Finite Group Databasebeta

← #11731797

Subgroups — #11731797

Order 896.

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

normal subgroups only

This group has 344 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×#384C211yesmaximal, characteristic
2#89758C211yesmaximal, characteristic
3#89755C211yesmaximal, characteristic
4#89754C212yesmaximal
5#89754C212yesmaximal
6#89753C212yesmaximal
7#89753C212yesmaximal
8#89750C212yesmaximal
9#89705C212yesmaximal
10C2×C7×C4⋊C4C2212yes—
11C2×C7×C4⋊C4C2212yes—
12C2×C7×C4⋊C4C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C42×C7C2211yescharacteristic
15C2×C4⋊Dic7C2211yescharacteristic
16C2×C4⋊Dic7C2211yescharacteristic
17C2×Dic7⋊C4C2212yes—
18C2×Dic7⋊C4C2212yes—
19C2×C4×Dic7C2212yes—
20C7⋊#95C2212yes—
21C7⋊#95C2212yes—
22C7⋊#95C2212yes—
23C7⋊#95C2212yes—
24C7⋊#95C2212yes—
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2211yescharacteristic
30C7⋊#95C2211yescharacteristic
31#2794—71nomaximal, Sylow
32C22×C4×C7C2312yes—
33C22×C4×C7C2311yescharacteristic
34C22×C4×C7C2311yescharacteristic
35C22×C4×C7C2312yes—
36C22×C4×C7C2311yescharacteristic
37C22×Dic7C2312yes—
38C22×Dic7C2312yes—
39C22×Dic7C2312yes—
40C22×Dic7C2312yes—
41C7×C4⋊C4—24no—
42C7×C4⋊C4—24no—
43C7×C4⋊C4—41no—
44C7×C4⋊C4—41no—
45C42×C7—22no—
46C4⋊Dic7—41no—
47C4⋊Dic7—41no—
48Dic7⋊C4—24no—
49Dic7⋊C4—24no—
50C4×Dic7—42no—
51#400—72no—
52#400—72no—
53#400—71no—
54#398—71no—
55#387—72no—
56#387—72no—
57#387—72no—
58#387—72no—
59#384D711yescharacteristic
60C23×C7C2411yescharacteristic
61C2×C4×C7—22no—
62C2×C4×C7—21no—
63C2×C4×C7—21no—
64C2×C4×C7—22no—
65C2×C4×C7—22no—
66C2×C4×C7—22no—
67C2×C4×C7—22no—
68C2×C4×C7—22no—
69C2×C4×C7—22no—
70C2×C4×C7—21no—
71C2×C4×C7—21no—
72C2×C4×C7#4112yes—
73C2×C4×C7#4114yes—
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—24no—
85C2×C4⋊C4—72no—
86C2×C4⋊C4—72no—
87C2×C4⋊C4—71no—
88C2×C4⋊C4—71no—
89C2×C4⋊C4C2×D712yes—
90C2×C4⋊C4C2×D712yes—
91C2×C4⋊C4C2×D711yescharacteristic
92C2×C4⋊C4C2×D711yescharacteristic
93C2×C42—72no—
94C2×C42C2×D711yescharacteristic
95#95—72no—
96#95—72no—
97#95—72no—
98#95—72no—
99#95—72no—
100#95—72no—
101#95—72no—
102#95—72no—
103#95—72no—
104#95—71no—
105#95—71no—
106C22×C7C2×#4111yescharacteristic
107C22×C7C2×#4112yes—
108C22×C7H(5,𝔽2)11yescharacteristic
109C22×C72−1+411yescharacteristic
110C22×C72−1+412yes—
111C4×C7—22no—
112C4×C7—24no—
113C4×C7—42no—
114C4×C7—41no—
115C4×C7—41no—
116Dic7—42no—
117Dic7—42no—
118Dic7—42no—
119Dic7—42no—
120C22×C4—72no—
121C22×C4—72no—
122C22×C4—72no—
123C22×C4—72no—
124C22×C4C22×D712yes—
125C22×C4C22×D711yescharacteristic
126C22×C4C22×D711yescharacteristic
127C22×C4C22×D711yescharacteristic
128C22×C4C22×D712yes—
129C4⋊C4—24no—
130C4⋊C4—24no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C4⋊C4—144no—
134C4⋊C4—144no—
135C4⋊C4—281no—
136C4⋊C4—281no—
137C42—22no—
138C42—282no—
139C2×C7#53911yescharacteristic
140C2×C7#54112yes—
141C2×C7#55212yes—
142C2×C7#55911yescharacteristic
143C2×C7#56311yescharacteristic
144C23C23×D711yescharacteristic
145C2×C4—22no—
146C2×C4—21no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—142no—
157C2×C4—142no—
158C2×C4—142no—
159C2×C4—142no—
160C2×C4—142no—
161C2×C4—142no—
162C2×C4—142no—
163C2×C4—142no—
164C2×C4—144no—
165C2×C4—142no—
166C2×C4—142no—
167C2×C4#115712yes—
168C2×C4#115914yes—
169C7#279411yescharacteristic, complemented, Sylow
170C22C2×#115711yescharacteristic
171C22C2×#115912yes—
172C22#654811yescharacteristic
173C22#655111yescharacteristic
174C22#655112yes—
175C4—24no—
176C4—22no—
177C4—42no—
178C4—41no—
179C4—41no—
180C4—282no—
181C4—282no—
182C4—282no—
183C4—282no—
184C2#9020412yes—
185C2#9020611yescharacteristic
186C2#9040411yescharacteristic
187C2#9042212yes—
188C2#9042611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#384C2
hypercenter#384D7
hypocenterC7#2794
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
169. C7#2794semidirect

Maximal subgroups

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

subgroupindex
C7×#3842
#897582
#897552
#897542
#897532
#897502
#897052
#27947