Finite Group Databasebeta

← #11734515

Subgroups — #11734515

Order 896.

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

normal subgroups only

This group has 428 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#89751C211yesmaximal, characteristic, complemented
4#89749C211yesmaximal, characteristic, complemented
5#89723C212yesmaximal, complemented
6#89719C212yesmaximal, complemented
7#89717C212yesmaximal, complemented
8#89716C212yesmaximal, complemented
9#89422C212yesmaximal, complemented
10C22×C7×D4C2211yescharacteristic
11C22×Dic14C2211yescharacteristic
12C2×C7×C4⋊C4C2211yescharacteristic
13C2×C7×C22⋊C4C2212yes—
14C2×C7×C22⋊C4C2212yes—
15C2×C22⋊Dic7C2212yes—
16C2×C22⋊Dic7C2212yes—
17C2×C22⋊Dic7C2212yes—
18C2×C22⋊Dic7C2212yes—
19C2×C4⋊Dic7C2211yescharacteristic
20C2×Dic7⋊C4C2212yes—
21C2×Dic7⋊C4C2212yes—
22C2×C4×Dic7C2212yes—
23C2×C4×Dic7C2211yescharacteristic
24C7×#95C2211yescharacteristic
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2211yescharacteristic
30C7⋊#95C2212yes—
31#2785—71nomaximal, Sylow
32C24×C7C2312yes—
33C2×C7×D4—42no—
34C2×C7×D4—41no—
35C22×C4×C7C2311yescharacteristic
36C22×C4×C7C2312yes—
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2311yescharacteristic
39C22×Dic7C2312yes—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C2×Dic14—42no—
44C2×Dic14—41no—
45C7×C4⋊C4—41no—
46C7×C22⋊C4—24no—
47C7×C22⋊C4—42no—
48C22⋊Dic7—24no—
49C22⋊Dic7—24no—
50C22⋊Dic7—42no—
51C22⋊Dic7—42no—
52C4⋊Dic7—41no—
53Dic7⋊C4—42no—
54Dic7⋊C4—42no—
55C4×Dic7—22no—
56C4×Dic7—24no—
57#395—71no—
58#394—72no—
59#394D711yescharacteristic
60#393—72no—
61#388—72no—
62#388—72no—
63#388—72no—
64#382—71no—
65#382—72no—
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—
97Dic14—44no—
98C22×Q8—71no—
99C22×D4C2×D711yescharacteristic
100C2×C4⋊C4—72no—
101C2×C4⋊C4—72no—
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—72no—
111C2×C42—71no—
112#95—72no—
113#95—72no—
114#95—72no—
115#95—72no—
116#95—71no—
117#95—72no—
118#95C2×D711yescharacteristic
119C22×C7—42no—
120C22×C7—42no—
121C22×C7—42no—
122C22×C7—42no—
123C22×C7—42no—
124C22×C7—42no—
125C22×C7—42no—
126C22×C7C2×#4111yescharacteristic
127C22×C7C2×#4112yes—
128C22×C7H(5,𝔽2)11yescharacteristic
129C22×C7H(5,𝔽2)12yes—
130C22×C72−1+411yescharacteristic
131C4×C7—22no—
132C4×C7—42no—
133C4×C7—41no—
134C4×C7—41no—
135Dic7—24no—
136Dic7—42no—
137Dic7—42no—
138Dic7—42no—
139C24C22×D712yes—
140C2×Q8—282no—
141C2×Q8—281no—
142C2×D4—41no—
143C2×D4—42no—
144C22×C4—72no—
145C22×C4—72no—
146C22×C4—72no—
147C22×C4—72no—
148C22×C4C22×D711yescharacteristic
149C22×C4C22×D712yes—
150C22×C4C22×D711yescharacteristic
151C22×C4C22×D711yescharacteristic
152C4⋊C4—41no—
153C4⋊C4—282no—
154C4⋊C4—282no—
155C4⋊C4—281no—
156C22⋊C4—24no—
157C22⋊C4—42no—
158C22⋊C4—144no—
159C22⋊C4—144no—
160C22⋊C4—282no—
161C22⋊C4—282no—
162C42—144no—
163C42—142no—
164C2×C7—82no—
165C2×C7#53911yescharacteristic
166C2×C7#53912yes—
167C2×C7#55512yes—
168C2×C7#55511yescharacteristic
169C2×C7H(3,𝔽4)11yescharacteristic
170C23—22no—
171C23—22no—
172C23—22no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23C23×D711yescharacteristic
178Q8—284no—
179D4—44no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—74no—
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—142no—
200C2×C4—142no—
201C2×C4#115912yes—
202C7#278511yescharacteristic, complemented, Sylow
203C22—42no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22C2×#115911yescharacteristic
211C22#654811yescharacteristic
212C22#654812yes—
213C22D7×#4112yes—
214C22#655411yescharacteristic
215C4—22no—
216C4—41no—
217C4—41no—
218C4—42no—
219C4—144no—
220C4—282no—
221C4—282no—
222C4—282no—
223C2—82no—
224C2#9028511yescharacteristic
225C2#9029311yescharacteristic
226C2#9030712yes—
227C2#9035112yes—
228C2#9036111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
prime radicalC22×C7×D4C22
hypercenter#394D7
hypocenterC7#2785
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
3. #89751C2semidirect
4. #89749C2semidirect
5. #89723C2semidirect
6. #89719C2semidirect
7. #89717C2semidirect
8. #89716C2semidirect
9. #89422C2semidirect
202. C7#2785semidirect

Maximal subgroups

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

subgroupindex
C7×#3942
#899942
#897512
#897492
#897232
#897192
#897172
#897162
#894222
#27857