Finite Group Databasebeta

← #501×D137 #31791304

Subgroups — #501×D137 #31791304

Order 17536.

384 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C4⋊Q8×D137C211yesmaximal, characteristic
2#125×D137C211yesmaximal, characteristic
3#22577813C211yesmaximal, characteristic, complemented
4#22577812C211yesmaximal, characteristic, complemented
5#22577811C212yesmaximal, complemented
6#22577810C212yesmaximal, complemented
7#22577809C211yesmaximal, characteristic, complemented
8#107×D137C212yesmaximal
9#106×D137C212yesmaximal
10#97×D137C211yesmaximal, characteristic
11#501×C137C211yesmaximal, characteristic, complemented
12C2×Q8×D137—21no—
13C2×C8×D137C2212yes—
14#17984088C2211yescharacteristic
15#17984087—21no—
16#17984087C2212yes—
17#17984086—21no—
18#17984080C2211yescharacteristic
19#17984079C2211yescharacteristic
20#17984079C2211yescharacteristic
21#17984078—21no—
22#17984078—21no—
23C4⋊C4×D137—21no—
24C4⋊C4×D137—21no—
25C4⋊C4×D137—21no—
26C4⋊C4×D137C2211yescharacteristic
27C4⋊C4×D137C2211yescharacteristic
28C4⋊C4×D137C2212yes—
29C42×D137C2211yescharacteristic
30#17984022C2212yes—
31#17984021C2212yes—
32#17984021C2212yes—
33#17984020C2212yes—
34#17984019C2212yes—
35#17984019C2212yes—
36#17983999C2211yescharacteristic
37#17983998C2212yes—
38C4⋊Q8×C137C2211yescharacteristic
39#125×C137C2211yescharacteristic
40#107×C137C2212yes—
41#106×C137C2212yes—
42#97×C137C2211yescharacteristic
43C2×Dic274—22no—
44C2×Dic274—21no—
45Q8×D137—44no—
46C2×C4×D137—21no—
47C2×C4×D137—21no—
48C2×C4×D137—22no—
49C2×C4×D137D411yescharacteristic
50C2×C4×D137D411yescharacteristic
51C2×C4×D137C2311yescharacteristic
52C2×#11764040C2312yes—
53C8×D137Q818yescomplemented
54C4⋊Dic137—21no—
55C4⋊Dic137—21no—
56C4⋊Dic137—21no—
57C4⋊Dic137C2311yescharacteristic
58C4⋊Dic137C2312yes—
59C4⋊Dic137C2311yescharacteristic
60#14555489—21no—
61#14555489—21no—
62#14555489—22no—
63#14555489—21no—
64#14555489—21no—
65#14555489—22no—
66#14555489—21no—
67#14555489—21no—
68C4×Dic137D411yescharacteristic
69C4×Dic137D411yescharacteristic
70C4×Dic137C2311yescharacteristic
71C2×Q8×C137—21no—
72C2×C8×C137C2312yes—
73C4⋊C4×C137—21no—
74C4⋊C4×C137—21no—
75C4⋊C4×C137—21no—
76C4⋊C4×C137C2311yescharacteristic
77C4⋊C4×C137C2312yes—
78C4⋊C4×C137C2311yescharacteristic
79C42×C137C2311yescharacteristic
80C22×D137C2×D411yescharacteristic
81Dic274—44no—
82Dic274—42no—
83C2×Dic137—21no—
84C2×Dic137—21no—
85C2×Dic137—22no—
86C2×Dic137C2×D411yescharacteristic
87C2×Dic137C2×D411yescharacteristic
88C2×Dic137C2×D411yescharacteristic
89C4×D137—22no—
90C4×D137—22no—
91C4×D137—42no—
92C4×D137—44no—
93C4×D137—42no—
94C4×D137C2×Q814yes—
95#11764040C2×Q814yescomplemented
96Q8×C137—42no—
97C2×C4×C137—21no—
98C2×C4×C137—21no—
99C2×C4×C137—22no—
100C2×C4×C137C2×D411yescharacteristic
101C2×C4×C137C2×D411yescharacteristic
102C2×C4×C137C2411yescharacteristic
103C8×C137C2×Q814yescomplemented
104C22×C137C22×D411yescharacteristic
105C2×D137C4⋊Q812yes—
106C2×D137AGL(1,ℤ/8)12yes—
107C2×D137#13712yes—
108C4×C137—21no—
109C4×C137—21no—
110C4×C137—42no—
111C4×C137—41no—
112C4×C137—41no—
113C4×C137C22×Q812yes—
114Dic137—21no—
115Dic137—21no—
116Dic137—41no—
117Dic137—42no—
118Dic137—41no—
119Dic137C4⋊Q812yes—
120C2×C137C2×C4⋊Q811yescharacteristic
121C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
122C2×C137C2×#13711yescharacteristic
123D137#50114yesdirect product factor
124C137C2×#50111yescharacteristic, complemented, Sylow
125C2×#501—1371nomaximal, Sylow
126C2×C4⋊Q8—1371no—
127C2×#125—1371no—
128#501—1371no—
129#501—1371no—
130#501—1372no—
131#501—1372no—
132#501—1371no—
133#501D13711yescharacteristic, direct product factor
134C2×#107—1372no—
135C2×#106—1372no—
136C2×#97—1371no—
137C22×Q8—2741no—
138C22×C8—1372no—
139C4⋊Q8—2741no—
140C4⋊Q8—2741no—
141C4⋊Q8—1372no—
142C4⋊Q8—1371no—
143C4⋊Q8C2×D13711yescharacteristic
144#125—2741no—
145#125—2741no—
146#125—1371no—
147#125—1371no—
148#125—1371no—
149#125C2×D13711yescharacteristic
150C2×C4⋊C4—2741no—
151C2×C4⋊C4—2741no—
152C2×C4⋊C4—2741no—
153C2×C4⋊C4—1371no—
154C2×C4⋊C4—1371no—
155C2×C4⋊C4—1372no—
156C2×C42—1371no—
157#107—1372no—
158#107—1372no—
159#107—1372no—
160#107C2×D13712yes—
161#106—1372no—
162#106—1372no—
163#106—1372no—
164#106C2×D13712yes—
165#97—1372no—
166#97—1371no—
167#97C2×D13711yescharacteristic
168C2×Q8—21no—
169C2×Q8—2742no—
170C2×Q8—2741no—
171C2×Q8—5484no—
172C22×C4—2741no—
173C22×C4—2741no—
174C22×C4—2742no—
175C22×C4—1371no—
176C22×C4—1371no—
177C22×C4—1371no—
178C2×C8—1372no—
179C2×C8—1378no—
180C2×C8C22×D13712yes—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—2741no—
185C4⋊C4—2741no—
186C4⋊C4—2742no—
187C4⋊C4—2741no—
188C4⋊C4—2741no—
189C4⋊C4—2742no—
190C4⋊C4—2741no—
191C4⋊C4—2741no—
192C4⋊C4—2741no—
193C4⋊C4—2741no—
194C4⋊C4—2741no—
195C4⋊C4—1371no—
196C4⋊C4—1372no—
197C4⋊C4—1371no—
198C4⋊C4C22×D13711yescharacteristic
199C4⋊C4C22×D13712yes—
200C4⋊C4C22×D13711yescharacteristic
201C42—1371no—
202C42—1371no—
203C42—1371no—
204C42C22×D13711yescharacteristic
205C23—1371no—
206Q8—42no—
207Q8—5482no—
208Q8—5484no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—2742no—
213C2×C4—2742no—
214C2×C4—2741no—
215C2×C4—2741no—
216C2×C4—2742no—
217C2×C4—5484no—
218C2×C4—5482no—
219C2×C4—5482no—
220C2×C4—1371no—
221C2×C4—1371no—
222C2×C4—1371no—
223C2×C4—1374no—
224C2×C4D4×D13711yescharacteristic
225C2×C4D4×D13711yescharacteristic
226C2×C4C23×D13711yescharacteristic
227C8—1374no—
228C8Q8×D13714yescomplemented
229C22—1372no—
230C22—1372no—
231C22—1372no—
232C22C2×D4×D13711yescharacteristic
233C4—21no—
234C4—21no—
235C4—42no—
236C4—41no—
237C4—41no—
238C4—2741no—
239C4—2741no—
240C4—5482no—
241C4—5481no—
242C4—5481no—
243C4—1372no—
244C4C2×Q8×D13712yes—
245C2—1374no—
246C2C4⋊Q8×D13711yescharacteristic
247C2AGL(1,ℤ/8)×D13711yescharacteristic
248C2#137×D13711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#501×C137C2
socleC22×C137C22×D4
prime radicalC22×D137C2×D4
hypercenter#501D137
hypocenterC137C2×#501
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
3. #22577813C2semidirect
4. #22577812C2semidirect
5. #22577811C2semidirect
6. #22577810C2semidirect
7. #22577809C2semidirect
11. #501×C137C2semidirect
53. C8×D137Q8semidirect
95. #11764040C2×Q8semidirect
103. C8×C137C2×Q8semidirect
123. D137#501direct (N × H)
124. C137C2×#501semidirect
133. #501D137direct (N × H)
228. C8Q8×D137semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1372
#125×D1372
#225778132
#225778122
#225778112
#225778102
#225778092
#107×D1372
#106×D1372
#97×D1372
#501×C1372
C2×#501137