Finite Group Databasebeta

← #30640112

Subgroups — #30640112

Order 16256.

248 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16861140C211yesmaximal, characteristic
2C2×Dic127⋊C8C211yesmaximal, characteristic
3C2×#16861134C211yesmaximal, characteristic
4C2×#16861132C211yesmaximal, characteristic
5#22549781C211yesmaximal, characteristic
6#22549772C211yesmaximal, characteristic
7#340×C127C211yesmaximal, characteristic
8C22×Dic254C2211yescharacteristic
9C22×C127⋊C8C2211yescharacteristic
10C2×C4⋊Dic127C2211yescharacteristic
11C2×Dic127⋊C4—21no—
12C2×C4×Dic127C2211yescharacteristic
13#16861140—22no—
14Dic127⋊C8—22no—
15#16861134—22no—
16#16861132—22no—
17C127⋊#95—21no—
18C127⋊#95—21no—
19C127⋊#95—21no—
20C22×C8×C127C2211yescharacteristic
21C2×C4⋊C4×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2211yescharacteristic
23C2×Dic254—21no—
24C2×Dic254—21no—
25C2×Dic254—21no—
26C2×Dic254—21no—
27C2×Dic254—21no—
28C2×Dic254Q812yes—
29C22×Dic127—21no—
30C22×Dic127—21no—
31C22×Dic127D411yescharacteristic
32C22×Dic127D411yescharacteristic
33C2×C127⋊C8—21no—
34C2×C127⋊C8—21no—
35C2×C127⋊C8D412yes—
36C4⋊Dic127—21no—
37C4⋊Dic127D412yes—
38Dic127⋊C4—42no—
39C4×Dic127—21no—
40C4×Dic127—21no—
41C22×C4×C127—21no—
42C22×C4×C127—21no—
43C22×C4×C127C2311yescharacteristic
44C2×C8×C127—21no—
45C2×C8×C127—21no—
46C2×C8×C127—21no—
47C4⋊C4×C127—21no—
48C4⋊C4×C127—21no—
49C4⋊C4×C127—21no—
50C4⋊C4×C127—21no—
51Dic254—22no—
52Dic254—22no—
53Dic254—41no—
54Dic254—41no—
55C2×Dic127—21no—
56C2×Dic127—21no—
57C2×Dic127—21no—
58C2×Dic127—22no—
59C2×Dic127—22no—
60C2×Dic127—21no—
61C2×Dic127—21no—
62C2×Dic127—41no—
63C2×Dic127—41no—
64C2×Dic127—41no—
65C2×Dic127—41no—
66C2×Dic127SD1612yes—
67C127⋊C8—22no—
68C23×C127C2×D411yescharacteristic
69C2×C4×C127—41no—
70C2×C4×C127—41no—
71C2×C4×C127—41no—
72C2×C4×C127—41no—
73C2×C4×C127—41no—
74C2×C4×C127—41no—
75C2×C4×C127C2×D411yescharacteristic
76C2×C4×C127C2×D411yescharacteristic
77C2×C4×C127C2×Q811yescharacteristic
78C2×C4×C127#4111yescharacteristic
79C2×C4×C127#4111yescharacteristic
80C2×C4×C127#4111yescharacteristic
81C8×C127—41no—
82C22×C127#12111yescharacteristic
83C22×C127#12111yescharacteristic
84C22×C127C22⋊Q811yescharacteristic
85C22×C127C2×SD1611yescharacteristic
86C22×C127#13511yescharacteristic
87C22×C127#13711yescharacteristic
88C22×C127#13711yescharacteristic
89C4×C127—81no—
90C4×C127—81no—
91C4×C127#12111yescharacteristic
92C4×C127C22⋊Q811yescharacteristic
93C4×C127C22⋊Q811yescharacteristic
94C4×C127#12611yescharacteristic
95Dic127—22no—
96Dic127—42no—
97Dic127—42no—
98Dic127—41no—
99C2×C127#39611yescharacteristic
100C2×C127#46111yescharacteristic
101C2×C127#46411yescharacteristic
102C2×C127#46511yescharacteristic
103C2×C127#47011yescharacteristic
104C2×C127#47511yescharacteristic
105C2×C127#47911yescharacteristic
106#2066—1271nomaximal, Sylow
107C127#206611yescharacteristic, complemented, Sylow
108C2×#106—1271no—
109C2×C4⋊C8—1271no—
110C2×#103—1271no—
111C2×#103—1271no—
112#391—1271no—
113#389—1271no—
114#340D12711yescharacteristic
115C22×Q8—1271no—
116C22×C8—1271no—
117C22×C8C2×D12711yescharacteristic
118C2×C4⋊C4—1271no—
119C2×C4⋊C4—2541no—
120C2×C4⋊C4C2×D12711yescharacteristic
121C2×C4⋊C4C2×D12711yescharacteristic
122C2×C42—1271no—
123#106—2542no—
124C4⋊C8—2542no—
125#103—2542no—
126#103—2542no—
127#95—2541no—
128#95—2541no—
129#95—2541no—
130C2×Q8—1272no—
131C2×Q8—2541no—
132C2×Q8—2541no—
133C2×Q8—2541no—
134C2×Q8—2541no—
135C2×Q8—2541no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—1271no—
139C22×C4—1271no—
140C22×C4—2541no—
141C22×C4—2541no—
142C22×C4C22×D12711yescharacteristic
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—1272no—
147C2×C8—2541no—
148C2×C8—2541no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—1272no—
154C4⋊C4—5082no—
155C4⋊C4—2541no—
156C42—2541no—
157C42—2541no—
158C23D4×D12711yescharacteristic
159Q8—5081no—
160Q8—5081no—
161Q8—2542no—
162Q8—2542no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—1272no—
170C2×C4—5081no—
171C2×C4—5081no—
172C2×C4—5081no—
173C2×C4—5081no—
174C2×C4—2542no—
175C2×C4—2541no—
176C2×C4—2541no—
177C2×C4—2541no—
178C2×C4—2542no—
179C2×C4—2541no—
180C2×C4—2541no—
181C2×C4D4×D12711yescharacteristic
182C2×C4D4×D12711yescharacteristic
183C2×C4Q8×D12711yescharacteristic
184C2×C4#1452324811yescharacteristic
185C2×C4#1452324811yescharacteristic
186C2×C4#1452324811yescharacteristic
187C8—41no—
188C8—2542no—
189C22#1686118211yescharacteristic
190C22#1686118211yescharacteristic
191C22#1686118711yescharacteristic
192C22SD16×D12711yescharacteristic
193C22#1686122811yescharacteristic
194C22#1686123811yescharacteristic
195C22#1686124111yescharacteristic
196C4—81no—
197C4—81no—
198C4—5082no—
199C4—5082no—
200C4—5081no—
201C4—2542no—
202C4#1686118211yescharacteristic
203C4#1686118711yescharacteristic
204C4#1686118711yescharacteristic
205C4#1686120311yescharacteristic
206C2#2254980011yescharacteristic
207C2#2255008011yescharacteristic
208C2#2255009711yescharacteristic
209C2#2255010411yescharacteristic
210C2#2255013311yescharacteristic
211C2#2255016011yescharacteristic
212C2#2255018211yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#340×C127C2
socle, prime radicalC23×C127C2×D4
hypercenter#340D127
hypocenterC127#2066
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
107. C127#2066semidirect

Maximal subgroups

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

subgroupindex
C2×#168611402
C2×Dic127⋊C82
C2×#168611342
C2×#168611322
#225497812
#225497722
#340×C1272
#2066127