Finite Group Databasebeta

← #487×D137 #31791097

Subgroups — #487×D137 #31791097

Order 17536.

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

normal subgroups only

This group has 424 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
2C4⋊Q8×D137C211yesmaximal, characteristic
3#22577755C211yesmaximal, characteristic, complemented
4#22577754C211yesmaximal, characteristic, complemented
5#22577753C211yesmaximal, characteristic, complemented
6#22577752C214yesmaximal, complemented
7#103×D137C214yesmaximal
8C4×C8×D137C211yesmaximal, characteristic
9#487×C137C211yesmaximal, characteristic, complemented
10C2×Q8×D137—21no—
11C2×Q8×D137C2212yes—
12C2×C8×D137C2212yes—
13#17984088C2211yescharacteristic
14#17984088C2211yescharacteristic
15#17984087—21no—
16#17984087—21no—
17#17984087—21no—
18#17984087C2212yes—
19#17984086—21no—
20#17984086C2212yes—
21C4⋊C4×D137—21no—
22C4⋊C4×D137—21no—
23C4⋊C4×D137—21no—
24C4⋊C4×D137C2212yes—
25C42×D137C2211yescharacteristic
26#17984013C2214yes—
27#17984012C2214yes—
28#17984011C2214yes—
29C4×#11764040C2211yescharacteristic
30C8×Dic137C2212yes—
31C4⋊Q8×C137C2211yescharacteristic
32C4⋊Q8×C137C2211yescharacteristic
33#103×C137C2214yes—
34C4×C8×C137C2211yescharacteristic
35C2×Dic274—22no—
36C2×Dic274—22no—
37C2×Dic274—21no—
38C2×Dic274C2312yes—
39Q8×D137—28no—
40Q8×D137—44no—
41C2×C4×D137—22no—
42C2×C4×D137—22no—
43C2×C4×D137D411yescharacteristic
44C2×C4×D137D411yescharacteristic
45C2×C4×D137C2311yescharacteristic
46C2×#11764040C2312yes—
47C8×D137—24no—
48C4⋊Dic137—21no—
49C4⋊Dic137—21no—
50C4⋊Dic137—21no—
51C4⋊Dic137C2312yes—
52#14555489—22no—
53#14555489—22no—
54#14555489—22no—
55#14555489—22no—
56C4×Dic137D411yescharacteristic
57C4×Dic137D411yescharacteristic
58C4×Dic137C2311yescharacteristic
59C2×Q8×C137—21no—
60C2×Q8×C137C2312yes—
61C2×C8×C137C2312yes—
62C4⋊C4×C137—21no—
63C4⋊C4×C137—21no—
64C4⋊C4×C137—21no—
65C4⋊C4×C137C2312yes—
66C42×C137C2311yescharacteristic
67C22×D137C2×D411yescharacteristic
68Dic274—24no—
69Dic274—42no—
70Dic274—44no—
71Dic274—44no—
72C2×Dic137—22no—
73C2×Dic137—22no—
74C2×Dic137C2×D411yescharacteristic
75C2×Dic137C2×D411yescharacteristic
76C2×Dic137C2×D411yescharacteristic
77C4×D137—44no—
78C4×D137—44no—
79C4×D137SD1614yes—
80C4×D137Q1614yes—
81C4×D137#4114yes—
82#11764040—22no—
83Q8×C137—24no—
84Q8×C137—42no—
85C2×C4×C137—22no—
86C2×C4×C137—22no—
87C2×C4×C137C2×D411yescharacteristic
88C2×C4×C137C2×D411yescharacteristic
89C2×C4×C137C2411yescharacteristic
90C8×C137—22no—
91C22×C137C22×D411yescharacteristic
92C2×D137#12412yes—
93C2×D137C2×SD1612yes—
94C2×D137C2×Q1612yes—
95C4×C137—42no—
96C4×C137—42no—
97C4×C137C2×SD1612yes—
98C4×C137C2×Q1612yes—
99C4×C137C2×#4112yes—
100Dic137—42no—
101Dic137—42no—
102Dic137#12412yes—
103Dic137C2×SD1612yes—
104Dic137C2×Q1612yes—
105C2×C137C2×#12411yescharacteristic
106C2×C137C22×SD1611yescharacteristic
107C2×C137C22×Q1611yescharacteristic
108D137#48714yesdirect product factor
109C137C2×#48711yescharacteristic, complemented, Sylow
110C2×#487—1371nomaximal, Sylow
111C2×C4⋊Q8—1371no—
112C2×C4⋊Q8—1371no—
113#487—1371no—
114#487—1371no—
115#487—1374no—
116#487—1371no—
117#487D13711yescharacteristic, direct product factor
118C2×#103—1374no—
119C2×C4×C8—1371no—
120C22×Q8—2741no—
121C22×Q8—1372no—
122C22×C8—1372no—
123C4⋊Q8—2741no—
124C4⋊Q8—2741no—
125C4⋊Q8—2741no—
126C4⋊Q8—2741no—
127C4⋊Q8—1372no—
128C4⋊Q8—1372no—
129C4⋊Q8—1371no—
130C4⋊Q8—1371no—
131C4⋊Q8C2×D13711yescharacteristic
132C4⋊Q8C2×D13711yescharacteristic
133C2×C4⋊C4—2741no—
134C2×C4⋊C4—2741no—
135C2×C4⋊C4—2741no—
136C2×C4⋊C4—1372no—
137C2×C42—1371no—
138#103—1374no—
139#103—1374no—
140#103—1374no—
141#103C2×D13714yes—
142C4×C8—1372no—
143C4×C8—1371no—
144C4×C8C2×D13711yescharacteristic
145C2×Q8—21no—
146C2×Q8—2742no—
147C2×Q8—2742no—
148C2×Q8—2741no—
149C2×Q8—2748no—
150C2×Q8—5484no—
151C2×Q8—1372no—
152C2×Q8C22×D13712yes—
153C22×C4—2742no—
154C22×C4—2742no—
155C22×C4—1371no—
156C22×C4—1371no—
157C22×C4—1371no—
158C2×C8—2744no—
159C2×C8—1372no—
160C2×C8C22×D13712yes—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—2742no—
165C4⋊C4—2742no—
166C4⋊C4—2742no—
167C4⋊C4—2742no—
168C4⋊C4—2741no—
169C4⋊C4—2741no—
170C4⋊C4—2741no—
171C4⋊C4—1372no—
172C4⋊C4C22×D13712yes—
173C42—1371no—
174C42—1371no—
175C42—1371no—
176C42C22×D13711yescharacteristic
177C23—1371no—
178Q8—24no—
179Q8—42no—
180Q8—2744no—
181Q8—5484no—
182Q8—5484no—
183Q8—5482no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—2742no—
187C2×C4—2742no—
188C2×C4—5484no—
189C2×C4—5484no—
190C2×C4—1371no—
191C2×C4—1371no—
192C2×C4—1371no—
193C2×C4—1374no—
194C2×C4—1374no—
195C2×C4—1374no—
196C2×C4D4×D13711yescharacteristic
197C2×C4D4×D13711yescharacteristic
198C2×C4C23×D13711yescharacteristic
199C8—22no—
200C8—2742no—
201C22—1372no—
202C22—1372no—
203C22—1372no—
204C22C2×D4×D13711yescharacteristic
205C4—42no—
206C4—42no—
207C4—5482no—
208C4—5482no—
209C4—1372no—
210C4—1372no—
211C4—1372no—
212C4SD16×D13712yes—
213C4Q16×D13712yes—
214C4#41×D13712yes—
215C2—1374no—
216C2#124×D13711yescharacteristic
217C2C2×SD16×D13711yescharacteristic
218C2C2×Q16×D13711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#487×C137C2
socleC22×C137C22×D4
prime radicalC22×D137C2×D4
hypercenter#487D137
hypocenterC137C2×#487
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
3. #22577755C2semidirect
4. #22577754C2semidirect
5. #22577753C2semidirect
6. #22577752C2semidirect
9. #487×C137C2semidirect
108. D137#487direct (N × H)
109. C137C2×#487semidirect
117. #487D137direct (N × H)

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1372
#225777552
#225777542
#225777532
#225777522
#103×D1372
C4×C8×D1372
#487×C1372
C2×#487137