Finite Group Databasebeta

← #11733859

Subgroups — #11733859

Order 896.

472 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 137 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×#390C211yesmaximal, characteristic
2#89992C212yesmaximal
3#89758C211yesmaximal, characteristic, complemented
4#89751C211yesmaximal, characteristic, complemented
5#89720C212yesmaximal, complemented
6#89719C212yesmaximal, complemented
7#89718C212yesmaximal, complemented
8#89714C212yesmaximal, complemented
9#89705C212yesmaximal, complemented
10C22×C7×D4C2211yescharacteristic
11C23×Dic7C2212yes—
12C22×Dic14C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C22⋊C4C2212yes—
15C2×C7×C22⋊C4C2212yes—
16C2×C42×C7C2211yescharacteristic
17C2×C22⋊Dic7C2212yes—
18C2×C22⋊Dic7C2212yes—
19C2×C22⋊Dic7C2212yes—
20C2×C4⋊Dic7C2211yescharacteristic
21C2×Dic7⋊C4C2212yes—
22C2×Dic7⋊C4C2212yes—
23C2×Dic7⋊C4C2212yes—
24C7⋊#95C2212yes—
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2211yescharacteristic
29C7⋊#95C2211yescharacteristic
30C7⋊#95C2212yes—
31#2784—71nomaximal, Sylow
32C24×C7C2312yes—
33C2×C7×D4—41no—
34C2×C7×D4—41no—
35C2×C7×D4Q814yescomplemented
36C22×C4×C7C2311yescharacteristic
37C22×C4×C7C2312yes—
38C22×C4×C7C2311yescharacteristic
39C22×C4×C7C2311yescharacteristic
40C22×Dic7—24no—
41C22×Dic7—42no—
42C22×Dic7—42no—
43C22×Dic7C2312yes—
44C22×Dic7C2312yes—
45C22×Dic7C2312yes—
46C22×Dic7C2312yes—
47C2×Dic14—42no—
48C2×Dic14—41no—
49C7×C4⋊C4—41no—
50C7×C22⋊C4—42no—
51C7×C22⋊C4—42no—
52C42×C7—41no—
53C22⋊Dic7—24no—
54C22⋊Dic7—42no—
55C22⋊Dic7—42no—
56C4⋊Dic7—41no—
57Dic7⋊C4—24no—
58Dic7⋊C4—42no—
59Dic7⋊C4—42no—
60#400—71no—
61#396—72no—
62#395—71no—
63#393—72no—
64#390D711yescharacteristic
65#387—72no—
66#386—72no—
67#385—72no—
68#380—72no—
69C23×C7—22no—
70C23×C7—22no—
71C23×C7—22no—
72C23×C7—22no—
73C23×C7—22no—
74C23×C7—22no—
75C23×C7C2×Q814yes—
76C23×C7C2411yescharacteristic
77C7×D4—44no—
78C2×C4×C7—21no—
79C2×C4×C7—21no—
80C2×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7—22no—
83C2×C4×C7—21no—
84C2×C4×C7—22no—
85C2×C4×C7—22no—
86C2×C4×C7—21no—
87C2×C4×C7C2×Q812yes—
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—22no—
97C2×Dic7—22no—
98C2×Dic7—22no—
99C2×Dic7—48no—
100C2×Dic7#4114yes—
101Dic14—44no—
102C22×Q8—71no—
103C22×D4C2×D711yescharacteristic
104C23×C4—72no—
105C2×C4⋊C4—72no—
106C2×C4⋊C4—72no—
107C2×C4⋊C4—72no—
108C2×C4⋊C4—71no—
109C2×C4⋊C4C2×D711yescharacteristic
110C2×C22⋊C4—72no—
111C2×C22⋊C4—72no—
112C2×C22⋊C4—72no—
113C2×C22⋊C4C2×D712yes—
114C2×C22⋊C4C2×D712yes—
115C2×C42C2×D711yescharacteristic
116#95—71no—
117#95—72no—
118#95—71no—
119#95—72no—
120#95—72no—
121#95—72no—
122#95—72no—
123C22×C7—24no—
124C22×C7—24no—
125C22×C7—24no—
126C22×C7—42no—
127C22×C7—42no—
128C22×C7—42no—
129C22×C7—42no—
130C22×C7C22×Q811yescharacteristic
131C22×C7C2×#4112yes—
132C22×C7H(5,𝔽2)11yescharacteristic
133C22×C7H(5,𝔽2)12yes—
134C22×C7H(5,𝔽2)11yescharacteristic
135C4×C7—22no—
136C4×C7—42no—
137C4×C7—41no—
138C4×C7—41no—
139Dic7—24no—
140Dic7—42no—
141Dic7—42no—
142Dic7—42no—
143C24C22×D712yes—
144C2×Q8—282no—
145C2×Q8—281no—
146C2×D4—41no—
147C2×D4—41no—
148C2×D4Dic1414yescomplemented
149C22×C4—72no—
150C22×C4—72no—
151C22×C4—72no—
152C22×C4—72no—
153C22×C4—144no—
154C22×C4—282no—
155C22×C4—282no—
156C22×C4C22×D711yescharacteristic
157C22×C4C22×D712yes—
158C22×C4C22×D711yescharacteristic
159C22×C4C22×D711yescharacteristic
160C4⋊C4—41no—
161C4⋊C4—144no—
162C4⋊C4—282no—
163C4⋊C4—282no—
164C4⋊C4—281no—
165C22⋊C4—42no—
166C22⋊C4—42no—
167C22⋊C4—144no—
168C22⋊C4—282no—
169C22⋊C4—282no—
170C42—41no—
171C2×C7—44no—
172C2×C7#53812yes—
173C2×C7#54311yescharacteristic
174C2×C7#55412yes—
175C2×C7#55511yescharacteristic
176C2×C7#56011yescharacteristic
177C23—22no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23C2×Dic1414yes—
184C23C23×D711yescharacteristic
185Q8—284no—
186D4—44no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—74no—
197C2×C4—142no—
198C2×C4—142no—
199C2×C4—142no—
200C2×C4—142no—
201C2×C4—142no—
202C2×C4—142no—
203C2×C4—142no—
204C2×C4—142no—
205C2×C4—142no—
206C2×C4—142no—
207C2×C4—142no—
208C2×C4—288no—
209C2×C4C2×Dic1412yes—
210C7#278411yescharacteristic, complemented, Sylow
211C22—24no—
212C22—24no—
213C22—24no—
214C22—42no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22C22×Dic1411yescharacteristic
219C22#654811yescharacteristic
220C22#654812yes—
221C22#654811yescharacteristic
222C22D7×#4112yes—
223C4—22no—
224C4—41no—
225C4—41no—
226C4—42no—
227C4—144no—
228C4—282no—
229C4—282no—
230C4—282no—
231C2—44no—
232C2#9017411yescharacteristic
233C2#9023012yes—
234C2#9030412yes—
235C2#9036111yescharacteristic
236C2#9041111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#390C2
prime radicalC22×C7×D4C22
hypercenter#390D7
hypocenterC7#2784
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
3. #89758C2semidirect
4. #89751C2semidirect
5. #89720C2semidirect
6. #89719C2semidirect
7. #89718C2semidirect
8. #89714C2semidirect
9. #89705C2semidirect
35. C2×C7×D4Q8semidirect
148. C2×D4Dic14semidirect
210. C7#2784semidirect

Maximal subgroups

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

subgroupindex
C7×#3902
#899922
#897582
#897512
#897202
#897192
#897182
#897142
#897052
#27847