Finite Group Databasebeta

← #11735372

Subgroups — #11735372

Order 896.

380 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 125 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×#6452C212yesmaximal
2C7×#398C211yesmaximal, characteristic
3#89756C212yesmaximal
4#89756C212yesmaximal
5#89755C211yesmaximal, characteristic
6#89749C211yesmaximal, characteristic
7#89432C212yesmaximal
8#89430C212yesmaximal
9#89429C212yesmaximal
10C2×C7×C4⋊C4C2212yes—
11C2×C7×C4⋊C4C2211yescharacteristic
12C2×C7×C4⋊C4C2211yescharacteristic
13C2×C4⋊Dic7C2212yes—
14C2×C4⋊Dic7C2211yescharacteristic
15C2×C4⋊Dic7C2211yescharacteristic
16C2×Dic7⋊C4C2212yes—
17C2×Dic7⋊C4C2212yes—
18C2×Dic7⋊C4C2212yes—
19C2×Dic7⋊C4C2212yes—
20C2×Dic7⋊C4C2212yes—
21C2×C4×Dic7C2211yescharacteristic
22#6452—28no—
23C7×#95C2212yes—
24C7×#95C2211yescharacteristic
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2211yescharacteristic
31C7⋊#95C2212yes—
32#2822—71nomaximal, Sylow
33C22×C4×C7C2311yescharacteristic
34C22×C4×C7C2311yescharacteristic
35C22×C4×C7C2312yes—
36C22×C4×C7C2312yes—
37C22×C4×C7C2311yescharacteristic
38C22×Dic7C2312yes—
39C22×Dic7C2312yes—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C7×C4⋊C4—24no—
43C7×C4⋊C4—41no—
44C7×C4⋊C4—41no—
45C4⋊Dic7—24no—
46C4⋊Dic7—41no—
47C4⋊Dic7—41no—
48Dic7⋊C4—24no—
49Dic7⋊C4—24no—
50Dic7⋊C4—24no—
51Dic7⋊C4—24no—
52Dic7⋊C4—42no—
53C4×Dic7D414yes—
54C2×#125—72no—
55#400—72no—
56#400—72no—
57#398—71no—
58#398—72no—
59#398—72no—
60#398—72no—
61#398D711yescharacteristic
62#382—71no—
63C23×C7C2411yescharacteristic
64C2×C4×C7—21no—
65C2×C4×C7—21no—
66C2×C4×C7—22no—
67C2×C4×C7—22no—
68C2×C4×C7—22no—
69C2×C4×C7—21no—
70C2×C4×C7—21no—
71C2×C4×C7—22no—
72C2×C4×C7—22no—
73C2×C4×C7—22no—
74C2×C4×C7—22no—
75C2×C4×C7—22no—
76C2×C4×C7C2×D412yes—
77C2×Dic7—22no—
78C2×Dic7—22no—
79C2×Dic7—22no—
80C2×Dic7—22no—
81C2×Dic7—22no—
82C2×Dic7—22no—
83C2×Dic7—22no—
84C2×Dic7—24no—
85C2×Dic7—22no—
86C2×Dic7—22no—
87C2×Dic7C2×D414yes—
88#125—148no—
89C2×C4⋊C4—72no—
90C2×C4⋊C4—72no—
91C2×C4⋊C4—72no—
92C2×C4⋊C4—72no—
93C2×C4⋊C4—72no—
94C2×C4⋊C4—72no—
95C2×C4⋊C4—71no—
96C2×C4⋊C4—71no—
97C2×C4⋊C4C2×D712yes—
98C2×C4⋊C4C2×D711yescharacteristic
99C2×C4⋊C4C2×D711yescharacteristic
100C2×C42—71no—
101#95—72no—
102#95—72no—
103#95—72no—
104#95—72no—
105#95—72no—
106#95—72no—
107#95—71no—
108#95C2×D712yes—
109#95C2×D711yescharacteristic
110C22×C7C22×D411yescharacteristic
111C22×C7H(5,𝔽2)11yescharacteristic
112C22×C72−1+412yes—
113C22×C72−1+412yes—
114C22×C72−1+411yescharacteristic
115C4×C7—22no—
116C4×C7—42no—
117C4×C7—42no—
118C4×C7—41no—
119C4×C7—41no—
120Dic7—24no—
121Dic7—42no—
122Dic7—42no—
123Dic7—42no—
124C22×C4—72no—
125C22×C4—72no—
126C22×C4—72no—
127C22×C4—72no—
128C22×C4C22×D711yescharacteristic
129C22×C4C22×D711yescharacteristic
130C22×C4C22×D712yes—
131C22×C4C22×D712yes—
132C22×C4C22×D711yescharacteristic
133C4⋊C4—24no—
134C4⋊C4—41no—
135C4⋊C4—41no—
136C4⋊C4—144no—
137C4⋊C4—144no—
138C4⋊C4—144no—
139C4⋊C4—144no—
140C4⋊C4—144no—
141C4⋊C4—282no—
142C4⋊C4—281no—
143C4⋊C4—281no—
144C42—74no—
145C2×C7#53612yes—
146C2×C7#53711yescharacteristic
147C2×C7#56311yescharacteristic
148C2×C7#56311yescharacteristic
149C2×C7#56412yes—
150C23C23×D711yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—74no—
164C2×C4—142no—
165C2×C4—142no—
166C2×C4—142no—
167C2×C4—142no—
168C2×C4—142no—
169C2×C4—142no—
170C2×C4—144no—
171C2×C4—142no—
172C2×C4—142no—
173C2×C4—142no—
174C2×C4D4×D712yes—
175C7#282211yescharacteristic, complemented, Sylow
176C22C2×D4×D711yescharacteristic
177C22#654811yescharacteristic
178C22#655112yes—
179C22#655411yescharacteristic
180C22#655412yes—
181C4—22no—
182C4—42no—
183C4—42no—
184C4—41no—
185C4—41no—
186C4—144no—
187C4—282no—
188C4—282no—
189C4—282no—
190C2#9028811yescharacteristic
191C2#9031912yes—
192C2#9034211yescharacteristic
193C2C7⋊#56412yes—
194C2#9042611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#398C2
hypercenter#398D7
hypocenterC7#2822
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
175. C7#2822semidirect

Maximal subgroups

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

subgroupindex
C2×#64522
C7×#3982
#897562
#897552
#897492
#894322
#894302
#894292
#28227