Finite Group Databasebeta

← #11731639

Subgroups — #11731639

Order 896.

488 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 157 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×#6435C211yesmaximal, characteristic
2C2×#6432C211yesmaximal, characteristic, complemented
3C7×#382C211yesmaximal, characteristic, complemented
4#89759C212yesmaximal
5#89757C212yesmaximal, complemented
6#89449C212yesmaximal
7#89445C212yesmaximal
8#89431C212yesmaximal, complemented
9#89428C212yesmaximal, complemented
10C22×C4×D7C2211yescharacteristic
11C22×Dic14C2212yes—
12C2×C7×C4⋊C4C2212yes—
13C2×C42×C7C2211yescharacteristic
14C2×(C2×D7)⋊C4C2211yescharacteristic
15C2×(C2×D7)⋊C4C2211yescharacteristic
16C2×(C2×D7)⋊C4C2214yes—
17C2×C4⋊Dic7C2212yes—
18C2×C4⋊Dic7C2212yes—
19C2×C4⋊Dic7C2214yes—
20C2×Dic7⋊C4C2211yescharacteristic
21C2×Dic7⋊C4C2211yescharacteristic
22C2×C4×Dic7C2211yescharacteristic
23#6435—24no—
24#6432—24no—
25C7×#95C2212yes—
26C7×#95C2212yes—
27C7⋊#95C2214yes—
28C7⋊#95C2214yes—
29#2628—71nomaximal, Sylow
30C23×D7C2311yescharacteristic
31C22×C4×C7C2314yes—
32C22×C4×C7C2311yescharacteristic
33C22×C4×C7C2311yescharacteristic
34C22×C4×C7C2311yescharacteristic
35C22×Dic7C2311yescharacteristic
36C22×Dic7C2314yes—
37C22×Dic7C2311yescharacteristic
38C22×Dic7C2311yescharacteristic
39C2×C4×D7—22no—
40C2×C4×D7—22no—
41C2×C4×D7Q814yescomplemented
42C2×Dic14—24no—
43C2×Dic14—44no—
44C7×C4⋊C4—24no—
45C42×C7D414yes—
46(C2×D7)⋊C4—22no—
47(C2×D7)⋊C4—22no—
48(C2×D7)⋊C4—44no—
49C4⋊Dic7—24no—
50C4⋊Dic7—24no—
51C4⋊Dic7—44no—
52Dic7⋊C4—22no—
53Dic7⋊C4—22no—
54C4×Dic7—22no—
55C2×C4⋊Q8—71no—
56C2×#117—71no—
57#400—72no—
58#399—72no—
59#395—72no—
60#393—72no—
61#389—72no—
62#382D711yescharacteristic, complemented
63#380—72no—
64C23×C7C2411yescharacteristic
65C22×D7—21no—
66C22×D7—21no—
67C22×D7—22no—
68C22×D7—22no—
69C22×D7C2×Q812yes—
70C2×C4×C7—22no—
71C2×C4×C7—22no—
72C2×C4×C7—24no—
73C2×C4×C7—24no—
74C2×C4×C7—24no—
75C2×C4×C7C2×D412yes—
76C2×C4×C7C2×D412yes—
77C2×C4×C7C2×D412yes—
78C2×C4×C7C2×Q812yes—
79C2×C4×C7#4112yes—
80C2×Dic7—21no—
81C2×Dic7—21no—
82C2×Dic7—21no—
83C2×Dic7—21no—
84C2×Dic7—24no—
85C2×Dic7—22no—
86C2×Dic7—24no—
87C2×Dic7—24no—
88C2×Dic7—22no—
89C2×Dic7C2×Q812yes—
90C4×D7—28no—
91Dic14—48no—
92C22×Q8—72no—
93C23×C4—71no—
94C4⋊Q8—144no—
95#117—144no—
96C2×C4⋊C4—71no—
97C2×C4⋊C4—71no—
98C2×C4⋊C4—72no—
99C2×C4⋊C4—72no—
100C2×C4⋊C4—74no—
101C2×C4⋊C4C2×D712yes—
102C2×C22⋊C4—71no—
103C2×C22⋊C4—71no—
104C2×C22⋊C4—74no—
105C2×C42—71no—
106C2×C42C2×D711yescharacteristic
107#95—74no—
108#95—74no—
109#95C2×D712yes—
110#95C2×D712yes—
111C22×C7C22×D411yescharacteristic
112C22×C7C22×Q811yescharacteristic
113C22×C7C2×#4111yescharacteristic
114C22×C7H(5,𝔽2)12yes—
115C22×C72−1+412yes—
116C2×D7—22no—
117C2×D7—24no—
118C2×D7—41no—
119C2×D7—41no—
120C2×D7—42no—
121C4×C7—22no—
122C4×C7—22no—
123C4×C7—44no—
124C4×C7C22⋊Q814yes—
125Dic7—22no—
126Dic7—44no—
127Dic7—41no—
128Dic7—41no—
129C24—71no—
130C2×Q8—144no—
131C2×Q8—284no—
132C22×C4—71no—
133C22×C4—74no—
134C22×C4—71no—
135C22×C4—74no—
136C22×C4—71no—
137C22×C4—142no—
138C22×C4—142no—
139C22×C4C22×D714yes—
140C22×C4C22×D711yescharacteristic
141C22×C4C22×D711yescharacteristic
142C22×C4C22×D711yescharacteristic
143C4⋊C4—24no—
144C4⋊C4—142no—
145C4⋊C4—142no—
146C4⋊C4—144no—
147C4⋊C4—144no—
148C4⋊C4—284no—
149C22⋊C4—142no—
150C22⋊C4—142no—
151C22⋊C4—284no—
152C42—142no—
153C42D2814yes—
154C2×C7C2×C22⋊Q811yescharacteristic
155C2×C7#53511yescharacteristic
156C2×C7#53611yescharacteristic
157C2×C7#54212yes—
158C2×C7#54412yes—
159D7—42no—
160C23—72no—
161C23—141no—
162C23—141no—
163C23—142no—
164C23—142no—
165C23C23×D711yescharacteristic
166Q8—288no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—72no—
173C2×C4—141no—
174C2×C4—141no—
175C2×C4—148no—
176C2×C4—141no—
177C2×C4—141no—
178C2×C4—144no—
179C2×C4—144no—
180C2×C4—142no—
181C2×C4—144no—
182C2×C4—142no—
183C2×C4C2×D2812yes—
184C2×C4C2×D2812yes—
185C2×C4C2×D2812yes—
186C2×C4#115912yes—
187C2×C4Q8×D712yes—
188C7#262811yescharacteristic, complemented, Sylow
189C22—144no—
190C22—142no—
191C22—282no—
192C22—281no—
193C22—281no—
194C22C22×D2811yescharacteristic
195C22C2×#115911yescharacteristic
196C22C2×Q8×D711yescharacteristic
197C22#655312yes—
198C22#655412yes—
199C4—22no—
200C4—22no—
201C4—44no—
202C4—142no—
203C4—284no—
204C4—281no—
205C4—281no—
206C4#646014yes—
207C2—282no—
208C2C2×#646011yescharacteristic
209C2#9021611yescharacteristic
210C2#9021711yescharacteristic
211C2#9037312yes—
212C2#9038012yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#382C2
prime radicalC23×D7C23
hypercenter#382D7
hypocenterC7#2628
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
2. C2×#6432C2semidirect
3. C7×#382C2semidirect
5. #89757C2semidirect
8. #89431C2semidirect
9. #89428C2semidirect
41. C2×C4×D7Q8semidirect
62. #382D7semidirect
188. C7#2628semidirect

Maximal subgroups

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

subgroupindex
C2×#64352
C2×#64322
C7×#3822
#897592
#897572
#894492
#894452
#894312
#894282
#26287