Finite Group Databasebeta

← #36323273

Subgroups — #36323273

Order 21376.

214 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 55 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#24923980C211yesmaximal, characteristic
2#24923648C211yesmaximal, characteristic, complemented
3#24923565C211yesmaximal, characteristic, complemented
4#24923560C211yesmaximal, characteristic
5#24922999C211yesmaximal, characteristic, complemented
6#24922996C211yesmaximal, characteristic
7#333×C167C211yesmaximal, characteristic, complemented
8C2×#14581940—22no—
9#19128832C2211yescharacteristic
10#19128823—21no—
11#19128812—21no—
12#19128808—22no—
13C4×Dic334C2211yescharacteristic
14C4×D668C2211yescharacteristic
15#19128794—21no—
16C4⋊C4×D167—21no—
17#19128766—21no—
18#19128764—21no—
19#19128757—21no—
20#19128754—21no—
21C4×C167⋊C8C2211yescharacteristic
22C4⋊Q8×C167C2211yescharacteristic
23C4⋊C8×C167C2211yescharacteristic
24C4⋊C8×C167C2211yescharacteristic
25C2×Dic334—21no—
26C2×Dic334D411yescharacteristic
27C2×D668D411yescharacteristic
28C2×C4×D167—22no—
29C2×C4×D167—21no—
30#14581940—44no—
31C2×C167⋊C8—21no—
32C4⋊Dic167—22no—
33C4⋊Dic167—21no—
34C4⋊Dic167—21no—
35C4⋊Dic167D411yescharacteristic
36C4⋊Dic167D411yescharacteristic
37Dic167⋊C4—22no—
38Dic167⋊C4—21no—
39(C2×D167)⋊C4—22no—
40(C2×D167)⋊C4—21no—
41C4×Dic167—21no—
42C2×Q8×C167D412yes—
43C2×C8×C167—21no—
44C2×C8×C167—21no—
45C4⋊C4×C167—21no—
46C4⋊C4×C167—21no—
47C42×C167C2311yescharacteristic
48C22×D167—21no—
49Dic334—21no—
50Dic334—21no—
51Dic334—42no—
52D668—21no—
53D668Q1612yescomplemented
54C2×Dic167—21no—
55C2×Dic167—21no—
56C2×Dic167—21no—
57C2×Dic167—41no—
58C4×D167—41no—
59C4×D167—41no—
60C4×D167—82no—
61C167⋊C8—22no—
62Q8×C167—42no—
63C2×C4×C167—22no—
64C2×C4×C167C2×D411yescharacteristic
65C2×C4×C167C2×D411yescharacteristic
66C2×C4×C167C2×D411yescharacteristic
67C8×C167—41no—
68C8×C167—41no—
69C22×C167#12011yescharacteristic
70C2×D167—22no—
71C2×D167—41no—
72C2×D167—41no—
73C4×C167—21no—
74C4×C167—42no—
75C4×C167C2×Q1611yescharacteristic
76C4×C167#13511yescharacteristic
77C4×C167#13711yescharacteristic
78C4×C167#13711yescharacteristic
79Dic167—41no—
80Dic167—41no—
81Dic167—41no—
82Dic167—81no—
83C2×C167#45111yescharacteristic
84C2×C167#45211yescharacteristic
85C2×C167#45311yescharacteristic
86D167—42no—
87C167#163211yescharacteristic, complemented, Sylow
88#1632—1671nomaximal, Sylow
89#554—1671no—
90#494—1671no—
91#477—1671no—
92#476—1671no—
93#333D16711yescharacteristic, complemented
94Q8⋊C8—1671no—
95D4⋊C8—1671no—
96C2×Q16—3342no—
97C4⋊Q8—1671no—
98C4⋊Q8C2×D16711yescharacteristic
99#125—3341no—
100C22⋊Q8—3341no—
101C22⋊Q8—3342no—
102C4×Q8—1671no—
103C4×D4—3341no—
104C4×D4—1671no—
105C2×C4⋊C4—3341no—
106#107—3341no—
107#106—3341no—
108C4⋊C8C2×D16711yescharacteristic
109C4⋊C8C2×D16711yescharacteristic
110#103—3341no—
111#102—3341no—
112C4×C8—1671no—
113C2×Q8—3341no—
114C2×Q8—1671no—
115C2×Q8D66812yes—
116C2×D4—1671no—
117C22×C4—3342no—
118C22×C4—3341no—
119Q16—6684no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—3341no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—3342no—
126C4⋊C4—3342no—
127C4⋊C4—3341no—
128C4⋊C4—3341no—
129C4⋊C4—3341no—
130C4⋊C4—1671no—
131C4⋊C4—1671no—
132C22⋊C4—3342no—
133C22⋊C4—3341no—
134C42—3341no—
135C42C22×D16711yescharacteristic
136C23—3341no—
137Q8—42no—
138Q8—3341no—
139Q8—3341no—
140Q8—6682no—
141D4—3341no—
142D4—1672no—
143C2×C4—22no—
144C2×C4—13362no—
145C2×C4—3341no—
146C2×C4—3341no—
147C2×C4—3341no—
148C2×C4—6681no—
149C2×C4—6681no—
150C2×C4—6681no—
151C2×C4D4×D16711yescharacteristic
152C2×C4D4×D16711yescharacteristic
153C2×C4C2×D66811yescharacteristic
154C8—41no—
155C8—41no—
156C8—3342no—
157C22—3342no—
158C22—6681no—
159C22—6681no—
160C22#1912880211yescharacteristic
161C4—21no—
162C4—42no—
163C4—13361no—
164C4—6681no—
165C4—6681no—
166C4—6681no—
167C4Q16×D16711yescharacteristic
168C4#1912885311yescharacteristic
169C4#1912886111yescharacteristic
170C4#1912886411yescharacteristic
171C2—6682no—
172C2#2492342611yescharacteristic
173C2#2492343411yescharacteristic
174C2#2492343911yescharacteristic
subgroupG/H
centerC22#19128802
commutator, cosocleC42×C167C23
FrattiniC42C22×D167
Fitting#333×C167C2
socleC22×C167#120
prime radicalC2×D668D4
hypercenter#333D167
hypocenterC167#1632
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. #24923648C2semidirect
3. #24923565C2semidirect
5. #24922999C2semidirect
7. #333×C167C2semidirect
53. D668Q16semidirect
87. C167#1632semidirect
93. #333D167semidirect

Maximal subgroups

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

subgroupindex
#249239802
#249236482
#249235652
#249235602
#249229992
#249229962
#333×C1672
#1632167