Finite Group Databasebeta

← #30640278

Subgroups — #30640278

Order 16256.

238 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 85 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×#16861142C211yesmaximal, characteristic
2C2×#16861140C211yesmaximal, characteristic
3C2×#16861133C211yesmaximal, characteristic
4C2×#16861133C211yesmaximal, characteristic
5#22549742C211yesmaximal, characteristic
6#22549631C211yesmaximal, characteristic
7#391×C127C211yesmaximal, characteristic
8C22×C127⋊C8C2211yescharacteristic
9C22×C127⋊C8C2211yescharacteristic
10C2×C4⋊Dic127C2211yescharacteristic
11C2×C4⋊Dic127C2211yescharacteristic
12#16861142—22no—
13#16861140—22no—
14#16861133—22no—
15#16861133—22no—
16C127⋊#95—22no—
17C22×Q8×C127C2211yescharacteristic
18C2×C4⋊C4×C127C2211yescharacteristic
19C2×C42×C127C2211yescharacteristic
20#95×C127—22no—
21C22×Dic127—21no—
22C22×Dic127—21no—
23C2×C127⋊C8—21no—
24C2×C127⋊C8—21no—
25C2×C127⋊C8—21no—
26C2×C127⋊C8—21no—
27C2×C127⋊C8D412yes—
28C2×C127⋊C8D412yes—
29C4⋊Dic127—21no—
30C4⋊Dic127—21no—
31C4⋊Dic127—21no—
32C4⋊Dic127—21no—
33C2×Q8×C127—21no—
34C2×Q8×C127—21no—
35C2×Q8×C127—21no—
36C2×Q8×C127—21no—
37C2×Q8×C127—21no—
38C2×Q8×C127Q812yes—
39C22×C4×C127—21no—
40C22×C4×C127—21no—
41C22×C4×C127D412yes—
42C22×C4×C127C2311yescharacteristic
43C4⋊C4×C127—21no—
44C4⋊C4×C127—21no—
45C42×C127—21no—
46C42×C127—21no—
47C2×Dic127—41no—
48C2×Dic127—41no—
49C2×Dic127—42no—
50C2×Dic127—42no—
51C127⋊C8—22no—
52C127⋊C8—22no—
53C23×C127C2×D411yescharacteristic
54Q8×C127—22no—
55Q8×C127—22no—
56Q8×C127—41no—
57Q8×C127—41no—
58C2×C4×C127—22no—
59C2×C4×C127—22no—
60C2×C4×C127—22no—
61C2×C4×C127—22no—
62C2×C4×C127—41no—
63C2×C4×C127—41no—
64C2×C4×C127—41no—
65C2×C4×C127—42no—
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127C2×D411yescharacteristic
68C2×C4×C127C2×Q811yescharacteristic
69C2×C4×C127#4111yescharacteristic
70C2×C4×C127#4111yescharacteristic
71C2×C4×C127#4111yescharacteristic
72C22×C127C22⋊Q811yescharacteristic
73C22×C127#12311yescharacteristic
74C22×C127#12711yescharacteristic
75C22×C127#13512yes—
76C22×C127#13712yes—
77C4×C127—42no—
78C4×C127—42no—
79C4×C127—81no—
80C4×C127C22⋊Q811yescharacteristic
81C4×C127C22⋊Q811yescharacteristic
82C4×C127#12311yescharacteristic
83C4×C127#12311yescharacteristic
84Dic127—81no—
85Dic127—81no—
86C2×C127#39911yescharacteristic
87C2×C127#47912yes—
88C2×C127#48512yes—
89C2×C127#49511yescharacteristic
90C2×C127#49711yescharacteristic
91#2097—1271nomaximal, Sylow
92C127#209711yescharacteristic, complemented, Sylow
93C2×#107—1271no—
94C2×#106—1271no—
95C2×#103—1271no—
96C2×#103—1271no—
97#391D12711yescharacteristic
98#382—1271no—
99#336—1271no—
100C22×Q8C2×D12711yescharacteristic
101C22×C8—1271no—
102C22×C8—1271no—
103C2×C4⋊C4—1271no—
104C2×C4⋊C4—1271no—
105C2×C4⋊C4C2×D12711yescharacteristic
106C2×C42C2×D12711yescharacteristic
107#107—2542no—
108#106—2542no—
109#103—2542no—
110#103—2542no—
111#95—22no—
112#95—2542no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8Dic25412yes—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—2541no—
122C22×C4—2541no—
123C22×C4C22⋊D12712yes—
124C22×C4C22×D12711yescharacteristic
125C2×C8—1272no—
126C2×C8—1272no—
127C2×C8—2541no—
128C2×C8—2541no—
129C2×C8—2541no—
130C2×C8—2541no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—2541no—
134C4⋊C4—2541no—
135C4⋊C4—2541no—
136C4⋊C4—2541no—
137C42—21no—
138C42—21no—
139C23C2×C22⋊D12711yescharacteristic
140Q8—22no—
141Q8—22no—
142Q8—41no—
143Q8—41no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—42no—
152C2×C4—5081no—
153C2×C4—5081no—
154C2×C4—5082no—
155C2×C4—5082no—
156C2×C4D4×D12711yescharacteristic
157C2×C4D4×D12711yescharacteristic
158C2×C4C2×Dic25411yescharacteristic
159C2×C4#1452324711yescharacteristic
160C2×C4#1452324711yescharacteristic
161C2×C4#1452324811yescharacteristic
162C8—2542no—
163C8—2542no—
164C22#1686118811yescharacteristic
165C22#1686119011yescharacteristic
166C22#1686120511yescharacteristic
167C22#1686122912yes—
168C22#1686124012yes—
169C4—42no—
170C4—42no—
171C4—81no—
172C4—10161no—
173C4—10161no—
174C4#1686118611yescharacteristic
175C4#1686118611yescharacteristic
176C4#1686119111yescharacteristic
177C4#1686119111yescharacteristic
178C2#2254981211yescharacteristic
179C2#2255018312yes—
180C2#2255021412yes—
181C2#2255025811yescharacteristic
182C2#2255026811yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#391×C127C2
socle, prime radicalC23×C127C2×D4
hypercenter#391D127
hypocenterC127#2097
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
92. C127#2097semidirect

Maximal subgroups

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

subgroupindex
C2×#168611422
C2×#168611402
C2×#168611332
#225497422
#225496312
#391×C1272
#2097127