Finite Group Databasebeta

← #36326164

Subgroups — #36326164

Order 21376.

236 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 45 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#24923451C211yesmaximal, characteristic, complemented
2#24923441C212yesmaximal, complemented
3#24923068C211yesmaximal, characteristic, complemented
4#24923066C212yesmaximal, complemented
5#555×C167C211yesmaximal, characteristic, complemented
6#19128880C2211yescharacteristic
7#19128863—21no—
8#19128858—22no—
9#19128760—22no—
10#19128760—21no—
11#19128751C2212yes—
12#19128751C2211yescharacteristic
13C4⋊Q8×C167C2211yescharacteristic
14#124×C167—21no—
15#124×C167C2212yes—
16#121×C167—22no—
17C4×D4×C167—21no—
18#117×C167—22no—
19#14581951—41no—
20#14581950—22no—
21#14581950—21no—
22Q8×D167—41no—
23#14581940—41no—
24#14581938—42no—
25#14581937—41no—
26#14581935—42no—
27#14581934—22no—
28#14581934—21no—
29C2×Q8×C167D411yescharacteristic
30C2×Q8×C167C2311yescharacteristic
31C2×D4×C167—21no—
32C2×D4×C167—21no—
33C2×D4×C167D412yes—
34C22×C4×C167—22no—
35C4⋊C4×C167—21no—
36C4⋊C4×C167—22no—
37C22⋊C4×C167—22no—
38C22⋊C4×C167—22no—
39C22⋊C4×C167—22no—
40C42×C167—21no—
41C42×C167D412yes—
42C42×C167D411yescharacteristic
43Dic334—22no—
44Dic334—21no—
45D668—22no—
46D668—21no—
47C22⋊D167—41no—
48C4×D167—42no—
49C4×D167—41no—
50C167⋊C8—42no—
51C167⋊C8—41no—
52C23×C167—22no—
53Q8×C167—41no—
54Q8×C167—41no—
55D4×C167—42no—
56D4×C167—42no—
57D4×C167—81no—
58C2×C4×C167—21no—
59C2×C4×C167—22no—
60C2×C4×C167—21no—
61C2×C4×C167—44no—
62C2×C4×C167C2×D411yescharacteristic
63C2×C4×C167C2×D412yes—
64C22×C167—42no—
65C22×C167—42no—
66C22×C167—42no—
67C22×C167#12011yescharacteristic
68C2×D167—41no—
69C4×C167—22no—
70C4×C167—22no—
71C4×C167—21no—
72C4×C167—42no—
73C4×C167—41no—
74Dic167—41no—
75C2×C167—21no—
76C2×C167—82no—
77C2×C167B(4,𝔽2)11yescharacteristic
78D167—81no—
79C167#220411yescharacteristic, complemented, Sylow
80#2204—1671nomaximal, Sylow
81#555D16711yescharacteristic, complemented
82#456—1671no—
83#454—1672no—
84#356—1671no—
85#355—1672no—
862−1+4—1671no—
87#137—3341no—
88AGL(1,ℤ/8)—3342no—
89C4⋊Q8C2×D16711yescharacteristic
90#124—21no—
91#124C2×D16712yes—
92#121—22no—
93C4×D4—21no—
94#117—22no—
95#104—3342no—
96#104—3341no—
97#101—1671no—
98#101—1672no—
99#41—3342no—
100#41—3341no—
101#41—6681no—
102C2×Q8—6681no—
103C2×Q8C22⋊D16711yescharacteristic
104C2×Q8C22×D16711yescharacteristic
105C2×D4—21no—
106C2×D4—21no—
107C2×D4C22⋊D16712yes—
108C22×C4—22no—
109Q16—6681no—
110SD16—6682no—
111SD16—6681no—
112D8—6682no—
113M16—3342no—
114M16—3341no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C42—21no—
121C42C22⋊D16712yes—
122C42C22⋊D16711yescharacteristic
123C23—22no—
124Q8—41no—
125Q8—41no—
126Q8—3342no—
127Q8—3341no—
128D4—42no—
129D4—42no—
130D4—81no—
131D4—3342no—
132D4—3341no—
133D4—6681no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—44no—
138C2×C4—6682no—
139C2×C4—6681no—
140C2×C4C2×C22⋊D16711yescharacteristic
141C2×C4C2×C22⋊D16712yes—
142C8—6682no—
143C8—6681no—
144C22—42no—
145C22—42no—
146C22—42no—
147C22—6681no—
148C22#1912880111yescharacteristic
149C4—22no—
150C4—22no—
151C4—21no—
152C4—42no—
153C4—41no—
154C4—6681no—
155C2—21no—
156C2—82no—
157C2—13361no—
158C2#2492345611yescharacteristic
subgroupG/H
centerC2#24923456
commutator, cosocleC2×Q8×C167C23
FrattiniC2×Q8C22×D167
Fitting#555×C167C2
socleC2×C167B(4,𝔽2)
hypercenter#555D167
hypocenterC167#2204
LCSG ▷ C2×Q8×C167 ▷ C22×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #24923451C2semidirect
2. #24923441C2semidirect
3. #24923068C2semidirect
4. #24923066C2semidirect
5. #555×C167C2semidirect
79. C167#2204semidirect
81. #555D167semidirect

Maximal subgroups

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

subgroupindex
#249234512
#249234412
#249230682
#249230662
#555×C1672
#2204167