Finite Group Databasebeta

← #31793437

Subgroups — #31793437

Order 17536.

350 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 115 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#22578123C211yesmaximal, characteristic, complemented
2#22578114C211yesmaximal, characteristic, complemented
3#22577756C212yesmaximal, complemented
4#22577755C211yesmaximal, characteristic, complemented
5#22577713C212yesmaximal, complemented
6#22577617C212yesmaximal
7#22577609C211yesmaximal, characteristic, complemented
8#22577527C211yesmaximal, characteristic, complemented
9#22577490C212yesmaximal, complemented
10C4×#14555495C211yesmaximal, characteristic
11#476×C137C211yesmaximal, characteristic, complemented
12C2×#14555499C2212yes—
13C2×#14555495C2211yescharacteristic
14#17984088C2211yescharacteristic
15#17984081—22no—
16#17984079C2212yes—
17#17984075C2212yes—
18#17984068—21no—
19#17984067—21no—
20#17984064—22no—
21C4×Dic274C2211yescharacteristic
22#17984053C2212yes—
23C4×D548C2211yescharacteristic
24#17984050C2212yes—
25#17984047—21no—
26C4⋊C4×D137—21no—
27#17984021C2211yescharacteristic
28#17984017C2212yes—
29#17984013C2212yes—
30#17984011C2212yes—
31#17984011C2212yes—
32#17984009C2211yescharacteristic
33#17984008C2211yescharacteristic
34C4×#11764040C2211yescharacteristic
35C8×Dic137C2212yes—
36C4⋊Q8×C137C2211yescharacteristic
37C4×D4×C137C2211yescharacteristic
38#106×C137C2212yes—
39C4⋊C8×C137C2211yescharacteristic
40#102×C137C2212yes—
41C2×Dic274D411yescharacteristic
42C2×Dic274C2312yes—
43C2×D548C2311yescharacteristic
44C2×C4×D137—22no—
45C2×C4×D137—21no—
46#14555499—24no—
47#14555495—22no—
48C2×#11764040C2311yescharacteristic
49C2×#11764040C2311yescharacteristic
50C4⋊Dic137—22no—
51C4⋊Dic137—21no—
52C4⋊Dic137—21no—
53C4⋊Dic137D411yescharacteristic
54C4⋊Dic137C2311yescharacteristic
55#14555489—22no—
56#14555489—22no—
57#14555489D412yes—
58C22⋊Dic137—22no—
59C22⋊Dic137—22no—
60#14555487—22no—
61#14555487—21no—
62C4×Dic137—22no—
63C4×Dic137C2312yes—
64C2×Q8×C137—21no—
65C2×D4×C137C2311yescharacteristic
66C22×C4×C137—21no—
67C2×C8×C137C2312yes—
68C4⋊C4×C137—21no—
69C4⋊C4×C137C2311yescharacteristic
70C4⋊C4×C137C2312yes—
71C22⋊C4×C137—21no—
72C42×C137C2311yescharacteristic
73C22×D137—21no—
74Dic274—22no—
75Dic274—22no—
76Dic274—22no—
77D548—21no—
78D548#4112yes—
79C2×Dic137—22no—
80C2×Dic137—21no—
81C2×Dic137C2×D412yes—
82C2×Dic137C2×D412yes—
83C4×D137—42no—
84C4×D137—82no—
85#11764040—21no—
86#11764040—21no—
87C23×C137—21no—
88Q8×C137—81no—
89D4×C137—21no—
90D4×C137—21no—
91C2×C4×C137—21no—
92C2×C4×C137—22no—
93C2×C4×C137—42no—
94C2×C4×C137C2×D411yescharacteristic
95C2×C4×C137C2×D411yescharacteristic
96C2×C4×C137C2411yescharacteristic
97C8×C137—22no—
98C22×C137—41no—
99C22×C137—41no—
100C22×C137—41no—
101C22×C137C22×D411yescharacteristic
102C2×D137—22no—
103C2×D137—41no—
104C2×D137—41no—
105C4×C137—21no—
106C4×C137—41no—
107C4×C137—42no—
108C4×C137#13512yes—
109C4×C137C2×#4111yescharacteristic
110C4×C1372−1+411yescharacteristic
111Dic137—22no—
112Dic137—22no—
113Dic137—42no—
114Dic137—41no—
115C2×C137—81no—
116C2×C137#54811yescharacteristic
117C2×C137C2×#13511yescharacteristic
118C2×C137#57711yescharacteristic
119D137—42no—
120C137#336011yescharacteristic, complemented, Sylow
121#3360—1371nomaximal, Sylow
122#556—1371no—
123#554—1371no—
124#488—1372no—
125#487—1371no—
126#479—1372no—
127#476D13711yescharacteristic, complemented
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—2742no—
138#125—1372no—
139#124—1372no—
140C22⋊Q8—2741no—
141C22⋊Q8—2741no—
142C22⋊Q8—2742no—
143C4×Q8—1372no—
144C4×Q8—1371no—
145C4×D4—1372no—
146C4×D4—1371no—
147C4×D4C2×D13711yescharacteristic
148#117—2741no—
149C2×C4⋊C4—2741no—
150#107—1371no—
151#106C2×D13712yes—
152C4⋊C8—1372no—
153C4⋊C8C2×D13711yescharacteristic
154#103—1372no—
155#103—1372no—
156#103—1372no—
157#102—1371no—
158#102—1371no—
159#102C2×D13712yes—
160C4×C8—1372no—
161C4×C8—1371no—
162C2×Q8—21no—
163C2×Q8—1372no—
164C2×Q8—1371no—
165C2×D4—1371no—
166C2×D4C22×D13711yescharacteristic
167C22×C4—21no—
168C22×C4—2742no—
169C22×C4—2741no—
170SD16—2744no—
171D8—2742no—
172C2×C8—1371no—
173C2×C8—1371no—
174C2×C8C22×D13712yes—
175C4⋊C4—21no—
176C4⋊C4—2742no—
177C4⋊C4—2742no—
178C4⋊C4—2742no—
179C4⋊C4—2741no—
180C4⋊C4—2741no—
181C4⋊C4—1371no—
182C4⋊C4—1372no—
183C4⋊C4—1371no—
184C4⋊C4C22×D13711yescharacteristic
185C4⋊C4C22×D13712yes—
186C22⋊C4—21no—
187C22⋊C4—2742no—
188C22⋊C4—2742no—
189C22⋊C4—2742no—
190C22⋊C4—2741no—
191C42—2742no—
192C42—1372no—
193C42C22×D13711yescharacteristic
194C23—21no—
195C23—2741no—
196Q8—81no—
197Q8—2742no—
198Q8—2742no—
199Q8—2742no—
200D4—21no—
201D4—21no—
202D4—2741no—
203D4—1372no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—2742no—
208C2×C4—2741no—
209C2×C4—5482no—
210C2×C4—10962no—
211C2×C4—1372no—
212C2×C4—1372no—
213C2×C4D4×D13711yescharacteristic
214C2×C4D4×D13711yescharacteristic
215C2×C4C23×D13711yescharacteristic
216C8—22no—
217C8—2741no—
218C8—2741no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—2742no—
223C22—5481no—
224C22—5481no—
225C22C2×D4×D13711yescharacteristic
226C4—21no—
227C4—41no—
228C4—42no—
229C4—2742no—
230C4—2742no—
231C4—5481no—
232C4—5482no—
233C4#1798410912yes—
234C4#41×D13711yescharacteristic
235C4#1798413611yescharacteristic
236C2—81no—
237C2—5482no—
238C2#2257806311yescharacteristic
239C2C2×#1798410911yescharacteristic
240C2#2257821311yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#476×C137C2
socleC22×C137C22×D4
prime radicalC2×#14555495C22
hypercenter#476D137
hypocenterC137#3360
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #22578123C2semidirect
2. #22578114C2semidirect
3. #22577756C2semidirect
4. #22577755C2semidirect
5. #22577713C2semidirect
7. #22577609C2semidirect
8. #22577527C2semidirect
9. #22577490C2semidirect
11. #476×C137C2semidirect
120. C137#3360semidirect
127. #476D137semidirect

Maximal subgroups

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

subgroupindex
#225781232
#225781142
#225777562
#225777552
#225777132
#225776172
#225776092
#225775272
#225774902
C4×#145554952
#476×C1372
#3360137