Finite Group Databasebeta

← #44558320

Subgroups — #44558320

Order 32128.

368 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 133 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#30631629C212yesmaximal
2#30631627C212yesmaximal
3#30631606C212yesmaximal
4#30631585C211yesmaximal, characteristic
5#30631585C211yesmaximal, characteristic
6#30631584C212yesmaximal
7#30631570C212yesmaximal
8#30631568C212yesmaximal
9#391×C251C211yesmaximal, characteristic
10C2×C4⋊Dic251C2211yescharacteristic
11C2×C4⋊Dic251C2211yescharacteristic
12C2×C4⋊Dic251C2211yescharacteristic
13C2×C4⋊Dic251C2211yescharacteristic
14C2×Dic251⋊C4C2212yes—
15C2×Dic251⋊C4C2212yes—
16C2×Dic251⋊C4C2212yes—
17C2×Dic251⋊C4C2212yes—
18C2×C4×Dic251C2212yes—
19C251⋊#95C2212yes—
20C251⋊#95C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2212yes—
26C22×Q8×C251C2211yescharacteristic
27C2×C4⋊C4×C251C2211yescharacteristic
28C2×C42×C251C2211yescharacteristic
29#95×C251C2212yes—
30#95×C251C2212yes—
31C22×Dic251C2312yes—
32C22×Dic251C2312yes—
33C22×Dic251C2312yes—
34C22×Dic251C2312yes—
35C4⋊Dic251—22no—
36C4⋊Dic251—41no—
37C4⋊Dic251—41no—
38C4⋊Dic251Q814yescomplemented
39Dic251⋊C4—24no—
40Dic251⋊C4—24no—
41Dic251⋊C4—42no—
42Dic251⋊C4—42no—
43C4×Dic251—42no—
44C2×Q8×C251—22no—
45C2×Q8×C251—41no—
46C2×Q8×C251—41no—
47C22×C4×C251C2312yes—
48C22×C4×C251C2312yes—
49C22×C4×C251C2311yescharacteristic
50C22×C4×C251C2311yescharacteristic
51C22×C4×C251C2311yescharacteristic
52C4⋊C4×C251—41no—
53C42×C251—41no—
54C2×Dic251—22no—
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—
66C23×C251C2411yescharacteristic
67Q8×C251—44no—
68C2×C4×C251—21no—
69C2×C4×C251—21no—
70C2×C4×C251—21no—
71C2×C4×C251—21no—
72C2×C4×C251—21no—
73C2×C4×C251—21no—
74C2×C4×C251—22no—
75C2×C4×C251—22no—
76C2×C4×C251—22no—
77C2×C4×C251—22no—
78C2×C4×C251—22no—
79C2×C4×C251—22no—
80C2×C4×C251C2×Q812yes—
81C2×C4×C251#4112yes—
82C2×C4×C251#4112yes—
83C22×C251C22×Q811yescharacteristic
84C22×C251C2×#4111yescharacteristic
85C22×C251C2×#4111yescharacteristic
86C22×C251H(5,𝔽2)11yescharacteristic
87C22×C251H(5,𝔽2)11yescharacteristic
88C22×C2512−1+411yescharacteristic
89C22×C2512−1+411yescharacteristic
90C4×C251—22no—
91C4×C251—22no—
92C4×C251—22no—
93C4×C251—42no—
94C4×C251—42no—
95Dic251—24no—
96Dic251—42no—
97Dic251—42no—
98Dic251—42no—
99C2×C251#53811yescharacteristic
100C2×C251#53911yescharacteristic
101C2×C251#54411yescharacteristic
102C2×C251#55411yescharacteristic
103C2×C251#55611yescharacteristic
104C2×C251#55711yescharacteristic
105C2×C251#56311yescharacteristic
106C251#280511yescharacteristic, complemented, Sylow
107#2805—2511nomaximal, Sylow
108#400—2512no—
109#400—2512no—
110#395—2512no—
111#391D25111yescharacteristic
112#389—2511no—
113#389—2511no—
114#389—2512no—
115#387—2512no—
116#387—2512no—
117C22×Q8C2×D25111yescharacteristic
118C2×C4⋊C4—2512no—
119C2×C4⋊C4—2511no—
120C2×C4⋊C4—2512no—
121C2×C4⋊C4—2511no—
122C2×C4⋊C4—2512no—
123C2×C4⋊C4—2512no—
124C2×C4⋊C4—2511no—
125C2×C4⋊C4—2511no—
126C2×C4⋊C4C2×D25111yescharacteristic
127C2×C42—2512no—
128C2×C42C2×D25111yescharacteristic
129#95—2512no—
130#95—2512no—
131#95—2512no—
132#95—2512no—
133#95—2512no—
134#95—2512no—
135#95—2512no—
136#95C2×D25112yes—
137#95C2×D25112yes—
138C2×Q8—22no—
139C2×Q8—41no—
140C2×Q8—41no—
141C22×C4—2512no—
142C22×C4—2512no—
143C22×C4—2512no—
144C22×C4—2512no—
145C22×C4C22×D25112yes—
146C22×C4C22×D25112yes—
147C22×C4C22×D25111yescharacteristic
148C22×C4C22×D25111yescharacteristic
149C22×C4C22×D25111yescharacteristic
150C4⋊C4—41no—
151C4⋊C4—10042no—
152C4⋊C4—10042no—
153C4⋊C4—10041no—
154C4⋊C4—10041no—
155C4⋊C4—5024no—
156C4⋊C4—5024no—
157C4⋊C4—5022no—
158C4⋊C4—2514no—
159C42—41no—
160C42—10042no—
161C23C23×D25111yescharacteristic
162Q8—44no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—5022no—
176C2×C4—5022no—
177C2×C4—5022no—
178C2×C4—5022no—
179C2×C4—5022no—
180C2×C4—5022no—
181C2×C4—5022no—
182C2×C4—5022no—
183C2×C4—5022no—
184C2×C4—5022no—
185C2×C4—5022no—
186C2×C4—2514no—
187C2×C4Q8×D25112yes—
188C2×C4#1686081012yes—
189C2×C4#1686081112yes—
190C22C2×Q8×D25111yescharacteristic
191C22C2×#1686081011yescharacteristic
192C22C2×#1686081111yescharacteristic
193C22#2254876011yescharacteristic
194C22#2254876011yescharacteristic
195C22#2254876211yescharacteristic
196C22#2254876211yescharacteristic
197C4—22no—
198C4—22no—
199C4—22no—
200C4—42no—
201C4—42no—
202C4—10042no—
203C4—10042no—
204C4—10042no—
205C4—5024no—
206C2#3063227511yescharacteristic
207C2#3063228611yescharacteristic
208C2#3063231611yescharacteristic
209C2#3063239811yescharacteristic
210C2#3063241011yescharacteristic
211C2#3063241511yescharacteristic
212C2#3063243711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, prime radical, cosocleC23×C251C24
Fitting#391×C251C2
hypercenter#391D251
hypocenterC251#2805
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
38. C4⋊Dic251Q8semidirect
106. C251#2805semidirect

Maximal subgroups

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

subgroupindex
#306316292
#306316272
#306316062
#306315852
#306315842
#306315702
#306315682
#391×C2512
#2805251