Finite Group Databasebeta

← #46838017

Subgroups — #46838017

Order 34688.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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#31774468C211yesmaximal, characteristic
2#31773881C211yesmaximal, characteristic
3#31773686C214yesmaximal, complemented
4#410×C271C211yesmaximal, characteristic
5C22×Dic542C411yescharacteristic, complemented
6C2×#16864638—21no—
7#22576158—21no—
8#22576147—21no—
9#22576141C412yescomplemented
10#22576141C2212yes—
11#22576139—21no—
12#22576139—21no—
13#22576119C411yescharacteristic, complemented
14#22576079—21no—
15#22576077C2212yes—
16C2×#41×C271C2211yescharacteristic
17#117×C271—21no—
18#99×C271—21no—
19#99×C271C2212yes—
20C2×Dic542—21no—
21C2×Dic542—21no—
22C2×Dic542—41no—
23C2×Dic542—41no—
24C22×Dic271C2×C412yes—
25#16864638—22no—
26#16864638—41no—
27C2×C271⋊C8—41no—
28C4⋊Dic271—21no—
29Dic271⋊C4—22no—
30Dic271⋊C4—22no—
31C22⋊Dic271—22no—
32C22⋊Dic271—22no—
33C22⋊Dic271C2×C412yes—
34C4×Dic271—21no—
35#41×C271—41no—
36#41×C271—41no—
37C2×Q8×C271D411yescharacteristic
38C2×D4×C271C2×C411yescharacteristic
39C2×D4×C271D411yescharacteristic
40C2×D4×C271C2311yescharacteristic
41C22×C4×C271C2×C411yescharacteristic
42C22×C4×C271D411yescharacteristic
43C22×C4×C271D411yescharacteristic
44C4⋊C4×C271—41no—
45C22⋊C4×C271—22no—
46C42×C271—41no—
47Dic542—42no—
48Dic542—41no—
49Dic542—41no—
50C2×Dic271—22no—
51C2×Dic271—22no—
52C2×Dic271—42no—
53C271⋊C8—42no—
54C23×C271C22⋊C411yescharacteristic
55C23×C271C22×C411yescharacteristic
56C23×C271C2×D411yescharacteristic
57Q8×C271—41no—
58D4×C271—41no—
59D4×C271—41no—
60D4×C271—41no—
61C2×C4×C271—21no—
62C2×C4×C271—21no—
63C2×C4×C271—21no—
64C2×C4×C271—21no—
65C2×C4×C271—21no—
66C2×C4×C271—21no—
67C2×C4×C271—42no—
68C2×C4×C271C22⋊C411yescharacteristic
69C2×C4×C271C22⋊C411yescharacteristic
70C2×C4×C271C22⋊C411yescharacteristic
71C2×C4×C271C2×D411yescharacteristic
72C22×C271—21no—
73C22×C271—21no—
74C22×C271—21no—
75C22×C271—41no—
76C22×C271—41no—
77C22×C271—41no—
78C22×C271C2×C22⋊C411yescharacteristic
79C4×C271—21no—
80C4×C271—21no—
81C4×C271—21no—
82C4×C271—82no—
83C4×C271#9911yescharacteristic
84C4×C271#9911yescharacteristic
85Dic271—42no—
86Dic271—42no—
87C2×C271—21no—
88C2×C271—41no—
89C2×C271—41no—
90C2×C271—41no—
91C2×C271C2×#9911yescharacteristic
92C271#212311yescharacteristic, complemented, Sylow
93#2123—2711nomaximal, Sylow
94#536—2711no—
95#413—2711no—
96#410D27111yescharacteristic
97#352—2714no—
98C2×#41C2×D27111yescharacteristic
99C22×Q8—2711no—
100C2×M16—5421no—
101C4⋊Q8—5421no—
102#124—5421no—
103#123—2712no—
104#123—2712no—
105C22⋊Q8—5421no—
106C22⋊Q8—5421no—
107#117—21no—
108#117—2711no—
109#101—5421no—
110#100—2712no—
111#99—21no—
112#99C2×D27112yes—
113#41—41no—
114#41—41no—
115C2×Q8—5421no—
116C2×Q8—5421no—
117C2×Q8—10841no—
118C2×Q8—10841no—
119C2×Q8D108411yescharacteristic
120C2×D4C4×D27111yescharacteristic
121C2×D4C22⋊D27111yescharacteristic
122C2×D4C22×D27111yescharacteristic
123C22×C4—2712no—
124C22×C4C4×D27111yescharacteristic
125C22×C4C22⋊D27111yescharacteristic
126C22×C4D108411yescharacteristic
127M16—5422no—
128M16—10841no—
129C2×C8—10841no—
130C4⋊C4—41no—
131C4⋊C4—5422no—
132C4⋊C4—5421no—
133C4⋊C4—5422no—
134C22⋊C4—22no—
135C22⋊C4—2712no—
136C22⋊C4—5422no—
137C22⋊C4—5422no—
138C42—41no—
139C42—5421no—
140C23(C2×D271)⋊C411yescharacteristic
141C23C2×C4×D27111yescharacteristic
142C23C2×C22⋊D27111yescharacteristic
143Q8—41no—
144Q8—10842no—
145Q8—10841no—
146Q8—10841no—
147D4—41no—
148D4—41no—
149D4—41no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—42no—
157C2×C4—5422no—
158C2×C4—5422no—
159C2×C4—10842no—
160C2×C4(C2×D271)⋊C411yescharacteristic
161C2×C4(C2×D271)⋊C411yescharacteristic
162C2×C4(C2×D271)⋊C411yescharacteristic
163C2×C4C2×D108411yescharacteristic
164C8—10842no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22C2×(C2×D271)⋊C411yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—82no—
176C4—10842no—
177C4—10842no—
178C4#2257607411yescharacteristic
179C4#2257607411yescharacteristic
180C2—21no—
181C2—41no—
182C2—41no—
183C2—41no—
184C2C2×#2257607411yescharacteristic
subgroupG/H
centerC2C2×#22576074
commutatorC23×C271C22×C4
FrattiniC2×D4C22×D271
Fitting#410×C271C2
socleC2×C271C2×#99
prime radicalC2×#41×C271C22
cosocleC2×D4×C271C23
hypercenter#410D271
hypocenterC271#2123
LCSG ▷ C23×C271 ▷ C22×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
3. #31773686C2semidirect
5. C22×Dic542C4semidirect
9. #22576141C4semidirect
13. #22576119C4semidirect
92. C271#2123semidirect

Maximal subgroups

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

subgroupindex
#317744682
#317738812
#317736862
#410×C2712
#2123271