Finite Group Databasebeta

← #30639383

Subgroups — #30639383

Order 16256.

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

normal subgroups only

This group has 212 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
1#22549916C211yesmaximal, characteristic
2#22549886C211yesmaximal, characteristic
3C2×#16861132C211yesmaximal, characteristic
4C2×#16861127C211yesmaximal, characteristic
5#22549834C211yesmaximal, characteristic
6#22549640C211yesmaximal, characteristic
7#343×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10C22×C127⋊C8C2211yescharacteristic
11#16861168C2211yescharacteristic
12#16861142—21no—
13#16861140—21no—
14#16861134—21no—
15#16861134—21no—
16#16861132—21no—
17#16861132—21no—
18#16861127C414yescomplemented
19#16861119—21no—
20C8×Dic127—21no—
21C2×M16×C127C2211yescharacteristic
22C2×M16×C127C2211yescharacteristic
23C2×C4⋊C4×C127C2211yescharacteristic
24C2×Dic254—21no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254C2×C412yes—
30C22×Dic127—21no—
31#14523232—21no—
32#14523232C2×C412yes—
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C2×C127⋊C8—21no—
36C2×C127⋊C8D411yescharacteristic
37C2×C127⋊C8D411yescharacteristic
38C4⋊Dic127D411yescharacteristic
39C4⋊Dic127D411yescharacteristic
40C22⋊Dic127—21no—
41C4×Dic127—21no—
42C22×C4×C127—21no—
43C22×C4×C127C2311yescharacteristic
44M16×C127—21no—
45M16×C127—21no—
46M16×C127—21no—
47M16×C127C2×C412yes—
48C2×C8×C127—21no—
49C2×C8×C127—21no—
50C4⋊C4×C127—21no—
51C4⋊C4×C127—21no—
52Dic254—22no—
53Dic254—22no—
54Dic254—41no—
55Dic254—41no—
56C2×Dic127—22no—
57C2×Dic127—21no—
58C2×Dic127—21no—
59C2×Dic127—41no—
60C2×Dic127—41no—
61C127⋊C8—22no—
62C127⋊C8—21no—
63C127⋊C8—21no—
64C23×C127#4111yescharacteristic
65C2×C4×C127—41no—
66C2×C4×C127—81no—
67C2×C4×C127C22×C411yescharacteristic
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127#4111yescharacteristic
71C2×C4×C127#4111yescharacteristic
72C2×C4×C127#4111yescharacteristic
73C8×C127—22no—
74C8×C127—41no—
75C22×C127—21no—
76C22×C127—21no—
77C22×C127#11711yescharacteristic
78C22×C127#12111yescharacteristic
79C22×C127#12611yescharacteristic
80C4×C127—81no—
81C4×C127C4×D411yescharacteristic
82C4×C127C4×D411yescharacteristic
83C4×C127#12311yescharacteristic
84C4×C127#12411yescharacteristic
85Dic127—42no—
86Dic127—41no—
87Dic127—41no—
88C2×C127—21no—
89C2×C127—21no—
90C2×C127#38811yescharacteristic
91C2×C127#47111yescharacteristic
92C2×C127#47311yescharacteristic
93#1930—1271nomaximal, Sylow
94C127#193011yescharacteristic, complemented, Sylow
95#428—1271no—
96#419—1271no—
97C2×#103—1271no—
98C2×#101—1271no—
99#405—1271no—
100#343D12711yescharacteristic
101#341—1271no—
102C22×Q8—1271no—
103C2×M16—1271no—
104C2×M16C2×D12711yescharacteristic
105C2×M16C2×D12711yescharacteristic
106C22×C8—1271no—
107#117—1271no—
108C2×C4⋊C4C2×D12711yescharacteristic
109#107—2541no—
110#106—2541no—
111#103—2541no—
112#103—2541no—
113#103—2541no—
114#103—2541no—
115#101—1274no—
116#97—2541no—
117C4×C8—2541no—
118C2×Q8—1272no—
119C2×Q8—2541no—
120C2×Q8—2541no—
121C2×Q8—2541no—
122C2×Q8—2541no—
123C2×Q8—2541no—
124C22×C4—21no—
125C22×C4—2541no—
126C22×C4C22×D12711yescharacteristic
127M16—21no—
128M16—21no—
129M16—21no—
130M16—1272no—
131M16—2541no—
132M16C4×D12712yes—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—1271no—
136C2×C8—1271no—
137C2×C8—2541no—
138C2×C8—2541no—
139C2×C8—2541no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—1271no—
143C4⋊C4—1271no—
144C22⋊C4—2541no—
145C42—2541no—
146C23#1452324811yescharacteristic
147Q8—5081no—
148Q8—5081no—
149Q8—2542no—
150Q8—2542no—
151C2×C4—41no—
152C2×C4—81no—
153C2×C4—5081no—
154C2×C4—5081no—
155C2×C4—2542no—
156C2×C4—2541no—
157C2×C4—2541no—
158C2×C4C2×C4×D12711yescharacteristic
159C2×C4D4×D12711yescharacteristic
160C2×C4D4×D12711yescharacteristic
161C2×C4#1452324711yescharacteristic
162C2×C4#1452324811yescharacteristic
163C2×C4#1452324911yescharacteristic
164C8—22no—
165C8—41no—
166C8—2541no—
167C8—2541no—
168C8—2542no—
169C22—21no—
170C22—21no—
171C22#1686116611yescharacteristic
172C22#1686118211yescharacteristic
173C22#1686120411yescharacteristic
174C4—81no—
175C4—5082no—
176C4—5081no—
177C4—5081no—
178C4(C22⋊D127)⋊C411yescharacteristic
179C4#1686117111yescharacteristic
180C4#1686119411yescharacteristic
181C4#1686119511yescharacteristic
182C2—21no—
183C2—21no—
184C2#2254976511yescharacteristic
185C2#2255013811yescharacteristic
186C2#2255015011yescharacteristic
subgroupG/H
centerC22#16861182
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#343×C127C2
socleC22×C127#121
prime radicalC23×C127#41
cosocleC22×C4×C127C23
hypercenter#343D127
hypocenterC127#1930
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
18. #16861127C4semidirect
94. C127#1930semidirect

Maximal subgroups

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

subgroupindex
#225499162
#225498862
C2×#168611322
C2×#168611272
#225498342
#225496402
#343×C1272
#1930127