Finite Group Databasebeta

← #1622×C167 #36319982

Subgroups — #1622×C167 #36319982

Order 21376.

236 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 72 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#554×C167C211yesmaximal, characteristic, complemented
2#492×C167C211yesmaximal, characteristic, complemented
3#460×C167C211yesmaximal, characteristic, complemented
4#459×C167C211yesmaximal, characteristic
5#333×C167C211yesmaximal, characteristic, complemented
6Q8⋊C8×C167C211yesmaximal, characteristic, complemented
7D4⋊C8×C167C211yesmaximal, characteristic, complemented
8C2×SD16×C167—22no—
9C2×SD16×C167—21no—
10C2×D8×C167—21no—
11C4⋊Q8×C167C2211yescharacteristic
12#127×C167C2211yescharacteristic
13#125×C167—21no—
14C22⋊Q8×C167—21no—
15C22⋊Q8×C167—22no—
16C4×Q8×C167C2211yescharacteristic
17C4×D4×C167—21no—
18C4×D4×C167C2211yescharacteristic
19C2×C4⋊C4×C167—21no—
20C4⋊C8×C167C2211yescharacteristic
21C4⋊C8×C167C2211yescharacteristic
22#102×C167—21no—
23#102×C167—21no—
24C4×C8×C167C2211yescharacteristic
25C2×Q8×C167D411yescharacteristic
26C2×Q8×C167D412yes—
27C2×D4×C167—21no—
28C2×D4×C167—21no—
29C2×D4×C167—21no—
30C2×D4×C167D411yescharacteristic
31C22×C4×C167—22no—
32C22×C4×C167—21no—
33SD16×C167—44no—
34SD16×C167—42no—
35D8×C167—42no—
36C2×C8×C167—21no—
37C2×C8×C167—21no—
38C2×C8×C167—21no—
39C4⋊C4×C167—22no—
40C4⋊C4×C167—22no—
41C4⋊C4×C167—21no—
42C4⋊C4×C167—21no—
43C4⋊C4×C167—21no—
44C4⋊C4×C167D411yescharacteristic
45C4⋊C4×C167D411yescharacteristic
46C22⋊C4×C167—22no—
47C22⋊C4×C167—21no—
48C42×C167—21no—
49C42×C167C2311yescharacteristic
50C23×C167—21no—
51C23×C167—41no—
52Q8×C167—21no—
53Q8×C167—21no—
54Q8×C167—42no—
55D4×C167—21no—
56D4×C167—42no—
57D4×C167—41no—
58D4×C167—41no—
59D4×C167—41no—
60D4×C167—41no—
61D4×C167SD1612yescomplemented
62C2×C4×C167—21no—
63C2×C4×C167—21no—
64C2×C4×C167—21no—
65C2×C4×C167—22no—
66C2×C4×C167—41no—
67C2×C4×C167—41no—
68C2×C4×C167—82no—
69C2×C4×C167C2×D411yescharacteristic
70C2×C4×C167C2×D411yescharacteristic
71C2×C4×C167C2×D411yescharacteristic
72C8×C167—22no—
73C8×C167—41no—
74C8×C167—41no—
75C22×C167—22no—
76C22×C167—41no—
77C22×C167—41no—
78C22×C167—81no—
79C22×C167—81no—
80C22×C167—81no—
81C22×C167#12011yescharacteristic
82C4×C167—21no—
83C4×C167—41no—
84C4×C167—41no—
85C4×C167—41no—
86C4×C167—42no—
87C4×C167C2×SD1611yescharacteristic
88C4×C167#13511yescharacteristic
89C4×C167AGL(1,ℤ/8)11yescharacteristic
90C4×C167#13711yescharacteristic
91C2×C167—42no—
92C2×C167—161no—
93C2×C167#44811yescharacteristic
94C2×C167#44911yescharacteristic
95C2×C167#45311yescharacteristic
96C167#162211yescharacteristic, direct product factor, Sylow
97#1622C16711yesmaximal, characteristic, direct product factor, Sylow
98#554C2×C16711yescharacteristic, complemented
99#492C2×C16711yescharacteristic, complemented
100#460C2×C16711yescharacteristic, complemented
101#459C2×C16711yescharacteristic
102#333C2×C16711yescharacteristic, complemented
103Q8⋊C8C2×C16711yescharacteristic, complemented
104D4⋊C8C2×C16711yescharacteristic, complemented
105C2×SD16—21no—
106C2×SD16—22no—
107C2×D8—21no—
108C4⋊Q8C22×C16711yescharacteristic
109#127C22×C16711yescharacteristic
110#125—21no—
111C22⋊Q8—21no—
112C22⋊Q8—22no—
113C4×Q8C22×C16711yescharacteristic
114C4×D4—21no—
115C4×D4C22×C16711yescharacteristic
116C2×C4⋊C4—21no—
117C4⋊C8C22×C16711yescharacteristic
118C4⋊C8C22×C16711yescharacteristic
119#102—21no—
120#102—21no—
121C4×C8C22×C16711yescharacteristic
122C2×Q8D4×C16711yescharacteristic
123C2×Q8D4×C16712yes—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4D4×C16711yescharacteristic
128C22×C4—22no—
129C22×C4—21no—
130SD16—44no—
131SD16—42no—
132D8—42no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4D4×C16711yescharacteristic
142C4⋊C4D4×C16711yescharacteristic
143C22⋊C4—22no—
144C22⋊C4—21no—
145C42—21no—
146C42C23×C16711yescharacteristic
147C23—21no—
148C23—41no—
149Q8—21no—
150Q8—21no—
151Q8—42no—
152D4—21no—
153D4—42no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4SD16×C16712yescomplemented
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—82no—
166C2×C4C2×D4×C16711yescharacteristic
167C2×C4C2×D4×C16711yescharacteristic
168C2×C4C2×D4×C16711yescharacteristic
169C8—22no—
170C8—41no—
171C8—41no—
172C22—22no—
173C22—41no—
174C22—41no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22#120×C16711yescharacteristic
179C4—21no—
180C4—41no—
181C4—41no—
182C4—41no—
183C4—42no—
184C4C2×SD16×C16711yescharacteristic
185C4#135×C16711yescharacteristic
186C4AGL(1,ℤ/8)×C16711yescharacteristic
187C4#137×C16711yescharacteristic
188C2—42no—
189C2—161no—
190C2#448×C16711yescharacteristic
191C2#449×C16711yescharacteristic
192C2#453×C16711yescharacteristic
subgroupG/H
center, socleC22×C167#120
commutator, Frattini, cosocleC42C23×C167
prime radical#459×C167C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C167 ◁ C42×C167 ◁ G

Complements

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

NH ≅ G/Ntype
1. #554×C167C2semidirect
2. #492×C167C2semidirect
3. #460×C167C2semidirect
5. #333×C167C2semidirect
6. Q8⋊C8×C167C2semidirect
7. D4⋊C8×C167C2semidirect
61. D4×C167SD16semidirect
96. C167#1622direct (N × H)
97. #1622C167direct (N × H)
98. #554C2×C167semidirect
99. #492C2×C167semidirect
100. #460C2×C167semidirect
102. #333C2×C167semidirect
103. Q8⋊C8C2×C167semidirect
104. D4⋊C8C2×C167semidirect
158. D4SD16×C167semidirect

Maximal subgroups

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

subgroupindex
#554×C1672
#492×C1672
#460×C1672
#459×C1672
#333×C1672
Q8⋊C8×C1672
D4⋊C8×C1672
#1622167