Finite Group Databasebeta

← #31793434

Subgroups — #31793434

Order 17536.

350 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 123 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
1#22578113C211yesmaximal, characteristic, complemented
2#22577757C212yesmaximal, complemented
3#22577753C211yesmaximal, characteristic, complemented
4#22577710C212yesmaximal, complemented
5#22577693C211yesmaximal, characteristic, complemented
6#22577615C212yesmaximal
7#22577605C211yesmaximal, characteristic, complemented
8Q8×#11764040C211yesmaximal, characteristic, complemented
9C4×#14555497C212yesmaximal, complemented
10C4×#14555495C211yesmaximal, characteristic
11#556×C137C211yesmaximal, characteristic, complemented
12C2×#14555497C2212yes—
13C2×#14555495C2211yescharacteristic
14#17984088C2211yescharacteristic
15#17984080C2212yes—
16#17984066—22no—
17#17984064—21no—
18C4×D548C2211yescharacteristic
19C4×D548C2212yes—
20C4⋊C4×D137—21no—
21#17984021C2211yescharacteristic
22#17984019C2212yes—
23C4⋊#11764040C2211yescharacteristic
24C4⋊#11764040C2212yes—
25#17984012C2212yes—
26#17984012C2212yes—
27#17984012C2212yes—
28#17984009C2212yes—
29#17984009C2211yescharacteristic
30#17984008C2211yescharacteristic
31C4×#11764040C2211yescharacteristic
32C4×#11764040C2212yes—
33C4⋊Q8×C137C2211yescharacteristic
34#126×C137—22no—
35#124×C137C2212yes—
36C22⋊Q8×C137—21no—
37C4×Q8×C137C2212yes—
38C4×Q8×C137C2211yescharacteristic
39C4×D4×C137C2211yescharacteristic
40#117×C137—21no—
41C2×Dic274—21no—
42C2×D548C2311yescharacteristic
43C2×C4×D137—21no—
44C2×C4×D137—22no—
45#14555497—24no—
46#14555495—22no—
47C2×#11764040C2311yescharacteristic
48C2×#11764040C2312yes—
49C2×#11764040C2311yescharacteristic
50C4⋊Dic137—21no—
51C4⋊Dic137C2312yes—
52C4⋊Dic137C2311yescharacteristic
53#14555489—22no—
54#14555489—22no—
55#14555487—22no—
56#14555487—21no—
57C2×Q8×C137D411yescharacteristic
58C2×Q8×C137C2312yes—
59C2×D4×C137C2311yescharacteristic
60C22×C4×C137—21no—
61C4⋊C4×C137—21no—
62C4⋊C4×C137—22no—
63C4⋊C4×C137—21no—
64C4⋊C4×C137D411yescharacteristic
65C4⋊C4×C137D412yes—
66C4⋊C4×C137C2311yescharacteristic
67C22⋊C4×C137—22no—
68C22⋊C4×C137—21no—
69C22⋊C4×C137—22no—
70C42×C137—22no—
71C42×C137C2312yes—
72C42×C137C2311yescharacteristic
73C22×D137—21no—
74Dic274—81no—
75D548—21no—
76D548#4112yes—
77C2×Dic137—21no—
78C2×Dic137—22no—
79C4×D137—42no—
80C4×D137—82no—
81#11764040—21no—
82#11764040—22no—
83#11764040—21no—
84C23×C137—21no—
85Q8×C137—22no—
86Q8×C137—22no—
87Q8×C137—22no—
88D4×C137—21no—
89D4×C137—21no—
90C2×C4×C137—21no—
91C2×C4×C137—22no—
92C2×C4×C137—42no—
93C2×C4×C137C2×D412yes—
94C2×C4×C137C2×D411yescharacteristic
95C2×C4×C137C2×D412yes—
96C2×C4×C137C2×D411yescharacteristic
97C2×C4×C137C2411yescharacteristic
98C22×C137—41no—
99C22×C137—41no—
100C22×C137—41no—
101C22×C137C22×D411yescharacteristic
102C2×D137—22no—
103C2×D137—41no—
104C2×D137—41no—
105C4×C137—22no—
106C4×C137—22no—
107C4×C137—21no—
108C4×C137—42no—
109C4×C137—41no—
110C4×C137#13512yes—
111C4×C137C2×#4111yescharacteristic
112C4×C1372−1+411yescharacteristic
113Dic137—41no—
114Dic137—42no—
115C2×C137—81no—
116C2×C137#54811yescharacteristic
117C2×C137C2×#13511yescharacteristic
118C2×C137#57711yescharacteristic
119D137—42no—
120C137#336011yescharacteristic, complemented, Sylow
121#3360—1371nomaximal, Sylow
122#556D13711yescharacteristic, complemented
123#554—1371no—
124#488—1372no—
125#487—1371no—
126#479—1372no—
127#476—1371no—
128#463—1372no—
129#461—1371no—
130C8×Q8—1371no—
131C4×SD16—1372no—
132C4×D8—1371no—
133C2×SD16—1372no—
134C2×D8—1371no—
135C4⋊Q8—1371no—
136C4⋊Q8C2×D13711yescharacteristic
137#126—22no—
138#125—1372no—
139#124C2×D13712yes—
140C22⋊Q8—21no—
141C22⋊Q8—2741no—
142C22⋊Q8—2742no—
143C4×Q8C2×D13712yes—
144C4×Q8C2×D13711yescharacteristic
145C4×D4—1371no—
146C4×D4—1372no—
147C4×D4C2×D13711yescharacteristic
148#117—21no—
149C2×C4⋊C4—2741no—
150#107—1371no—
151#106—1372no—
152C4⋊C8—1371no—
153C4⋊C8—1372no—
154#103—1372no—
155#103—1372no—
156#103—1372no—
157#102—1371no—
158#102—1372no—
159#102—1371no—
160C4×C8—1371no—
161C4×C8—1372no—
162C2×Q8—2741no—
163C2×Q8C22⋊D13711yescharacteristic
164C2×Q8C22×D13712yes—
165C2×D4—1371no—
166C2×D4C22×D13711yescharacteristic
167C22×C4—21no—
168C22×C4—2741no—
169C22×C4—2742no—
170SD16—2744no—
171D8—2742no—
172C2×C8—1371no—
173C2×C8—1372no—
174C2×C8—1371no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—2742no—
179C4⋊C4—2742no—
180C4⋊C4—2741no—
181C4⋊C4—1372no—
182C4⋊C4—1371no—
183C4⋊C4C22⋊D13711yescharacteristic
184C4⋊C4C22⋊D13712yes—
185C4⋊C4C22×D13711yescharacteristic
186C22⋊C4—22no—
187C22⋊C4—21no—
188C22⋊C4—22no—
189C22⋊C4—2741no—
190C22⋊C4—2742no—
191C42—22no—
192C42C22×D13711yescharacteristic
193C42C22×D13712yes—
194C23—21no—
195C23—2741no—
196Q8—22no—
197Q8—22no—
198Q8—22no—
199Q8—10961no—
200D4—21no—
201D4—21no—
202D4—2741no—
203D4—1372no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—2741no—
208C2×C4—2742no—
209C2×C4—5482no—
210C2×C4—10962no—
211C2×C4C2×C22⋊D13712yes—
212C2×C4C2×C22⋊D13712yes—
213C2×C4C2×C22⋊D13711yescharacteristic
214C2×C4C2×C22⋊D13711yescharacteristic
215C2×C4C23×D13711yescharacteristic
216C8—2741no—
217C8—2742no—
218C8—2741no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—2742no—
223C22—5481no—
224C22—5481no—
225C22C22×C22⋊D13711yescharacteristic
226C4—22no—
227C4—22no—
228C4—21no—
229C4—41no—
230C4—42no—
231C4—5481no—
232C4—5482no—
233C4#1798410812yes—
234C4#41×D13711yescharacteristic
235C4#1798413511yescharacteristic
236C2—81no—
237C2—5482no—
238C2#2257806011yescharacteristic
239C2C2×#1798410811yescharacteristic
240C2#2257821111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#556×C137C2
socleC22×C137C22×D4
prime radicalC2×#14555495C22
hypercenter#556D137
hypocenterC137#3360
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #22578113C2semidirect
2. #22577757C2semidirect
3. #22577753C2semidirect
4. #22577710C2semidirect
5. #22577693C2semidirect
7. #22577605C2semidirect
8. Q8×#11764040C2semidirect
9. C4×#14555497C2semidirect
11. #556×C137C2semidirect
120. C137#3360semidirect
122. #556D137semidirect

Maximal subgroups

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

subgroupindex
#225781132
#225777572
#225777532
#225777102
#225776932
#225776152
#225776052
Q8×#117640402
C4×#145554972
C4×#145554952
#556×C1372
#3360137