Finite Group Databasebeta

← #30640179

Subgroups — #30640179

Order 16256.

198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 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×#16861140C211yesmaximal, characteristic, complemented
2#22549878C211yesmaximal, characteristic
3#22549875C211yesmaximal, characteristic
4#22549844C211yesmaximal, characteristic
5#22549644C211yesmaximal, characteristic, complemented
6#22549640C211yesmaximal, characteristic, complemented
7#343×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C22×C127⋊C8C2211yescharacteristic
11C2×C4⋊Dic127C2211yescharacteristic
12#16861140—22no—
13#16861134—21no—
14#16861132—21no—
15#16861131—21no—
16#16861129—21no—
17(C2×D127)⋊C8—22no—
18C2×M16×C127C2211yescharacteristic
19C2×M16×C127C2211yescharacteristic
20C2×C4⋊C4×C127C2211yescharacteristic
21#14523248—21no—
22#14523248—21no—
23#14523248—21no—
24#14523248—21no—
25C2×Dic254D411yescharacteristic
26C2×D508D411yescharacteristic
27C2×C22⋊D127—21no—
28C22×Dic127—21no—
29C2×C4×D127—21no—
30#14523232—21no—
31#14523232—21no—
32C2×C127⋊C8—21no—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C2×C127⋊C8D412yes—
36C4⋊Dic127—21no—
37C4⋊Dic127—21no—
38C22×C4×C127—21no—
39C22×C4×C127C2311yescharacteristic
40M16×C127—21no—
41M16×C127—21no—
42M16×C127—21no—
43M16×C127—21no—
44C2×C8×C127—21no—
45C2×C8×C127—21no—
46C4⋊C4×C127—21no—
47C4⋊C4×C127—21no—
48C22×D127—21no—
49Dic254—21no—
50Dic254—21no—
51D508—21no—
52D508—21no—
53C22⋊D127—41no—
54C22⋊D127—41no—
55C2×Dic127—21no—
56C2×Dic127—41no—
57C2×Dic127—81no—
58C4×D127—41no—
59C4×D127—41no—
60C127⋊C8—22no—
61C127⋊C8—41no—
62C23×C127#4111yescharacteristic
63C2×C4×C127—41no—
64C2×C4×C127—81no—
65C2×C4×C127C2×D411yescharacteristic
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127#4111yescharacteristic
68C2×C4×C127#4111yescharacteristic
69C2×C4×C127#4111yescharacteristic
70C2×C4×C127#4111yescharacteristic
71C8×C127—41no—
72C8×C127—41no—
73C22×C127—21no—
74C22×C127—21no—
75C22×C127#12111yescharacteristic
76C22×C127#12611yescharacteristic
77C22×C127#12611yescharacteristic
78C2×D127—41no—
79C2×D127—41no—
80C2×D127—41no—
81C4×C127—81no—
82C4×C127#12311yescharacteristic
83C4×C127#12311yescharacteristic
84C4×C127#12311yescharacteristic
85C4×C127#12411yescharacteristic
86Dic127—41no—
87Dic127—81no—
88C2×C127—21no—
89C2×C127—21no—
90C2×C127#39711yescharacteristic
91C2×C127#47111yescharacteristic
92C2×C127#47111yescharacteristic
93D127—81no—
94#2079—1271nomaximal, Sylow
95C127#207911yescharacteristic, complemented, Sylow
96C2×#106—1271no—
97#417—1271no—
98#417—1271no—
99#408—1271no—
100#343—1271no—
101#343D12711yescharacteristic, complemented
102#341—1271no—
103C2×#41—1271no—
104C2×M16—1271no—
105C2×M16C2×D12711yescharacteristic
106C2×M16C2×D12711yescharacteristic
107C22×C8—1271no—
108C2×C4⋊C4—1271no—
109C2×C4⋊C4C2×D12711yescharacteristic
110#106—2542no—
111#103—2541no—
112#103—2541no—
113#102—2541no—
114#102—2541no—
115C22⋊C8—2542no—
116#41—2541no—
117#41—2541no—
118#41—2541no—
119#41—2541no—
120C2×Q8—1271no—
121C2×D4—1271no—
122C2×D4—2541no—
123C22×C4—21no—
124C22×C4—2541no—
125C22×C4—2541no—
126C22×C4C22×D12711yescharacteristic
127M16—21no—
128M16—21no—
129M16—21no—
130M16—21no—
131M16—2541no—
132M16—2541no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—1272no—
136C2×C8—2541no—
137C2×C8—2541no—
138C2×C8—2541no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—2541no—
142C4⋊C4—2541no—
143C23—2541no—
144C23#1452324811yescharacteristic
145Q8—2541no—
146Q8—2541no—
147D4—5081no—
148D4—5081no—
149D4—2541no—
150D4—2541no—
151C2×C4—41no—
152C2×C4—81no—
153C2×C4—10161no—
154C2×C4—5081no—
155C2×C4—5081no—
156C2×C4—5081no—
157C2×C4—2541no—
158C2×C4D4×D12711yescharacteristic
159C2×C4D4×D12711yescharacteristic
160C2×C4#1452324711yescharacteristic
161C2×C4#1452324811yescharacteristic
162C2×C4#1452324811yescharacteristic
163C2×C4#1452324911yescharacteristic
164C8—41no—
165C8—41no—
166C8—5081no—
167C8—2542no—
168C22—21no—
169C22—21no—
170C22—5081no—
171C22—5081no—
172C22—5081no—
173C22#1686118211yescharacteristic
174C22#1686120311yescharacteristic
175C22#1686120411yescharacteristic
176C4—81no—
177C4—10161no—
178C4—5081no—
179C4#1686119111yescharacteristic
180C4#1686119411yescharacteristic
181C4#1686119411yescharacteristic
182C4#1686119511yescharacteristic
183C2—21no—
184C2—21no—
185C2—10161no—
186C2#2254980511yescharacteristic
187C2#2255013811yescharacteristic
188C2#2255014211yescharacteristic
subgroupG/H
centerC22#16861182
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#343×C127C2
socleC22×C127#121
prime radicalC2×#14523248C22
hypercenter#343D127
hypocenterC127#2079
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. C2×#16861140C2semidirect
5. #22549644C2semidirect
6. #22549640C2semidirect
7. #343×C127C2semidirect
95. C127#2079semidirect
101. #343D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611402
#225498782
#225498752
#225498442
#225496442
#225496402
#343×C1272
#2079127