Finite Group Databasebeta

← #31784060

Subgroups — #31784060

Order 17536.

528 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 159 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×Q8×Dic137C211yesmaximal, characteristic, complemented
2C2×#17984046C211yesmaximal, characteristic
3#22577342C211yesmaximal, characteristic, complemented
4#22577340C211yesmaximal, characteristic
5#22577335C212yesmaximal, complemented
6#22577325C212yesmaximal
7#22577314C211yesmaximal, characteristic
8C4⋊C4×Dic137C212yesmaximal, complemented
9#22577254C211yesmaximal, characteristic, complemented
10#22577250C212yesmaximal
11#395×C137C211yesmaximal, characteristic, complemented
12C22×C4×D137C2211yescharacteristic
13Q8×Dic137—24no—
14#17984046—24no—
15C2×C4⋊Dic137C2212yes—
16C2×C4⋊Dic137C2212yes—
17C2×C4⋊Dic137C2211yescharacteristic
18C2×C4⋊Dic137C2211yescharacteristic
19C2×#14555489C2212yes—
20C2×#14555487C2212yes—
21C2×#14555487C2212yes—
22C2×#14555487C2211yescharacteristic
23C2×#14555487C2211yescharacteristic
24C2×C4×Dic137C2212yes—
25C2×C4×Dic137C2212yes—
26C2×C4×Dic137C2211yescharacteristic
27C2×C4×Dic137C2211yescharacteristic
28#17983995C2212yes—
29#17983995C2211yescharacteristic
30#17983995C2212yes—
31#17983995C2212yes—
32C22×Q8×C137C2211yescharacteristic
33C2×C4⋊C4×C137C2212yes—
34C2×C4⋊C4×C137C2211yescharacteristic
35#95×C137C2212yes—
36#95×C137C2211yescharacteristic
37C23×D137C2311yescharacteristic
38C22×Dic137C2312yes—
39C22×Dic137C2311yescharacteristic
40C22×Dic137C2312yes—
41C22×Dic137C2311yescharacteristic
42C22×Dic137C2311yescharacteristic
43C2×C4×D137—22no—
44C2×C4×D137—22no—
45C2×C4×D137Q814yescomplemented
46C4⋊Dic137—22no—
47C4⋊Dic137—24no—
48C4⋊Dic137—22no—
49C4⋊Dic137—24no—
50#14555489—24no—
51#14555487—22no—
52#14555487—22no—
53#14555487—24no—
54#14555487—24no—
55C4×Dic137—22no—
56C4×Dic137—24no—
57C4×Dic137—24no—
58C4×Dic137D414yes—
59C2×Q8×C137—22no—
60C2×Q8×C137—24no—
61C22×C4×C137C2311yescharacteristic
62C22×C4×C137C2311yescharacteristic
63C22×C4×C137C2311yescharacteristic
64C22×C4×C137C2312yes—
65C22×C4×C137C2312yes—
66C4⋊C4×C137—22no—
67C4⋊C4×C137—24no—
68C22×D137—21no—
69C22×D137—22no—
70C22×D137—22no—
71C22×D137—21no—
72C22×D137C2×Q812yes—
73C2×Dic137—22no—
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×Dic137C2×D412yes—
83C2×Dic137C2×D412yes—
84C2×Dic137C2×Q812yes—
85C2×Dic137#4112yes—
86C4×D137—28no—
87C23×C137C2411yescharacteristic
88Q8×C137—28no—
89C2×C4×C137—21no—
90C2×C4×C137—22no—
91C2×C4×C137—22no—
92C2×C4×C137—22no—
93C2×C4×C137—22no—
94C2×C4×C137—22no—
95C2×C4×C137—21no—
96C2×C4×C137—21no—
97C2×C4×C137—22no—
98C2×C4×C137C2×D412yes—
99C2×C4×C137C2×Q812yes—
100C2×C4×C137#4114yes—
101C2×C4×C137#4114yes—
102C22×C137C22×D411yescharacteristic
103C22×C137C22×Q811yescharacteristic
104C22×C137C2×#4111yescharacteristic
105C22×C137C2×#4112yes—
106C22×C137C2×#4112yes—
107C2×D137—22no—
108C2×D137—24no—
109C2×D137—41no—
110C2×D137—42no—
111C2×D137—41no—
112C4×C137—22no—
113C4×C137—22no—
114C4×C137—24no—
115C4×C137—24no—
116C4×C137—41no—
117Dic137—22no—
118Dic137—42no—
119Dic137—41no—
120Dic137—42no—
121Dic137C22⋊Q814yes—
122C2×C137C2×C22⋊Q811yescharacteristic
123C2×C137#52511yescharacteristic
124C2×C137#52912yes—
125C2×C137#53211yescharacteristic
126C2×C137#53312yes—
127D137—42no—
128C137#239611yescharacteristic, complemented, Sylow
129#2396—1371nomaximal, Sylow
130C2×C4×Q8—1371no—
131C2×#117—1371no—
132#400—1371no—
133#399—1371no—
134#398—1372no—
135#396—1372no—
136#395D13711yescharacteristic, complemented
137#393—1371no—
138C4×C4⋊C4—1372no—
139C4×C4⋊C4—1371no—
140C4×C22⋊C4—1372no—
141C22×Q8C2×D13711yescharacteristic
142C23×C4—1371no—
143C4×Q8—2744no—
144#117—2744no—
145C2×C4⋊C4—1371no—
146C2×C4⋊C4—1372no—
147C2×C4⋊C4—1371no—
148C2×C4⋊C4—1372no—
149C2×C4⋊C4—1372no—
150C2×C4⋊C4C2×D13711yescharacteristic
151C2×C4⋊C4C2×D13712yes—
152C2×C22⋊C4—1371no—
153C2×C22⋊C4—1371no—
154C2×C22⋊C4—1372no—
155C2×C22⋊C4—1372no—
156C2×C42—1371no—
157C2×C42—1372no—
158C2×C42—1372no—
159C2×C42—1371no—
160#95—1372no—
161#95—1372no—
162#95—1371no—
163#95—1372no—
164#95C2×D13712yes—
165#95C2×D13711yescharacteristic
166C24—1371no—
167C2×Q8—22no—
168C2×Q8—24no—
169C22×C4—2742no—
170C22×C4—2742no—
171C22×C4—1372no—
172C22×C4—1371no—
173C22×C4—1372no—
174C22×C4—1371no—
175C22×C4—1374no—
176C22×C4—1371no—
177C22×C4C22×D13711yescharacteristic
178C22×C4C22×D13711yescharacteristic
179C22×C4C22×D13711yescharacteristic
180C22×C4C22×D13712yes—
181C22×C4C22×D13712yes—
182C4⋊C4—22no—
183C4⋊C4—24no—
184C4⋊C4—2742no—
185C4⋊C4—2744no—
186C4⋊C4—2742no—
187C4⋊C4—2744no—
188C4⋊C4—2744no—
189C22⋊C4—2742no—
190C22⋊C4—2742no—
191C22⋊C4—2744no—
192C22⋊C4—2744no—
193C42—2744no—
194C42—2742no—
195C42—2744no—
196C42—1374no—
197C23—2741no—
198C23—2741no—
199C23—2742no—
200C23—2742no—
201C23—1372no—
202C23C23×D13711yescharacteristic
203Q8—28no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—2748no—
214C2×C4—2742no—
215C2×C4—2741no—
216C2×C4—2742no—
217C2×C4—2742no—
218C2×C4—2742no—
219C2×C4—2742no—
220C2×C4—2742no—
221C2×C4—2742no—
222C2×C4—2742no—
223C2×C4—1372no—
224C2×C4—1372no—
225C2×C4—1372no—
226C2×C4—1372no—
227C2×C4D4×D13712yes—
228C2×C4Q8×D13712yes—
229C2×C4#1455550914yes—
230C2×C4#1455551114yes—
231C22—2744no—
232C22—2742no—
233C22—5482no—
234C22—5481no—
235C22—5481no—
236C22C2×D4×D13711yescharacteristic
237C22C2×Q8×D13711yescharacteristic
238C22#41×D13711yescharacteristic
239C22C2×#1455550912yes—
240C22C2×#1455551112yes—
241C4—24no—
242C4—22no—
243C4—22no—
244C4—24no—
245C4—41no—
246C4—2742no—
247C4—5482no—
248C4—5481no—
249C4—5482no—
250C4—1374no—
251C2—5482no—
252C2C22⋊Q8×D13711yescharacteristic
253C2#2257790911yescharacteristic
254C2#2257793912yes—
255C2#2257795411yescharacteristic
256C2#2257795812yes—
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
Fitting#395×C137C2
prime radicalC23×D137C23
hypercenter#395D137
hypocenterC137#2396
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic137C2semidirect
3. #22577342C2semidirect
5. #22577335C2semidirect
8. C4⋊C4×Dic137C2semidirect
9. #22577254C2semidirect
11. #395×C137C2semidirect
45. C2×C4×D137Q8semidirect
128. C137#2396semidirect
136. #395D137semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1372
C2×#179840462
#225773422
#225773402
#225773352
#225773252
#225773142
C4⋊C4×Dic1372
#225772542
#225772502
#395×C1372
#2396137