Finite Group Databasebeta

← #31792236

Subgroups — #31792236

Order 17536.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#22578015C211yesmaximal, characteristic, complemented
2#22578008C211yesmaximal, characteristic
3#22577711C212yesmaximal, complemented
4#22577704C212yesmaximal, complemented
5#22577649C212yesmaximal, complemented
6#22577632C212yesmaximal, complemented
7#22577609C212yesmaximal, complemented
8#22577604C212yesmaximal, complemented
9#424×C137C211yesmaximal, characteristic, complemented
10C2×#14555499C2212yes—
11C2×#14555499C2212yes—
12#17984088C2211yescharacteristic
13#17984085C2211yescharacteristic
14#17984081—22no—
15#17984080C2211yescharacteristic
16#17984080C2211yescharacteristic
17#17984078—22no—
18#17984073—22no—
19#17984067C2212yes—
20#17984062C2212yes—
21#17984061—22no—
22C4×Dic274C2212yes—
23C4×D548C2212yes—
24#17984022C2212yes—
25#17984020C2212yes—
26#17984013C2212yes—
27#17984013C2212yes—
28#17984010C2212yes—
29#17984010C2212yes—
30C2×M16×C137C2211yescharacteristic
31C22×C8×C137C2211yescharacteristic
32#117×C137C2211yescharacteristic
33C4⋊C8×C137C2212yes—
34C4⋊C8×C137C2212yes—
35C2×Dic274C2312yes—
36C2×D548—22no—
37C2×D548C2312yes—
38C2×C22⋊D137—22no—
39C2×C4×D137—22no—
40#14555499—42no—
41#14555499—42no—
42C4⋊Dic137—22no—
43C4⋊Dic137C2312yes—
44C4⋊Dic137C2312yes—
45#14555489—22no—
46#14555489—22no—
47#14555489—22no—
48#14555489—22no—
49C22⋊Dic137—22no—
50#14555487—22no—
51#14555487—22no—
52C4×Dic137—22no—
53C22×C4×C137C2311yescharacteristic
54M16×C137—41no—
55C2×C8×C137—41no—
56C2×C8×C137C2312yes—
57C2×C8×C137C2312yes—
58C4⋊C4×C137D412yes—
59C22⋊C4×C137D412yes—
60C42×C137C2311yescharacteristic
61C42×C137C2311yescharacteristic
62C22×D137—22no—
63Dic274—22no—
64Dic274—22no—
65D548—22no—
66D548—22no—
67D548—82no—
68C22⋊D137—82no—
69C2×Dic137—22no—
70C2×Dic137—22no—
71C2×Dic137—22no—
72C4×D137—82no—
73C23×C137C2×D411yescharacteristic
74C2×C4×C137—21no—
75C2×C4×C137—21no—
76C2×C4×C137C2×D412yes—
77C2×C4×C137C2×D412yes—
78C2×C4×C137C2×D411yescharacteristic
79C2×C4×C137C2411yescharacteristic
80C8×C137—22no—
81C8×C137—22no—
82C22×C137—21no—
83C22×C137—22no—
84C22×C137C22×D411yescharacteristic
85C2×D137—42no—
86C2×D137—42no—
87C2×D137—42no—
88C4×C137—21no—
89C4×C137—22no—
90C4×C137—22no—
91C4×C137H(5,𝔽2)11yescharacteristic
92C4×C1372−1+411yescharacteristic
93Dic137—42no—
94Dic137—42no—
95Dic137—42no—
96C2×C137—41no—
97C2×C137#53711yescharacteristic
98C2×C137#57712yes—
99D137—82no—
100C137#326011yescharacteristic, complemented, Sylow
101#3260—1371nomaximal, Sylow
102#541—1371no—
103#540—1371no—
104#479—1372no—
105#478—1372no—
106#468—1372no—
107#465—1372no—
108#461—1372no—
109#460—1372no—
110#424D13711yescharacteristic, complemented
111C2×SD16—1372no—
112C2×SD16—1372no—
113C2×M16C2×D13711yescharacteristic
114C22×C8C2×D13711yescharacteristic
115C4⋊Q8—1371no—
116#127—1371no—
117#126—2742no—
118#125—2742no—
119#125—1371no—
120#125—1371no—
121#123—2742no—
122C22⋊Q8—1372no—
123#121—2742no—
124#121—1372no—
125C4×Q8—1372no—
126C4×D4—1372no—
127#117C2×D13711yescharacteristic
128#107—1372no—
129#106—1372no—
130C4⋊C8C2×D13712yes—
131C4⋊C8C2×D13712yes—
132#103—1372no—
133#103—1372no—
134#102—1372no—
135#102—1372no—
136C2×Q8—1372no—
137C2×D4—2742no—
138C2×D4—2742no—
139C2×D4—1372no—
140C22×C4—2742no—
141C22×C4C22×D13711yescharacteristic
142SD16—5482no—
143SD16—5482no—
144M16—41no—
145C2×C8—41no—
146C2×C8C22×D13712yes—
147C2×C8C22×D13712yes—
148C4⋊C4—2742no—
149C4⋊C4—2742no—
150C4⋊C4—2742no—
151C4⋊C4—2742no—
152C4⋊C4—2742no—
153C4⋊C4—1372no—
154C4⋊C4—1372no—
155C4⋊C4D54812yes—
156C22⋊C4—2742no—
157C22⋊C4—2742no—
158C22⋊C4—2742no—
159C22⋊C4D54812yes—
160C42—2742no—
161C42C22×D13711yescharacteristic
162C42C22×D13711yescharacteristic
163C23—2742no—
164C23C2×D54811yescharacteristic
165Q8—2742no—
166Q8—2742no—
167D4—2742no—
168D4—2742no—
169D4—10962no—
170D4—10962no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—2742no—
174C2×C4—2742no—
175C2×C4—2742no—
176C2×C4—10962no—
177C2×C4C2×D54812yes—
178C2×C4C2×D54812yes—
179C2×C4C2×D54811yescharacteristic
180C2×C4C23×D13711yescharacteristic
181C8—22no—
182C8—22no—
183C22—22no—
184C22—21no—
185C22—5482no—
186C22—5482no—
187C22—5482no—
188C22C22×D54811yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—5482no—
193C4—5482no—
194C4—5482no—
195C4#1798413311yescharacteristic
196C4#1798413611yescharacteristic
197C2—41no—
198C2—10962no—
199C2#2257798411yescharacteristic
200C2#2257821512yes—
subgroupG/H
centerC22C22×D548
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#424×C137C2
socleC22×C137C22×D4
prime radical#22578008C2
hypercenter#424D137
hypocenterC137#3260
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #22578015C2semidirect
3. #22577711C2semidirect
4. #22577704C2semidirect
5. #22577649C2semidirect
6. #22577632C2semidirect
7. #22577609C2semidirect
8. #22577604C2semidirect
9. #424×C137C2semidirect
100. C137#3260semidirect
110. #424D137semidirect

Maximal subgroups

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

subgroupindex
#225780152
#225780082
#225777112
#225777042
#225776492
#225776322
#225776092
#225776042
#424×C1372
#3260137