Finite Group Databasebeta

← #11734639

Subgroups — #11734639

Order 896.

412 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C7×#395C211yesmaximal, characteristic
2#90002C212yesmaximal
3#89757C212yesmaximal
4#89755C211yesmaximal, characteristic
5#89750C212yesmaximal
6#89746C211yesmaximal, characteristic
7#89428C212yesmaximal
8#89427C212yesmaximal
9#89421C212yesmaximal
10C22×C7×Q8C2211yescharacteristic
11C22×Dic14C2211yescharacteristic
12C2×C7×C4⋊C4C2212yes—
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C4⋊Dic7C2212yes—
15C2×C4⋊Dic7C2212yes—
16C2×C4⋊Dic7C2211yescharacteristic
17C2×Dic7⋊C4C2212yes—
18C2×Dic7⋊C4C2212yes—
19C2×C4×Dic7C2212yes—
20C2×C4×Dic7C2211yescharacteristic
21C2×C4×Dic7C2212yes—
22C7×#95C2212yes—
23C7×#95C2211yescharacteristic
24C7⋊#95C2212yes—
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2211yescharacteristic
31#2586—71nomaximal, Sylow
32C2×C7×Q8—24no—
33C2×C7×Q8—41no—
34C22×C4×C7C2311yescharacteristic
35C22×C4×C7C2311yescharacteristic
36C22×C4×C7C2312yes—
37C22×C4×C7C2312yes—
38C22×C4×C7C2311yescharacteristic
39C22×Dic7C2312yes—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C2×Dic14—22no—
44C2×Dic14—41no—
45C2×Dic14—41no—
46C7×C4⋊C4—24no—
47C7×C4⋊C4—41no—
48C4⋊Dic7—42no—
49C4⋊Dic7—41no—
50C4⋊Dic7—42no—
51Dic7⋊C4—24no—
52Dic7⋊C4Q818yescomplemented
53C4×Dic7—24no—
54C4×Dic7—41no—
55C4×Dic7—42no—
56#398—71no—
57#395—72no—
58#395D711yescharacteristic
59#391—72no—
60#391—71no—
61#389—72no—
62#387—72no—
63#387—72no—
64C4×C4⋊C4—72no—
65C23×C7C2411yescharacteristic
66C7×Q8—44no—
67C2×C4×C7—22no—
68C2×C4×C7—22no—
69C2×C4×C7—22no—
70C2×C4×C7—21no—
71C2×C4×C7—22no—
72C2×C4×C7—21no—
73C2×C4×C7—22no—
74C2×C4×C7—22no—
75C2×C4×C7—21no—
76C2×C4×C7—21no—
77C2×C4×C7C2×Q814yes—
78C2×C4×C7#4114yes—
79C2×C4×C7#4112yes—
80C2×Dic7—22no—
81C2×Dic7—22no—
82C2×Dic7—22no—
83C2×Dic7—22no—
84C2×Dic7—22no—
85C2×Dic7—22no—
86C2×Dic7—22no—
87C2×Dic7—22no—
88C2×Dic7—22no—
89C2×Dic7—22no—
90C2×Dic7C2×Q814yes—
91C2×Dic7C2×Q814yes—
92Dic14—44no—
93C22×Q8—71no—
94C22×Q8C2×D711yescharacteristic
95C2×C4⋊C4—72no—
96C2×C4⋊C4—72no—
97C2×C4⋊C4—72no—
98C2×C4⋊C4—71no—
99C2×C4⋊C4—72no—
100C2×C4⋊C4C2×D712yes—
101C2×C4⋊C4C2×D711yescharacteristic
102C2×C42—72no—
103C2×C42—71no—
104C2×C42—72no—
105#95—72no—
106#95—72no—
107#95—71no—
108#95—72no—
109#95—72no—
110#95—72no—
111#95—72no—
112#95C2×D712yes—
113#95C2×D711yescharacteristic
114C22×C7C22×Q812yes—
115C22×C7C2×#4112yes—
116C22×C7C2×#4111yescharacteristic
117C22×C7H(5,𝔽2)11yescharacteristic
118C22×C7H(5,𝔽2)11yescharacteristic
119C4×C7—22no—
120C4×C7—24no—
121C4×C7—24no—
122C4×C7—41no—
123C4×C7—41no—
124Dic7—24no—
125Dic7—24no—
126Dic7—42no—
127Dic7—42no—
128C2×Q8—24no—
129C2×Q8—41no—
130C2×Q8—142no—
131C2×Q8—281no—
132C2×Q8—281no—
133C22×C4—72no—
134C22×C4—72no—
135C22×C4—72no—
136C22×C4—72no—
137C22×C4C22×D711yescharacteristic
138C22×C4C22×D711yescharacteristic
139C22×C4C22×D712yes—
140C22×C4C22×D712yes—
141C22×C4C22×D711yescharacteristic
142C4⋊C4—24no—
143C4⋊C4—41no—
144C4⋊C4—78no—
145C4⋊C4—144no—
146C4⋊C4—282no—
147C4⋊C4—282no—
148C4⋊C4—281no—
149C42—144no—
150C42—282no—
151C42—281no—
152C2×C7#53312yes—
153C2×C7#53811yescharacteristic
154C2×C7#55111yescharacteristic
155C2×C7#55412yes—
156C2×C7Q8211yescharacteristic
157C23C23×D711yescharacteristic
158Q8—44no—
159Q8—284no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—74no—
171C2×C4—74no—
172C2×C4—142no—
173C2×C4—142no—
174C2×C4—142no—
175C2×C4—142no—
176C2×C4—142no—
177C2×C4—142no—
178C2×C4—142no—
179C2×C4—142no—
180C2×C4—142no—
181C2×C4—142no—
182C2×C4#115914yes—
183C2×C4Q8×D714yes—
184C2×C4#116112yes—
185C7#258611yescharacteristic, complemented, Sylow
186C22C2×#115912yes—
187C22#654811yescharacteristic
188C22C2×Q8×D712yes—
189C22C2×#116111yescharacteristic
190C22#655311yescharacteristic
191C4—24no—
192C4—24no—
193C4—22no—
194C4—41no—
195C4—41no—
196C4—144no—
197C4—144no—
198C4—282no—
199C4—282no—
200C2#9027811yescharacteristic
201C2#9032612yes—
202C2#9033511yescharacteristic
203C2#9041311yescharacteristic
204C2#9041912yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#395C2
hypercenter#395D7
hypocenterC7#2586
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
52. Dic7⋊C4Q8semidirect
185. C7#2586semidirect

Maximal subgroups

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

subgroupindex
C7×#3952
#900022
#897572
#897552
#897502
#897462
#894282
#894272
#894212
#25867