Finite Group Databasebeta

← #31793479

Subgroups — #31793479

Order 17536.

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

normal subgroups only

This group has 336 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#22578128C211yesmaximal, characteristic, complemented
2#22578127C211yesmaximal, characteristic
3#22577759C212yesmaximal
4#22577708C212yesmaximal, complemented
5#22577680C211yesmaximal, characteristic
6#22577628C212yesmaximal
7#22577610C211yesmaximal, characteristic, complemented
8C8×Dic274C211yesmaximal, characteristic, complemented
9#22577495C212yesmaximal, complemented
10C4×#14555499C211yesmaximal, characteristic
11#487×C137C211yesmaximal, characteristic, complemented
12C2×#14555500C2212yes—
13C2×#14555499C2211yescharacteristic
14#17984086—21no—
15#17984083—22no—
16#17984079C2212yes—
17#17984078—22no—
18#17984077C2212yes—
19#17984068—21no—
20Q8×Dic137—21no—
21Q8×Dic137C2212yes—
22C4×Dic274C2211yescharacteristic
23C4×Dic274C2211yescharacteristic
24#17984053C2212yes—
25C4×D548C2211yescharacteristic
26#17984046—21no—
27#17984021C2212yes—
28#17984020C2211yescharacteristic
29C4⋊#11764040C2211yescharacteristic
30#17984017C2212yes—
31#17984013C2211yescharacteristic
32#17984012C2212yes—
33#17984010C2211yescharacteristic
34#17984008C2212yes—
35C8×Dic137C2212yes—
36C4⋊Q8×C137C2211yescharacteristic
37C4⋊Q8×C137C2211yescharacteristic
38#103×C137C2212yes—
39#103×C137C2212yes—
40C4×C8×C137C2211yescharacteristic
41C2×Dic274D411yescharacteristic
42C2×Dic274C2311yescharacteristic
43C2×D548C2311yescharacteristic
44C2×C4×D137—21no—
45#14555500—24no—
46#14555499—22no—
47C2×#11764040C2312yes—
48C4⋊Dic137—22no—
49C4⋊Dic137—22no—
50C4⋊Dic137D411yescharacteristic
51C4⋊Dic137C2311yescharacteristic
52C4⋊Dic137C2311yescharacteristic
53#14555489—22no—
54#14555489—22no—
55#14555489—21no—
56#14555489D412yes—
57#14555487—22no—
58#14555487—22no—
59#14555487—21no—
60C4×Dic137—22no—
61C4×Dic137—22no—
62C4×Dic137—21no—
63C4×Dic137C2312yes—
64C2×Q8×C137—21no—
65C2×Q8×C137C2312yes—
66C2×C8×C137C2311yescharacteristic
67C2×C8×C137C2311yescharacteristic
68C4⋊C4×C137—21no—
69C4⋊C4×C137—21no—
70C4⋊C4×C137—21no—
71C4⋊C4×C137C2312yes—
72C42×C137C2311yescharacteristic
73C22×D137—21no—
74Dic274—22no—
75Dic274—21no—
76Dic274#4112yes—
77D548—21no—
78D548—21no—
79C2×Dic137—21no—
80C2×Dic137—21no—
81C2×Dic137—21no—
82C2×Dic137C2×D412yes—
83C2×Dic137C2×D412yes—
84C4×D137—42no—
85#11764040—22no—
86Q8×C137—22no—
87Q8×C137—22no—
88Q8×C137—42no—
89C2×C4×C137—22no—
90C2×C4×C137—22no—
91C2×C4×C137C2×D411yescharacteristic
92C2×C4×C137C2×D411yescharacteristic
93C2×C4×C137C2411yescharacteristic
94C8×C137—21no—
95C8×C137—21no—
96C22×C137C22×D411yescharacteristic
97C2×D137—41no—
98C2×D137—41no—
99C2×D137—41no—
100C4×C137—21no—
101C4×C137—42no—
102C4×C137—42no—
103C4×C137#13512yes—
104C4×C137C2×#4111yescharacteristic
105C4×C1372−1+411yescharacteristic
106Dic137—22no—
107Dic137—22no—
108Dic137—22no—
109Dic137—41no—
110Dic137—41no—
111C2×C137#54811yescharacteristic
112C2×C137C2×#13511yescharacteristic
113C2×C137#57811yescharacteristic
114D137—81no—
115C137#336411yescharacteristic, complemented, Sylow
116#3364—1371nomaximal, Sylow
117#557—1371no—
118#556—1371no—
119#488—1372no—
120#487D13711yescharacteristic, complemented
121#479—1372no—
122#474—1371no—
123#464—1372no—
124#462—1371no—
125C8×Q8—1371no—
126C4×Q16—1372no—
127C4×SD16—1371no—
128C2×Q16—1372no—
129C2×SD16—1371no—
130C4⋊Q8—2741no—
131C4⋊Q8C2×D13711yescharacteristic
132C4⋊Q8C2×D13711yescharacteristic
133#126—2742no—
134#125—2742no—
135#125—1372no—
136#124—1372no—
137C22⋊Q8—2741no—
138C4×Q8—2741no—
139C4×Q8—1372no—
140C4×Q8—1372no—
141C4×Q8—1371no—
142C4×Q8—1371no—
143C4×D4—1371no—
144#117—2741no—
145#107—1372no—
146#106—1371no—
147C4⋊C8—1372no—
148C4⋊C8—1371no—
149#103—1372no—
150#103—1371no—
151#103C2×D13712yes—
152#103C2×D13712yes—
153#102—1371no—
154#102—1372no—
155C4×C8—1372no—
156C4×C8C2×D13711yescharacteristic
157C2×Q8—21no—
158C2×Q8—1371no—
159C2×Q8—1371no—
160C2×Q8C22×D13712yes—
161C2×D4—1371no—
162C22×C4—2741no—
163Q16—2744no—
164SD16—2742no—
165C2×C8—1372no—
166C2×C8C22×D13711yescharacteristic
167C2×C8C22×D13711yescharacteristic
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—2742no—
172C4⋊C4—2742no—
173C4⋊C4—2742no—
174C4⋊C4—2742no—
175C4⋊C4—2741no—
176C4⋊C4—1371no—
177C4⋊C4—1371no—
178C4⋊C4—1372no—
179C4⋊C4—1371no—
180C4⋊C4C22×D13712yes—
181C22⋊C4—2742no—
182C22⋊C4—2742no—
183C22⋊C4—2741no—
184C42—2742no—
185C42—2742no—
186C42—2741no—
187C42—1372no—
188C42C22×D13711yescharacteristic
189C23—2741no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—2742no—
194Q8—2741no—
195Q8—1372no—
196D4—2741no—
197D4—2741no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—2741no—
201C2×C4—2741no—
202C2×C4—2741no—
203C2×C4—5482no—
204C2×C4—1372no—
205C2×C4—1372no—
206C2×C4D4×D13711yescharacteristic
207C2×C4D4×D13711yescharacteristic
208C2×C4C23×D13711yescharacteristic
209C8—21no—
210C8—21no—
211C8—2742no—
212C22—5481no—
213C22—5481no—
214C22—5481no—
215C22C2×D4×D13711yescharacteristic
216C4—21no—
217C4—42no—
218C4—42no—
219C4—2742no—
220C4—2742no—
221C4—2742no—
222C4—5481no—
223C4—5481no—
224C4#1798410912yes—
225C4#41×D13711yescharacteristic
226C4#1798413511yescharacteristic
227C2—10961no—
228C2#2257806111yescharacteristic
229C2C2×#1798410911yescharacteristic
230C2#2257821611yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#487×C137C2
socleC22×C137C22×D4
prime radicalC2×D548C23
hypercenter#487D137
hypocenterC137#3364
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #22578128C2semidirect
4. #22577708C2semidirect
7. #22577610C2semidirect
8. C8×Dic274C2semidirect
9. #22577495C2semidirect
11. #487×C137C2semidirect
115. C137#3364semidirect
120. #487D137semidirect

Maximal subgroups

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

subgroupindex
#225781282
#225781272
#225777592
#225777082
#225776802
#225776282
#225776102
C8×Dic2742
#225774952
C4×#145554992
#487×C1372
#3364137