Finite Group Databasebeta

← #31784079

Subgroups — #31784079

Order 17536.

504 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 167 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
2C2×#17984053C211yesmaximal, characteristic
3#22577336C212yesmaximal
4#22577335C211yesmaximal, characteristic
5#22577333C211yesmaximal, characteristic
6#22577324C212yesmaximal
7#22577322C211yesmaximal, characteristic
8C4⋊C4×Dic137C212yesmaximal
9#22577254C212yesmaximal
10#22577248C211yesmaximal, characteristic
11#395×C137C211yesmaximal, characteristic
12C22×Dic274C2211yescharacteristic
13Q8×Dic137—24no—
14#17984053—24no—
15C2×C4⋊Dic137C2212yes—
16C2×C4⋊Dic137C2212yes—
17C2×C4⋊Dic137C2211yescharacteristic
18C2×#14555489C2212yes—
19C2×#14555489C2212yes—
20C2×#14555489C2211yescharacteristic
21C2×#14555489C2211yescharacteristic
22C2×C4×Dic137C2211yescharacteristic
23C2×C4×Dic137C2212yes—
24C2×C4×Dic137C2211yescharacteristic
25C2×C4×Dic137C2211yescharacteristic
26C2×C4×Dic137C2212yes—
27#17983995C2211yescharacteristic
28#17983995C2212yes—
29#17983995C2212yes—
30#17983995C2212yes—
31#17983995C2212yes—
32C22×Q8×C137C2211yescharacteristic
33C2×C4⋊C4×C137C2212yes—
34C2×C4⋊C4×C137C2211yescharacteristic
35#95×C137C2211yescharacteristic
36#95×C137C2212yes—
37C2×Dic274—22no—
38C2×Dic274—22no—
39C2×Dic274—22no—
40C22×Dic137C2311yescharacteristic
41C22×Dic137C2312yes—
42C22×Dic137C2311yescharacteristic
43C22×Dic137C2312yes—
44C22×Dic137C2311yescharacteristic
45C22×Dic137C2311yescharacteristic
46C4⋊Dic137—24no—
47C4⋊Dic137—24no—
48C4⋊Dic137—22no—
49#14555489—22no—
50#14555489—22no—
51#14555489—24no—
52#14555489—24no—
53C4×Dic137—22no—
54C4×Dic137—22no—
55C4×Dic137—24no—
56C4×Dic137D414yes—
57C4×Dic137Q818yescomplemented
58C2×Q8×C137—22no—
59C2×Q8×C137—24no—
60C22×C4×C137C2311yescharacteristic
61C22×C4×C137C2311yescharacteristic
62C22×C4×C137C2312yes—
63C22×C4×C137C2311yescharacteristic
64C22×C4×C137C2312yes—
65C4⋊C4×C137—22no—
66C4⋊C4×C137—24no—
67Dic274—28no—
68C2×Dic137—21no—
69C2×Dic137—22no—
70C2×Dic137—22no—
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×Dic137C2×D412yes—
80C2×Dic137C2×D412yes—
81C2×Dic137C2×Q814yes—
82C2×Dic137C2×Q814yes—
83C2×Dic137#4112yes—
84C23×C137C2411yescharacteristic
85Q8×C137—28no—
86C2×C4×C137—21no—
87C2×C4×C137—22no—
88C2×C4×C137—22no—
89C2×C4×C137—22no—
90C2×C4×C137—22no—
91C2×C4×C137—22no—
92C2×C4×C137—21no—
93C2×C4×C137—21no—
94C2×C4×C137—22no—
95C2×C4×C137C2×D412yes—
96C2×C4×C137C2×Q814yes—
97C2×C4×C137#4114yes—
98C2×C4×C137#4112yes—
99C22×C137C22×D411yescharacteristic
100C22×C137C22×Q812yes—
101C22×C137C2×#4111yescharacteristic
102C22×C137C2×#4112yes—
103C22×C137C2×#4111yescharacteristic
104C4×C137—22no—
105C4×C137—24no—
106C4×C137—22no—
107C4×C137—24no—
108C4×C137—41no—
109Dic137—22no—
110Dic137—22no—
111Dic137—24no—
112Dic137—41no—
113Dic137—42no—
114Dic137C4⋊Q814yes—
115C2×C137#52511yescharacteristic
116C2×C137#52511yescharacteristic
117C2×C137C2×C4⋊Q811yescharacteristic
118C2×C137#53312yes—
119C2×C137#53312yes—
120C137#239911yescharacteristic, complemented, Sylow
121#2399—1371nomaximal, Sylow
122C2×C4×Q8—1371no—
123C2×C4×Q8—1371no—
124#398—1371no—
125#398—1372no—
126#398—1371no—
127#395—1371no—
128#395—1372no—
129#395D13711yescharacteristic
130C4×C4⋊C4—1372no—
131C4×C4⋊C4—1372no—
132#376—1371no—
133C22×Q8—1371no—
134C22×Q8C2×D13711yescharacteristic
135C4×Q8—2744no—
136C4×Q8—2744no—
137C2×C4⋊C4—1371no—
138C2×C4⋊C4—1371no—
139C2×C4⋊C4—1372no—
140C2×C4⋊C4—1371no—
141C2×C4⋊C4—1372no—
142C2×C4⋊C4—1372no—
143C2×C4⋊C4—1372no—
144C2×C4⋊C4C2×D13711yescharacteristic
145C2×C4⋊C4C2×D13712yes—
146C2×C42—1372no—
147C2×C42—1371no—
148C2×C42—1371no—
149C2×C42—1371no—
150C2×C42—1372no—
151#95—1371no—
152#95—1372no—
153#95—1372no—
154#95—1372no—
155#95—1372no—
156#95C2×D13711yescharacteristic
157#95C2×D13712yes—
158C2×Q8—22no—
159C2×Q8—24no—
160C2×Q8—2742no—
161C2×Q8—2742no—
162C2×Q8—2742no—
163C22×C4—1372no—
164C22×C4—1371no—
165C22×C4—1372no—
166C22×C4—1371no—
167C22×C4—1371no—
168C22×C4—1371no—
169C22×C4C22×D13711yescharacteristic
170C22×C4C22×D13711yescharacteristic
171C22×C4C22×D13712yes—
172C22×C4C22×D13712yes—
173C22×C4C22×D13711yescharacteristic
174C4⋊C4—22no—
175C4⋊C4—24no—
176C4⋊C4—2742no—
177C4⋊C4—2744no—
178C4⋊C4—2744no—
179C4⋊C4—2742no—
180C4⋊C4—2742no—
181C4⋊C4—2744no—
182C4⋊C4—2744no—
183C42—2742no—
184C42—2744no—
185C42—2742no—
186C42—1374no—
187C42—1378no—
188C23C23×D13711yescharacteristic
189Q8—28no—
190Q8—2748no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—2741no—
201C2×C4—2741no—
202C2×C4—2742no—
203C2×C4—2742no—
204C2×C4—2741no—
205C2×C4—2741no—
206C2×C4—2741no—
207C2×C4—2742no—
208C2×C4—2742no—
209C2×C4—2742no—
210C2×C4—2742no—
211C2×C4—1372no—
212C2×C4—1374no—
213C2×C4—1372no—
214C2×C4—1372no—
215C2×C4—1374no—
216C2×C4D4×D13712yes—
217C2×C4Q8×D13714yes—
218C2×C4#1455550914yes—
219C2×C4#1455551112yes—
220C22C2×D4×D13711yescharacteristic
221C22C2×Q8×D13712yes—
222C22#41×D13711yescharacteristic
223C22C2×#1455550912yes—
224C22C2×#1455551111yescharacteristic
225C4—24no—
226C4—22no—
227C4—24no—
228C4—22no—
229C4—41no—
230C4—2744no—
231C4—2742no—
232C4—2742no—
233C4—5481no—
234C4—5482no—
235C4—1374no—
236C2#2257790911yescharacteristic
237C2#2257791611yescharacteristic
238C2C4⋊Q8×D13711yescharacteristic
239C2#2257795512yes—
240C2#2257795812yes—
subgroupG/H
center, FrattiniC23C23×D137
commutator, socle, prime radical, cosocleC23×C137C24
Fitting#395×C137C2
hypercenter#395D137
hypocenterC137#2399
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
57. C4×Dic137Q8semidirect
120. C137#2399semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1372
C2×#179840532
#225773362
#225773352
#225773332
#225773242
#225773222
C4⋊C4×Dic1372
#225772542
#225772482
#395×C1372
#2399137