Finite Group Databasebeta

← #46838037

Subgroups — #46838037

Order 34688.

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

normal subgroups only

This group has 238 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#31774467C211yesmaximal, characteristic
2#31773869C212yesmaximal
3#31773692C211yesmaximal, characteristic, complemented
4#31773691C212yesmaximal, complemented
5#354×C271C211yesmaximal, characteristic, complemented
6C2×#16864653C2211yescharacteristic
7C2×Q8×D271C411yescharacteristic, complemented
8#22576160C411yescharacteristic, complemented
9#22576158C411yescharacteristic, complemented
10#22576147C2211yescharacteristic
11#22576142—22no—
12#22576137—22no—
13#22576117C411yescharacteristic, complemented
14#22576116—22no—
15#22576075C2212yes—
16#22576074—22no—
17#124×C271C2211yescharacteristic
18#99×C271C2212yes—
19#16864653—42no—
20C2×Dic542—21no—
21C2×Dic542C2×C411yescharacteristic
22C2×Dic542C2×C411yescharacteristic
23Q8×D271—42no—
24C2×C22⋊D271D412yes—
25C22×Dic271D412yes—
26C2×C4×D271—21no—
27C2×C4×D271C2×C411yescharacteristic
28C4⋊Dic271—22no—
29Dic271⋊C4—21no—
30Dic271⋊C4—22no—
31Dic271⋊C4—42no—
32C22⋊Dic271—22no—
33C22⋊Dic271—22no—
34(C2×D271)⋊C4—21no—
35C4×Dic271—42no—
36C4×Dic271C2×C411yescharacteristic
37C2×Q8×C271C2×C411yescharacteristic
38C2×D4×C271C2311yescharacteristic
39C22⋊C4×C271—22no—
40C22⋊C4×C271—22no—
41C42×C271C2×C411yescharacteristic
42C22×D271C22⋊C411yescharacteristic
43Dic542—41no—
44Dic542—42no—
45Dic542—41no—
46C22⋊D271—42no—
47C2×Dic271—22no—
48C2×Dic271—22no—
49C2×Dic271—21no—
50C2×Dic271—21no—
51C2×Dic271—42no—
52C2×Dic271C22⋊C412yes—
53C2×Dic271C22⋊C411yescharacteristic
54C4×D271—22no—
55C4×D271—42no—
56C23×C271C2×D412yes—
57Q8×C271—41no—
58D4×C271—41no—
59C2×C4×C271—21no—
60C2×C4×C271—21no—
61C2×C4×C271—42no—
62C2×C4×C271C22×C411yescharacteristic
63C22×C271—22no—
64C22×C271—42no—
65C22×C271C2×C22⋊C411yescharacteristic
66C2×D271—21no—
67C2×D271—41no—
68C4×C271—21no—
69C4×C271—41no—
70C4×C271—41no—
71C4×C271—82no—
72Dic271—22no—
73Dic271—41no—
74Dic271—41no—
75Dic271—82no—
76Dic271#9912yes—
77C2×C271—21no—
78C2×C271—42no—
79C2×C271C2×#9911yescharacteristic
80D271—41no—
81C271#212911yescharacteristic, complemented, Sylow
82#2129—2711nomaximal, Sylow
83#536—2711no—
84#410—2712no—
85#354—2711no—
86#354—2712no—
87#354D27111yescharacteristic, complemented
88C2×#41—2711no—
89C22×Q8—2711no—
90C4⋊Q8—2711no—
91C4⋊Q8—2711no—
92#124—2711no—
93#124C2×D27111yescharacteristic
94#123—5422no—
95C22⋊Q8—5422no—
96#117—2711no—
97#117—5422no—
98#99—2712no—
99#99—5422no—
100#99C2×D27112yes—
101#41—10842no—
102C2×Q8—2711no—
103C2×Q8—2711no—
104C2×Q8—5421no—
105C2×Q8—10842no—
106C2×Q8C4×D27111yescharacteristic
107C2×D4—2712no—
108C2×D4C22×D27111yescharacteristic
109C22×C4—2712no—
110C22×C4—2711no—
111C22×C4—5421no—
112C4⋊C4—5421no—
113C4⋊C4—5422no—
114C4⋊C4—5422no—
115C4⋊C4—10842no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—5421no—
119C22⋊C4—5422no—
120C22⋊C4—5422no—
121C42—2711no—
122C42—10842no—
123C42C4×D27111yescharacteristic
124C23—2711no—
125C23D4×D27112yes—
126Q8—41no—
127Q8—10841no—
128Q8—10841no—
129Q8—10842no—
130D4—41no—
131D4—10842no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—42no—
135C2×C4—2711no—
136C2×C4—2712no—
137C2×C4—5422no—
138C2×C4—5422no—
139C2×C4—5422no—
140C2×C4—5421no—
141C2×C4—5421no—
142C2×C4—10842no—
143C2×C4—10842no—
144C2×C4C2×C4×D27111yescharacteristic
145C22—22no—
146C22—42no—
147C22—5421no—
148C22—10841no—
149C22C22⋊C4×D27111yescharacteristic
150C4—21no—
151C4—41no—
152C4—41no—
153C4—82no—
154C4—2712no—
155C4—5422no—
156C4—10841no—
157C4—10841no—
158C4—21682no—
159C2—21no—
160C2—42no—
161C2—10841no—
162C2#99×D27111yescharacteristic
subgroupG/H
centerC2#99×D271
commutatorC2×C4×C271C22×C4
FrattiniC2×D4C22×D271
Fitting#354×C271C2
socleC2×C271C2×#99
prime radicalC2×#16864653C22
cosocleC2×D4×C271C23
hypercenter#354D271
hypocenterC271#2129
LCSG ▷ C2×C4×C271 ▷ C22×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #31773692C2semidirect
4. #31773691C2semidirect
5. #354×C271C2semidirect
7. C2×Q8×D271C4semidirect
8. #22576160C4semidirect
9. #22576158C4semidirect
13. #22576117C4semidirect
81. C271#2129semidirect
87. #354D271semidirect

Maximal subgroups

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

subgroupindex
#317744672
#317738692
#317736922
#317736912
#354×C2712
#2129271