Finite Group Databasebeta

← #36325832

Subgroups — #36325832

Order 21376.

230 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 65 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#24923840C211yesmaximal, characteristic
2#24923253C211yesmaximal, characteristic
3#24923067C214yesmaximal, complemented
4#413×C167C211yesmaximal, characteristic
5C22×Dic334C411yescharacteristic
6C2×#14581934—21no—
7#19128830C2212yes—
8#19128819C412yes—
9#19128813—21no—
10#19128813—21no—
11#19128811—21no—
12#19128811—21no—
13#19128791C411yescharacteristic
14#19128751C2212yes—
15#19128749—21no—
16C2×#41×C167C2211yescharacteristic
17C2×M16×C167—21no—
18#101×C167C2212yes—
19#100×C167—21no—
20C2×Dic334—21no—
21C2×Dic334—41no—
22C2×Dic334—41no—
23C2×Dic334C2×C412yes—
24C22×Dic167—21no—
25#14581934—22no—
26#14581934—41no—
27C2×C167⋊C8—41no—
28C4⋊Dic167—21no—
29Dic167⋊C4—22no—
30Dic167⋊C4—22no—
31C22⋊Dic167—21no—
32C22⋊Dic167—22no—
33C22⋊Dic167—22no—
34C4×Dic167C2×C412yes—
35#41×C167—41no—
36#41×C167—41no—
37C2×Q8×C167C2311yescharacteristic
38C2×D4×C167C2×C411yescharacteristic
39C2×D4×C167D411yescharacteristic
40C2×D4×C167D411yescharacteristic
41C22×C4×C167C2×C411yescharacteristic
42C22×C4×C167D411yescharacteristic
43C22×C4×C167D411yescharacteristic
44M16×C167—22no—
45M16×C167—41no—
46C2×C8×C167—41no—
47Dic334—41no—
48Dic334—41no—
49Dic334—42no—
50C2×Dic167—22no—
51C2×Dic167—22no—
52C2×Dic167—42no—
53C167⋊C8—42no—
54C23×C167C22⋊C411yescharacteristic
55C23×C167C22⋊C411yescharacteristic
56C23×C167C22⋊C411yescharacteristic
57Q8×C167—41no—
58D4×C167—41no—
59D4×C167—41no—
60D4×C167—41no—
61C2×C4×C167—21no—
62C2×C4×C167—21no—
63C2×C4×C167—21no—
64C2×C4×C167—21no—
65C2×C4×C167—21no—
66C2×C4×C167—21no—
67C2×C4×C167C22⋊C411yescharacteristic
68C2×C4×C167C22×C411yescharacteristic
69C2×C4×C167C2×D411yescharacteristic
70C2×C4×C167C2×D411yescharacteristic
71C8×C167—42no—
72C22×C167—21no—
73C22×C167—21no—
74C22×C167—21no—
75C22×C167—41no—
76C22×C167—41no—
77C22×C167—41no—
78C22×C167C2×C22⋊C411yescharacteristic
79C4×C167—21no—
80C4×C167—21no—
81C4×C167—21no—
82C4×C167#9911yescharacteristic
83C4×C167#9911yescharacteristic
84Dic167—42no—
85Dic167—42no—
86C2×C167—21no—
87C2×C167—41no—
88C2×C167—41no—
89C2×C167—41no—
90C2×C167C2×#9911yescharacteristic
91C167#213511yescharacteristic, complemented, Sylow
92#2135—1671nomaximal, Sylow
93#536—1671no—
94#413—1671no—
95#413D16711yescharacteristic
96#356—1674no—
97C2×#41C2×D16711yescharacteristic
98C22×Q8—1671no—
99C2×M16—21no—
100C2×M16—3341no—
101C4⋊Q8—1672no—
102#124—1672no—
103#123—3341no—
104#123—3341no—
105C22⋊Q8—3341no—
106C22⋊Q8—3341no—
107#117—1671no—
108#101—1672no—
109#101C2×D16712yes—
110#100—21no—
111#100—3341no—
112#41—41no—
113#41—41no—
114C2×Q8—3341no—
115C2×Q8—6681no—
116C2×Q8—6681no—
117C2×Q8—1672no—
118C2×Q8C22×D16711yescharacteristic
119C2×D4C4×D16711yescharacteristic
120C2×D4C22⋊D16711yescharacteristic
121C2×D4D66811yescharacteristic
122C22×C4—3341no—
123C22×C4C4×D16711yescharacteristic
124C22×C4C22⋊D16711yescharacteristic
125C22×C4D66811yescharacteristic
126M16—22no—
127M16—41no—
128M16—3342no—
129M16—6681no—
130C2×C8—41no—
131C2×C8—6681no—
132C4⋊C4—3341no—
133C4⋊C4—3342no—
134C4⋊C4—3342no—
135C22⋊C4—3341no—
136C22⋊C4—3342no—
137C22⋊C4—3342no—
138C42—1672no—
139C23(C2×D167)⋊C411yescharacteristic
140C23(C2×D167)⋊C411yescharacteristic
141C23(C2×D167)⋊C411yescharacteristic
142Q8—41no—
143Q8—6682no—
144Q8—6681no—
145Q8—6681no—
146D4—41no—
147D4—41no—
148D4—41no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—3342no—
156C2×C4—3342no—
157C2×C4—6682no—
158C2×C4(C2×D167)⋊C411yescharacteristic
159C2×C4C2×C4×D16711yescharacteristic
160C2×C4C2×C22⋊D16711yescharacteristic
161C2×C4C2×D66811yescharacteristic
162C8—42no—
163C8—6682no—
164C22—21no—
165C22—21no—
166C22—21no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22C2×(C2×D167)⋊C411yescharacteristic
171C4—21no—
172C4—21no—
173C4—21no—
174C4—6682no—
175C4—6682no—
176C4#1912874611yescharacteristic
177C4#1912874611yescharacteristic
178C2—21no—
179C2—41no—
180C2—41no—
181C2—41no—
182C2C2×#1912874611yescharacteristic
subgroupG/H
centerC2C2×#19128746
commutatorC2×C4×C167C22×C4
FrattiniC2×Q8C22×D167
Fitting#413×C167C2
socleC2×C167C2×#99
prime radicalC2×#41×C167C22
cosocleC2×Q8×C167C23
hypercenter#413D167
hypocenterC167#2135
LCSG ▷ C2×C4×C167 ▷ C22×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
3. #24923067C2semidirect
91. C167#2135semidirect

Maximal subgroups

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

subgroupindex
#249238402
#249232532
#249230672
#413×C1672
#2135167