Finite Group Databasebeta

← #36324554

Subgroups — #36324554

Order 21376.

342 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 85 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#24924036C211yesmaximal, characteristic, complemented
2#24923776C211yesmaximal, characteristic
3#24923232C212yesmaximal, complemented
4#24923041C212yesmaximal, complemented
5#413×C167C211yesmaximal, characteristic, complemented
6C22×C4×D167C2211yescharacteristic
7#19128839—24no—
8C2×#14581934—22no—
9C2×#14581934C2212yes—
10#19128813C412yes—
11#19128808—21no—
12#19128803—21no—
13#19128800C412yes—
14C4×C22⋊D167—22no—
15#19128791C2211yescharacteristic
16(C2×D167)⋊C8—24no—
17C2×#41×C167C2211yescharacteristic
18C2×M16×C167C2212yes—
19#101×C167—21no—
20#100×C167—21no—
21C23×D167C2×C411yescharacteristic
22C2×C22⋊D167—22no—
23C22×Dic167C2×C411yescharacteristic
24C2×C4×D167—22no—
25C2×C4×D167—21no—
26C2×C4×D167—21no—
27C2×C4×D167—22no—
28#14581934—24no—
29#14581934—22no—
30#14581934—22no—
31C2×C167⋊C8—24no—
32C2×C167⋊C8D414yescomplemented
33C4⋊Dic167—21no—
34Dic167⋊C4—22no—
35C22⋊Dic167—22no—
36C22⋊Dic167C2×C412yes—
37(C2×D167)⋊C4—22no—
38C4×Dic167—21no—
39#41×C167—22no—
40#41×C167—22no—
41C2×Q8×C167D411yescharacteristic
42C2×D4×C167C2×C412yes—
43C2×D4×C167D411yescharacteristic
44C22×C4×C167D412yes—
45C22×C4×C167C2311yescharacteristic
46M16×C167—22no—
47M16×C167—22no—
48C2×C8×C167—22no—
49C22×D167—21no—
50C22×D167—22no—
51C22×D167—42no—
52C22⋊D167—44no—
53C2×Dic167—22no—
54C2×Dic167—21no—
55C2×Dic167—22no—
56C4×D167—41no—
57C4×D167—42no—
58C4×D167—41no—
59C167⋊C8—24no—
60C23×C167C22⋊C412yes—
61C23×C167C22×C411yescharacteristic
62Q8×C167—22no—
63D4×C167—24no—
64D4×C167—22no—
65C2×C4×C167—22no—
66C2×C4×C167—22no—
67C2×C4×C167C22⋊C412yes—
68C2×C4×C167C2×D412yes—
69C2×C4×C167C2×D411yescharacteristic
70C2×C4×C167C2×D411yescharacteristic
71C2×C4×C167#4112yes—
72C8×C167—42no—
73C22×C167—22no—
74C22×C167—41no—
75C22×C167—42no—
76C22×C167C2×C22⋊C411yescharacteristic
77C22×C167C4×D412yes—
78C2×D167—22no—
79C2×D167—24no—
80C2×D167—41no—
81C2×D167—42no—
82C2×D167—41no—
83C4×C167—22no—
84C4×C167#12011yescharacteristic
85C4×C167#12112yes—
86C4×C167#12311yescharacteristic
87Dic167—42no—
88Dic167—41no—
89C2×C167—21no—
90C2×C167—22no—
91C2×C167—42no—
92C2×C167#38611yescharacteristic
93D167—42no—
94C167#189111yescharacteristic, complemented, Sylow
95#1891—1671nomaximal, Sylow
96#568—1671no—
97#525—1671no—
98#413D16711yescharacteristic, complemented
99#407—1672no—
100#344—1672no—
101C2×#41C2×D16711yescharacteristic
102C23×C4—1671no—
103#131—3344no—
104C2×M16—3342no—
105C2×M16—1672no—
106C2×M16C2×D16712yes—
107#123—1672no—
108C22⋊Q8—3341no—
109#121—3341no—
110#120—1672no—
111C4×D4—3342no—
112#117—1671no—
113#101—21no—
114#100—21no—
115C22⋊C8—3344no—
116C24—1671no—
117#41—22no—
118#41—22no—
119C2×Q8C22⋊D16711yescharacteristic
120C2×D4—3342no—
121C2×D4C4×D16712yes—
122C2×D4C22⋊D16711yescharacteristic
123C22×C4—3342no—
124C22×C4—3341no—
125C22×C4—3341no—
126C22×C4—3342no—
127C22×C4—1671no—
128C22×C4D66812yes—
129C22×C4C22×D16711yescharacteristic
130M16—22no—
131M16—22no—
132M16—3344no—
133M16—3342no—
134M16—3342no—
135C2×C8—22no—
136C2×C8—3344no—
137C2×C8—1674no—
138C4⋊C4—3342no—
139C4⋊C4—3341no—
140C22⋊C4—3342no—
141C22⋊C4—3342no—
142C22⋊C4—1672no—
143C42—3341no—
144C23—3341no—
145C23—3342no—
146C23—6682no—
147C23(C2×D167)⋊C412yes—
148C23C2×C4×D16711yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—6684no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—3342no—
156C2×C4—3341no—
157C2×C4—3342no—
158C2×C4—6681no—
159C2×C4—6682no—
160C2×C4—6681no—
161C2×C4(C2×D167)⋊C412yes—
162C2×C4D4×D16712yes—
163C2×C4C2×C22⋊D16711yescharacteristic
164C2×C4C2×D66811yescharacteristic
165C2×C4#1458194912yes—
166C8—42no—
167C8—3344no—
168C22—22no—
169C22—41no—
170C22—42no—
171C22—3342no—
172C22—3344no—
173C22—6681no—
174C22—6682no—
175C22—6681no—
176C22C2×(C2×D167)⋊C411yescharacteristic
177C22(C22⋊D167)⋊C412yes—
178C4—22no—
179C4—6682no—
180C4—6681no—
181C4#1912880211yescharacteristic
182C4#1912880412yes—
183C4#1912881611yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—6682no—
188C2#2492314611yescharacteristic
subgroupG/H
centerC4#19128802
commutatorC23×C167C22×C4
FrattiniC22×C4C22×D167
Fitting#413×C167C2
socleC2×C167#386
prime radical#24923776C2
cosocleC22×C4×C167C23
hypercenter#413D167
hypocenterC167#1891
LCSG ▷ C23×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #24924036C2semidirect
3. #24923232C2semidirect
4. #24923041C2semidirect
5. #413×C167C2semidirect
32. C2×C167⋊C8D4semidirect
94. C167#1891semidirect
98. #413D167semidirect

Maximal subgroups

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

subgroupindex
#249240362
#249237762
#249232322
#249230412
#413×C1672
#1891167