Finite Group Databasebeta

← #11735091

Subgroups — #11735091

Order 896.

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

normal subgroups only

This group has 444 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
1C2×#6441C212yesmaximal
2C7×#397C211yesmaximal, characteristic
3#89751C212yesmaximal, complemented
4#89725C211yesmaximal, characteristic
5#89724C212yesmaximal
6#89715C211yesmaximal, characteristic
7#89432C212yesmaximal, complemented
8#89430C212yesmaximal, complemented
9#89428C212yesmaximal, complemented
10C23×Dic7C2211yescharacteristic
11C22×Dic14C2212yes—
12C2×C7×C4⋊C4C2211yescharacteristic
13C2×C7×C22⋊C4C2212yes—
14C2×C7×C22⋊C4C2211yescharacteristic
15C2×C22⋊Dic7C2212yes—
16C2×C22⋊Dic7C2211yescharacteristic
17C2×C4⋊Dic7C2212yes—
18C2×C4⋊Dic7C2211yescharacteristic
19C2×Dic7⋊C4C2212yes—
20C2×Dic7⋊C4C2212yes—
21C2×Dic7⋊C4C2212yes—
22#6441—28no—
23C7×#95C2212yes—
24C7×#95C2211yescharacteristic
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2211yescharacteristic
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2212yes—
31C7⋊#95C2212yes—
32#2817—71nomaximal, Sylow
33C24×C7C2311yescharacteristic
34C22×C4×C7C2311yescharacteristic
35C22×C4×C7C2311yescharacteristic
36C22×C4×C7C2312yes—
37C22×C4×C7C2312yes—
38C22×Dic7—41no—
39C22×Dic7—41no—
40C22×Dic7D414yes—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C22×Dic7C2312yes—
44C22×Dic7C2312yes—
45C2×Dic14—24no—
46C2×Dic14—42no—
47C2×Dic14—42no—
48C7×C4⋊C4—41no—
49C7×C22⋊C4—24no—
50C7×C22⋊C4—41no—
51C22⋊Dic7—24no—
52C22⋊Dic7—41no—
53C4⋊Dic7—24no—
54C4⋊Dic7—41no—
55Dic7⋊C4—24no—
56Dic7⋊C4—24no—
57Dic7⋊C4—42no—
58C2×C22⋊Q8—72no—
59#400—72no—
60#400—72no—
61#397—72no—
62#397—71no—
63#397D711yescharacteristic
64#395—72no—
65#395—72no—
66#381—71no—
67C23×C7—21no—
68C23×C7—21no—
69C23×C7—22no—
70C23×C7—22no—
71C23×C7C2×D412yes—
72C23×C7C2411yescharacteristic
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×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7—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—22no—
92C2×Dic7—22no—
93C2×Dic7—22no—
94C2×Dic7—44no—
95C2×Dic7C2×D414yes—
96Dic14—48no—
97C22×Q8—72no—
98C23×C4—71no—
99C22⋊Q8—148no—
100C2×C4⋊C4—72no—
101C2×C4⋊C4—72no—
102C2×C4⋊C4—72no—
103C2×C4⋊C4—72no—
104C2×C4⋊C4—71no—
105C2×C4⋊C4C2×D711yescharacteristic
106C2×C22⋊C4—72no—
107C2×C22⋊C4—71no—
108C2×C22⋊C4C2×D712yes—
109C2×C22⋊C4C2×D711yescharacteristic
110#95—72no—
111#95—72no—
112#95—72no—
113#95—72no—
114#95—72no—
115#95—72no—
116#95—71no—
117#95C2×D712yes—
118#95C2×D711yescharacteristic
119C22×C7—22no—
120C22×C7—24no—
121C22×C7—42no—
122C22×C7—41no—
123C22×C7—41no—
124C22×C7C22×D411yescharacteristic
125C22×C7H(5,𝔽2)11yescharacteristic
126C22×C7H(5,𝔽2)11yescharacteristic
127C22×C72−1+412yes—
128C22×C72−1+412yes—
129C4×C7—42no—
130C4×C7—42no—
131C4×C7—41no—
132C4×C7—41no—
133Dic7—24no—
134Dic7—42no—
135Dic7—42no—
136Dic7—42no—
137C24C22×D711yescharacteristic
138C2×Q8—144no—
139C2×Q8—282no—
140C2×Q8—282no—
141C22×C4—72no—
142C22×C4—74no—
143C22×C4—72no—
144C22×C4—72no—
145C22×C4—72no—
146C22×C4—281no—
147C22×C4—281no—
148C22×C4C22×D711yescharacteristic
149C22×C4C22×D711yescharacteristic
150C22×C4C22×D712yes—
151C22×C4C22×D712yes—
152C4⋊C4—41no—
153C4⋊C4—144no—
154C4⋊C4—144no—
155C4⋊C4—144no—
156C4⋊C4—282no—
157C4⋊C4—281no—
158C22⋊C4—24no—
159C22⋊C4—41no—
160C22⋊C4—144no—
161C22⋊C4—281no—
162C2×C7—42no—
163C2×C7#53411yescharacteristic
164C2×C7#53612yes—
165C2×C7#56312yes—
166C2×C7#56311yescharacteristic
167C2×C7#56311yescharacteristic
168C23—21no—
169C23—21no—
170C23—22no—
171C23—22no—
172C23D4×D712yes—
173C23C23×D711yescharacteristic
174Q8—288no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—74no—
186C2×C4—142no—
187C2×C4—142no—
188C2×C4—142no—
189C2×C4—142no—
190C2×C4—142no—
191C2×C4—142no—
192C2×C4—142no—
193C2×C4—142no—
194C2×C4—142no—
195C2×C4—142no—
196C2×C4—142no—
197C2×C4—284no—
198C7#281711yescharacteristic, complemented, Sylow
199C22—22no—
200C22—24no—
201C22—42no—
202C22—41no—
203C22—41no—
204C22C2×D4×D711yescharacteristic
205C22#654811yescharacteristic
206C22#655311yescharacteristic
207C22#655412yes—
208C22#655412yes—
209C4—42no—
210C4—42no—
211C4—41no—
212C4—41no—
213C4—144no—
214C4—282no—
215C4—282no—
216C4—282no—
217C2—42no—
218C2#9029611yescharacteristic
219C2#9033912yes—
220C2#9034211yescharacteristic
221C2#9036311yescharacteristic
222C2#9040312yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#397C2
prime radicalC24×C7C23
hypercenter#397D7
hypocenterC7#2817
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
3. #89751C2semidirect
7. #89432C2semidirect
8. #89430C2semidirect
9. #89428C2semidirect
198. C7#2817semidirect

Maximal subgroups

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

subgroupindex
C2×#64412
C7×#3972
#897512
#897252
#897242
#897152
#894322
#894302
#894282
#28177