Finite Group Databasebeta

← #355×D127 #30640406

Subgroups — #355×D127 #30640406

Order 16256.

260 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 57 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#124×D127C211yesmaximal, characteristic
2#101×D127C212yesmaximal, complemented
3#22549673C211yesmaximal, characteristic, complemented
4#22549672C212yesmaximal, complemented
5#355×C127C211yesmaximal, characteristic, complemented
6C2×Q8×D127C2211yescharacteristic
7C2×D4×D127C411yescharacteristic
8M16×D127—22no—
9#16861198C2211yescharacteristic
10#16861197C411yescharacteristic
11#16861196C411yescharacteristic
12#16861195—22no—
13C22⋊C4×D127—22no—
14C42×D127C411yescharacteristic
15#16861128C2212yescomplemented
16#16861127—22no—
17#124×C127C2211yescharacteristic
18#101×C127C2212yescomplemented
19C23×D127—21no—
20C2×Dic254C2×C411yescharacteristic
21C2×Dic254D412yes—
22Q8×D127—41no—
23Q8×D127—41no—
24C2×D508C2×C411yescharacteristic
25C2×C22⋊D127—21no—
26D4×D127—41no—
27D4×D127—41no—
28C2×C4×D127—21no—
29C2×C4×D127C2×C411yescharacteristic
30C2×C4×D127D412yes—
31#14523232—22no—
32#14523231—42no—
33C8×D127—42no—
34C22⋊Dic127—22no—
35(C2×D127)⋊C4—22no—
36(C2×D127)⋊C4—22no—
37C4×Dic127—21no—
38C4×Dic127C2×C411yescharacteristic
39C2×Q8×C127C2311yescharacteristic
40C2×D4×C127C2×C411yescharacteristic
41M16×C127—22no—
42C22⋊C4×C127—22no—
43C42×C127C2×C411yescharacteristic
44C22×D127—21no—
45C22×D127—44no—
46C22×D127—41no—
47C22×D127—41no—
48C22×D127C22⋊C411yescharacteristic
49Dic254—41no—
50Dic254—42no—
51D508—41no—
52C22⋊D127—42no—
53C2×Dic127—21no—
54C2×Dic127C22⋊C411yescharacteristic
55C2×Dic127C22⋊C412yes—
56C4×D127—22no—
57C4×D127—22no—
58C4×D127—21no—
59C4×D127—21no—
60C4×D127—42no—
61C127⋊C8—42no—
62C23×C127—21no—
63Q8×C127—41no—
64D4×C127—41no—
65C2×C4×C127—21no—
66C2×C4×C127C22×C411yescharacteristic
67C2×C4×C127C2×D412yes—
68C8×C127—42no—
69C22×C127—42no—
70C22×C127—41no—
71C22×C127C2×C22⋊C411yescharacteristic
72C2×D127—22no—
73C2×D127—42no—
74C2×D127—41no—
75C2×D127—82no—
76C2×D127—82no—
77C2×D127#9911yescharacteristic
78C2×D127#9911yescharacteristic
79C4×C127—22no—
80C4×C127—21no—
81C4×C127—41no—
82Dic127—21no—
83Dic127—22no—
84Dic127—41no—
85C2×C127—21no—
86C2×C127—81no—
87C2×C127C2×#9911yescharacteristic
88D127—21no—
89D127—81no—
90D127#35512yesdirect product factor
91C2×#355—1271nomaximal, Sylow
92C127C2×#35511yescharacteristic, complemented, Sylow
93C2×#124—1271no—
94C2×#101—1272no—
95#355—1271no—
96#355—1272no—
97#355D12711yescharacteristic, direct product factor
98C22×Q8—1271no—
99C22×D4—1271no—
100C2×M16—2542no—
101#124—1271no—
102#124—1271no—
103#124—1271no—
104#124—2542no—
105#124C2×D12711yescharacteristic
106C2×C22⋊C4—2542no—
107C2×C42—1271no—
108#101—1272no—
109#101—2542no—
110#101C2×D12712yescomplemented
111C24—2541no—
112C2×Q8—1271no—
113C2×Q8—1272no—
114C2×Q8—5081no—
115C2×Q8—5081no—
116C2×Q8C22×D12711yescharacteristic
117C2×D4—1271no—
118C2×D4—5081no—
119C2×D4—5081no—
120C2×D4—2541no—
121C2×D4C4×D12711yescharacteristic
122C22×C4—1271no—
123C22×C4—1272no—
124C22×C4—2541no—
125M16—22no—
126M16—5082no—
127M16—2542no—
128C2×C8—5082no—
129C22⋊C4—22no—
130C22⋊C4—2542no—
131C22⋊C4—2542no—
132C22⋊C4—2542no—
133C42—1271no—
134C42—2541no—
135C42C4×D12711yescharacteristic
136C23—21no—
137C23—1271no—
138C23—5081no—
139C23—5081no—
140C23—5084no—
141C23—2541no—
142Q8—41no—
143Q8—5082no—
144Q8—5081no—
145D4—41no—
146D4—5081no—
147D4—5082no—
148C2×C4—21no—
149C2×C4—1271no—
150C2×C4—1272no—
151C2×C4—5082no—
152C2×C4—2542no—
153C2×C4—2542no—
154C2×C4—2541no—
155C2×C4—2541no—
156C2×C4—2541no—
157C2×C4C2×C4×D12711yescharacteristic
158C2×C4D4×D12712yes—
159C8—42no—
160C8—5082no—
161C22—42no—
162C22—41no—
163C22—1271no—
164C22—1271no—
165C22—10162no—
166C22—10162no—
167C22—5082no—
168C22—5081no—
169C22—2542no—
170C22C22⋊C4×D12711yescharacteristic
171C4—22no—
172C4—21no—
173C4—41no—
174C4—5081no—
175C4—2541no—
176C4—2542no—
177C2—21no—
178C2—81no—
179C2—1272no—
180C2—10161no—
181C2—2541no—
182C2#99×D12711yescharacteristic
subgroupG/H
centerC2#99×D127
commutatorC2×C4×C127C22×C4
FrattiniC2×Q8C22×D127
Fitting#355×C127C2
socleC2×C127C2×#99
prime radicalC2×D4×D127C4
cosocleC2×Q8×C127C23
hypercenter#355D127
hypocenterC127C2×#355
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #101×D127C2semidirect
3. #22549673C2semidirect
4. #22549672C2semidirect
5. #355×C127C2semidirect
15. #16861128C22semidirect
18. #101×C127C22semidirect
90. D127#355direct (N × H)
92. C127C2×#355semidirect
97. #355D127direct (N × H)
110. #101C2×D127semidirect

Maximal subgroups

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

subgroupindex
#124×D1272
#101×D1272
#225496732
#225496722
#355×C1272
C2×#355127