Finite Group Databasebeta

← #11731698

Subgroups — #11731698

Order 896.

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

normal subgroups only

This group has 448 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×#382C211yesmaximal, characteristic, complemented
2#89763C211yesmaximal, characteristic
3#89761C212yesmaximal, complemented
4#89758C211yesmaximal, characteristic, complemented
5#89712C212yesmaximal, complemented
6#89447C212yesmaximal, complemented
7#89440C212yesmaximal, complemented
8#89437C212yesmaximal
9#89430C212yesmaximal, complemented
10C22×D28C2211yescharacteristic
11C22×C4×D7C2212yes—
12C2×C7×C4⋊C4C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C42×C7C2211yescharacteristic
15C2×(C2×D7)⋊C4C2212yes—
16C2×(C2×D7)⋊C4C2212yes—
17C2×(C2×D7)⋊C4C2212yes—
18C2×(C2×D7)⋊C4C2212yes—
19C2×(C2×D7)⋊C4C2212yes—
20C2×C4⋊Dic7C2211yescharacteristic
21C2×Dic7⋊C4C2212yes—
22C2×Dic7⋊C4C2212yes—
23C7×#95C2212yes—
24C7×#95C2212yes—
25C7⋊#95C2211yescharacteristic
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2211yescharacteristic
30C7⋊#95C2212yes—
31#2782—71nomaximal, Sylow
32C23×D7C2312yes—
33C22×C4×C7C2311yescharacteristic
34C22×C4×C7C2312yes—
35C22×C4×C7C2312yes—
36C22×C4×C7C2311yescharacteristic
37C22×C4×C7C2311yescharacteristic
38C22×Dic7C2312yes—
39C22×Dic7C2312yes—
40C22×Dic7C2312yes—
41C2×D28—24no—
42C2×D28—41no—
43C2×C4×D7—42no—
44C2×C4×D7—42no—
45C2×C4×D7—42no—
46C7×C4⋊C4—41no—
47C7×C4⋊C4—41no—
48C42×C7—22no—
49(C2×D7)⋊C4—24no—
50(C2×D7)⋊C4—24no—
51(C2×D7)⋊C4—24no—
52(C2×D7)⋊C4—42no—
53(C2×D7)⋊C4—42no—
54C4⋊Dic7—41no—
55Dic7⋊C4—42no—
56Dic7⋊C4—42no—
57#400—71no—
58#400—72no—
59#397—72no—
60#394—71no—
61#388—72no—
62#386—72no—
63#385—72no—
64#385—72no—
65#382D711yescharacteristic, complemented
66C23×C7C2411yescharacteristic
67C22×D7—22no—
68C22×D7—22no—
69C22×D7—22no—
70C22×D7—22no—
71C22×D7—22no—
72C22×D7—22no—
73C22×D7#4114yes—
74C2×C4×C7—21no—
75C2×C4×C7—21no—
76C2×C4×C7—22no—
77C2×C4×C7—22no—
78C2×C4×C7—22no—
79C2×C4×C7—22no—
80C2×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7—21no—
83C2×C4×C7—22no—
84C2×C4×C7—22no—
85C2×C4×C7—21no—
86C2×C4×C7#4112yes—
87C2×Dic7—22no—
88C2×Dic7—22no—
89C2×Dic7—22no—
90C2×Dic7—22no—
91C2×Dic7—22no—
92C2×Dic7—22no—
93C2×Dic7—22no—
94C2×Dic7—22no—
95C2×Dic7—22no—
96D28—44no—
97C4×D7—48no—
98C22×D4—71no—
99C23×C4—72no—
100C2×C4⋊C4—72no—
101C2×C4⋊C4—72no—
102C2×C4⋊C4—71no—
103C2×C4⋊C4C2×D711yescharacteristic
104C2×C4⋊C4C2×D711yescharacteristic
105C2×C22⋊C4—72no—
106C2×C22⋊C4—72no—
107C2×C22⋊C4—72no—
108C2×C22⋊C4—72no—
109C2×C22⋊C4—72no—
110C2×C42C2×D711yescharacteristic
111#95—72no—
112#95—72no—
113#95—72no—
114#95—72no—
115#95—71no—
116#95—71no—
117#95C2×D712yes—
118#95C2×D712yes—
119C22×C7C2×#4111yescharacteristic
120C22×C7C2×#4112yes—
121C22×C7H(5,𝔽2)11yescharacteristic
122C22×C7H(5,𝔽2)12yes—
123C22×C72−1+411yescharacteristic
124C2×D7—24no—
125C2×D7—24no—
126C2×D7—24no—
127C2×D7—42no—
128C2×D7—42no—
129C2×D7—42no—
130C2×D7—42no—
131C4×C7—22no—
132C4×C7—41no—
133C4×C7—41no—
134C4×C7—42no—
135C4×C7—42no—
136Dic7—42no—
137Dic7—42no—
138Dic7—42no—
139C24—72no—
140C2×D4—144no—
141C2×D4—281no—
142C22×C4—72no—
143C22×C4—72no—
144C22×C4—72no—
145C22×C4—282no—
146C22×C4—282no—
147C22×C4—282no—
148C22×C4C22×D711yescharacteristic
149C22×C4C22×D712yes—
150C22×C4C22×D711yescharacteristic
151C22×C4C22×D712yes—
152C22×C4C22×D711yescharacteristic
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—282no—
156C4⋊C4—282no—
157C4⋊C4—281no—
158C22⋊C4—144no—
159C22⋊C4—144no—
160C22⋊C4—144no—
161C22⋊C4—282no—
162C22⋊C4—282no—
163C42—22no—
164C2×C7#53912yes—
165C2×C7#53911yescharacteristic
166C2×C7#55111yescharacteristic
167C2×C7#55312yes—
168C2×C7H(3,𝔽4)11yescharacteristic
169D7—44no—
170C23—74no—
171C23—142no—
172C23—142no—
173C23—142no—
174C23—142no—
175C23—142no—
176C23—142no—
177C23C23×D711yescharacteristic
178D4—284no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—142no—
192C2×C4—142no—
193C2×C4—142no—
194C2×C4—142no—
195C2×C4—142no—
196C2×C4—142no—
197C2×C4—142no—
198C2×C4—142no—
199C2×C4—142no—
200C2×C4—288no—
201C2×C4#115712yes—
202C7#278211yescharacteristic, complemented, Sylow
203C22—144no—
204C22—144no—
205C22—144no—
206C22—282no—
207C22—282no—
208C22—282no—
209C22—282no—
210C22C2×#115711yescharacteristic
211C22#654811yescharacteristic
212C22#655111yescharacteristic
213C22D7×#4112yes—
214C22#655312yes—
215C4—22no—
216C4—41no—
217C4—41no—
218C4—42no—
219C4—42no—
220C4—282no—
221C4—282no—
222C4—282no—
223C2—284no—
224C2#9020611yescharacteristic
225C2#9022112yes—
226C2#9037011yescharacteristic
227C2#9037411yescharacteristic
228C2#9038312yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#382C2
prime radicalC22×D28C22
hypercenter#382D7
hypocenterC7#2782
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
1. C7×#382C2semidirect
3. #89761C2semidirect
4. #89758C2semidirect
5. #89712C2semidirect
6. #89447C2semidirect
7. #89440C2semidirect
9. #89430C2semidirect
65. #382D7semidirect
202. C7#2782semidirect

Maximal subgroups

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

subgroupindex
C7×#3822
#897632
#897612
#897582
#897122
#894472
#894402
#894372
#894302
#27827