Finite Group Databasebeta

← #36323367

Subgroups — #36323367

Order 21376.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 53 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#24923985C211yesmaximal, characteristic
2#24923619C211yesmaximal, characteristic, complemented
3#24923550C211yesmaximal, characteristic, complemented
4#24923475C211yesmaximal, characteristic, complemented
5#24922996C211yesmaximal, characteristic, complemented
6#24922996C211yesmaximal, characteristic, complemented
7#333×C167C211yesmaximal, characteristic, complemented
8C2×#14581939—21no—
9#19128832C2211yescharacteristic
10#19128821—22no—
11#19128807—21no—
12#19128807—21no—
13C4×D668C2211yescharacteristic
14C4×D668C2211yescharacteristic
15#19128790—21no—
16#19128790—21no—
17#19128766—21no—
18#19128757—21no—
19#19128756—22no—
20#19128754—21no—
21C4×C167⋊C8C2211yescharacteristic
22C4⋊Q8×C167C2211yescharacteristic
23C4⋊C8×C167C2211yescharacteristic
24C4⋊C8×C167C2211yescharacteristic
25C2×Dic334—21no—
26C2×D668—22no—
27C2×D668D411yescharacteristic
28C2×D668D411yescharacteristic
29C2×C4×D167—21no—
30C2×C4×D167—21no—
31#14581939—42no—
32C2×C167⋊C8—21no—
33C4⋊Dic167—21no—
34C4⋊Dic167—21no—
35C4⋊Dic167D411yescharacteristic
36C4⋊Dic167D411yescharacteristic
37(C2×D167)⋊C4—22no—
38(C2×D167)⋊C4—22no—
39(C2×D167)⋊C4—21no—
40(C2×D167)⋊C4—21no—
41C4×Dic167—22no—
42C2×Q8×C167D412yes—
43C2×C8×C167—21no—
44C2×C8×C167—21no—
45C4⋊C4×C167—21no—
46C4⋊C4×C167—21no—
47C42×C167C2311yescharacteristic
48C22×D167—21no—
49C22×D167—21no—
50Dic334—41no—
51Dic334—41no—
52D668—21no—
53D668—21no—
54D668—21no—
55D668—21no—
56D668—82no—
57C2×Dic167—21no—
58C2×Dic167—21no—
59C2×Dic167—41no—
60C4×D167—41no—
61C4×D167—41no—
62C4×D167—41no—
63C4×D167—41no—
64C167⋊C8—41no—
65Q8×C167—42no—
66C2×C4×C167—22no—
67C2×C4×C167C2×D411yescharacteristic
68C2×C4×C167C2×D411yescharacteristic
69C2×C4×C167C2×D411yescharacteristic
70C8×C167—41no—
71C8×C167—41no—
72C22×C167#12011yescharacteristic
73C2×D167—41no—
74C2×D167—41no—
75C2×D167—41no—
76C2×D167—41no—
77C2×D167—41no—
78C2×D167—41no—
79C4×C167—21no—
80C4×C167—42no—
81C4×C167#13511yescharacteristic
82C4×C167#13511yescharacteristic
83C4×C167AGL(1,ℤ/8)11yescharacteristic
84C4×C167#13711yescharacteristic
85Dic167—41no—
86Dic167—41no—
87Dic167—81no—
88C2×C167#44911yescharacteristic
89C2×C167#45211yescharacteristic
90C2×C167#45511yescharacteristic
91D167—81no—
92D167—81no—
93C167#164611yescharacteristic, complemented, Sylow
94#1646—1671nomaximal, Sylow
95#555—1671no—
96#487—1671no—
97#474—1671no—
98#461—1671no—
99#333D16711yescharacteristic, complemented
100D4⋊C8—1671no—
101D4⋊C8—1671no—
102C2×SD16—3341no—
103C4⋊Q8—1671no—
104C4⋊Q8C2×D16711yescharacteristic
105#124—3342no—
106#121—3341no—
107#121—3341no—
108C4×D4—1671no—
109C4×D4—1671no—
110#117—3341no—
111#117—3341no—
112#107—3341no—
113C4⋊C8C2×D16711yescharacteristic
114C4⋊C8C2×D16711yescharacteristic
115#103—3342no—
116#103—3341no—
117#102—3341no—
118C4×C8—1671no—
119C2×Q8—3341no—
120C2×Q8D66812yes—
121C2×D4—3342no—
122C2×D4—1671no—
123C2×D4—1671no—
124C22×C4—3341no—
125C22×C4—3341no—
126SD16—6682no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—3341no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—3341no—
133C4⋊C4—3341no—
134C4⋊C4—1671no—
135C4⋊C4—1671no—
136C22⋊C4—3342no—
137C22⋊C4—3342no—
138C22⋊C4—3341no—
139C22⋊C4—3341no—
140C42—3342no—
141C42C22×D16711yescharacteristic
142C23—3341no—
143C23—3341no—
144Q8—42no—
145Q8—6681no—
146Q8—6681no—
147D4—13362no—
148D4—3341no—
149D4—3341no—
150D4—3341no—
151D4—3341no—
152C2×C4—22no—
153C2×C4—3341no—
154C2×C4—3341no—
155C2×C4—6681no—
156C2×C4—6681no—
157C2×C4—6681no—
158C2×C4—6681no—
159C2×C4—6681no—
160C2×C4D4×D16711yescharacteristic
161C2×C4D4×D16711yescharacteristic
162C2×C4C2×D66811yescharacteristic
163C8—41no—
164C8—41no—
165C8—6681no—
166C22—6681no—
167C22—6681no—
168C22—6681no—
169C22—6681no—
170C22—6681no—
171C22—6681no—
172C22#1912880211yescharacteristic
173C4—21no—
174C4—42no—
175C4—13361no—
176C4—6681no—
177C4—6681no—
178C4#1912885311yescharacteristic
179C4#1912885511yescharacteristic
180C4#1912885911yescharacteristic
181C4#1912886411yescharacteristic
182C2—13361no—
183C2—13361no—
184C2#2492341711yescharacteristic
185C2#2492343411yescharacteristic
186C2#2492344911yescharacteristic
subgroupG/H
centerC22#19128802
commutator, cosocleC42×C167C23
FrattiniC42C22×D167
Fitting#333×C167C2
socleC22×C167#120
prime radical#24923985C2
hypercenter#333D167
hypocenterC167#1646
LCSG ▷ C42×C167 ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #24923619C2semidirect
3. #24923550C2semidirect
4. #24923475C2semidirect
5. #24922996C2semidirect
6. #24922996C2semidirect
7. #333×C167C2semidirect
93. C167#1646semidirect
99. #333D167semidirect

Maximal subgroups

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

subgroupindex
#249239852
#249236192
#249235502
#249234752
#249229962
#333×C1672
#1646167