Finite Group Databasebeta

← #11735718

Subgroups — #11735718

Order 896.

344 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 121 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×#400C211yesmaximal, characteristic
2#89756C212yesmaximal
3#89755C211yesmaximal, characteristic
4#89749C211yesmaximal, characteristic
5#89432C212yesmaximal
6#89427C212yesmaximal
7#89425C212yesmaximal
8#89423C212yesmaximal
9#89419C212yesmaximal
10C2×C7×C4⋊C4C2211yescharacteristic
11C2×C7×C4⋊C4C2211yescharacteristic
12C2×C4⋊Dic7C2212yes—
13C2×C4⋊Dic7C2211yescharacteristic
14C2×C4⋊Dic7C2211yescharacteristic
15C2×Dic7⋊C4C2212yes—
16C2×Dic7⋊C4C2212yes—
17C2×Dic7⋊C4C2212yes—
18C2×C4×Dic7C2212yes—
19C2×C4×Dic7C2211yescharacteristic
20C7×#95C2212yes—
21C7×#95C2211yescharacteristic
22C7×#95C2212yes—
23C7⋊#95C2212yes—
24C7⋊#95C2212yes—
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2211yescharacteristic
30C7⋊#95C2212yes—
31#2769—71nomaximal, Sylow
32C22×C4×C7C2311yescharacteristic
33C22×C4×C7C2312yes—
34C22×C4×C7C2311yescharacteristic
35C22×C4×C7C2312yes—
36C22×C4×C7C2311yescharacteristic
37C22×Dic7C2312yes—
38C22×Dic7C2312yes—
39C22×Dic7C2312yes—
40C22×Dic7C2312yes—
41C7×C4⋊C4—41no—
42C7×C4⋊C4—41no—
43C4⋊Dic7—24no—
44C4⋊Dic7—41no—
45C4⋊Dic7—41no—
46Dic7⋊C4—24no—
47Dic7⋊C4—24no—
48Dic7⋊C4—42no—
49C4×Dic7—22no—
50C4×Dic7—24no—
51#400—72no—
52#400D711yescharacteristic
53#398—71no—
54#398—72no—
55#389—72no—
56#387—72no—
57#387—72no—
58#383—72no—
59#382—71no—
60C23×C7C2411yescharacteristic
61C2×C4×C7—21no—
62C2×C4×C7—21no—
63C2×C4×C7—22no—
64C2×C4×C7—22no—
65C2×C4×C7—22no—
66C2×C4×C7—22no—
67C2×C4×C7—21no—
68C2×C4×C7—21no—
69C2×C4×C7—22no—
70C2×C4×C7—22no—
71C2×C4×C7—22no—
72C2×C4×C7—22no—
73C2×C4×C7#4112yes—
74C2×Dic7—22no—
75C2×Dic7—22no—
76C2×Dic7—22no—
77C2×Dic7—22no—
78C2×Dic7—22no—
79C2×Dic7—22no—
80C2×Dic7—22no—
81C2×Dic7—22no—
82C2×Dic7—22no—
83C2×Dic7—22no—
84C2×Dic7—22no—
85C2×Dic7#4114yes—
86C2×C4⋊C4—72no—
87C2×C4⋊C4—72no—
88C2×C4⋊C4—72no—
89C2×C4⋊C4—72no—
90C2×C4⋊C4—71no—
91C2×C4⋊C4—71no—
92C2×C4⋊C4C2×D711yescharacteristic
93C2×C4⋊C4C2×D711yescharacteristic
94C2×C42—71no—
95C2×C42—72no—
96#95—72no—
97#95—72no—
98#95—72no—
99#95—72no—
100#95—72no—
101#95—72no—
102#95—71no—
103#95—72no—
104#95C2×D711yescharacteristic
105#95C2×D712yes—
106#95C2×D712yes—
107C22×C7C2×#4111yescharacteristic
108C22×C7C2×#4112yes—
109C22×C7H(5,𝔽2)11yescharacteristic
110C22×C72−1+411yescharacteristic
111C22×C72−1+412yes—
112C4×C7—22no—
113C4×C7—42no—
114C4×C7—42no—
115C4×C7—41no—
116C4×C7—41no—
117Dic7—24no—
118Dic7—42no—
119Dic7—42no—
120Dic7—42no—
121C22×C4—72no—
122C22×C4—72no—
123C22×C4—72no—
124C22×C4—72no—
125C22×C4C22×D711yescharacteristic
126C22×C4C22×D712yes—
127C22×C4C22×D711yescharacteristic
128C22×C4C22×D712yes—
129C22×C4C22×D711yescharacteristic
130C4⋊C4—41no—
131C4⋊C4—41no—
132C4⋊C4—144no—
133C4⋊C4—144no—
134C4⋊C4—144no—
135C4⋊C4—282no—
136C4⋊C4—281no—
137C4⋊C4—281no—
138C42—144no—
139C42—142no—
140C2×C7#53911yescharacteristic
141C2×C7#54112yes—
142C2×C7#55212yes—
143C2×C7#55511yescharacteristic
144C2×C7#56311yescharacteristic
145C23C23×D711yescharacteristic
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—74no—
159C2×C4—142no—
160C2×C4—142no—
161C2×C4—142no—
162C2×C4—142no—
163C2×C4—142no—
164C2×C4—142no—
165C2×C4—142no—
166C2×C4—142no—
167C2×C4—142no—
168C2×C4—142no—
169C2×C4—142no—
170C2×C4#116112yes—
171C7#276911yescharacteristic, complemented, Sylow
172C22#654811yescharacteristic
173C22C2×#116111yescharacteristic
174C22D7×#4112yes—
175C22#655412yes—
176C22#655411yescharacteristic
177C4—22no—
178C4—42no—
179C4—41no—
180C4—41no—
181C4—42no—
182C4—144no—
183C4—282no—
184C4—282no—
185C4—282no—
186C2#9033611yescharacteristic
187C2#9034211yescharacteristic
188C2#9036111yescharacteristic
189C2#9038512yes—
190C2#9039212yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#400C2
hypercenter#400D7
hypocenterC7#2769
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
171. C7#2769semidirect

Maximal subgroups

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

subgroupindex
C7×#4002
#897562
#897552
#897492
#894322
#894272
#894252
#894232
#894192
#27697