Finite Group Databasebeta

← #30640178

Subgroups — #30640178

Order 16256.

198 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 63 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#22549877C211yesmaximal, characteristic
3#22549843C211yesmaximal, characteristic
4#22549645C211yesmaximal, characteristic, complemented
5#22549644C211yesmaximal, characteristic, complemented
6#22549640C211yesmaximal, characteristic, complemented
7#417×C127C211yesmaximal, characteristic
8C2×#14523232C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C22×C127⋊C8C2211yescharacteristic
11C2×C4⋊Dic127C2211yescharacteristic
12#16861140—22no—
13#16861133—21no—
14#16861130—21no—
15C22⋊(C127⋊C8)—22no—
16C2×#41×C127C2211yescharacteristic
17C2×M16×C127C2211yescharacteristic
18C2×C4⋊C4×C127C2211yescharacteristic
19#103×C127—21no—
20#102×C127—21no—
21C22×Dic127—21no—
22#14523232—21no—
23#14523232—21no—
24#14523232—21no—
25#14523232—21no—
26C2×C127⋊C8—21no—
27C2×C127⋊C8—21no—
28C2×C127⋊C8—21no—
29C2×C127⋊C8—21no—
30C2×C127⋊C8D412yes—
31C4⋊Dic127—21no—
32C4⋊Dic127—21no—
33#41×C127—21no—
34#41×C127—21no—
35#41×C127—21no—
36#41×C127—21no—
37C2×Q8×C127D411yescharacteristic
38C2×D4×C127—21no—
39C2×D4×C127D411yescharacteristic
40C22×C4×C127—21no—
41C22×C4×C127—21no—
42C22×C4×C127C2311yescharacteristic
43M16×C127—21no—
44M16×C127—21no—
45C2×C8×C127—21no—
46C4⋊C4×C127—21no—
47C4⋊C4×C127—21no—
48C2×Dic127—41no—
49C2×Dic127—81no—
50C127⋊C8—22no—
51C127⋊C8—41no—
52C127⋊C8—41no—
53C23×C127—21no—
54C23×C127#4111yescharacteristic
55Q8×C127—21no—
56Q8×C127—21no—
57D4×C127—21no—
58D4×C127—21no—
59D4×C127—41no—
60D4×C127—41no—
61C2×C4×C127—21no—
62C2×C4×C127—41no—
63C2×C4×C127—41no—
64C2×C4×C127—41no—
65C2×C4×C127—81no—
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127C2×D411yescharacteristic
68C2×C4×C127#4111yescharacteristic
69C2×C4×C127#4111yescharacteristic
70C2×C4×C127#4111yescharacteristic
71C2×C4×C127#4111yescharacteristic
72C8×C127—41no—
73C22×C127—21no—
74C22×C127—21no—
75C22×C127—41no—
76C22×C127—41no—
77C22×C127—41no—
78C22×C127#12111yescharacteristic
79C22×C127#12611yescharacteristic
80C22×C127#12611yescharacteristic
81C4×C127—41no—
82C4×C127—81no—
83C4×C127#12311yescharacteristic
84C4×C127#12311yescharacteristic
85C4×C127#12311yescharacteristic
86C4×C127#12411yescharacteristic
87Dic127—81no—
88C2×C127—21no—
89C2×C127—21no—
90C2×C127—81no—
91C2×C127#39711yescharacteristic
92C2×C127#47111yescharacteristic
93C2×C127#47111yescharacteristic
94#2079—1271nomaximal, Sylow
95C127#207911yescharacteristic, complemented, Sylow
96C2×#106—1271no—
97#417—1271no—
98#417D12711yescharacteristic
99#408—1271no—
100#343—1271no—
101#343—1271no—
102#341—1271no—
103C2×#41C2×D12711yescharacteristic
104C2×M16—1271no—
105C2×M16—1271no—
106C2×M16C2×D12711yescharacteristic
107C22×C8—1271no—
108C2×C4⋊C4—1271no—
109C2×C4⋊C4C2×D12711yescharacteristic
110#106—2542no—
111#103—21no—
112#103—2541no—
113#102—21no—
114#102—2541no—
115C22⋊C8—2542no—
116#41—21no—
117#41—21no—
118#41—21no—
119#41—21no—
120C2×Q8C22⋊D12711yescharacteristic
121C2×D4—21no—
122C2×D4C22⋊D12711yescharacteristic
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—2541no—
126C22×C4C22×D12711yescharacteristic
127M16—21no—
128M16—21no—
129M16—2541no—
130M16—2541no—
131M16—2541no—
132M16—2541no—
133C2×C8—21no—
134C2×C8—1272no—
135C2×C8—2541no—
136C2×C8—2541no—
137C2×C8—2541no—
138C2×C8—2541no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—2541no—
142C4⋊C4—2541no—
143C23—21no—
144C23#1452324711yescharacteristic
145Q8—21no—
146Q8—21no—
147D4—21no—
148D4—21no—
149D4—41no—
150D4—41no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—81no—
156C2×C4—10161no—
157C2×C4—5081no—
158C2×C4D4×D12711yescharacteristic
159C2×C4C2×C22⋊D12711yescharacteristic
160C2×C4#1452324711yescharacteristic
161C2×C4#1452324711yescharacteristic
162C2×C4#1452324811yescharacteristic
163C2×C4#1452324811yescharacteristic
164C8—41no—
165C8—5081no—
166C8—5081no—
167C8—2542no—
168C22—21no—
169C22—21no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22#1686118111yescharacteristic
174C22#1686120211yescharacteristic
175C22#1686120211yescharacteristic
176C4—41no—
177C4—81no—
178C4—10161no—
179C4#1686119011yescharacteristic
180C4#1686119111yescharacteristic
181C4#1686119211yescharacteristic
182C4#1686119511yescharacteristic
183C2—21no—
184C2—21no—
185C2—81no—
186C2#2254980411yescharacteristic
187C2#2255013711yescharacteristic
188C2#2255014111yescharacteristic
subgroupG/H
centerC22#16861181
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#417×C127C2
socleC22×C127#121
prime radicalC2×#41×C127C22
hypercenter#417D127
hypocenterC127#2079
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#16861140C2semidirect
4. #22549645C2semidirect
5. #22549644C2semidirect
6. #22549640C2semidirect
95. C127#2079semidirect

Maximal subgroups

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

subgroupindex
C2×#168611402
#225498772
#225498432
#225496452
#225496442
#225496402
#417×C1272
#2079127