Finite Group Databasebeta

← #46838054

Subgroups — #46838054

Order 34688.

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

normal subgroups only

This group has 262 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#31773880C212yesmaximal, complemented
3#31773694C211yesmaximal, characteristic, complemented
4#31773693C212yesmaximal, complemented
5#355×C271C211yesmaximal, characteristic, complemented
6C2×#16864655C2211yescharacteristic
7C2×D4×D271C411yescharacteristic
8M16×D271—22no—
9#22576157C411yescharacteristic
10#22576156C411yescharacteristic
11#22576149C2211yescharacteristic
12#22576133—22no—
13#22576130—22no—
14#22576117C411yescharacteristic
15#22576079C2212yescomplemented
16#22576076—22no—
17#124×C271C2211yescharacteristic
18#101×C271C2212yescomplemented
19C23×D271—21no—
20#16864655—42no—
21C2×D1084—22no—
22C2×D1084C2×C411yescharacteristic
23C2×D1084C2×C411yescharacteristic
24C2×D1084D412yes—
25C2×C22⋊D271—22no—
26C2×C22⋊D271—21no—
27D4×D271—42no—
28C2×C4×D271C2×C411yescharacteristic
29C2×C4×D271D412yes—
30#16864638—22no—
31#16864637—42no—
32C8×D271—42no—
33Dic271⋊C4—21no—
34(C2×D271)⋊C4—21no—
35(C2×D271)⋊C4—22no—
36C4×Dic271C2×C411yescharacteristic
37C2×Q8×C271C2311yescharacteristic
38C2×D4×C271C2×C411yescharacteristic
39M16×C271—22no—
40C22⋊C4×C271—22no—
41C42×C271C2×C411yescharacteristic
42C22×D271—21no—
43C22×D271—42no—
44C22×D271—82no—
45C22×D271C22⋊C411yescharacteristic
46C22×D271C22⋊C412yes—
47D1084—41no—
48D1084—42no—
49D1084—41no—
50D1084—82no—
51C22⋊D271—42no—
52C22⋊D271—82no—
53C2×Dic271—21no—
54C2×Dic271C22⋊C411yescharacteristic
55C4×D271—22no—
56C4×D271—22no—
57C4×D271—22no—
58C271⋊C8—42no—
59C23×C271—21no—
60Q8×C271—41no—
61D4×C271—41no—
62C2×C4×C271—21no—
63C2×C4×C271C22×C411yescharacteristic
64C2×C4×C271C2×D412yes—
65C8×C271—42no—
66C22×C271—42no—
67C22×C271—41no—
68C22×C271C2×C22⋊C411yescharacteristic
69C2×D271—21no—
70C2×D271—22no—
71C2×D271—42no—
72C2×D271—41no—
73C2×D271—42no—
74C2×D271—41no—
75C2×D271—84no—
76C4×C271—22no—
77C4×C271—21no—
78C4×C271—41no—
79Dic271—41no—
80Dic271#9912yes—
81C2×C271—21no—
82C2×C271—81no—
83C2×C271C2×#9911yescharacteristic
84D271—42no—
85D271—41no—
86D271—81no—
87C271#213411yescharacteristic, complemented, Sylow
88#2134—2711nomaximal, Sylow
89#535—2711no—
90#413—2712no—
91#355—2711no—
92#355—2712no—
93#355D27111yescharacteristic, complemented
94C2×#41—2711no—
95C22×D4—2711no—
96C2×M16—5422no—
97#127—2711no—
98#127—2711no—
99#124—2711no—
100#124C2×D27111yescharacteristic
101#121—5422no—
102#120—5422no—
103#117—2711no—
104#101—2712no—
105#101C2×D27112yescomplemented
106#100—5422no—
107C24—5421no—
108#41—10842no—
109C2×Q8C22×D27111yescharacteristic
110C2×D4—2711no—
111C2×D4—2711no—
112C2×D4—2712no—
113C2×D4—5422no—
114C2×D4—5421no—
115C2×D4—5422no—
116C2×D4—10842no—
117C2×D4C4×D27111yescharacteristic
118C22×C4—2711no—
119C22×C4—2712no—
120M16—22no—
121M16—5422no—
122M16—10842no—
123C2×C8—10842no—
124C4⋊C4—5421no—
125C22⋊C4—22no—
126C22⋊C4—5421no—
127C22⋊C4—5422no—
128C42—2711no—
129C42C4×D27111yescharacteristic
130C23—21no—
131C23—2711no—
132C23—2712no—
133C23—5421no—
134C23—10842no—
135C23—21682no—
136Q8—41no—
137D4—41no—
138D4—10842no—
139D4—10841no—
140D4—10841no—
141D4—10842no—
142D4—21682no—
143D4—21682no—
144C2×C4—21no—
145C2×C4—2711no—
146C2×C4—5422no—
147C2×C4—5422no—
148C2×C4—5422no—
149C2×C4—5421no—
150C2×C4C2×C4×D27111yescharacteristic
151C2×C4D4×D27112yes—
152C8—42no—
153C8—10842no—
154C22—42no—
155C22—41no—
156C22—5421no—
157C22—5422no—
158C22—10842no—
159C22—10841no—
160C22—10842no—
161C22—10841no—
162C22—21684no—
163C22C22⋊C4×D27111yescharacteristic
164C4—22no—
165C4—21no—
166C4—41no—
167C4—2712no—
168C4—10841no—
169C2—21no—
170C2—81no—
171C2—10841no—
172C2—10842no—
173C2—21681no—
174C2#99×D27111yescharacteristic
subgroupG/H
centerC2#99×D271
commutatorC2×C4×C271C22×C4
FrattiniC2×Q8C22×D271
Fitting#355×C271C2
socleC2×C271C2×#99
prime radical#31774465C2
cosocleC2×Q8×C271C23
hypercenter#355D271
hypocenterC271#2134
LCSG ▷ C2×C4×C271 ▷ C22×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #31773880C2semidirect
3. #31773694C2semidirect
4. #31773693C2semidirect
5. #355×C271C2semidirect
15. #22576079C22semidirect
18. #101×C271C22semidirect
87. C271#2134semidirect
93. #355D271semidirect
105. #101C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317744652
#317738802
#317736942
#317736932
#355×C2712
#2134271