Finite Group Databasebeta

← #30640011

Subgroups — #30640011

Order 16256.

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

normal subgroups only

This group has 254 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
1C2×#16861239C211yesmaximal, characteristic
2#22549925C211yesmaximal, characteristic, complemented
3#22549884C211yesmaximal, characteristic, complemented
4#22549860C211yesmaximal, characteristic
5C2×#16861127C211yesmaximal, characteristic, complemented
6#22549646C211yesmaximal, characteristic, complemented
7#421×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9#16861239—22no—
10#16861239—22no—
11C2×#14523238—21no—
12C2×#14523236—21no—
13C2×#14523232C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15C2×#14523232C2211yescharacteristic
16#16861145—22no—
17#16861132—22no—
18#16861128—21no—
19#16861127—22no—
20#16861126—21no—
21C2×#41×C127C2211yescharacteristic
22C2×M16×C127C2211yescharacteristic
23#117×C127C2211yescharacteristic
24#104×C127—22no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254—21no—
29C2×Dic254—21no—
30C2×Dic254D412yes—
31C22×Dic127—21no—
32#14523238—24no—
33#14523236—24no—
34#14523232—21no—
35#14523232—21no—
36#14523232—21no—
37#14523232—21no—
38#14523232—21no—
39#14523232D412yes—
40C2×C127⋊C8—21no—
41C2×C127⋊C8—21no—
42C2×C127⋊C8—21no—
43#41×C127—21no—
44#41×C127—21no—
45#41×C127—21no—
46#41×C127—21no—
47C2×Q8×C127D411yescharacteristic
48C2×D4×C127—21no—
49C2×D4×C127D411yescharacteristic
50C22×C4×C127—21no—
51C22×C4×C127C2311yescharacteristic
52M16×C127—21no—
53M16×C127D412yes—
54C2×C8×C127—21no—
55C4⋊C4×C127—21no—
56C22⋊C4×C127—21no—
57C42×C127—21no—
58Dic254—22no—
59Dic254—22no—
60Dic254—41no—
61Dic254—41no—
62C2×Dic127—22no—
63C2×Dic127—41no—
64C2×Dic127—41no—
65C127⋊C8—22no—
66C127⋊C8—41no—
67C127⋊C8—41no—
68C23×C127—21no—
69C23×C127#4111yescharacteristic
70Q8×C127—21no—
71Q8×C127—21no—
72D4×C127—21no—
73D4×C127—21no—
74D4×C127—41no—
75D4×C127—41no—
76C2×C4×C127—21no—
77C2×C4×C127—41no—
78C2×C4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2×D411yescharacteristic
84C2×C4×C127#4111yescharacteristic
85C2×C4×C127#4111yescharacteristic
86C8×C127—22no—
87C22×C127—41no—
88C22×C127—41no—
89C22×C127—81no—
90C22×C127#12111yescharacteristic
91C22×C127#12311yescharacteristic
92C22×C127#12411yescharacteristic
93C4×C127—41no—
94C4×C127—81no—
95C4×C127#12011yescharacteristic
96C4×C127#12111yescharacteristic
97C4×C127#12111yescharacteristic
98C4×C127#12311yescharacteristic
99Dic127—42no—
100C2×C127—21no—
101C2×C127—21no—
102C2×C127—21no—
103C2×C127—81no—
104C2×C127#39411yescharacteristic
105#2049—1271nomaximal, Sylow
106C127#204911yescharacteristic, complemented, Sylow
107C2×#137—1271no—
108#430—1271no—
109#421D12711yescharacteristic
110#419—1271no—
111#413—1271no—
112C2×#101—1271no—
113#344—1271no—
114C2×#41C2×D12711yescharacteristic
115C22×Q8—1271no—
116#137—2542no—
117#137—2542no—
118C2×Q16—2541no—
119C2×SD16—2541no—
120C2×M16—1271no—
121C2×M16—1271no—
122C2×M16—1271no—
123C2×M16C2×D12711yescharacteristic
124#117C2×D12711yescharacteristic
125#108—2542no—
126#104—22no—
127#103—2542no—
128#101—2541no—
129#101—2542no—
130#100—2541no—
131#41—21no—
132#41—21no—
133#41—21no—
134#41—21no—
135C2×Q8—1272no—
136C2×Q8—2541no—
137C2×Q8—2541no—
138C2×Q8—2541no—
139C2×Q8—2541no—
140C2×Q8—2541no—
141C2×Q8D50811yescharacteristic
142C2×D4—21no—
143C2×D4D50811yescharacteristic
144C22×C4—21no—
145C22×C4—2541no—
146C22×C4C22×D12711yescharacteristic
147Q16—2544no—
148SD16—2544no—
149M16—21no—
150M16—1272no—
151M16—2541no—
152M16—2541no—
153M16—2541no—
154M16—2541no—
155M16—2541no—
156M16C22⋊D12712yes—
157C2×C8—21no—
158C2×C8—2541no—
159C2×C8—2541no—
160C2×C8—2541no—
161C4⋊C4—21no—
162C22⋊C4—21no—
163C42—21no—
164C23—21no—
165C23#1452324711yescharacteristic
166Q8—21no—
167Q8—21no—
168Q8—5081no—
169Q8—5081no—
170Q8—2542no—
171Q8—2542no—
172D4—21no—
173D4—21no—
174D4—41no—
175D4—41no—
176C2×C4—21no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—5081no—
181C2×C4—5081no—
182C2×C4—2542no—
183C2×C4D4×D12711yescharacteristic
184C2×C4D4×D12711yescharacteristic
185C2×C4C2×C22⋊D12711yescharacteristic
186C2×C4C2×D50811yescharacteristic
187C2×C4#1452324711yescharacteristic
188C2×C4#1452324811yescharacteristic
189C8—22no—
190C8—5081no—
191C8—5081no—
192C8—2542no—
193C22—41no—
194C22—41no—
195C22—81no—
196C22#1686118011yescharacteristic
197C22#1686119311yescharacteristic
198C22#1686119511yescharacteristic
199C4—41no—
200C4—81no—
201C4—5082no—
202C4#1686117911yescharacteristic
203C4#1686118111yescharacteristic
204C4#1686118311yescharacteristic
205C4#1686119111yescharacteristic
206C2—21no—
207C2—21no—
208C2—21no—
209C2—81no—
210C2#2254979211yescharacteristic
subgroupG/H
centerC2#22549792
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#421×C127C2
socleC2×C127#394
prime radicalC2×#41×C127C22
hypercenter#421D127
hypocenterC127#2049
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #22549925C2semidirect
3. #22549884C2semidirect
5. C2×#16861127C2semidirect
6. #22549646C2semidirect
106. C127#2049semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
#225499252
#225498842
#225498602
C2×#168611272
#225496462
#421×C1272
#2049127