Finite Group Databasebeta

← #31793503

Subgroups — #31793503

Order 17536.

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

normal subgroups only

This group has 372 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
1Q8×D548C211yesmaximal, characteristic, complemented
2#22577762C212yesmaximal, complemented
3#22577753C211yesmaximal, characteristic, complemented
4#22577698C212yesmaximal, complemented
5#22577683C211yesmaximal, characteristic, complemented
6#22577615C212yesmaximal
7#22577605C211yesmaximal, characteristic, complemented
8#22577534C211yesmaximal, characteristic, complemented
9#22577501C212yesmaximal, complemented
10C4×#14555495C211yesmaximal, characteristic
11#556×C137C211yesmaximal, characteristic, complemented
12C2×Q8×D137—21no—
13C2×#14555497C2212yes—
14C2×#14555495C2211yescharacteristic
15#17984088C2211yescharacteristic
16#17984088C2212yes—
17#17984068—21no—
18#17984068—22no—
19C4×D548C2211yescharacteristic
20C4×D548C2212yes—
21#17984021C2212yes—
22#17984021C2211yescharacteristic
23C4⋊#11764040C2211yescharacteristic
24C4⋊#11764040C2212yes—
25#17984012C2212yes—
26#17984012C2212yes—
27#17984012C2212yes—
28#17984009C2212yes—
29#17984009C2211yescharacteristic
30#17984008C2211yescharacteristic
31#17983999C2212yes—
32C4×#11764040C2211yescharacteristic
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—22no—
42C2×Dic274—21no—
43Q8×D137—44no—
44C2×D548C2311yescharacteristic
45C2×C4×D137—22no—
46C2×C4×D137—21no—
47#14555497—24no—
48#14555495—22no—
49C2×#11764040C2311yescharacteristic
50C2×#11764040C2312yes—
51C2×#11764040C2311yescharacteristic
52C4⋊Dic137—22no—
53C4⋊Dic137—21no—
54C4⋊Dic137C2312yes—
55C4⋊Dic137C2311yescharacteristic
56#14555487—22no—
57#14555487—21no—
58C2×Q8×C137D411yescharacteristic
59C2×Q8×C137C2312yes—
60C2×D4×C137C2311yescharacteristic
61C22×C4×C137—21no—
62C4⋊C4×C137—21no—
63C4⋊C4×C137—22no—
64C4⋊C4×C137—21no—
65C4⋊C4×C137D411yescharacteristic
66C4⋊C4×C137D412yes—
67C4⋊C4×C137C2311yescharacteristic
68C22⋊C4×C137—22no—
69C22⋊C4×C137—21no—
70C22⋊C4×C137—22no—
71C42×C137—22no—
72C42×C137C2311yescharacteristic
73C42×C137C2312yes—
74C22×D137—21no—
75Dic274—44no—
76Dic274—42no—
77D548—21no—
78D548#4112yes—
79C2×Dic137—22no—
80C2×Dic137—21no—
81C4×D137—44no—
82C4×D137—42no—
83#11764040—21no—
84#11764040—22no—
85#11764040—21no—
86C23×C137—21no—
87Q8×C137—22no—
88Q8×C137—22no—
89Q8×C137—22no—
90D4×C137—21no—
91D4×C137—21no—
92C2×C4×C137—21no—
93C2×C4×C137—22no—
94C2×C4×C137—42no—
95C2×C4×C137C2×D412yes—
96C2×C4×C137C2×D411yescharacteristic
97C2×C4×C137C2×D412yes—
98C2×C4×C137C2×D411yescharacteristic
99C2×C4×C137C2411yescharacteristic
100C22×C137—41no—
101C22×C137—41no—
102C22×C137—41no—
103C22×C137C22×D411yescharacteristic
104C2×D137—22no—
105C2×D137—41no—
106C2×D137—41no—
107C4×C137—22no—
108C4×C137—22no—
109C4×C137—21no—
110C4×C137—42no—
111C4×C137—41no—
112C4×C137AGL(1,ℤ/8)12yes—
113C4×C137C2×#4111yescharacteristic
114C4×C1372−1+411yescharacteristic
115Dic137—42no—
116Dic137—41no—
117C2×C137—81no—
118C2×C137#54811yescharacteristic
119C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C137#57811yescharacteristic
121D137—42no—
122C137#336711yescharacteristic, complemented, Sylow
123#3367—1371nomaximal, Sylow
124#556D13711yescharacteristic, complemented
125D4×Q8—1371no—
126#489—1372no—
127#487—1371no—
128#477—1372no—
129#474—1371no—
130#463—1372no—
131#461—1371no—
132#446—1371no—
133#440—1372no—
134C4×D8—1371no—
135C22×Q8—2741no—
136C2×SD16—1372no—
137C2×D8—1371no—
138C4⋊Q8—1371no—
139C4⋊Q8—1372no—
140C4⋊Q8C2×D13711yescharacteristic
141#126—22no—
142#124C2×D13712yes—
143C22⋊Q8—21no—
144C22⋊Q8—2741no—
145C22⋊Q8—2742no—
146C4×Q8C2×D13712yes—
147C4×Q8C2×D13711yescharacteristic
148C4×D4—1371no—
149C4×D4—1372no—
150C4×D4C2×D13711yescharacteristic
151#117—21no—
152#107—1371no—
153#107—1372no—
154C4⋊C8—1371no—
155C4⋊C8—1372no—
156#103—1372no—
157#103—1372no—
158#103—1372no—
159#102—1371no—
160#102—1372no—
161#102—1371no—
162#97—1372no—
163C4×C8—1371no—
164C2×Q8—2742no—
165C2×Q8—2741no—
166C2×Q8—5484no—
167C2×Q8C22⋊D13711yescharacteristic
168C2×Q8C22×D13712yes—
169C2×D4—1371no—
170C2×D4C22×D13711yescharacteristic
171C22×C4—21no—
172C22×C4—2742no—
173C22×C4—2741no—
174SD16—2744no—
175D8—2742no—
176C2×C8—1371no—
177C2×C8—1371no—
178C2×C8—1372no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—2742no—
183C4⋊C4—2741no—
184C4⋊C4—1371no—
185C4⋊C4—1372no—
186C4⋊C4C22⋊D13711yescharacteristic
187C4⋊C4C22⋊D13712yes—
188C4⋊C4C22×D13711yescharacteristic
189C22⋊C4—22no—
190C22⋊C4—21no—
191C22⋊C4—22no—
192C22⋊C4—2742no—
193C22⋊C4—2741no—
194C42—22no—
195C42C22×D13711yescharacteristic
196C42C22×D13712yes—
197C23—21no—
198C23—2741no—
199Q8—22no—
200Q8—22no—
201Q8—22no—
202Q8—5484no—
203Q8—5482no—
204D4—21no—
205D4—21no—
206D4—2741no—
207D4—1372no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—2742no—
212C2×C4—2741no—
213C2×C4—5484no—
214C2×C4—5482no—
215C2×C4C2×C22⋊D13712yes—
216C2×C4C2×C22⋊D13711yescharacteristic
217C2×C4C2×C22⋊D13712yes—
218C2×C4C2×C22⋊D13711yescharacteristic
219C2×C4C23×D13711yescharacteristic
220C8—2741no—
221C8—2742no—
222C8—2741no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—2742no—
227C22—5481no—
228C22—5481no—
229C22C22×C22⋊D13711yescharacteristic
230C4—22no—
231C4—22no—
232C4—21no—
233C4—41no—
234C4—42no—
235C4—5482no—
236C4—5481no—
237C4#1798411312yes—
238C4#41×D13711yescharacteristic
239C4#1798413511yescharacteristic
240C2—81no—
241C2—5482no—
242C2#2257806011yescharacteristic
243C2C2×#1798411311yescharacteristic
244C2#2257821811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#556×C137C2
socleC22×C137C22×D4
prime radicalC2×#14555495C22
hypercenter#556D137
hypocenterC137#3367
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. Q8×D548C2semidirect
2. #22577762C2semidirect
3. #22577753C2semidirect
4. #22577698C2semidirect
5. #22577683C2semidirect
7. #22577605C2semidirect
8. #22577534C2semidirect
9. #22577501C2semidirect
11. #556×C137C2semidirect
122. C137#3367semidirect
124. #556D137semidirect

Maximal subgroups

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

subgroupindex
Q8×D5482
#225777622
#225777532
#225776982
#225776832
#225776152
#225776052
#225775342
#225775012
C4×#145554952
#556×C1372
#3367137