Finite Group Databasebeta

← #36323200

Subgroups — #36323200

Order 21376.

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

normal subgroups only

This group has 278 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#24923939C211yesmaximal, characteristic, complemented
2#24923466C211yesmaximal, characteristic, complemented
3#24923461C211yesmaximal, characteristic, complemented
4#24923013C211yesmaximal, characteristic, complemented
5#24923001C211yesmaximal, characteristic, complemented
6#24922998C211yesmaximal, characteristic, complemented
7#493×C167C211yesmaximal, characteristic, complemented
8C2×#14581949—21no—
9C2×#14581938—21no—
10C2×#14581935—21no—
11#19128828—21no—
12#19128804—22no—
13#19128803—21no—
14C4×Dic334C2211yescharacteristic
15C4×D668C2211yescharacteristic
16D4×Dic167—21no—
17C4⋊(C167⋊C8)C2211yescharacteristic
18C4⋊(C167⋊C8)C2211yescharacteristic
19#19128753—21no—
20#19128753—21no—
21C2×D8×C167—22no—
22#127×C167C2211yescharacteristic
23#127×C167C2211yescharacteristic
24C4×C8×C167C2211yescharacteristic
25#14581949—42no—
26#14581949—42no—
27C2×Dic334D411yescharacteristic
28C2×D668D411yescharacteristic
29C2×C22⋊D167—22no—
30C2×C22⋊D167—22no—
31C22×Dic167—22no—
32C2×C4×D167—21no—
33#14581938—42no—
34#14581935—42no—
35C2×C167⋊C8—21no—
36C2×C167⋊C8—21no—
37C4⋊Dic167D411yescharacteristic
38C4⋊Dic167D411yescharacteristic
39Dic167⋊C4—21no—
40C22⋊Dic167—22no—
41(C2×D167)⋊C4—21no—
42C4×Dic167—21no—
43C2×D4×C167—21no—
44C2×D4×C167—21no—
45C2×D4×C167—21no—
46C2×D4×C167—21no—
47C2×D4×C167—21no—
48C2×D4×C167D412yes—
49D8×C167—44no—
50C2×C8×C167—21no—
51C42×C167C2311yescharacteristic
52C22×D167—21no—
53Dic334—21no—
54Dic334D812yescomplemented
55D668—21no—
56D668—21no—
57C22⋊D167—44no—
58C22⋊D167—82no—
59C2×Dic167—21no—
60C2×Dic167—21no—
61C2×Dic167—21no—
62C2×Dic167—44no—
63C4×D167—41no—
64C4×D167—41no—
65C167⋊C8—41no—
66C167⋊C8—41no—
67C23×C167—22no—
68C23×C167—41no—
69D4×C167—42no—
70D4×C167—42no—
71D4×C167—41no—
72D4×C167—41no—
73D4×C167—41no—
74D4×C167—41no—
75D4×C167—41no—
76D4×C167—41no—
77D4×C167—41no—
78D4×C167—41no—
79C2×C4×C167C2×D411yescharacteristic
80C2×C4×C167C2×D411yescharacteristic
81C2×C4×C167C2×D411yescharacteristic
82C8×C167—22no—
83C22×C167—42no—
84C22×C167—42no—
85C22×C167—42no—
86C22×C167—81no—
87C22×C167—81no—
88C22×C167—81no—
89C22×C167#12011yescharacteristic
90C2×D167—41no—
91C2×D167—41no—
92C2×D167—41no—
93C4×C167—21no—
94C4×C167C2×D811yescharacteristic
95C4×C167#13511yescharacteristic
96C4×C167AGL(1,ℤ/8)11yescharacteristic
97C4×C167AGL(1,ℤ/8)11yescharacteristic
98Dic167—22no—
99Dic167—41no—
100Dic167—41no—
101C2×C167—82no—
102C2×C167—161no—
103C2×C167#44711yescharacteristic
104C2×C167#44911yescharacteristic
105C2×C167#45311yescharacteristic
106D167—81no—
107C167#162111yescharacteristic, complemented, Sylow
108#1621—1671nomaximal, Sylow
109#550—1671no—
110#493D16711yescharacteristic, complemented
111#460—1671no—
112#459—1671no—
113#331—1671no—
114Q8⋊C8—1671no—
115D4⋊C8—1671no—
116C2×#41—3341no—
117C2×SD16—3341no—
118C2×D8—22no—
119C2×D8—3341no—
120#127—3341no—
121#127C2×D16711yescharacteristic
122#127C2×D16711yescharacteristic
123#121—3342no—
124#121—3341no—
125C4×Q8—1671no—
126C4×D4—3341no—
127C4×D4—1671no—
128C4⋊C8—1671no—
129C4⋊C8—1671no—
130#102—3341no—
131#102—3341no—
132C4×C8C2×D16711yescharacteristic
133#41—6682no—
134#41—6682no—
135C2×Q8—1671no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—3342no—
142C2×D4—3342no—
143C2×D4—1671no—
144C2×D4C22⋊D16712yes—
145C22×C4—3342no—
146C22×C4—3341no—
147SD16—6682no—
148D8—44no—
149D8—6682no—
150C2×C8—21no—
151C2×C8—3341no—
152C2×C8—3341no—
153C4⋊C4—3341no—
154C4⋊C4—1671no—
155C4⋊C4—1671no—
156C22⋊C4—3342no—
157C22⋊C4—3341no—
158C42—3341no—
159C42C22×D16711yescharacteristic
160C23—22no—
161C23—41no—
162C23—3341no—
163Q8—3341no—
164Q8—1672no—
165D4—42no—
166D4—42no—
167D4—41no—
168D4—41no—
169D4—41no—
170D4—41no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—13362no—
176D4—3341no—
177D4—3341no—
178D4—6684no—
179C2×C4—3341no—
180C2×C4—3341no—
181C2×C4—3341no—
182C2×C4—6684no—
183C2×C4—6681no—
184C2×C4—6681no—
185C2×C4D4×D16711yescharacteristic
186C2×C4D4×D16711yescharacteristic
187C2×C4C2×C22⋊D16711yescharacteristic
188C8—22no—
189C8—6681no—
190C8—6681no—
191C22—42no—
192C22—42no—
193C22—42no—
194C22—81no—
195C22—81no—
196C22—81no—
197C22—6681no—
198C22—6681no—
199C22—6681no—
200C22#1912880011yescharacteristic
201C4—21no—
202C4—3342no—
203C4—6681no—
204C4—6681no—
205C4D8×D16711yescharacteristic
206C4#1912885111yescharacteristic
207C4#1912885611yescharacteristic
208C4#1912885611yescharacteristic
209C2—82no—
210C2—161no—
211C2—13361no—
212C2#2492340111yescharacteristic
213C2#2492341111yescharacteristic
214C2#2492343611yescharacteristic
subgroupG/H
centerC22#19128800
commutator, cosocleC42×C167C23
FrattiniC42C22×D167
Fitting#493×C167C2
socleC22×C167#120
hypercenter#493D167
hypocenterC167#1621
LCSG ▷ C42×C167 ▷ C22×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. #24923939C2semidirect
2. #24923466C2semidirect
3. #24923461C2semidirect
4. #24923013C2semidirect
5. #24923001C2semidirect
6. #24922998C2semidirect
7. #493×C167C2semidirect
54. Dic334D8semidirect
107. C167#1621semidirect
110. #493D167semidirect

Maximal subgroups

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

subgroupindex
#249239392
#249234662
#249234612
#249230132
#249230012
#249229982
#493×C1672
#1621167