Finite Group Databasebeta

← #46838032

Subgroups — #46838032

Order 34688.

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

normal subgroups only

This group has 286 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#31774465C211yesmaximal, characteristic
2#99×D271C212yesmaximal, complemented
3#31773692C211yesmaximal, characteristic, complemented
4#31773689C212yesmaximal, complemented
5#354×C271C211yesmaximal, characteristic, complemented
6C2×#16864655C411yescharacteristic, complemented
7C2×D4×D271C2211yescharacteristic
8#22576157C411yescharacteristic, complemented
9#22576156C2211yescharacteristic
10#22576149C411yescharacteristic, complemented
11#22576133—22no—
12#22576130—22no—
13#22576117C411yescharacteristic, complemented
14C22⋊C4×D271—22no—
15#22576075C2212yes—
16#22576074—22no—
17#124×C271C2211yescharacteristic
18#99×C271C2212yes—
19C23×D271D412yes—
20#16864655—42no—
21C2×D1084—21no—
22C2×D1084—22no—
23C2×D1084C2×C411yescharacteristic
24C2×D1084C2×C411yescharacteristic, complemented
25C2×C22⋊D271—22no—
26C2×C22⋊D271D412yes—
27D4×D271—42no—
28C2×C4×D271—21no—
29C2×C4×D271—42no—
30C2×C4×D271C2×C411yescharacteristic
31Dic271⋊C4—21no—
32C22⋊Dic271—22no—
33(C2×D271)⋊C4—21no—
34(C2×D271)⋊C4—22no—
35(C2×D271)⋊C4—42no—
36C4×Dic271C2×C411yescharacteristic, complemented
37C2×Q8×C271C2×C411yescharacteristic, complemented
38C2×D4×C271C2311yescharacteristic
39C22⋊C4×C271—22no—
40C22⋊C4×C271—22no—
41C42×C271C2×C411yescharacteristic, complemented
42C22×D271—22no—
43C22×D271—22no—
44C22×D271—21no—
45C22×D271—82no—
46C22×D271C22⋊C412yes—
47C22×D271C22⋊C411yescharacteristic
48D1084—41no—
49D1084—41no—
50D1084—42no—
51D1084—82no—
52C22⋊D271—42no—
53C22⋊D271—82no—
54C2×Dic271—21no—
55C2×Dic271—42no—
56C2×Dic271C22⋊C411yescharacteristic
57C4×D271—22no—
58C4×D271—42no—
59C4×D271—84no—
60C23×C271C2×D412yes—
61Q8×C271—41no—
62D4×C271—41no—
63C2×C4×C271—21no—
64C2×C4×C271—21no—
65C2×C4×C271—42no—
66C2×C4×C271C22×C411yescharacteristic
67C22×C271—22no—
68C22×C271—42no—
69C22×C271C2×C22⋊C411yescharacteristic
70C2×D271—22no—
71C2×D271—41no—
72C2×D271—44no—
73C2×D271—42no—
74C2×D271—42no—
75C2×D271—41no—
76C2×D271—82no—
77C2×D271#9912yes—
78C4×C271—21no—
79C4×C271—41no—
80C4×C271—41no—
81C4×C271—82no—
82Dic271—21no—
83Dic271—41no—
84Dic271—82no—
85C2×C271—21no—
86C2×C271—42no—
87C2×C271C2×#9911yescharacteristic
88D271—22no—
89D271—42no—
90D271—81no—
91C271#212711yescharacteristic, complemented, Sylow
92#2127—2711nomaximal, Sylow
93#535—2711no—
94C2×#99—2712no—
95#354—2711no—
96#354D27111yescharacteristic, complemented
97#353—2712no—
98C2×#41—2711no—
99C22×D4—2711no—
100#127—2711no—
101#127—2711no—
102#124—2711no—
103#124C2×D27111yescharacteristic
104#121—5422no—
105#120—5422no—
106#117—2711no—
107C2×C22⋊C4—5422no—
108#99—2712no—
109#99—5422no—
110#99C2×D27112yes—
111C24—2712no—
112#41—10842no—
113C2×Q8C4×D27111yescharacteristic, complemented
114C2×D4—2712no—
115C2×D4—2711no—
116C2×D4—2711no—
117C2×D4—5421no—
118C2×D4—5422no—
119C2×D4—5422no—
120C2×D4—10842no—
121C2×D4C22×D27111yescharacteristic
122C22×C4—2711no—
123C22×C4—5421no—
124C22×C4—10842no—
125C4⋊C4—5421no—
126C22⋊C4—22no—
127C22⋊C4—22no—
128C22⋊C4—5421no—
129C22⋊C4—5422no—
130C22⋊C4—5422no—
131C22⋊C4—10842no—
132C42—2711no—
133C42C4×D27111yescharacteristic, complemented
134C23—2711no—
135C23—2712no—
136C23—5422no—
137C23—5422no—
138C23—5421no—
139C23—21682no—
140C23D4×D27112yes—
141Q8—41no—
142D4—41no—
143D4—10841no—
144D4—10841no—
145D4—10842no—
146D4—10842no—
147D4—21682no—
148D4—21682no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—42no—
152C2×C4—2711no—
153C2×C4—5422no—
154C2×C4—5421no—
155C2×C4—10842no—
156C2×C4—10842no—
157C2×C4—21684no—
158C2×C4C2×C4×D27111yescharacteristic
159C22—22no—
160C22—42no—
161C22—2712no—
162C22—5422no—
163C22—10841no—
164C22—10842no—
165C22—10842no—
166C22—10844no—
167C22—10841no—
168C22—21682no—
169C22C22⋊C4×D27111yescharacteristic
170C4—21no—
171C4—41no—
172C4—41no—
173C4—82no—
174C4—5421no—
175C4—10841no—
176C4—21682no—
177C2—21no—
178C2—42no—
179C2—5422no—
180C2—10842no—
181C2—21681no—
182C2#99×D27111yescharacteristic
subgroupG/H
centerC2#99×D271
commutatorC2×C4×C271C22×C4
FrattiniC2×D4C22×D271
Fitting#354×C271C2
socleC2×C271C2×#99
prime radical#31774465C2
cosocleC2×D4×C271C23
hypercenter#354D271
hypocenterC271#2127
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
2. #99×D271C2semidirect
3. #31773692C2semidirect
4. #31773689C2semidirect
5. #354×C271C2semidirect
6. C2×#16864655C4semidirect
8. #22576157C4semidirect
10. #22576149C4semidirect
13. #22576117C4semidirect
24. C2×D1084C2×C4semidirect
36. C4×Dic271C2×C4semidirect
37. C2×Q8×C271C2×C4semidirect
41. C42×C271C2×C4semidirect
91. C271#2127semidirect
96. #354D271semidirect
113. C2×Q8C4×D271semidirect
133. C42C4×D271semidirect

Maximal subgroups

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

subgroupindex
#317744652
#99×D2712
#317736922
#317736892
#354×C2712
#2127271