Finite Group Databasebeta

← #30647101

Subgroups — #30647101

Order 16256.

524 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 153 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×#16861200C211yesmaximal, characteristic, complemented
2C2×#16861196C211yesmaximal, characteristic
3#22549782C211yesmaximal, characteristic, complemented
4#22549781C211yesmaximal, characteristic, complemented
5#22549779C211yesmaximal, characteristic
6#22549769C214yesmaximal, complemented
7#22549764C214yesmaximal, complemented
8C42×Dic127C211yesmaximal, characteristic, complemented
9#390×C127C211yesmaximal, characteristic
10C22×Dic254C2211yescharacteristic
11#16861200—24no—
12#16861196—24no—
13C2×C4⋊Dic127C2212yes—
14C2×C4⋊Dic127C2211yescharacteristic
15C2×Dic127⋊C4C2212yes—
16C2×C22⋊Dic127C2212yes—
17C2×C22⋊Dic127C2214yes—
18C2×C22⋊Dic127C2212yes—
19C2×C4×Dic127C2212yes—
20C2×C4×Dic127C2212yes—
21C2×C4×Dic127C2211yescharacteristic
22C2×C4×Dic127C2211yescharacteristic
23C127⋊#95C2214yes—
24C127⋊#95C2214yes—
25C22×D4×C127C2211yescharacteristic
26C2×C4⋊C4×C127C2211yescharacteristic
27C2×C22⋊C4×C127C2214yes—
28C2×C42×C127C2211yescharacteristic
29C2×Dic254—22no—
30C2×Dic254—22no—
31C2×Dic254—22no—
32C22×Dic127C2312yes—
33C22×Dic127C2312yes—
34C22×Dic127C2311yescharacteristic
35C22×Dic127C2311yescharacteristic
36C22×Dic127C2312yes—
37C4⋊Dic127—24no—
38C4⋊Dic127—22no—
39Dic127⋊C4—24no—
40C22⋊Dic127—24no—
41C22⋊Dic127—24no—
42C22⋊Dic127—28no—
43C4×Dic127—24no—
44C4×Dic127—24no—
45C4×Dic127—22no—
46C4×Dic127D414yes—
47C24×C127C2312yes—
48C2×D4×C127—22no—
49C2×D4×C127—22no—
50C2×D4×C127—22no—
51C22×C4×C127C2312yes—
52C22×C4×C127C2312yes—
53C22×C4×C127C2311yescharacteristic
54C4⋊C4×C127—22no—
55C22⋊C4×C127—28no—
56C42×C127—22no—
57Dic254—44no—
58C2×Dic127—22no—
59C2×Dic127—22no—
60C2×Dic127—22no—
61C2×Dic127—22no—
62C2×Dic127—22no—
63C2×Dic127—22no—
64C2×Dic127—24no—
65C2×Dic127—22no—
66C2×Dic127—22no—
67C2×Dic127C2×D412yes—
68C2×Dic127C2×D412yes—
69C2×Dic127#4114yes—
70C23×C127—22no—
71C23×C127—24no—
72C23×C127—22no—
73C23×C127—22no—
74C23×C127—24no—
75C23×C127C2411yescharacteristic
76D4×C127—44no—
77C2×C4×C127—22no—
78C2×C4×C127—22no—
79C2×C4×C127—22no—
80C2×C4×C127—24no—
81C2×C4×C127C2×D412yes—
82C2×C4×C127#4114yes—
83C2×C4×C127#4112yes—
84C2×C4×C127#4112yes—
85C22×C127—42no—
86C22×C127—44no—
87C22×C127—44no—
88C22×C127—42no—
89C22×C127—42no—
90C22×C127C22×D411yescharacteristic
91C22×C127C2×#4112yes—
92C22×C127C2×#4111yescharacteristic
93C22×C127C2×#4112yes—
94C22×C127C2×#4111yescharacteristic
95C4×C127—24no—
96C4×C127—42no—
97C4×C127#12414yes—
98Dic127—24no—
99Dic127—22no—
100Dic127—22no—
101Dic127—42no—
102Dic127—42no—
103C2×C127—82no—
104C2×C127C2×#12411yescharacteristic
105C2×C127#52912yes—
106C2×C127#52912yes—
107C2×C127#53211yescharacteristic
108C2×C127#53211yescharacteristic
109#2861—1271nomaximal, Sylow
110C127#286111yescharacteristic, complemented, Sylow
111C2×#125—1271no—
112C2×#124—1271no—
113#391—1271no—
114#391—1271no—
115#390—1271no—
116#390D12711yescharacteristic
117#388—1274no—
118#388—1274no—
119C43—1271no—
120C22×Q8—1271no—
121C22×D4C2×D12711yescharacteristic
122#125—2544no—
123#124—2544no—
124C2×C4⋊C4—1272no—
125C2×C4⋊C4—1272no—
126C2×C4⋊C4—1271no—
127C2×C4⋊C4C2×D12711yescharacteristic
128C2×C22⋊C4—1272no—
129C2×C22⋊C4—1272no—
130C2×C22⋊C4—1274no—
131C2×C22⋊C4C2×D12714yes—
132C2×C42—1271no—
133C2×C42—1272no—
134C2×C42—1272no—
135C2×C42—1271no—
136C2×C42C2×D12711yescharacteristic
137#95—1274no—
138#95—1274no—
139C24C22×D12712yes—
140C2×Q8—2542no—
141C2×Q8—2542no—
142C2×Q8—2542no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—22no—
146C22×C4—1272no—
147C22×C4—1271no—
148C22×C4—1271no—
149C22×C4—1272no—
150C22×C4—1272no—
151C22×C4C22×D12712yes—
152C22×C4C22×D12712yes—
153C22×C4C22×D12711yescharacteristic
154C4⋊C4—22no—
155C4⋊C4—2544no—
156C4⋊C4—2544no—
157C4⋊C4—2542no—
158C22⋊C4—28no—
159C22⋊C4—2548no—
160C22⋊C4—2544no—
161C22⋊C4—2544no—
162C42—22no—
163C42—1274no—
164C42—2544no—
165C42—2544no—
166C42—2542no—
167C23—22no—
168C23—24no—
169C23—24no—
170C23—22no—
171C23—22no—
172C23C23×D12711yescharacteristic
173Q8—5084no—
174D4—44no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—1272no—
180C2×C4—1272no—
181C2×C4—1274no—
182C2×C4—2542no—
183C2×C4—2542no—
184C2×C4—2544no—
185C2×C4—2542no—
186C2×C4—2542no—
187C2×C4—2542no—
188C2×C4—2542no—
189C2×C4—2542no—
190C2×C4—2542no—
191C2×C4D4×D12712yes—
192C2×C4#1452324714yes—
193C2×C4#1452324712yes—
194C2×C4#1452324812yes—
195C22—42no—
196C22—44no—
197C22—44no—
198C22—42no—
199C22—42no—
200C22C2×D4×D12711yescharacteristic
201C22#41×D12712yes—
202C22C2×#1452324711yescharacteristic
203C22C2×#1452324712yes—
204C22C2×#1452324811yescharacteristic
205C4—24no—
206C4—42no—
207C4—5082no—
208C4—5082no—
209C4—2544no—
210C4—2542no—
211C4—2542no—
212C4#1686119514yes—
213C2—82no—
214C2C2×#1686119511yescharacteristic
215C2#2255040712yes—
216C2#2255040812yes—
217C2#2255042411yescharacteristic
218C2#2255042511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#390×C127C2
prime radicalC22×D4×C127C22
hypercenter#390D127
hypocenterC127#2861
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#16861200C2semidirect
3. #22549782C2semidirect
4. #22549781C2semidirect
6. #22549769C2semidirect
7. #22549764C2semidirect
8. C42×Dic127C2semidirect
110. C127#2861semidirect

Maximal subgroups

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

subgroupindex
C2×#168612002
C2×#168611962
#225497822
#225497812
#225497792
#225497692
#225497642
C42×Dic1272
#390×C1272
#2861127