Finite Group Databasebeta

← #30639777

Subgroups — #30639777

Order 16256.

370 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 75 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×#16861239C212yesmaximal, complemented
2C2×#16861136C212yesmaximal, complemented
3#22549781C211yesmaximal, characteristic, complemented
4#22549645C211yesmaximal, characteristic, complemented
5#537×C127C211yesmaximal, characteristic
6C22×Dic254C2211yescharacteristic
7#16861239—24no—
8#16861239—24no—
9C2×#14523238—22no—
10C2×#14523236—22no—
11C2×#14523232C2212yes—
12C2×C4×Dic127C2211yescharacteristic
13#16861136—24no—
14C127⋊#95—22no—
15C2×#41×C127C2212yes—
16C22⋊Q8×C127—22no—
17#121×C127—22no—
18#121×C127—22no—
19C2×C4⋊C4×C127C2211yescharacteristic
20C2×Dic254—22no—
21C2×Dic254—22no—
22C2×Dic254—21no—
23C2×Dic254D412yescomplemented
24C22×Dic127—21no—
25C22×Dic127—21no—
26#14523238—28no—
27#14523236—28no—
28#14523232—22no—
29#14523232D414yescomplemented
30C2×C127⋊C8—22no—
31C4×Dic127—22no—
32#41×C127—22no—
33#41×C127—22no—
34#41×C127—22no—
35#41×C127—22no—
36C2×Q8×C127D412yes—
37C2×D4×C127—22no—
38C2×D4×C127—22no—
39C2×D4×C127D412yes—
40C22×C4×C127—22no—
41C22×C4×C127D412yes—
42C22×C4×C127C2311yescharacteristic
43C4⋊C4×C127—22no—
44C4⋊C4×C127—22no—
45C22⋊C4×C127—24no—
46Dic254—24no—
47Dic254—42no—
48C2×Dic127—22no—
49C2×Dic127—22no—
50C2×Dic127—41no—
51C2×Dic127—41no—
52C2×Dic127—41no—
53C2×Dic127—41no—
54C127⋊C8—24no—
55C23×C127—22no—
56C23×C127C2×D411yescharacteristic
57Q8×C127—22no—
58Q8×C127—22no—
59D4×C127—22no—
60D4×C127—22no—
61D4×C127—42no—
62D4×C127—42no—
63D4×C127—82no—
64C2×C4×C127—22no—
65C2×C4×C127—22no—
66C2×C4×C127—42no—
67C2×C4×C127—42no—
68C2×C4×C127—42no—
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D412yes—
71C2×C4×C127C2×D412yes—
72C2×C4×C127#4111yescharacteristic
73C22×C127—21no—
74C22×C127—21no—
75C22×C127—42no—
76C22×C127—42no—
77C22×C127—42no—
78C22×C127#12011yescharacteristic
79C22×C127#12111yescharacteristic
80C22×C127#12711yescharacteristic
81C4×C127—42no—
82C4×C127—42no—
83C4×C127#12012yes—
84C4×C127#12112yes—
85Dic127—42no—
86Dic127—42no—
87C2×C127—21no—
88C2×C127—21no—
89C2×C127—82no—
90C2×C127#39211yescharacteristic
91C2×C127#45511yescharacteristic
92C2×C127#45611yescharacteristic
93#2007—1271nomaximal, Sylow
94C127#200711yescharacteristic, complemented, Sylow
95C2×#137—1272no—
96#537D12711yescharacteristic
97C2×#104—1272no—
98#391—1271no—
99#343—1271no—
100C2×#41C2×D12712yes—
101C22×Q8—1271no—
102#137—2544no—
103#137—2544no—
104C2×Q16—2542no—
105C2×SD16—2542no—
106C2×M16—1272no—
107C22⋊Q8—22no—
108#121—22no—
109#121—22no—
110C2×C4⋊C4C2×D12711yescharacteristic
111C2×C42—1271no—
112#104—2544no—
113#95—2542no—
114#41—22no—
115#41—22no—
116#41—22no—
117#41—22no—
118C2×Q8—1272no—
119C2×Q8—2542no—
120C2×Q8—2542no—
121C2×Q8—2541no—
122C2×Q8C22⋊D12712yes—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4C22⋊D12712yes—
126C22×C4—22no—
127C22×C4—2541no—
128C22×C4—2541no—
129C22×C4C22⋊D12712yes—
130C22×C4C22×D12711yescharacteristic
131Q16—2548no—
132SD16—2548no—
133M16—1274no—
134M16—2542no—
135C2×C8—2542no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C22⋊C4—24no—
139C42—2542no—
140C23—22no—
141C23C2×C22⋊D12711yescharacteristic
142Q8—22no—
143Q8—22no—
144Q8—5082no—
145Q8—2544no—
146D4—22no—
147D4—22no—
148D4—42no—
149D4—42no—
150D4—82no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—5081no—
157C2×C4—5081no—
158C2×C4—5081no—
159C2×C4—5081no—
160C2×C4—2542no—
161C2×C4—2542no—
162C2×C4D4×D12711yescharacteristic
163C2×C4D4×D12712yes—
164C2×C4C2×C22⋊D12712yes—
165C2×C4#1452324711yescharacteristic
166C8—2544no—
167C22—21no—
168C22—21no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22#1686117811yescharacteristic
173C22#1686118011yescharacteristic
174C22#1686120511yescharacteristic
175C4—42no—
176C4—42no—
177C4—5082no—
178C4—5082no—
179C4#1686117712yes—
180C4#1686118112yes—
181C2—21no—
182C2—21no—
183C2—82no—
184C2#2254978411yescharacteristic
185C2#2255005411yescharacteristic
186C2#2255005911yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting, prime radical#537×C127C2
socleC22×C127#120
hypercenter#537D127
hypocenterC127#2007
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#16861239C2semidirect
2. C2×#16861136C2semidirect
3. #22549781C2semidirect
4. #22549645C2semidirect
23. C2×Dic254D4semidirect
29. #14523232D4semidirect
94. C127#2007semidirect

Maximal subgroups

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

subgroupindex
C2×#168612392
C2×#168611362
#225497812
#225496452
#537×C1272
#2007127