—
243C22—24no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22C2×D4×D13711yescharacteristic
249C22#41×D13711yescharacteristic
250C22#41×D13712yes—
251C22C2×#1455550912yes—
252C22C2×#1455550911yescharacteristic
253C4—22no—
254C4—22no—
255C4—42no—
256C4—41no—
257C4—2742no—
258C4—2742no—
259C4—5482no—
260C4—5481no—
261C4—5482no—
262C4—1374no—
263C2—44no—
264C2#2257790911yescharacteristic
265C2#2257791311yescharacteristic
266C2#124×D13711yescharacteristic
267C2#2257793012yes—
268C2#2257793412yes—
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
Fitting#394×C137C2
prime radicalC22×D4×C137C22
hypercenter#394D137
hypocenterC137#2398
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#17984053C2semidirect
3. #22577343C2semidirect
4. #22577342C2semidirect
5. #22577331C2semidirect
7. #22577315C2semidirect
8. C22⋊C4×Dic137C2semidirect
9. C22⋊C4×Dic137C2semidirect
10. #22577248C2semidirect
134. C137#2398semidirect

Maximal subgroups

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

subgroupindex
C2×#179840532
C2×D4×Dic1372
#225773432
#225773422
#225773312
#225773172
#225773152
C22⋊C4×Dic1372
#225772482
#394×C1372
#2398137