Finite Group Databasebeta

← #11735062

Subgroups — #11735062

Order 896.

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

normal subgroups only

This group has 392 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×#397C211yesmaximal, characteristic
2#89744C212yesmaximal, complemented
3#89723C212yesmaximal
4#89721C211yesmaximal, characteristic
5#89718C212yesmaximal
6#89715C211yesmaximal, characteristic
7#89432C212yesmaximal, complemented
8#89432C212yesmaximal, complemented
9#89425C212yesmaximal, complemented
10C23×Dic7C2211yescharacteristic
11C2×C7×C4⋊C4C2211yescharacteristic
12C2×C7×C22⋊C4C2212yes—
13C2×C7×C22⋊C4C2211yescharacteristic
14C2×C22⋊Dic7C2212yes—
15C2×C22⋊Dic7C2211yescharacteristic
16C2×C4⋊Dic7C2212yes—
17C2×C4⋊Dic7C2211yescharacteristic
18C2×Dic7⋊C4C2212yes—
19C2×Dic7⋊C4C2212yes—
20C2×Dic7⋊C4C2212yes—
21C2×C4×Dic7C2212yes—
22C7×#95C2212yes—
23C7×#95C2211yescharacteristic
24C7⋊#95C2211yescharacteristic
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2212yes—
31#2771—71nomaximal, Sylow
32C24×C7C2311yescharacteristic
33C22×C4×C7C2312yes—
34C22×C4×C7C2311yescharacteristic
35C22×C4×C7C2311yescharacteristic
36C22×C4×C7C2312yes—
37C22×Dic7—22no—
38C22×Dic7—41no—
39C22×Dic7—41no—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C22×Dic7C2312yes—
44C7×C4⋊C4—41no—
45C7×C22⋊C4—24no—
46C7×C22⋊C4—41no—
47C22⋊Dic7—24no—
48C22⋊Dic7—41no—
49C4⋊Dic7—24no—
50C4⋊Dic7—41no—
51Dic7⋊C4—24no—
52Dic7⋊C4—42no—
53Dic7⋊C4—42no—
54C4×Dic7—42no—
55#400—72no—
56#400—72no—
57#399—71no—
58#397D711yescharacteristic
59#389—72no—
60#388—72no—
61#387—72no—
62#385—72no—
63#381—71no—
64C23×C7—22no—
65C23×C7—21no—
66C23×C7—21no—
67C23×C7—22no—
68C23×C7#4112yes—
69C23×C7C2411yescharacteristic
70C2×C4×C7—22no—
71C2×C4×C7—22no—
72C2×C4×C7—22no—
73C2×C4×C7—22no—
74C2×C4×C7—22no—
75C2×C4×C7—22no—
76C2×C4×C7—21no—
77C2×C4×C7—21no—
78C2×C4×C7—22no—
79C2×C4×C7—22no—
80C2×Dic7—22no—
81C2×Dic7—22no—
82C2×Dic7—22no—
83C2×Dic7—22no—
84C2×Dic7—22no—
85C2×Dic7—22no—
86C2×Dic7—22no—
87C2×Dic7—22no—
88C2×Dic7—22no—
89C2×Dic7—22no—
90C2×Dic7—22no—
91C2×Dic7—44no—
92C2×Dic7#4114yes—
93C23×C4—71no—
94C2×C4⋊C4—72no—
95C2×C4⋊C4—72no—
96C2×C4⋊C4—72no—
97C2×C4⋊C4—72no—
98C2×C4⋊C4—71no—
99C2×C4⋊C4C2×D711yescharacteristic
100C2×C22⋊C4—72no—
101C2×C22⋊C4—71no—
102C2×C22⋊C4C2×D712yes—
103C2×C22⋊C4C2×D711yescharacteristic
104C2×C42—72no—
105#95—72no—
106#95—72no—
107#95—72no—
108#95—72no—
109#95—71no—
110#95—72no—
111#95—72no—
112#95C2×D712yes—
113#95C2×D711yescharacteristic
114C22×C7—22no—
115C22×C7—24no—
116C22×C7—42no—
117C22×C7—41no—
118C22×C7—41no—
119C22×C7C2×#4111yescharacteristic
120C22×C7C2×#4112yes—
121C22×C7H(5,𝔽2)11yescharacteristic
122C22×C7H(5,𝔽2)11yescharacteristic
123C22×C72−1+412yes—
124C4×C7—42no—
125C4×C7—42no—
126C4×C7—41no—
127C4×C7—41no—
128Dic7—24no—
129Dic7—42no—
130Dic7—42no—
131Dic7—42no—
132C24C22×D711yescharacteristic
133C22×C4—72no—
134C22×C4—72no—
135C22×C4—72no—
136C22×C4—72no—
137C22×C4—142no—
138C22×C4—281no—
139C22×C4—281no—
140C22×C4C22×D712yes—
141C22×C4C22×D711yescharacteristic
142C22×C4C22×D711yescharacteristic
143C22×C4C22×D712yes—
144C4⋊C4—41no—
145C4⋊C4—144no—
146C4⋊C4—144no—
147C4⋊C4—282no—
148C4⋊C4—282no—
149C4⋊C4—281no—
150C22⋊C4—24no—
151C22⋊C4—41no—
152C22⋊C4—144no—
153C22⋊C4—281no—
154C42—282no—
155C2×C7—42no—
156C2×C7#53811yescharacteristic
157C2×C7#53912yes—
158C2×C7#55212yes—
159C2×C7#55911yescharacteristic
160C2×C7#56311yescharacteristic
161C23—21no—
162C23—22no—
163C23—21no—
164C23—22no—
165C23#115912yes—
166C23C23×D711yescharacteristic
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—74no—
178C2×C4—142no—
179C2×C4—142no—
180C2×C4—142no—
181C2×C4—142no—
182C2×C4—142no—
183C2×C4—142no—
184C2×C4—142no—
185C2×C4—142no—
186C2×C4—142no—
187C2×C4—142no—
188C2×C4—142no—
189C2×C4—284no—
190C7#277111yescharacteristic, complemented, Sylow
191C22—22no—
192C22—24no—
193C22—42no—
194C22—41no—
195C22—41no—
196C22C2×#115911yescharacteristic
197C22#654811yescharacteristic
198C22D7×#4112yes—
199C22#655311yescharacteristic
200C22#655412yes—
201C4—42no—
202C4—42no—
203C4—41no—
204C4—41no—
205C4—144no—
206C4—282no—
207C4—282no—
208C4—282no—
209C2—42no—
210C2#9030811yescharacteristic
211C2#9034112yes—
212C2#9035511yescharacteristic
213C2#9036311yescharacteristic
214C2#9039812yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#397C2
prime radicalC24×C7C23
hypercenter#397D7
hypocenterC7#2771
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #89744C2semidirect
7. #89432C2semidirect
8. #89432C2semidirect
9. #89425C2semidirect
190. C7#2771semidirect

Maximal subgroups

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

subgroupindex
C7×#3972
#897442
#897232
#897212
#897182
#897152
#894322
#894252
#27717