Finite Group Databasebeta

← #11735783

Subgroups — #11735783

Order 896.

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

normal subgroups only

This group has 348 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
1C7×#400C211yesmaximal, characteristic
2#89755C212yesmaximal
3#89751C211yesmaximal, characteristic
4#89748C211yesmaximal, characteristic
5#89433C212yesmaximal
6#89430C212yesmaximal
7#89429C212yesmaximal
8#89428C212yesmaximal
9#89423C212yesmaximal
10C22×Dic14C2211yescharacteristic
11C2×C7×C4⋊C4C2211yescharacteristic
12C2×C7×C4⋊C4C2211yescharacteristic
13C2×C4⋊Dic7C2212yes—
14C2×C4⋊Dic7C2212yes—
15C2×C4⋊Dic7C2211yescharacteristic
16C2×Dic7⋊C4C2212yes—
17C2×Dic7⋊C4C2212yes—
18C2×Dic7⋊C4C2212yes—
19C2×Dic7⋊C4C2212yes—
20C2×C4×Dic7C2211yescharacteristic
21C7×#95C2212yes—
22C7×#95C2212yes—
23C7×#95C2211yescharacteristic
24C7⋊#95C2212yes—
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2211yescharacteristic
30C7⋊#95C2212yes—
31#2843—71nomaximal, Sylow
32C22×C4×C7C2312yes—
33C22×C4×C7C2312yes—
34C22×C4×C7C2311yescharacteristic
35C22×C4×C7C2311yescharacteristic
36C22×C4×C7C2311yescharacteristic
37C22×Dic7C2312yes—
38C22×Dic7C2312yes—
39C22×Dic7C2312yes—
40C22×Dic7C2312yes—
41C2×Dic14—42no—
42C2×Dic14—41no—
43C7×C4⋊C4—41no—
44C7×C4⋊C4—41no—
45C4⋊Dic7—42no—
46C4⋊Dic7—41no—
47C4⋊Dic7—42no—
48Dic7⋊C4—42no—
49Dic7⋊C4—42no—
50Dic7⋊C4—42no—
51Dic7⋊C4—42no—
52C4×Dic7Q814yescomplemented
53#400—72no—
54#400D711yescharacteristic
55#398—72no—
56#398—72no—
57#398—72no—
58#395—71no—
59#395—72no—
60#384—71no—
61#383—72no—
62C23×C7C2411yescharacteristic
63C2×C4×C7—21no—
64C2×C4×C7—21no—
65C2×C4×C7—22no—
66C2×C4×C7—22no—
67C2×C4×C7—22no—
68C2×C4×C7—22no—
69C2×C4×C7—21no—
70C2×C4×C7—21no—
71C2×C4×C7—24no—
72C2×C4×C7—22no—
73C2×C4×C7—22no—
74C2×C4×C7C2×Q812yes—
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—22no—
86C2×Dic7C2×Q814yes—
87Dic14—44no—
88C22×Q8—71no—
89C2×C4⋊C4—72no—
90C2×C4⋊C4—72no—
91C2×C4⋊C4—72no—
92C2×C4⋊C4—72no—
93C2×C4⋊C4—71no—
94C2×C4⋊C4—72no—
95C2×C4⋊C4—72no—
96C2×C4⋊C4C2×D711yescharacteristic
97C2×C4⋊C4C2×D711yescharacteristic
98C2×C42—71no—
99#95—72no—
100#95—72no—
101#95—72no—
102#95—72no—
103#95—72no—
104#95—71no—
105#95—72no—
106#95C2×D712yes—
107#95C2×D712yes—
108#95C2×D711yescharacteristic
109C22×C7C22×Q811yescharacteristic
110C22×C7H(5,𝔽2)11yescharacteristic
111C22×C7H(5,𝔽2)12yes—
112C22×C72−1+412yes—
113C22×C72−1+411yescharacteristic
114C4×C7—22no—
115C4×C7—41no—
116C4×C7—41no—
117C4×C7—42no—
118C4×C7—42no—
119Dic7—24no—
120Dic7—42no—
121Dic7—42no—
122Dic7—42no—
123C2×Q8—281no—
124C2×Q8—282no—
125C22×C4—72no—
126C22×C4—72no—
127C22×C4—72no—
128C22×C4—72no—
129C22×C4C22×D712yes—
130C22×C4C22×D712yes—
131C22×C4C22×D711yescharacteristic
132C22×C4C22×D711yescharacteristic
133C22×C4C22×D711yescharacteristic
134C4⋊C4—41no—
135C4⋊C4—41no—
136C4⋊C4—282no—
137C4⋊C4—282no—
138C4⋊C4—282no—
139C4⋊C4—282no—
140C4⋊C4—281no—
141C4⋊C4—282no—
142C4⋊C4—282no—
143C42—74no—
144C2×C7#54412yes—
145C2×C7#54411yescharacteristic
146C2×C7#56011yescharacteristic
147C2×C7#56312yes—
148C2×C7#56311yescharacteristic
149C23C23×D711yescharacteristic
150Q8—284no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—24no—
162C2×C4—74no—
163C2×C4—142no—
164C2×C4—142no—
165C2×C4—142no—
166C2×C4—142no—
167C2×C4—142no—
168C2×C4—142no—
169C2×C4—142no—
170C2×C4—142no—
171C2×C4—142no—
172C2×C4—142no—
173C2×C4—142no—
174C2×C4Q8×D712yes—
175C7#284311yescharacteristic, complemented, Sylow
176C22#654811yescharacteristic
177C22C2×Q8×D711yescharacteristic
178C22#655312yes—
179C22#655412yes—
180C22#655411yescharacteristic
181C4—22no—
182C4—42no—
183C4—41no—
184C4—41no—
185C4—42no—
186C4—144no—
187C4—282no—
188C4—282no—
189C4—282no—
190C2#9031411yescharacteristic
191C2#9035711yescharacteristic
192C2#9036311yescharacteristic
193C2#9038012yes—
194C2#9040312yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#400C2
hypercenter#400D7
hypocenterC7#2843
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
52. C4×Dic7Q8semidirect
175. C7#2843semidirect

Maximal subgroups

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

subgroupindex
C7×#4002
#897552
#897512
#897482
#894332
#894302
#894292
#894282
#894232
#28437