Finite Group Databasebeta

← #46837410

Subgroups — #46837410

Order 34688.

370 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C2×#22576191C212yesmaximal, complemented
2#31774478C211yesmaximal, characteristic
3C2×#22576088C212yesmaximal, complemented
4#31773666C211yesmaximal, characteristic, complemented
5#391×C271C211yesmaximal, characteristic, complemented
6C2×#16864654C2212yes—
7#22576191—24no—
8#22576191—24no—
9C2×#16864644—22no—
10C2×#16864641—22no—
11C2×#16864638C2212yes—
12#22576140—22no—
13#22576135—22no—
14#22576134—22no—
15#22576088—24no—
16C22×Q8×C271C2211yescharacteristic
17C2×C4⋊C4×C271C2211yescharacteristic
18C2×C42×C271C2211yescharacteristic
19#95×C271—22no—
20#16864654—22no—
21#16864654—22no—
22#16864654—22no—
23#16864654—22no—
24C2×Dic542D412yes—
25C2×D1084—22no—
26C2×D1084D412yes—
27C2×C22⋊D271—22no—
28C2×C4×D271—22no—
29#16864644—28no—
30#16864641—28no—
31#16864638—22no—
32#16864638D414yescomplemented
33C2×C271⋊C8—22no—
34C4⋊Dic271—22no—
35(C2×D271)⋊C4—24no—
36C2×Q8×C271—21no—
37C2×Q8×C271—22no—
38C2×Q8×C271—22no—
39C2×Q8×C271D412yescomplemented
40C22×C4×C271—21no—
41C22×C4×C271—21no—
42C22×C4×C271D412yes—
43C22×C4×C271C2311yescharacteristic
44C4⋊C4×C271—22no—
45C42×C271—22no—
46C22×D271—22no—
47Dic542—22no—
48Dic542—22no—
49D1084—22no—
50D1084—22no—
51D1084—82no—
52C22⋊D271—42no—
53C22⋊D271—42no—
54C2×Dic271—22no—
55C4×D271—42no—
56C4×D271—42no—
57C271⋊C8—24no—
58C23×C271C2×D411yescharacteristic
59Q8×C271—24no—
60Q8×C271—42no—
61C2×C4×C271—22no—
62C2×C4×C271—22no—
63C2×C4×C271—22no—
64C2×C4×C271—41no—
65C2×C4×C271—41no—
66C2×C4×C271—41no—
67C2×C4×C271—41no—
68C2×C4×C271—42no—
69C2×C4×C271C2×D412yes—
70C2×C4×C271C2×D412yes—
71C2×C4×C271C2×D411yescharacteristic
72C2×C4×C271#4111yescharacteristic
73C22×C271—21no—
74C22×C271—21no—
75C22×C271#12011yescharacteristic
76C22×C271#12111yescharacteristic
77C22×C271#12711yescharacteristic
78C2×D271—42no—
79C2×D271—42no—
80C2×D271—42no—
81C4×C271—42no—
82C4×C271—42no—
83C4×C271—42no—
84C4×C271#12012yes—
85C4×C271#12112yes—
86Dic271—42no—
87C2×C271—21no—
88C2×C271—21no—
89C2×C271#39211yescharacteristic
90C2×C271#45511yescharacteristic
91C2×C271#45611yescharacteristic
92D271—82no—
93C271#200711yescharacteristic, complemented, Sylow
94#2007—2711nomaximal, Sylow
95C2×#137—2712no—
96#537—2711no—
97C2×#104—2712no—
98#391D27111yescharacteristic, complemented
99#343—2711no—
100C2×#41—2712no—
101C22×Q8C2×D27111yescharacteristic
102#137—5424no—
103#137—5424no—
104C2×Q16—5422no—
105C2×SD16—5422no—
106C2×M16—2712no—
107C22⋊Q8—5422no—
108#121—5422no—
109#121—5422no—
110C2×C4⋊C4C2×D27111yescharacteristic
111C2×C42C2×D27111yescharacteristic
112#104—5424no—
113#95—22no—
114#41—5422no—
115#41—5422no—
116#41—5422no—
117#41—5422no—
118C2×Q8—21no—
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—2712no—
122C2×Q8D108412yescomplemented
123C2×D4—2712no—
124C2×D4—5422no—
125C2×D4—5422no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—5422no—
129C22×C4C22⋊D27112yes—
130C22×C4C22×D27111yescharacteristic
131Q16—5428no—
132SD16—5428no—
133M16—2714no—
134M16—5422no—
135C2×C8—5422no—
136C4⋊C4—22no—
137C4⋊C4—5422no—
138C22⋊C4—5424no—
139C42—22no—
140C23—5422no—
141C23C2×C22⋊D27111yescharacteristic
142Q8—24no—
143Q8—42no—
144Q8—5422no—
145Q8—5422no—
146D4—5422no—
147D4—5422no—
148D4—10842no—
149D4—10842no—
150D4—21682no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—5422no—
160C2×C4—10842no—
161C2×C4—10842no—
162C2×C4D4×D27112yes—
163C2×C4D4×D27112yes—
164C2×C4C2×D108411yescharacteristic
165C2×C4#1686465411yescharacteristic
166C8—5424no—
167C22—21no—
168C22—21no—
169C22—10842no—
170C22—10842no—
171C22—10842no—
172C22#2257612811yescharacteristic
173C22#2257613411yescharacteristic
174C22#2257615611yescharacteristic
175C4—42no—
176C4—42no—
177C4—42no—
178C4—10842no—
179C4#2257613012yes—
180C4#2257613312yes—
181C2—21no—
182C2—21no—
183C2—21682no—
184C2#3177380611yescharacteristic
185C2#3177407411yescharacteristic
186C2#3177407911yescharacteristic
subgroupG/H
centerC22#22576128
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#391×C271C2
socleC22×C271#120
prime radical#31774478C2
hypercenter#391D271
hypocenterC271#2007
LCSG ▷ C22×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#22576191C2semidirect
3. C2×#22576088C2semidirect
4. #31773666C2semidirect
5. #391×C271C2semidirect
32. #16864638D4semidirect
39. C2×Q8×C271D4semidirect
93. C271#2007semidirect
98. #391D271semidirect
122. C2×Q8D1084semidirect

Maximal subgroups

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

subgroupindex
C2×#225761912
#317744782
C2×#225760882
#317736662
#391×C2712
#2007271