Finite Group Databasebeta

← #31793462

Subgroups — #31793462

Order 17536.

336 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 125 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
1Q8×Dic274C211yesmaximal, characteristic
2#22578128C211yesmaximal, characteristic
3#22577753C211yesmaximal, characteristic
4#22577752C212yesmaximal
5#22577699C211yesmaximal, characteristic
6#22577696C212yesmaximal
7#22577611C212yesmaximal
8#22577527C211yesmaximal, characteristic
9Q16×Dic137C212yesmaximal
10C4×#14555500C211yesmaximal, characteristic
11#462×C137C211yesmaximal, characteristic
12C2×#14555500C2211yescharacteristic
13#17984088—21no—
14#17984088C2211yescharacteristic
15#17984087—22no—
16#17984087C2212yes—
17#17984086C2212yes—
18#17984079—21no—
19#17984078—22no—
20Q8×Dic137C2212yes—
21Q8×Dic137C2212yes—
22C4×Dic274—21no—
23C4×Dic274C2211yescharacteristic
24C4×Dic274C2211yescharacteristic
25#17984053—21no—
26#17984022C2212yes—
27#17984021C2211yescharacteristic
28#17984017C2212yes—
29#17984013C2212yes—
30#17984012C2212yes—
31#17984012C2211yescharacteristic
32#17984012C2212yes—
33#17984011C2211yescharacteristic
34C4×#11764040C2211yescharacteristic
35C8×Dic137C2212yes—
36C2×Q16×C137C2212yes—
37C4⋊Q8×C137C2211yescharacteristic
38C4×Q8×C137C2211yescharacteristic
39C4⋊C8×C137C2211yescharacteristic
40#103×C137C2212yes—
41C2×Dic274—22no—
42C2×Dic274—21no—
43C2×Dic274D411yescharacteristic
44C2×Dic274C2311yescharacteristic
45#14555500—22no—
46C2×#11764040C2311yescharacteristic
47C2×#11764040C2311yescharacteristic
48C4⋊Dic137—22no—
49C4⋊Dic137—22no—
50C4⋊Dic137—21no—
51C4⋊Dic137D411yescharacteristic
52C4⋊Dic137C2312yes—
53C4⋊Dic137C2311yescharacteristic
54#14555489—22no—
55#14555489—22no—
56#14555489—22no—
57#14555489—21no—
58#14555489D412yes—
59C4×Dic137—22no—
60C4×Dic137—22no—
61C4×Dic137—21no—
62C4×Dic137C2312yes—
63C2×Q8×C137C2312yes—
64C2×Q8×C137C2311yescharacteristic
65Q16×C137—24no—
66C2×C8×C137C2312yes—
67C4⋊C4×C137—21no—
68C4⋊C4×C137—21no—
69C4⋊C4×C137—21no—
70C4⋊C4×C137C2311yescharacteristic
71C42×C137—21no—
72C42×C137C2311yescharacteristic
73Dic274—21no—
74Dic274—21no—
75Dic274—82no—
76Dic274—81no—
77Dic274Q1614yescomplemented
78C2×Dic137—21no—
79C2×Dic137—21no—
80C2×Dic137—22no—
81C2×Dic137C2×D412yes—
82C2×Dic137C2×D412yes—
83#11764040—21no—
84#11764040—21no—
85Q8×C137—21no—
86Q8×C137—22no—
87Q8×C137—22no—
88Q8×C137#4112yes—
89C2×C4×C137—21no—
90C2×C4×C137—21no—
91C2×C4×C137—22no—
92C2×C4×C137C2×D411yescharacteristic
93C2×C4×C137C2×D411yescharacteristic
94C2×C4×C137C2411yescharacteristic
95C8×C137—22no—
96C22×C137C22×D411yescharacteristic
97C4×C137—22no—
98C4×C137—21no—
99C4×C137—42no—
100C4×C137—41no—
101C4×C137C2×Q1612yes—
102C4×C137C2×#4111yescharacteristic
103C4×C1372−1+411yescharacteristic
104Dic137—22no—
105Dic137—41no—
106Dic137—41no—
107Dic137—42no—
108Dic137C2×Q1614yes—
109C2×C137#54811yescharacteristic
110C2×C137C22×Q1611yescharacteristic
111C2×C137#57711yescharacteristic
112C137#336311yescharacteristic, complemented, Sylow
113#3363—1371nomaximal, Sylow
114Q82—1371no—
115#557—1371no—
116#487—1371no—
117#487—1372no—
118#477—1371no—
119#477—1372no—
120#462—1372no—
121#462D13711yescharacteristic
122C8×Q8—1371no—
123C4×Q16—1372no—
124C4×Q16—1371no—
125C2×Q16—1371no—
126C2×Q16C2×D13712yes—
127C4⋊Q8—2741no—
128C4⋊Q8—2742no—
129C4⋊Q8—1372no—
130C4⋊Q8—1372no—
131C4⋊Q8—1371no—
132C4⋊Q8C2×D13711yescharacteristic
133#125—2742no—
134#125—2741no—
135C4×Q8—2741no—
136C4×Q8—2741no—
137C4×Q8—1372no—
138C4×Q8—1372no—
139C4×Q8—1371no—
140C4×Q8—1371no—
141C4×Q8C2×D13711yescharacteristic
142#107—1372no—
143#107—1371no—
144C4⋊C8—1372no—
145C4⋊C8C2×D13711yescharacteristic
146#103—1371no—
147#103—1372no—
148#103—1372no—
149#103—1372no—
150#103—1371no—
151#103C2×D13712yes—
152C4×C8—1372no—
153C4×C8—1371no—
154C2×Q8—2742no—
155C2×Q8—2741no—
156C2×Q8—1371no—
157C2×Q8—1371no—
158C2×Q8C22×D13712yes—
159C2×Q8C22×D13711yescharacteristic
160Q16—24no—
161Q16—2742no—
162C2×C8—1371no—
163C2×C8—1371no—
164C2×C8C22×D13712yes—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—2742no—
169C4⋊C4—2742no—
170C4⋊C4—2742no—
171C4⋊C4—2742no—
172C4⋊C4—2742no—
173C4⋊C4—2741no—
174C4⋊C4—2741no—
175C4⋊C4—1372no—
176C4⋊C4—1371no—
177C4⋊C4—1372no—
178C4⋊C4—1371no—
179C4⋊C4C22×D13711yescharacteristic
180C42—21no—
181C42—2742no—
182C42—2742no—
183C42—2741no—
184C42—1372no—
185C42C22×D13711yescharacteristic
186Q8—21no—
187Q8—22no—
188Q8—22no—
189Q8—2741no—
190Q8—2741no—
191Q8—10961no—
192Q8—10962no—
193Q8—1374no—
194Q8#1455550912yes—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—2741no—
199C2×C4—2741no—
200C2×C4—2742no—
201C2×C4—1372no—
202C2×C4—1372no—
203C2×C4D4×D13711yescharacteristic
204C2×C4D4×D13711yescharacteristic
205C2×C4C23×D13711yescharacteristic
206C8—22no—
207C8—2741no—
208C8—2741no—
209C22C2×D4×D13711yescharacteristic
210C4—21no—
211C4—22no—
212C4—41no—
213C4—42no—
214C4—2742no—
215C4—5482no—
216C4—5481no—
217C4—5481no—
218C4—1374no—
219C4Q16×D13712yes—
220C4C2×#1455550911yescharacteristic
221C4#1798413511yescharacteristic
222C2#2257805911yescharacteristic
223C2C2×Q16×D13711yescharacteristic
224C2#2257821211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#462×C137C2
socle, prime radicalC22×C137C22×D4
hypercenter#462D137
hypocenterC137#3363
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
77. Dic274Q16semidirect
112. C137#3363semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic2742
#225781282
#225777532
#225777522
#225776992
#225776962
#225776112
#225775272
Q16×Dic1372
C4×#145555002
#462×C1372
#3363137