Finite Group Databasebeta

← #36325831

Subgroups — #36325831

Order 21376.

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

normal subgroups only

This group has 230 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#24923844C211yesmaximal, characteristic
2#24923252C212yesmaximal
3#24923068C211yesmaximal, characteristic, complemented
4#24923067C212yesmaximal, complemented
5#356×C167C211yesmaximal, characteristic, complemented
6C2×#14581951C2211yescharacteristic
7C2×Q8×D167C411yescharacteristic
8M16×D167—22no—
9#19128830C2211yescharacteristic
10#19128821C411yescharacteristic
11#19128820C411yescharacteristic
12#19128817—22no—
13#19128810—22no—
14#19128789C411yescharacteristic
15#19128751C2212yes—
16#19128748—22no—
17C4⋊Q8×C167C2211yescharacteristic
18#101×C167C2212yes—
19#14581951—42no—
20C2×Dic334—21no—
21C2×Dic334C2×C411yescharacteristic
22Q8×D167—42no—
23C2×D668C2×C411yescharacteristic
24C2×D668D412yes—
25C2×C4×D167—21no—
26C2×C4×D167C2×C411yescharacteristic
27C2×C4×D167D412yes—
28#14581934—22no—
29#14581933—42no—
30C8×D167—42no—
31Dic167⋊C4—21no—
32Dic167⋊C4—22no—
33(C2×D167)⋊C4—21no—
34(C2×D167)⋊C4—22no—
35(C2×D167)⋊C4—22no—
36C4×Dic167C2×C411yescharacteristic
37C2×Q8×C167C2×C411yescharacteristic
38C2×Q8×C167C2311yescharacteristic
39M16×C167—22no—
40C4⋊C4×C167—22no—
41C42×C167C2×C411yescharacteristic
42C22×D167C22⋊C412yes—
43C22×D167C22⋊C411yescharacteristic
44Dic334—42no—
45Dic334—41no—
46D668—41no—
47D668—42no—
48C2×Dic167—21no—
49C2×Dic167—21no—
50C2×Dic167C22⋊C411yescharacteristic
51C4×D167—22no—
52C4×D167—22no—
53C4×D167—22no—
54C4×D167—42no—
55C167⋊C8—42no—
56Q8×C167—41no—
57Q8×C167—41no—
58C2×C4×C167—21no—
59C2×C4×C167—21no—
60C2×C4×C167C22×C411yescharacteristic
61C2×C4×C167C2×D412yes—
62C8×C167—42no—
63C22×C167C2×C22⋊C411yescharacteristic
64C2×D167—21no—
65C2×D167—22no—
66C2×D167—42no—
67C2×D167—41no—
68C4×C167—22no—
69C4×C167—21no—
70C4×C167—41no—
71C4×C167—41no—
72Dic167—41no—
73Dic167—41no—
74Dic167#9912yes—
75C2×C167—21no—
76C2×C167C2×#9911yescharacteristic
77D167—41no—
78D167—42no—
79C167#213511yescharacteristic, complemented, Sylow
80#2135—1671nomaximal, Sylow
81#536—1671no—
82#413—1672no—
83#356—1671no—
84#356—1672no—
85#356D16711yescharacteristic, complemented
86C2×#41—1671no—
87C22×Q8—1671no—
88C2×M16—3342no—
89C4⋊Q8—1671no—
90C4⋊Q8C2×D16711yescharacteristic
91#124—1671no—
92#124—1671no—
93#123—3342no—
94C22⋊Q8—3342no—
95#117—1671no—
96#101—1672no—
97#101C2×D16712yes—
98#100—3342no—
99#41—6682no—
100C2×Q8—3341no—
101C2×Q8—6682no—
102C2×Q8—1671no—
103C2×Q8C4×D16711yescharacteristic
104C2×Q8C22×D16711yescharacteristic
105C2×D4—1671no—
106C2×D4—1672no—
107C22×C4—3341no—
108C22×C4—1671no—
109C22×C4—1672no—
110M16—22no—
111M16—3342no—
112M16—6682no—
113C2×C8—6682no—
114C4⋊C4—22no—
115C4⋊C4—3341no—
116C4⋊C4—3342no—
117C22⋊C4—3341no—
118C22⋊C4—3342no—
119C22⋊C4—3342no—
120C42—1671no—
121C42C4×D16711yescharacteristic
122C23—1672no—
123C23—1671no—
124Q8—41no—
125Q8—41no—
126Q8—6681no—
127Q8—6682no—
128D4—6682no—
129D4—6681no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—3342no—
133C2×C4—3342no—
134C2×C4—3342no—
135C2×C4—3341no—
136C2×C4—3341no—
137C2×C4—6682no—
138C2×C4—1671no—
139C2×C4C2×C4×D16711yescharacteristic
140C2×C4D4×D16712yes—
141C8—42no—
142C8—6682no—
143C22—3341no—
144C22—3342no—
145C22—6682no—
146C22—6681no—
147C22C22⋊C4×D16711yescharacteristic
148C4—22no—
149C4—21no—
150C4—41no—
151C4—41no—
152C4—6681no—
153C4—6681no—
154C4—1672no—
155C2—21no—
156C2—6681no—
157C2—6682no—
158C2#99×D16711yescharacteristic
subgroupG/H
centerC2#99×D167
commutatorC2×C4×C167C22×C4
FrattiniC2×Q8C22×D167
Fitting#356×C167C2
socleC2×C167C2×#99
prime radicalC2×#14581951C22
cosocleC2×Q8×C167C23
hypercenter#356D167
hypocenterC167#2135
LCSG ▷ C2×C4×C167 ▷ C22×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
3. #24923068C2semidirect
4. #24923067C2semidirect
5. #356×C167C2semidirect
79. C167#2135semidirect
85. #356D167semidirect

Maximal subgroups

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

subgroupindex
#249238442
#249232522
#249230682
#249230672
#356×C1672
#2135167