Finite Group Databasebeta

← #354×D127 #30640389

Subgroups — #354×D127 #30640389

Order 16256.

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

normal subgroups only

This group has 284 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#99×D127C212yesmaximal
3#22549671C211yesmaximal, characteristic, complemented
4#22549670C212yesmaximal, complemented
5#354×C127C211yesmaximal, characteristic, complemented
6C2×Q8×D127C411yescharacteristic, complemented
7C2×D4×D127C2211yescharacteristic
8#16861198C411yescharacteristic, complemented
9#16861197C411yescharacteristic, complemented
10#16861196C2211yescharacteristic
11#16861195—22no—
12C22⋊C4×D127—22no—
13C22⋊C4×D127—22no—
14C42×D127C411yescharacteristic, complemented
15#16861124C2212yes—
16#16861123—22no—
17#124×C127C2211yescharacteristic
18#99×C127C2212yes—
19C23×D127D412yes—
20C2×Dic254—21no—
21C2×Dic254C2×C411yescharacteristic, complemented
22Q8×D127—41no—
23Q8×D127—41no—
24C2×D508C2×C411yescharacteristic
25C2×C22⋊D127D412yes—
26D4×D127—41no—
27D4×D127—41no—
28C2×C4×D127—21no—
29C2×C4×D127—21no—
30C2×C4×D127—42no—
31C2×C4×D127C2×C411yescharacteristic
32C22⋊Dic127—22no—
33C22⋊Dic127—22no—
34(C2×D127)⋊C4—22no—
35(C2×D127)⋊C4—22no—
36(C2×D127)⋊C4—42no—
37C4×Dic127—21no—
38C4×Dic127C2×C411yescharacteristic, complemented
39C2×Q8×C127C2×C411yescharacteristic, complemented
40C2×D4×C127C2311yescharacteristic
41C22⋊C4×C127—22no—
42C22⋊C4×C127—22no—
43C42×C127C2×C411yescharacteristic, complemented
44C22×D127—22no—
45C22×D127—22no—
46C22×D127—44no—
47C22×D127C22⋊C411yescharacteristic
48C22×D127C22⋊C412yes—
49Dic254—42no—
50Dic254—41no—
51D508—41no—
52C22⋊D127—42no—
53C2×Dic127—21no—
54C2×Dic127—21no—
55C2×Dic127—42no—
56C2×Dic127C22⋊C411yescharacteristic
57C4×D127—21no—
58C4×D127—21no—
59C4×D127—42no—
60C4×D127—41no—
61C4×D127—41no—
62C4×D127—84no—
63C23×C127C2×D412yes—
64Q8×C127—41no—
65D4×C127—41no—
66C2×C4×C127—21no—
67C2×C4×C127—21no—
68C2×C4×C127—42no—
69C2×C4×C127C22×C411yescharacteristic
70C22×C127—22no—
71C22×C127—42no—
72C22×C127C2×C22⋊C411yescharacteristic
73C2×D127—22no—
74C2×D127—22no—
75C2×D127—44no—
76C2×D127—42no—
77C2×D127—82no—
78C2×D127#9911yescharacteristic
79C2×D127#9911yescharacteristic
80C4×C127—21no—
81C4×C127—41no—
82C4×C127—41no—
83C4×C127—82no—
84Dic127—21no—
85Dic127—41no—
86Dic127—41no—
87Dic127—82no—
88C2×C127—21no—
89C2×C127—42no—
90C2×C127C2×#9911yescharacteristic
91D127—21no—
92D127—42no—
93D127#35412yesdirect product factor
94C2×#354—1271nomaximal, Sylow
95C127C2×#35411yescharacteristic, complemented, Sylow
96C2×#124—1271no—
97C2×#99—1272no—
98#354—1271no—
99#354—1272no—
100#354D12711yescharacteristic, direct product factor
101C22×Q8—1271no—
102C22×D4—1271no—
103#124—1271no—
104#124—1271no—
105#124—1271no—
106#124—2542no—
107#124C2×D12711yescharacteristic
108C2×C22⋊C4—2542no—
109C2×C22⋊C4—2542no—
110C2×C42—1271no—
111#99—1272no—
112#99—2542no—
113#99C2×D12712yes—
114C24—1272no—
115C2×Q8—1271no—
116C2×Q8—5081no—
117C2×Q8—5081no—
118C2×Q8—2541no—
119C2×Q8C4×D12711yescharacteristic, complemented
120C2×D4—1272no—
121C2×D4—1271no—
122C2×D4—5081no—
123C2×D4—5081no—
124C2×D4C22×D12711yescharacteristic
125C22×C4—1271no—
126C22×C4—5082no—
127C22×C4—2541no—
128C22×C4—2541no—
129C22⋊C4—22no—
130C22⋊C4—22no—
131C22⋊C4—5082no—
132C22⋊C4—2542no—
133C22⋊C4—2542no—
134C22⋊C4—2542no—
135C22⋊C4—2542no—
136C42—1271no—
137C42—2541no—
138C42C4×D12711yescharacteristic, complemented
139C23—1271no—
140C23—1272no—
141C23—5084no—
142C23—2542no—
143C23—2542no—
144C23D4×D12712yes—
145Q8—41no—
146Q8—5081no—
147Q8—5082no—
148D4—41no—
149D4—5081no—
150D4—5082no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—42no—
154C2×C4—1271no—
155C2×C4—10164no—
156C2×C4—5081no—
157C2×C4—5081no—
158C2×C4—5082no—
159C2×C4—5082no—
160C2×C4—2541no—
161C2×C4—2541no—
162C2×C4—2541no—
163C2×C4—2541no—
164C2×C4C2×C4×D12711yescharacteristic
165C22—22no—
166C22—42no—
167C22—1271no—
168C22—1271no—
169C22—10162no—
170C22—5082no—
171C22—5084no—
172C22—2542no—
173C22—2542no—
174C22C22⋊C4×D12711yescharacteristic
175C4—21no—
176C4—41no—
177C4—41no—
178C4—82no—
179C4—10162no—
180C4—5081no—
181C4—5081no—
182C4—2541no—
183C2—21no—
184C2—42no—
185C2—1272no—
186C2—5082no—
187C2—2541no—
188C2#99×D12711yescharacteristic
subgroupG/H
centerC2#99×D127
commutatorC2×C4×C127C22×C4
FrattiniC2×D4C22×D127
Fitting#354×C127C2
socleC2×C127C2×#99
prime radicalC2×D4×D127C22
cosocleC2×D4×C127C23
hypercenter#354D127
hypocenterC127C2×#354
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #22549671C2semidirect
4. #22549670C2semidirect
5. #354×C127C2semidirect
6. C2×Q8×D127C4semidirect
8. #16861198C4semidirect
9. #16861197C4semidirect
14. C42×D127C4semidirect
21. C2×Dic254C2×C4semidirect
38. C4×Dic127C2×C4semidirect
39. C2×Q8×C127C2×C4semidirect
43. C42×C127C2×C4semidirect
93. D127#354direct (N × H)
95. C127C2×#354semidirect
100. #354D127direct (N × H)
119. C2×Q8C4×D127semidirect
138. C42C4×D127semidirect

Maximal subgroups

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

subgroupindex
#124×D1272
#99×D1272
#225496712
#225496702
#354×C1272
C2×#354127