Finite Group Databasebeta

← #30639629

Subgroups — #30639629

Order 16256.

284 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 93 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#22549914C211yesmaximal, characteristic, complemented
2#22549893C212yesmaximal, complemented
3#22549850C212yesmaximal, complemented
4#22549831C211yesmaximal, characteristic, complemented
5C2×#135×C127C211yesmaximal, characteristic
6C2×#14523232C2211yescharacteristic
7#16861168C2212yes—
8#16861168C2211yescharacteristic
9#16861143—21no—
10#16861141—21no—
11#16861136—24no—
12#16861124—24no—
13#16861119—21no—
14C8×Dic127—21no—
15C2×#41×C127C2212yes—
16#135×C127—21no—
17#135×C127—21no—
18#135×C127—21no—
19#135×C127—21no—
20C2×Q16×C127C411yescharacteristic, complemented
21C2×SD16×C127C412yescomplemented
22C2×D8×C127C411yescharacteristic, complemented
23C22×C8×C127C2211yescharacteristic
24#14523232—21no—
25#14523232D412yes—
26C2×C127⋊C8—21no—
27C4⋊Dic127—22no—
28C4⋊Dic127—21no—
29C22⋊Dic127—22no—
30C22⋊Dic127D412yes—
31C4×Dic127—21no—
32C4×Dic127—22no—
33#41×C127—22no—
34#41×C127—22no—
35#41×C127—22no—
36#41×C127—22no—
37C2×Q8×C127C2×C412yes—
38C2×D4×C127—22no—
39C2×D4×C127C2×C412yes—
40C22×C4×C127—22no—
41C22×C4×C127C2311yescharacteristic
42Q16×C127—21no—
43Q16×C127—21no—
44SD16×C127—22no—
45SD16×C127—22no—
46D8×C127—21no—
47D8×C127—21no—
48C2×C8×C127C2×C411yescharacteristic
49C2×C8×C127C2×C411yescharacteristic
50C2×C8×C127D411yescharacteristic
51C2×C8×C127D411yescharacteristic
52C2×C8×C127D411yescharacteristic
53C2×C8×C127D411yescharacteristic
54C2×Dic127—22no—
55C2×Dic127—42no—
56C127⋊C8—22no—
57C23×C127—22no—
58C23×C127C2×D411yescharacteristic
59Q8×C127—22no—
60Q8×C127—22no—
61D4×C127—22no—
62D4×C127—22no—
63D4×C127—42no—
64D4×C127—42no—
65C2×C4×C127—22no—
66C2×C4×C127—42no—
67C2×C4×C127—42no—
68C2×C4×C127C22×C411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C2×C4×C127C2×D411yescharacteristic
72C2×C4×C127#4111yescharacteristic
73C2×C4×C127#4111yescharacteristic
74C8×C127C22⋊C411yescharacteristic
75C8×C127C22⋊C411yescharacteristic
76C8×C127C22⋊C411yescharacteristic
77C8×C127C22⋊C411yescharacteristic
78C22×C127—41no—
79C22×C127—42no—
80C22×C127—82no—
81C22×C127C4×D411yescharacteristic
82C22×C127#12111yescharacteristic
83C22×C127#12711yescharacteristic
84C4×C127—42no—
85C4×C127C2×C22⋊C411yescharacteristic
86C4×C127C4×D411yescharacteristic
87C4×C127#12111yescharacteristic
88C4×C127#12411yescharacteristic
89Dic127—42no—
90Dic127—82no—
91C2×C127—21no—
92C2×C127—21no—
93C2×C127—21no—
94C2×C127—82no—
95C2×C127#39011yescharacteristic
96#1976—1271nomaximal, Sylow
97C127#197611yescharacteristic, complemented, Sylow
98C2×#135D12711yescharacteristic
99#427—1271no—
100#421—1272no—
101#410—1272no—
102#405—1271no—
103C2×#41C2×D12712yes—
104#135—21no—
105#135—21no—
106#135—21no—
107#135—21no—
108C2×Q16Dic12711yescharacteristic, complemented
109C2×SD16Dic12712yescomplemented
110C2×D8Dic12711yescharacteristic, complemented
111C2×M16—1271no—
112C22×C8C2×D12711yescharacteristic
113#117—1271no—
114#117—1272no—
115#107—2541no—
116#106—2541no—
117#104—2544no—
118#99—2544no—
119#97—2541no—
120C4×C8—2541no—
121#41—22no—
122#41—22no—
123#41—22no—
124#41—22no—
125C2×Q8C2×Dic12712yes—
126C2×D4—22no—
127C2×D4C2×Dic12712yes—
128C22×C4—22no—
129C22×C4C22×D12711yescharacteristic
130Q16—21no—
131Q16—21no—
132SD16—22no—
133SD16—22no—
134D8—21no—
135D8—21no—
136M16—1272no—
137M16—2541no—
138C2×C8—2541no—
139C2×C8C2×Dic12711yescharacteristic
140C2×C8C2×Dic12711yescharacteristic
141C2×C8C22⋊D12711yescharacteristic
142C2×C8C22⋊D12711yescharacteristic
143C2×C8C22⋊D12711yescharacteristic
144C2×C8C22⋊D12711yescharacteristic
145C4⋊C4—2542no—
146C4⋊C4—2541no—
147C22⋊C4—1272no—
148C22⋊C4—2542no—
149C42—2542no—
150C42—2541no—
151C23—22no—
152C23D4×D12711yescharacteristic
153Q8—22no—
154Q8—22no—
155D4—22no—
156D4—22no—
157D4—42no—
158D4—42no—
159C2×C4—22no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—5082no—
163C2×C4—2542no—
164C2×C4C22×Dic12711yescharacteristic
165C2×C4D4×D12711yescharacteristic
166C2×C4C2×C22⋊D12711yescharacteristic
167C2×C4C2×C22⋊D12711yescharacteristic
168C2×C4#1452324711yescharacteristic
169C2×C4#1452324711yescharacteristic
170C8—2542no—
171C8C22⋊Dic12711yescharacteristic
172C8C22⋊Dic12711yescharacteristic
173C8C22⋊Dic12711yescharacteristic
174C8C22⋊Dic12711yescharacteristic
175C22—41no—
176C22—42no—
177C22—82no—
178C22D4×Dic12711yescharacteristic
179C22#1686118011yescharacteristic
180C22#1686120511yescharacteristic
181C4—42no—
182C4—10162no—
183C4—5082no—
184C4C2×C22⋊Dic12711yescharacteristic
185C4D4×Dic12711yescharacteristic
186C4#1686118011yescharacteristic
187C4#1686119611yescharacteristic
188C2—21no—
189C2—21no—
190C2—21no—
191C2—82no—
192C2#2254977911yescharacteristic
subgroupG/H
centerC4#16861180
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting, prime radicalC2×#135×C127C2
socleC2×C127#390
cosocleC22×C4×C127C23
hypercenterC2×#135D127
hypocenterC127#1976
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. #22549914C2semidirect
2. #22549893C2semidirect
3. #22549850C2semidirect
4. #22549831C2semidirect
20. C2×Q16×C127C4semidirect
21. C2×SD16×C127C4semidirect
22. C2×D8×C127C4semidirect
97. C127#1976semidirect
108. C2×Q16Dic127semidirect
109. C2×SD16Dic127semidirect
110. C2×D8Dic127semidirect

Maximal subgroups

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

subgroupindex
#225499142
#225498932
#225498502
#225498312
C2×#135×C1272
#1976127