Finite Group Databasebeta

← #31796874

Subgroups — #31796874

Order 17536.

212 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 58 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#22578414C211yesmaximal, characteristic
2#22578396C211yesmaximal, characteristic
3C2×#17984151C211yesmaximal, characteristic
4#22578369C211yesmaximal, characteristic
5#22578282C211yesmaximal, characteristic
6#22578278C211yesmaximal, characteristic
7#101×D137C211yesmaximal, characteristic
8D137⋊M16C2211yescharacteristic
9C2×#14555518C2211yescharacteristic
10#17984167C2211yescharacteristic
11C2×C4⋊#10587739C2211yescharacteristic
12#17984157—21no—
13#17984156—21no—
14#17984151—21no—
15#17984151—21no—
16#17984151—21no—
17#17984151—21no—
18#17984142—21no—
19C8×#10587739—21no—
20C2×Q8×D137C2211yescharacteristic
21M16×D137C2211yescharacteristic
22M16×D137C2211yescharacteristic
23#17984007C411yescharacteristic, complemented
24#17984006C411yescharacteristic, complemented
25#17984006C411yescharacteristic, complemented
26#101×C137C411yescharacteristic, complemented
27C22×#10587739—21no—
28#14555520—21no—
29#14555519—21no—
30#14555518—21no—
31#14555518—21no—
32#14555518D411yescharacteristic
33#14555518D411yescharacteristic
34C2×#11764047—21no—
35C4⋊#10587739—21no—
36C4⋊#10587739—21no—
37C4⋊#10587739D411yescharacteristic
38C4⋊#10587739D411yescharacteristic
39C22⋊#10587739—21no—
40C4×#10587739—21no—
41C2×Dic274—21no—
42C2×Dic274C2×C411yescharacteristic
43Q8×D137—21no—
44Q8×D137—21no—
45Q8×D137—21no—
46Q8×D137—21no—
47C2×C4×D137—21no—
48C2×C4×D137C2311yescharacteristic
49#14555494C2×C411yescharacteristic
50#14555494C2×C411yescharacteristic
51#14555493—21no—
52#14555493—21no—
53C8×D137—21no—
54C8×D137—21no—
55C2×Q8×C137C2×C411yescharacteristic
56M16×C137C2×C411yescharacteristic
57M16×C137C2×C411yescharacteristic
58C2×#10587739—21no—
59C2×#10587739—21no—
60C2×#10587739—41no—
61C2×#10587739—81no—
62#11764047—21no—
63#11764047—21no—
64#11764047—41no—
65C22×D137#4111yescharacteristic
66Dic274—21no—
67Dic274—21no—
68Dic274—41no—
69Dic274—41no—
70C2×Dic137—21no—
71C2×Dic137#4111yescharacteristic
72C4×D137—41no—
73C4×D137—41no—
74C4×D137C2×D411yescharacteristic
75C4×D137C2×D411yescharacteristic
76C4×D137#4111yescharacteristic
77C4×D137#4111yescharacteristic
78#11764040—21no—
79#11764040—21no—
80Q8×C137—21no—
81Q8×C137—21no—
82C2×C4×C137—21no—
83C2×C4×C137C22×C411yescharacteristic
84C8×C137—21no—
85C8×C137—21no—
86C22×C137#11711yescharacteristic
87C2×D137—21no—
88C2×D137—21no—
89C2×D137#12111yescharacteristic
90C2×D137#12611yescharacteristic
91#10587739—41no—
92#10587739—41no—
93#10587739—81no—
94C4×C137—41no—
95C4×C137C4×D411yescharacteristic
96C4×C137C4×D411yescharacteristic
97Dic137—41no—
98Dic137#12311yescharacteristic
99Dic137#12411yescharacteristic
100C2×C137—21no—
101C2×C137#38811yescharacteristic
102D137—21no—
103D137#47111yescharacteristic
104D137#47311yescharacteristic
105C137#193011yescharacteristic, complemented, Sylow
106#1930—1371nomaximal, Sylow
107#428—1371no—
108#419—1371no—
109C2×#103—1371no—
110C2×#101—1371no—
111#405—1371no—
112#343—1371no—
113#341—1371no—
114C22×Q8—1371no—
115C2×M16—1371no—
116C2×M16—1371no—
117C2×M16—1371no—
118C22×C8—1371no—
119#117—1371no—
120C2×C4⋊C4—1371no—
121#107—2741no—
122#106—2741no—
123#103—2741no—
124#103—2741no—
125#103—2741no—
126#103—2741no—
127#101—1371no—
128#101—1371no—
129#101—1371no—
130#101#1058773911yescharacteristic, complemented
131#97—2741no—
132C4×C8—2741no—
133C2×Q8—2741no—
134C2×Q8—2741no—
135C2×Q8—2741no—
136C2×Q8—2741no—
137C2×Q8—2741no—
138C2×Q8—1371no—
139C2×Q8C2×#1058773911yescharacteristic
140C22×C4—2741no—
141C22×C4—2741no—
142C22×C4—1371no—
143M16—2741no—
144M16—2741no—
145M16—2741no—
146M16—2741no—
147M16—1371no—
148M16—1371no—
149M16C2×#1058773911yescharacteristic
150M16C2×#1058773911yescharacteristic
151C2×C8—2741no—
152C2×C8—2741no—
153C2×C8—2741no—
154C2×C8—2741no—
155C2×C8—2741no—
156C2×C8—1371no—
157C2×C8—1371no—
158C4⋊C4—2741no—
159C4⋊C4—2741no—
160C4⋊C4—1371no—
161C4⋊C4—1371no—
162C22⋊C4—2741no—
163C42—2741no—
164C23—1371no—
165Q8—21no—
166Q8—21no—
167Q8—2741no—
168Q8—2741no—
169Q8—5481no—
170Q8—5481no—
171C2×C4—21no—
172C2×C4—2741no—
173C2×C4—2741no—
174C2×C4—2741no—
175C2×C4—5481no—
176C2×C4—5481no—
177C2×C4—5481no—
178C2×C4—10961no—
179C2×C4—1371no—
180C2×C4—1371no—
181C2×C4—1371no—
182C2×C4—1371no—
183C2×C4—1371no—
184C2×C4C22×#1058773911yescharacteristic
185C8—21no—
186C8—21no—
187C8—2741no—
188C8—2741no—
189C8—2741no—
190C8—2741no—
191C8—5481no—
192C22—2741no—
193C22—2741no—
194C22—1371no—
195C22—1371no—
196C22#1798416711yescharacteristic
197C4—41no—
198C4—5481no—
199C4—5481no—
200C4—5481no—
201C4—10961no—
202C4—1371no—
203C4—1371no—
204C4D4×#1058773911yescharacteristic
205C4D4×#1058773911yescharacteristic
206C2—21no—
207C2—2741no—
208C2—1371no—
209C2—1371no—
210C2#2257835311yescharacteristic
subgroupG/H
centerC2#22578353
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#101×C137C4
socleC2×C137#388
prime radicalC22×D137#41
cosocleC2×C4×D137C23
hypercenter#101#10587739
hypocenterC137#1930
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
23. #17984007C4semidirect
24. #17984006C4semidirect
25. #17984006C4semidirect
26. #101×C137C4semidirect
105. C137#1930semidirect
130. #101#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784142
#225783962
C2×#179841512
#225783692
#225782822
#225782782
#101×D1372
#1930137