Finite Group Databasebeta

← #30639148

Subgroups — #30639148

Order 16256.

342 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 89 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#22550383C211yesmaximal, characteristic
2#22549860C211yesmaximal, characteristic, complemented
3#22549839C212yesmaximal, complemented
4#22549647C212yesmaximal, complemented
5#568×C127C211yesmaximal, characteristic, complemented
6C22×C4×D127C2211yescharacteristic
7C2×#14523232C2212yes—
8#16861190C412yes—
9#16861185—21no—
10#16861180—21no—
11#16861177C412yes—
12C4×C22⋊D127—22no—
13#16861168C2211yescharacteristic
14#16861128—21no—
15#16861126—21no—
16(C2×D127)⋊C8—24no—
17C2×#41×C127C2211yescharacteristic
18#131×C127—24no—
19C2×M16×C127—22no—
20C2×M16×C127C2212yes—
21C23×D127C2×C411yescharacteristic
22C2×C22⋊D127—22no—
23C22×Dic127C2×C411yescharacteristic
24C2×C4×D127—21no—
25C2×C4×D127—22no—
26C2×C4×D127—21no—
27C2×C4×D127—22no—
28#14523232—22no—
29#14523232—22no—
30C2×C127⋊C8—22no—
31C4⋊Dic127—21no—
32Dic127⋊C4—22no—
33C22⋊Dic127—22no—
34C22⋊Dic127C2×C412yes—
35(C2×D127)⋊C4—22no—
36C4×Dic127—21no—
37#41×C127—22no—
38#41×C127—22no—
39C2×Q8×C127D411yescharacteristic
40C2×D4×C127C2×C412yes—
41C2×D4×C127D411yescharacteristic
42C22×C4×C127D412yes—
43C22×C4×C127C2311yescharacteristic
44M16×C127—24no—
45M16×C127—22no—
46M16×C127—22no—
47C2×C8×C127—24no—
48C2×C8×C127D414yescomplemented
49C22×D127—21no—
50C22×D127—22no—
51C22×D127—42no—
52C22⋊D127—44no—
53C2×Dic127—22no—
54C2×Dic127—21no—
55C2×Dic127—22no—
56C4×D127—41no—
57C4×D127—42no—
58C4×D127—41no—
59C127⋊C8—42no—
60C23×C127C22⋊C412yes—
61C23×C127C22×C411yescharacteristic
62Q8×C127—22no—
63D4×C127—24no—
64D4×C127—22no—
65C2×C4×C127—22no—
66C2×C4×C127—22no—
67C2×C4×C127C22⋊C412yes—
68C2×C4×C127C2×D412yes—
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C2×C4×C127#4112yes—
72C8×C127—24no—
73C22×C127—22no—
74C22×C127—41no—
75C22×C127—42no—
76C22×C127C2×C22⋊C411yescharacteristic
77C22×C127C4×D412yes—
78C2×D127—22no—
79C2×D127—24no—
80C2×D127—41no—
81C2×D127—42no—
82C2×D127—41no—
83C4×C127—22no—
84C4×C127#12011yescharacteristic
85C4×C127#12112yes—
86C4×C127#12311yescharacteristic
87Dic127—42no—
88Dic127—41no—
89C2×C127—21no—
90C2×C127—22no—
91C2×C127—42no—
92C2×C127#38611yescharacteristic
93D127—42no—
94#1891—1271nomaximal, Sylow
95C127#189111yescharacteristic, complemented, Sylow
96#568D12711yescharacteristic, complemented
97#525—1271no—
98#413—1271no—
99#407—1272no—
100#344—1272no—
101C2×#41C2×D12711yescharacteristic
102C23×C4—1271no—
103#131—24no—
104C2×M16—22no—
105C2×M16—1272no—
106C2×M16C2×D12712yes—
107#123—1272no—
108C22⋊Q8—2541no—
109#121—2541no—
110#120—1272no—
111C4×D4—2542no—
112#117—1271no—
113#101—2541no—
114#100—2541no—
115C22⋊C8—2544no—
116C24—1271no—
117#41—22no—
118#41—22no—
119C2×Q8D50811yescharacteristic
120C2×D4—2542no—
121C2×D4C4×D12712yes—
122C2×D4D50811yescharacteristic
123C22×C4—1271no—
124C22×C4—2542no—
125C22×C4—2541no—
126C22×C4—2541no—
127C22×C4—2542no—
128C22×C4C22⋊D12712yes—
129C22×C4C22×D12711yescharacteristic
130M16—24no—
131M16—22no—
132M16—22no—
133M16—2542no—
134M16—2542no—
135C2×C8—24no—
136C2×C8—2542no—
137C2×C8C22⋊D12714yescomplemented
138C4⋊C4—2542no—
139C4⋊C4—2541no—
140C22⋊C4—1272no—
141C22⋊C4—2542no—
142C22⋊C4—2542no—
143C42—2541no—
144C23—5082no—
145C23—2541no—
146C23—2542no—
147C23(C2×D127)⋊C412yes—
148C23C2×C4×D12711yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—5084no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—5081no—
156C2×C4—5081no—
157C2×C4—5082no—
158C2×C4—2542no—
159C2×C4—2541no—
160C2×C4—2542no—
161C2×C4(C2×D127)⋊C412yes—
162C2×C4C2×C22⋊D12711yescharacteristic
163C2×C4C2×C22⋊D12712yes—
164C2×C4C2×D50811yescharacteristic
165C2×C4#1452324812yes—
166C8—24no—
167C8—5082no—
168C22—22no—
169C22—41no—
170C22—42no—
171C22—5081no—
172C22—5082no—
173C22—5081no—
174C22—2542no—
175C22—2544no—
176C22C2×(C2×D127)⋊C411yescharacteristic
177C22C4×C22⋊D12712yes—
178C4—22no—
179C4—5082no—
180C4—5081no—
181C4#1686117811yescharacteristic
182C4#1686118312yes—
183C4#1686119211yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—5082no—
188C2#2254975511yescharacteristic
subgroupG/H
centerC4#16861178
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#568×C127C2
socleC2×C127#386
prime radical#22550383C2
cosocleC22×C4×C127C23
hypercenter#568D127
hypocenterC127#1891
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #22549860C2semidirect
3. #22549839C2semidirect
4. #22549647C2semidirect
5. #568×C127C2semidirect
48. C2×C8×C127D4semidirect
95. C127#1891semidirect
96. #568D127semidirect
137. C2×C8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225503832
#225498602
#225498392
#225496472
#568×C1272
#1891127