Finite Group Databasebeta

← #11734517

Subgroups — #11734517

Order 896.

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

normal subgroups only

This group has 420 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×#394C211yesmaximal, characteristic
2#89994C212yesmaximal
3#89758C211yesmaximal, characteristic, complemented
4#89748C211yesmaximal, characteristic, complemented
5#89723C212yesmaximal, complemented
6#89720C212yesmaximal, complemented
7#89716C212yesmaximal, complemented
8#89716C212yesmaximal, complemented
9#89424C212yesmaximal, complemented
10C22×C7×D4C2211yescharacteristic
11C2×C7×C4⋊C4C2211yescharacteristic
12C2×C7×C22⋊C4C2212yes—
13C2×C7×C22⋊C4C2212yes—
14C2×C22⋊Dic7C2212yes—
15C2×C22⋊Dic7C2212yes—
16C2×C22⋊Dic7C2212yes—
17C2×C22⋊Dic7C2212yes—
18C2×C4⋊Dic7C2211yescharacteristic
19C2×C4⋊Dic7C2211yescharacteristic
20C2×Dic7⋊C4C2212yes—
21C2×Dic7⋊C4C2212yes—
22C2×Dic7⋊C4C2212yes—
23C2×C4×Dic7C2211yescharacteristic
24C2×C4×Dic7C2212yes—
25C7×#95C2211yescharacteristic
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2211yescharacteristic
30C7⋊#95C2212yes—
31#2795—71nomaximal, Sylow
32C24×C7C2312yes—
33C2×C7×D4—42no—
34C2×C7×D4—41no—
35C22×C4×C7C2312yes—
36C22×C4×C7C2311yescharacteristic
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2311yescharacteristic
39C22×Dic7C2312yes—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C7×C4⋊C4—41no—
44C7×C22⋊C4—24no—
45C7×C22⋊C4—42no—
46C22⋊Dic7—24no—
47C22⋊Dic7—24no—
48C22⋊Dic7—42no—
49C22⋊Dic7—42no—
50C4⋊Dic7—41no—
51C4⋊Dic7—41no—
52Dic7⋊C4—24no—
53Dic7⋊C4—42no—
54Dic7⋊C4—42no—
55C4×Dic7—22no—
56C4×Dic7—42no—
57#400—71no—
58#396—72no—
59#394—72no—
60#394D711yescharacteristic
61#388—72no—
62#388—72no—
63#388—72no—
64#387—72no—
65#384—71no—
66C23×C7—22no—
67C23×C7—22no—
68C23×C7—22no—
69C23×C7—22no—
70C23×C7—22no—
71C23×C7—22no—
72C23×C7—22no—
73C23×C7C2411yescharacteristic
74C7×D4—44no—
75C2×C4×C7—21no—
76C2×C4×C7—21no—
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—21no—
84C2×C4×C7#4112yes—
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—22no—
95C2×Dic7—22no—
96C2×Dic7#4114yes—
97C22×D4C2×D711yescharacteristic
98C2×C4⋊C4—72no—
99C2×C4⋊C4—72no—
100C2×C4⋊C4—72no—
101C2×C4⋊C4—71no—
102C2×C4⋊C4—71no—
103C2×C4⋊C4C2×D711yescharacteristic
104C2×C22⋊C4—72no—
105C2×C22⋊C4—72no—
106C2×C22⋊C4—72no—
107C2×C22⋊C4—72no—
108C2×C22⋊C4C2×D712yes—
109C2×C22⋊C4C2×D712yes—
110C2×C42—71no—
111C2×C42—72no—
112#95—72no—
113#95—72no—
114#95—72no—
115#95—71no—
116#95—72no—
117#95C2×D711yescharacteristic
118C22×C7—42no—
119C22×C7—42no—
120C22×C7—42no—
121C22×C7—42no—
122C22×C7—42no—
123C22×C7—42no—
124C22×C7—42no—
125C22×C7C2×#4111yescharacteristic
126C22×C7C2×#4112yes—
127C22×C7H(5,𝔽2)11yescharacteristic
128C22×C7H(5,𝔽2)12yes—
129C22×C72−1+411yescharacteristic
130C4×C7—22no—
131C4×C7—42no—
132C4×C7—41no—
133C4×C7—41no—
134Dic7—24no—
135Dic7—42no—
136Dic7—42no—
137Dic7—42no—
138C24C22×D712yes—
139C2×D4—41no—
140C2×D4—42no—
141C22×C4—72no—
142C22×C4—72no—
143C22×C4—72no—
144C22×C4—72no—
145C22×C4C22×D712yes—
146C22×C4C22×D711yescharacteristic
147C22×C4C22×D711yescharacteristic
148C22×C4C22×D711yescharacteristic
149C4⋊C4—41no—
150C4⋊C4—144no—
151C4⋊C4—282no—
152C4⋊C4—282no—
153C4⋊C4—281no—
154C4⋊C4—281no—
155C22⋊C4—24no—
156C22⋊C4—42no—
157C22⋊C4—144no—
158C22⋊C4—144no—
159C22⋊C4—282no—
160C22⋊C4—282no—
161C42—142no—
162C42—282no—
163C2×C7—82no—
164C2×C7#53911yescharacteristic
165C2×C7#54212yes—
166C2×C7#55312yes—
167C2×C7#55511yescharacteristic
168C2×C7#56011yescharacteristic
169C23—22no—
170C23—22no—
171C23—22no—
172C23—22no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23C23×D711yescharacteristic
177D4—44no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—74no—
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—142no—
198C2×C4—142no—
199C2×C4#115912yes—
200C7#279511yescharacteristic, complemented, Sylow
201C22—42no—
202C22—42no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22C2×#115911yescharacteristic
209C22#654811yescharacteristic
210C22#654812yes—
211C22D7×#4112yes—
212C22#655411yescharacteristic
213C4—22no—
214C4—41no—
215C4—41no—
216C4—42no—
217C4—144no—
218C4—282no—
219C4—282no—
220C4—282no—
221C2—82no—
222C2#9028611yescharacteristic
223C2#9029311yescharacteristic
224C2#9030512yes—
225C2#9035312yes—
226C2#9036111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
prime radicalC22×C7×D4C22
hypercenter#394D7
hypocenterC7#2795
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
3. #89758C2semidirect
4. #89748C2semidirect
5. #89723C2semidirect
6. #89720C2semidirect
7. #89716C2semidirect
8. #89716C2semidirect
9. #89424C2semidirect
200. C7#2795semidirect

Maximal subgroups

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

subgroupindex
C7×#3942
#899942
#897582
#897482
#897232
#897202
#897162
#894242
#27957