Finite Group Databasebeta

← #31784058

Subgroups — #31784058

Order 17536.

528 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 155 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×#17984044C211yesmaximal, characteristic
3#22577341C211yesmaximal, characteristic, complemented
4#22577334C212yesmaximal, complemented
5#22577326C212yesmaximal
6#22577324C211yesmaximal, characteristic, complemented
7#22577315C211yesmaximal, characteristic
8C4⋊C4×Dic137C212yesmaximal, complemented
9#22577254C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic137C212yesmaximal
11#399×C137C211yesmaximal, characteristic
12C22×Dic274C2211yescharacteristic
13C23×Dic137C2211yescharacteristic
14#17984053—24no—
15#17984044—24no—
16C2×C4⋊Dic137C2212yes—
17C2×#14555489C2212yes—
18C2×#14555489C2211yescharacteristic
19C2×#14555489C2212yes—
20C2×#14555489C2211yescharacteristic
21C2×C22⋊Dic137C2212yes—
22C2×C22⋊Dic137C2211yescharacteristic
23C2×C4×Dic137C2212yes—
24C2×C4×Dic137C2212yes—
25C2×C4×Dic137C2211yescharacteristic
26C2×C4×Dic137C2211yescharacteristic
27#17983995C2212yes—
28#17983995C2212yes—
29#17983995C2211yescharacteristic
30#17983995C2212yes—
31#17983995C2212yes—
32C2×C4⋊C4×C137C2212yes—
33C2×C4⋊C4×C137C2211yescharacteristic
34C2×C22⋊C4×C137C2212yes—
35C2×C22⋊C4×C137C2211yescharacteristic
36#95×C137C2211yescharacteristic
37C2×Dic274—24no—
38C2×Dic274—22no—
39C22×Dic137—22no—
40C22×Dic137—22no—
41C22×Dic137Q814yescomplemented
42C22×Dic137C2311yescharacteristic
43C22×Dic137C2312yes—
44C22×Dic137C2311yescharacteristic
45C22×Dic137C2311yescharacteristic
46C22×Dic137C2312yes—
47C22×Dic137C2311yescharacteristic
48C4⋊Dic137—24no—
49#14555489—22no—
50#14555489—24no—
51#14555489—22no—
52#14555489—24no—
53C22⋊Dic137—22no—
54C22⋊Dic137—24no—
55C4×Dic137—22no—
56C4×Dic137—24no—
57C4×Dic137—24no—
58C4×Dic137D414yes—
59C24×C137C2311yescharacteristic
60C22×C4×C137C2311yescharacteristic
61C22×C4×C137C2312yes—
62C22×C4×C137C2311yescharacteristic
63C22×C4×C137C2312yes—
64C4⋊C4×C137—22no—
65C4⋊C4×C137—24no—
66C22⋊C4×C137—22no—
67C22⋊C4×C137—24no—
68Dic274—28no—
69C2×Dic137—22no—
70C2×Dic137—21no—
71C2×Dic137—22no—
72C2×Dic137—21no—
73C2×Dic137—22no—
74C2×Dic137—28no—
75C2×Dic137—21no—
76C2×Dic137—22no—
77C2×Dic137—22no—
78C2×Dic137—22no—
79C2×Dic137—22no—
80C2×Dic137C2×D412yes—
81C2×Dic137C2×D412yes—
82C2×Dic137C2×Q812yes—
83C2×Dic137C2×Q812yes—
84C2×Dic137#4114yes—
85C2×Dic137#4112yes—
86C23×C137—21no—
87C23×C137—22no—
88C23×C137—22no—
89C23×C137—21no—
90C23×C137C2×Q812yes—
91C23×C137C2411yescharacteristic
92C2×C4×C137—22no—
93C2×C4×C137—22no—
94C2×C4×C137—22no—
95C2×C4×C137—21no—
96C2×C4×C137—22no—
97C2×C4×C137—22no—
98C2×C4×C137—22no—
99C2×C4×C137—22no—
100C2×C4×C137C2×D412yes—
101C2×C4×C137#4114yes—
102C22×C137—22no—
103C22×C137—24no—
104C22×C137—41no—
105C22×C137—42no—
106C22×C137—41no—
107C22×C137C22×D411yescharacteristic
108C22×C137C22×Q811yescharacteristic
109C22×C137C2×#4111yescharacteristic
110C22×C137C2×#4112yes—
111C22×C137C2×#4112yes—
112C4×C137—22no—
113C4×C137—24no—
114C4×C137—41no—
115C4×C137—42no—
116Dic137—22no—
117Dic137—24no—
118Dic137—22no—
119Dic137—41no—
120Dic137—42no—
121Dic137C22⋊Q814yes—
122C2×C137—42no—
123C2×C137C2×C22⋊Q811yescharacteristic
124C2×C137#52511yescharacteristic
125C2×C137#52912yes—
126C2×C137#53211yescharacteristic
127C2×C137#53312yes—
128C137#239611yescharacteristic, complemented, Sylow
129#2396—1371nomaximal, Sylow
130C2×C4×Q8—1371no—
131C2×#117—1371no—
132#400—1371no—
133#399D13711yescharacteristic
134#398—1372no—
135#396—1372no—
136#395—1371no—
137#393—1371no—
138C4×C4⋊C4—1372no—
139C4×C4⋊C4—1371no—
140C4×C22⋊C4—1372no—
141C22×Q8—1371no—
142C23×C4—1371no—
143C4×Q8—2744no—
144#117—2744no—
145C2×C4⋊C4—1371no—
146C2×C4⋊C4—1371no—
147C2×C4⋊C4—1372no—
148C2×C4⋊C4—1372no—
149C2×C4⋊C4—1372no—
150C2×C4⋊C4C2×D13711yescharacteristic
151C2×C4⋊C4C2×D13712yes—
152C2×C22⋊C4—1371no—
153C2×C22⋊C4—1372no—
154C2×C22⋊C4C2×D13711yescharacteristic
155C2×C22⋊C4C2×D13712yes—
156C2×C42—1371no—
157C2×C42—1372no—
158C2×C42—1372no—
159C2×C42—1371no—
160#95—1372no—
161#95—1372no—
162#95—1372no—
163#95—1371no—
164#95—1372no—
165#95C2×D13711yescharacteristic
166C24C22×D13711yescharacteristic
167C2×Q8—2744no—
168C2×Q8—2742no—
169C22×C4—2742no—
170C22×C4—2742no—
171C22×C4—1372no—
172C22×C4—1371no—
173C22×C4—1374no—
174C22×C4—1371no—
175C22×C4—1372no—
176C22×C4—1371no—
177C22×C4—1371no—
178C22×C4C22×D13711yescharacteristic
179C22×C4C22×D13712yes—
180C22×C4C22×D13711yescharacteristic
181C22×C4C22×D13712yes—
182C4⋊C4—22no—
183C4⋊C4—24no—
184C4⋊C4—2742no—
185C4⋊C4—2744no—
186C4⋊C4—2744no—
187C4⋊C4—2742no—
188C4⋊C4—2744no—
189C22⋊C4—22no—
190C22⋊C4—24no—
191C22⋊C4—2744no—
192C22⋊C4—2742no—
193C42—2744no—
194C42—2742no—
195C42—2744no—
196C42—1374no—
197C23—21no—
198C23—22no—
199C23—21no—
200C23—22no—
201C23Q8×D13712yes—
202C23C23×D13711yescharacteristic
203Q8—2748no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—2742no—
213C2×C4—2741no—
214C2×C4—2741no—
215C2×C4—2742no—
216C2×C4—2742no—
217C2×C4—2748no—
218C2×C4—2741no—
219C2×C4—2742no—
220C2×C4—2742no—
221C2×C4—2742no—
222C2×C4—2742no—
223C2×C4—1372no—
224C2×C4—1372no—
225C2×C4—1374no—
226C2×C4—1372no—
227C2×C4—1372no—
228C2×C4—1372no—
229C2×C4D4×D13712yes—
230C2×C4#1455550914yes—
231C22—24no—
232C22—22no—
233C22—42no—
234C22—41no—
235C22—41no—
236C22C2×D4×D13711yescharacteristic
237C22C2×Q8×D13711yescharacteristic
238C22#41×D13711yescharacteristic
239C22#41×D13712yes—
240C22C2×#1455550912yes—
241C4—24no—
242C4—22no—
243C4—41no—
244C4—42no—
245C4—2742no—
246C4—2742no—
247C4—2744no—
248C4—5482no—
249C4—5481no—
250C4—1374no—
251C2—42no—
252C2C22⋊Q8×D13711yescharacteristic
253C2#2257791111yescharacteristic
254C2#2257793012yes—
255C2#2257795111yescharacteristic
256C2#2257795512yes—
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, cosocleC23×C137C24
Fitting#399×C137C2
prime radicalC24×C137C23
hypercenter#399D137
hypocenterC137#2396
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#17984053C2semidirect
3. #22577341C2semidirect
4. #22577334C2semidirect
6. #22577324C2semidirect
8. C4⋊C4×Dic137C2semidirect
9. #22577254C2semidirect
41. C22×Dic137Q8semidirect
128. C137#2396semidirect

Maximal subgroups

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

subgroupindex
C2×#179840532
C2×#179840442
#225773412
#225773342
#225773262
#225773242
#225773152
C4⋊C4×Dic1372
#225772542
C22⋊C4×Dic1372
#399×C1372
#2396137