Finite Group Databasebeta

← #44558253

Subgroups — #44558253

Order 32128.

376 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 129 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#30631627C212yesmaximal
2#30631621C212yesmaximal
3#30631607C211yesmaximal, characteristic
4#30631592C211yesmaximal, characteristic
5#30631591C212yesmaximal
6#30631591C212yesmaximal
7#30631574C212yesmaximal
8#30631568C212yesmaximal
9#400×C251C211yesmaximal, characteristic
10C22×Dic502C2211yescharacteristic
11C22×Dic502C2211yescharacteristic
12C2×C4⋊Dic251C2212yes—
13C2×Dic251⋊C4C2212yes—
14C2×Dic251⋊C4C2212yes—
15C2×Dic251⋊C4C2212yes—
16C2×C4×Dic251C2212yes—
17C2×C4×Dic251C2211yescharacteristic
18C251⋊#95C2212yes—
19C251⋊#95C2212yes—
20C251⋊#95C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2211yescharacteristic
26C2×C4⋊C4×C251C2211yescharacteristic
27C2×C4⋊C4×C251C2211yescharacteristic
28#95×C251C2212yes—
29#95×C251C2212yes—
30#95×C251C2211yescharacteristic
31C2×Dic502—24no—
32C2×Dic502—41no—
33C2×Dic502—41no—
34C2×Dic502—41no—
35C2×Dic502Q814yescomplemented
36C22×Dic251C2312yes—
37C22×Dic251C2312yes—
38C22×Dic251C2312yes—
39C22×Dic251C2312yes—
40C4⋊Dic251—42no—
41Dic251⋊C4—24no—
42Dic251⋊C4—24no—
43Dic251⋊C4—42no—
44C4×Dic251—41no—
45C4×Dic251—42no—
46C22×C4×C251C2311yescharacteristic
47C22×C4×C251C2312yes—
48C22×C4×C251C2312yes—
49C22×C4×C251C2311yescharacteristic
50C22×C4×C251C2311yescharacteristic
51C4⋊C4×C251—41no—
52C4⋊C4×C251—41no—
53Dic502—44no—
54Dic502—44no—
55C2×Dic251—22no—
56C2×Dic251—22no—
57C2×Dic251—22no—
58C2×Dic251—22no—
59C2×Dic251—22no—
60C2×Dic251—22no—
61C2×Dic251—22no—
62C2×Dic251—22no—
63C2×Dic251—22no—
64C2×Dic251—22no—
65C2×Dic251C2×Q814yes—
66C2×Dic251#4114yes—
67C23×C251C2411yescharacteristic
68C2×C4×C251—21no—
69C2×C4×C251—21no—
70C2×C4×C251—22no—
71C2×C4×C251—22no—
72C2×C4×C251—22no—
73C2×C4×C251—21no—
74C2×C4×C251—22no—
75C2×C4×C251—21no—
76C2×C4×C251—22no—
77C2×C4×C251—22no—
78C2×C4×C251—24no—
79C2×C4×C251C2×Q812yes—
80C22×C251C22×Q811yescharacteristic
81C22×C251C2×#4112yes—
82C22×C251H(5,𝔽2)11yescharacteristic
83C22×C251H(5,𝔽2)11yescharacteristic
84C22×C2512−1+412yes—
85C4×C251—22no—
86C4×C251—42no—
87C4×C251—41no—
88C4×C251—41no—
89C4×C251—42no—
90Dic251—24no—
91Dic251—24no—
92Dic251—42no—
93Dic251—42no—
94C2×C251#54212yes—
95C2×C251#54311yescharacteristic
96C2×C251#55111yescharacteristic
97C2×C251#55712yes—
98C2×C251#56311yescharacteristic
99C251#279911yescharacteristic, complemented, Sylow
100#2799—2511nomaximal, Sylow
101#400—2512no—
102#400D25111yescharacteristic
103#398—2512no—
104#395—2511no—
105#391—2511no—
106#391—2512no—
107#391—2512no—
108#387—2512no—
109#387—2512no—
110C22×Q8—2511no—
111C22×Q8—2511no—
112C2×C4⋊C4—2512no—
113C2×C4⋊C4—2512no—
114C2×C4⋊C4—2512no—
115C2×C4⋊C4—2512no—
116C2×C4⋊C4C2×D25111yescharacteristic
117C2×C4⋊C4C2×D25111yescharacteristic
118C2×C42—2511no—
119C2×C42—2512no—
120#95—2512no—
121#95—2512no—
122#95—2512no—
123#95—2512no—
124#95—2512no—
125#95—2512no—
126#95—2512no—
127#95—2511no—
128#95C2×D25112yes—
129#95C2×D25112yes—
130#95C2×D25111yescharacteristic
131C2×Q8—10041no—
132C2×Q8—10041no—
133C2×Q8—10041no—
134C2×Q8—5024no—
135C2×Q8—2514no—
136C22×C4—2512no—
137C22×C4—2512no—
138C22×C4—2512no—
139C22×C4—2512no—
140C22×C4C22×D25112yes—
141C22×C4C22×D25111yescharacteristic
142C22×C4C22×D25111yescharacteristic
143C22×C4C22×D25112yes—
144C22×C4C22×D25111yescharacteristic
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—10042no—
148C4⋊C4—10042no—
149C4⋊C4—5024no—
150C4⋊C4—5024no—
151C42—10041no—
152C42—10042no—
153C23C23×D25111yescharacteristic
154Q8—10044no—
155Q8—10044no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—24no—
167C2×C4—5022no—
168C2×C4—5022no—
169C2×C4—5022no—
170C2×C4—5022no—
171C2×C4—5022no—
172C2×C4—5022no—
173C2×C4—5022no—
174C2×C4—5022no—
175C2×C4—5022no—
176C2×C4—5022no—
177C2×C4—2514no—
178C2×C4—2514no—
179C2×C4Q8×D25112yes—
180C22C2×Q8×D25111yescharacteristic
181C22#41×D25112yes—
182C22#2254876011yescharacteristic
183C22#2254876111yescharacteristic
184C22#2254876212yes—
185C4—22no—
186C4—42no—
187C4—41no—
188C4—41no—
189C4—42no—
190C4—10042no—
191C4—10042no—
192C4—5024no—
193C4—5024no—
194C2#3063230312yes—
195C2#3063231211yescharacteristic
196C2#3063236511yescharacteristic
197C2#3063241412yes—
198C2#3063243711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, prime radical, cosocleC23×C251C24
Fitting#400×C251C2
hypercenter#400D251
hypocenterC251#2799
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
35. C2×Dic502Q8semidirect
99. C251#2799semidirect

Maximal subgroups

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

subgroupindex
#306316272
#306316212
#306316072
#306315922
#306315912
#306315742
#306315682
#400×C2512
#2799251