Finite Group Databasebeta

← #11735375

Subgroups — #11735375

Order 896.

332 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 117 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×#398C211yesmaximal, characteristic
2#89758C211yesmaximal, characteristic
3#89756C212yesmaximal
4#89753C212yesmaximal
5#89751C212yesmaximal
6#89746C211yesmaximal, characteristic
7C7⋊#401C212yesmaximal
8#89432C212yesmaximal
9#89431C212yesmaximal
10C22×Dic14C2211yescharacteristic
11C2×C7×C4⋊C4C2212yes—
12C2×C7×C4⋊C4C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C4⋊Dic7C2212yes—
15C2×C4⋊Dic7C2211yescharacteristic
16C2×Dic7⋊C4C2212yes—
17C2×Dic7⋊C4C2212yes—
18C2×C4×Dic7C2211yescharacteristic
19C7×#95C2212yes—
20C7×#95C2211yescharacteristic
21C7⋊#95C2211yescharacteristic
22C7⋊#95C2212yes—
23C7⋊#95C2212yes—
24C7⋊#95C2212yes—
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2212yes—
31#2833—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—
41C2×Dic14—22no—
42C2×Dic14—41no—
43C2×Dic14—41no—
44C7×C4⋊C4—24no—
45C7×C4⋊C4—41no—
46C7×C4⋊C4—41no—
47C4⋊Dic7—42no—
48C4⋊Dic7—41no—
49Dic7⋊C4—42no—
50Dic7⋊C4—42no—
51C4×Dic7—41no—
52#401—72no—
53#400—71no—
54#400—72no—
55#400—72no—
56#398—72no—
57#398D711yescharacteristic
58#395—72no—
59#391—71no—
60#387—72no—
61C23×C7C2411yescharacteristic
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—21no—
69C2×C4×C7—21no—
70C2×C4×C7—22no—
71C2×C4×C7—22no—
72C2×C4×C7—22no—
73C2×C4×C7—22no—
74C2×C4×C7#4112yes—
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—24no—
84C2×Dic7—22no—
85C2×Dic7—22no—
86Dic14—44no—
87C22×Q8—71no—
88C2×C4⋊C4—72no—
89C2×C4⋊C4—72no—
90C2×C4⋊C4—72no—
91C2×C4⋊C4—71no—
92C2×C4⋊C4C2×D712yes—
93C2×C4⋊C4C2×D711yescharacteristic
94C2×C4⋊C4C2×D711yescharacteristic
95C2×C42—71no—
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—72no—
105#95—71no—
106#95C2×D712yes—
107#95C2×D711yescharacteristic
108C22×C7C2×#4111yescharacteristic
109C22×C7H(5,𝔽2)11yescharacteristic
110C22×C7H(5,𝔽2)11yescharacteristic
111C22×C72−1+412yes—
112C22×C72−1+412yes—
113C4×C7—22no—
114C4×C7—42no—
115C4×C7—42no—
116C4×C7—41no—
117C4×C7—41no—
118Dic7—42no—
119Dic7—42no—
120Dic7—42no—
121Dic7—42no—
122C2×Q8—142no—
123C2×Q8—281no—
124C2×Q8—281no—
125C22×C4—72no—
126C22×C4—72no—
127C22×C4—72no—
128C22×C4—72no—
129C22×C4C22×D712yes—
130C22×C4C22×D711yescharacteristic
131C22×C4C22×D711yescharacteristic
132C22×C4C22×D712yes—
133C22×C4C22×D711yescharacteristic
134C4⋊C4—24no—
135C4⋊C4—41no—
136C4⋊C4—41no—
137C4⋊C4—282no—
138C4⋊C4—282no—
139C4⋊C4—281no—
140C4⋊C4—282no—
141C42—281no—
142C2×C7#53811yescharacteristic
143C2×C7#54112yes—
144C2×C7#56312yes—
145C2×C7#56311yescharacteristic
146C2×C7#56311yescharacteristic
147C23C23×D711yescharacteristic
148Q8—284no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—22no—
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—142no—
162C2×C4—142no—
163C2×C4—142no—
164C2×C4—142no—
165C2×C4—142no—
166C2×C4—144no—
167C2×C4—142no—
168C2×C4—142no—
169C2×C4—142no—
170C2×C4—142no—
171C2×C4—142no—
172C2×C4#115912yes—
173C7#283311yescharacteristic, complemented, Sylow
174C22C2×#115911yescharacteristic
175C22#654811yescharacteristic
176C22#655112yes—
177C22#655311yescharacteristic
178C22#655412yes—
179C4—22no—
180C4—42no—
181C4—42no—
182C4—41no—
183C4—41no—
184C4—282no—
185C4—282no—
186C4—282no—
187C4—282no—
188C2#9030811yescharacteristic
189C2#9031612yes—
190C2#9034211yescharacteristic
191C2#9039012yes—
192C2#9042611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#398C2
hypercenter#398D7
hypocenterC7#2833
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
173. C7#2833semidirect

Maximal subgroups

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

subgroupindex
C7×#3982
#897582
#897562
#897532
#897512
#897462
C7⋊#4012
#894322
#894312
#28337