Finite Group Databasebeta

← #36324217

Subgroups — #36324217

Order 21376.

316 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C2×C4⋊C4×D167C211yesmaximal, characteristic
2#24923232C212yesmaximal
3#24923038C211yesmaximal, characteristic, complemented
4#24923037C212yesmaximal, complemented
5#343×C167C211yesmaximal, characteristic, complemented
6C22×C4×D167C412yes—
7C22×C4×D167C2211yescharacteristic
8C2×#14581934C2212yes—
9C2×C4⋊Dic167C2211yescharacteristic
10C2×Dic167⋊C4C412yes—
11C4⋊C4×D167—22no—
12C4⋊C4×D167—22no—
13(C2×D167)⋊C8—22no—
14(C2×D167)⋊C8—22no—
15C2×M16×C167C2212yes—
16C2×C4⋊C4×C167C2211yescharacteristic
17C23×D167C2×C411yescharacteristic
18C22×Dic167C2×C411yescharacteristic
19C22×Dic167C2×C412yes—
20C2×C4×D167—22no—
21C2×C4×D167—22no—
22C2×C4×D167—21no—
23C2×C4×D167—22no—
24C2×C4×D167—21no—
25C2×C4×D167—44no—
26C2×C4×D167D412yes—
27C2×C4×D167D412yes—
28#14581934—22no—
29#14581934—22no—
30C2×C167⋊C8—22no—
31C4⋊Dic167—22no—
32Dic167⋊C4—24no—
33C22×C4×C167C2×C412yes—
34C22×C4×C167C2311yescharacteristic
35M16×C167—22no—
36M16×C167—22no—
37C2×C8×C167—22no—
38C4⋊C4×C167—22no—
39C22×D167—21no—
40C22×D167—21no—
41C22×D167—22no—
42C22×D167—22no—
43C22×D167C22⋊C411yescharacteristic
44C22×D167C22⋊C411yescharacteristic
45C2×Dic167—21no—
46C2×Dic167—21no—
47C2×Dic167—22no—
48C2×Dic167—42no—
49C2×Dic167C22⋊C412yes—
50C4×D167—22no—
51C4×D167—22no—
52C4×D167—22no—
53C4×D167—22no—
54C4×D167—44no—
55C4×D167—44no—
56C167⋊C8—42no—
57C23×C167C22×C411yescharacteristic
58C2×C4×C167—22no—
59C2×C4×C167—42no—
60C2×C4×C167C2×D412yes—
61C2×C4×C167#4111yescharacteristic
62C2×C4×C167#4112yes—
63C2×C4×C167#4111yescharacteristic
64C8×C167—42no—
65C22×C167—21no—
66C22×C167—21no—
67C22×C167C2×C22⋊C411yescharacteristic
68C22×C167#11711yescharacteristic
69C22×C167#11711yescharacteristic
70C2×D167—24no—
71C2×D167—24no—
72C2×D167—21no—
73C2×D167—21no—
74C2×D167—21no—
75C2×D167—21no—
76C2×D167#10012yes—
77C2×D167#10112yes—
78C4×C167—42no—
79C4×C167#12312yes—
80C4×C167#12312yes—
81Dic167—22no—
82Dic167—42no—
83C2×C167—21no—
84C2×C167—21no—
85C2×C167#38111yescharacteristic
86C2×C167C2×#10011yescharacteristic
87C2×C167C2×#10111yescharacteristic
88D167—22no—
89D167—22no—
90C167#182111yescharacteristic, complemented, Sylow
91#1821—1671nomaximal, Sylow
92C22×C4⋊C4—1671no—
93#407—1672no—
94#343—1672no—
95#343—1671no—
96#343D16711yescharacteristic, complemented
97C23×C4—1671no—
98C23×C4—1672no—
99C2×M16—1672no—
100C2×M16C2×D16712yes—
101C2×C4⋊C4—3342no—
102C2×C4⋊C4—3342no—
103C2×C4⋊C4—1672no—
104C2×C4⋊C4—1671no—
105C2×C4⋊C4C2×D16711yescharacteristic
106C22⋊C8—3342no—
107C22⋊C8—3342no—
108C24—1671no—
109C22×C4—3342no—
110C22×C4—3342no—
111C22×C4—3341no—
112C22×C4—3342no—
113C22×C4—3341no—
114C22×C4—6684no—
115C22×C4—1671no—
116C22×C4—1672no—
117C22×C4—1672no—
118C22×C4—1672no—
119C22×C4C4×D16712yes—
120C22×C4C22×D16711yescharacteristic
121M16—22no—
122M16—22no—
123M16—3342no—
124M16—3342no—
125C2×C8—22no—
126C2×C8—3342no—
127C4⋊C4—22no—
128C4⋊C4—3344no—
129C4⋊C4—3342no—
130C23—3342no—
131C23—3342no—
132C23—3341no—
133C23—3341no—
134C23—1671no—
135C23—1671no—
136C23C2×C4×D16711yescharacteristic
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—3341no—
140C2×C4—3341no—
141C2×C4—3342no—
142C2×C4—3342no—
143C2×C4—3342no—
144C2×C4—3342no—
145C2×C4—3342no—
146C2×C4—6684no—
147C2×C4—6684no—
148C2×C4—6682no—
149C2×C4—1672no—
150C2×C4D4×D16712yes—
151C2×C4#1458194911yescharacteristic
152C2×C4#1458195012yes—
153C2×C4#1458195111yescharacteristic
154C8—42no—
155C8—6682no—
156C22—21no—
157C22—21no—
158C22—3344no—
159C22—3344no—
160C22—3341no—
161C22—3341no—
162C22—3341no—
163C22—3341no—
164C22—1672no—
165C22—1672no—
166C22C22⋊C4×D16711yescharacteristic
167C22#1912878911yescharacteristic
168C22#1912879011yescharacteristic
169C4—42no—
170C4—3342no—
171C4—6682no—
172C4#1912881412yes—
173C4#1912881712yes—
174C2—21no—
175C2—21no—
176C2—3342no—
177C2—3342no—
178C2#2492313111yescharacteristic
179C2#100×D16711yescharacteristic
180C2#101×D16711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D167
commutatorC23×C167C22×C4
FrattiniC22×C4C22×D167
Fitting#343×C167C2
socleC22×C167C2×C22⋊C4
prime radicalC23×D167C2×C4
cosocleC22×C4×C167C23
hypercenter#343D167
hypocenterC167#1821
LCSG ▷ C23×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
3. #24923038C2semidirect
4. #24923037C2semidirect
5. #343×C167C2semidirect
90. C167#1821semidirect
96. #343D167semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D1672
#249232322
#249230382
#249230372
#343×C1672
#1821167