Finite Group Databasebeta

← #31796660

Subgroups — #31796660

Order 17536.

292 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 84 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#22578404C211yesmaximal, characteristic
2C2×#17984151C212yesmaximal
3#22578347C211yesmaximal, characteristic
4#22578276C212yesmaximal
5C4×Q8×D137C211yesmaximal, characteristic
6C2×#14555518C2212yes—
7C2×C4⋊#10587739C2212yes—
8#17984155—21no—
9C4⋊#11764047—21no—
10#17984151—24no—
11(C2×#10587739)⋊C4—21no—
12(C2×#10587739)⋊C4—21no—
13C2×Q8×D137C2211yescharacteristic
14Q8×Dic137C411yescharacteristic, complemented
15C4×Dic274—21no—
16C4×Dic274C411yescharacteristic, complemented
17#17984053—21no—
18#17984053C411yescharacteristic, complemented
19C4⋊C4×D137—21no—
20C4⋊C4×D137C2211yescharacteristic
21C42×D137—21no—
22C42×D137C2211yescharacteristic
23C4×Q8×C137C411yescharacteristic, complemented
24C22×#10587739—22no—
25#14555518—22no—
26#14555518—22no—
27C2×#11764047—22no—
28C4⋊#10587739—22no—
29C4⋊#10587739—22no—
30C2×Dic274—21no—
31C2×Dic274C2×C411yescharacteristic
32Q8×D137—21no—
33Q8×D137—21no—
34Q8×D137D412yes—
35Q8×D137Q812yes—
36C2×C4×D137—21no—
37C2×C4×D137—21no—
38C2×C4×D137D411yescharacteristic
39C2×C4×D137D411yescharacteristic
40C2×C4×D137C2311yescharacteristic
41C4⋊Dic137—21no—
42C4⋊Dic137C2×C411yescharacteristic
43#14555489—21no—
44#14555489—21no—
45#14555489—21no—
46C4×Dic137—21no—
47C4×Dic137—21no—
48C4×Dic137—21no—
49C4×Dic137C2×C411yescharacteristic
50C4×Dic137D411yescharacteristic
51C4×Dic137D411yescharacteristic
52C2×Q8×C137C2×C411yescharacteristic
53C4⋊C4×C137—21no—
54C4⋊C4×C137C2×C411yescharacteristic
55C42×C137—21no—
56C42×C137C2×C411yescharacteristic
57C2×#10587739—42no—
58C2×#10587739—42no—
59C2×#10587739—42no—
60#11764047—42no—
61C22×D137C2×D411yescharacteristic
62Dic274—21no—
63Dic274—22no—
64Dic274—22no—
65Dic274C4⋊C412yes—
66C2×Dic137—21no—
67C2×Dic137—21no—
68C2×Dic137C22⋊C411yescharacteristic
69C2×Dic137C22⋊C411yescharacteristic
70C2×Dic137C2×D411yescharacteristic
71C4×D137—22no—
72C4×D137—22no—
73C4×D137—22no—
74C4×D137—22no—
75C4×D137—42no—
76C4×D137C2×D411yescharacteristic
77C4×D137C2×Q811yescharacteristic
78C4×D137#4111yescharacteristic
79C4×D137#4111yescharacteristic
80Q8×C137—21no—
81Q8×C137C4⋊C412yes—
82C2×C4×C137—21no—
83C2×C4×C137—21no—
84C2×C4×C137C22⋊C411yescharacteristic
85C2×C4×C137C22⋊C411yescharacteristic
86C2×C4×C137C22×C411yescharacteristic
87C22×C137C2×C22⋊C411yescharacteristic
88C2×D137#12111yescharacteristic
89C2×D137C22⋊Q811yescharacteristic
90C2×D137#13512yes—
91C2×D137#13712yes—
92#10587739—82no—
93C4×C137—22no—
94C4×C137—21no—
95C4×C137—21no—
96C4×C137—41no—
97C4×C137C2×C4⋊C411yescharacteristic
98C4×C137#11711yescharacteristic
99Dic137—22no—
100Dic137—21no—
101Dic137—21no—
102Dic137—41no—
103Dic137#12111yescharacteristic
104Dic137C22⋊Q811yescharacteristic
105C2×C137#38011yescharacteristic
106C2×C137#41611yescharacteristic
107C2×C137#41911yescharacteristic
108D137#46412yes—
109D137#47912yes—
110C137#180511yescharacteristic, complemented, Sylow
111#1805—1371nomaximal, Sylow
112C2×C4×Q8—1371no—
113C2×C4⋊C8—1371no—
114C2×#103—1372no—
115#382—1371no—
116#340—1372no—
117C22×Q8—1371no—
118C22×C8—1372no—
119C4×Q8—2741no—
120C4×Q8—2741no—
121C4×Q8—1371no—
122C4×Q8—1371no—
123C4×Q8—1371no—
124C4×Q8#1058773911yescharacteristic, complemented
125C2×C4⋊C4—2741no—
126C2×C4⋊C4—1372no—
127C2×C4⋊C4—1371no—
128C2×C42—2741no—
129C2×C42—1371no—
130C4⋊C8—2741no—
131C4⋊C8—2741no—
132#103—2744no—
133#95—2741no—
134#95—2741no—
135C2×Q8—2741no—
136C2×Q8—2741no—
137C2×Q8—2741no—
138C2×Q8—1372no—
139C2×Q8—1372no—
140C2×Q8—1371no—
141C2×Q8C2×#1058773911yescharacteristic
142C22×C4—2741no—
143C22×C4—2741no—
144C22×C4—2742no—
145C22×C4—1371no—
146C22×C4—1371no—
147C22×C4—1371no—
148C2×C8—2742no—
149C2×C8—2742no—
150C2×C8—2742no—
151C4⋊C4—21no—
152C4⋊C4—2741no—
153C4⋊C4—2741no—
154C4⋊C4—2741no—
155C4⋊C4—2741no—
156C4⋊C4—2742no—
157C4⋊C4—2742no—
158C4⋊C4—1371no—
159C4⋊C4C2×#1058773911yescharacteristic
160C42—21no—
161C42—2741no—
162C42—2741no—
163C42—2741no—
164C42—1371no—
165C42—1371no—
166C42—1371no—
167C42C2×#1058773911yescharacteristic
168C23—1371no—
169Q8—21no—
170Q8—2742no—
171Q8—2742no—
172Q8—2741no—
173Q8—1372no—
174Q8C4⋊#1058773912yes—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—2742no—
178C2×C4—2742no—
179C2×C4—2742no—
180C2×C4—2742no—
181C2×C4—2741no—
182C2×C4—2741no—
183C2×C4—5482no—
184C2×C4—5482no—
185C2×C4—5482no—
186C2×C4—5482no—
187C2×C4—1371no—
188C2×C4—1371no—
189C2×C4—1371no—
190C2×C4—1371no—
191C2×C4—1371no—
192C2×C4—1371no—
193C2×C4—1371no—
194C2×C4C22⋊#1058773911yescharacteristic
195C2×C4C22⋊#1058773911yescharacteristic
196C2×C4C22×#1058773911yescharacteristic
197C8—5482no—
198C22—1371no—
199C22—1371no—
200C22—1372no—
201C22—1372no—
202C22C2×C22⋊#1058773911yescharacteristic
203C4—22no—
204C4—21no—
205C4—21no—
206C4—41no—
207C4—2742no—
208C4—2741no—
209C4—2741no—
210C4—5481no—
211C4—10962no—
212C4—1371no—
213C4—1371no—
214C4C2×C4⋊#1058773911yescharacteristic
215C4#1798416711yescharacteristic
216C2—1372no—
217C2—1372no—
218C2#2257834511yescharacteristic
219C2#2257839311yescharacteristic
220C2#2257839611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4×Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×D137C2×D4
cosocleC2×C4×D137C23
hypercenterC4×Q8#10587739
hypocenterC137#1805
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
14. Q8×Dic137C4semidirect
16. C4×Dic274C4semidirect
18. #17984053C4semidirect
23. C4×Q8×C137C4semidirect
110. C137#1805semidirect
124. C4×Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784042
C2×#179841512
#225783472
#225782762
C4×Q8×D1372
#1805137