Finite Group Databasebeta

← #11734708

Subgroups — #11734708

Order 896.

592 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 149 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×#6509C212yesmaximal, complemented
2C2×#6459C212yesmaximal, complemented
3C7×#395C211yesmaximal, characteristic, complemented
4#89766C212yesmaximal, complemented
5#89760C211yesmaximal, characteristic
6#89751C211yesmaximal, characteristic, complemented
7#89448C212yesmaximal, complemented
8#89437C212yesmaximal
9#89419C212yesmaximal, complemented
10C22×C7×Q8C2211yescharacteristic
11C22×D28C2211yescharacteristic
12C22×C4×D7C2212yes—
13C22×Dic14C2211yescharacteristic
14C2×C7×C4⋊C4C2212yes—
15C2×C7×C4⋊C4C2211yescharacteristic
16#6509—28no—
17C2×(C2×D7)⋊C4C2212yes—
18C2×(C2×D7)⋊C4C2212yes—
19C2×(C2×D7)⋊C4C2212yes—
20C2×(C2×D7)⋊C4C2212yes—
21C2×(C2×D7)⋊C4C2212yes—
22C2×C4⋊Dic7C2212yes—
23C2×Dic7⋊C4C2212yes—
24C2×Dic7⋊C4C2212yes—
25C2×Dic7⋊C4C2212yes—
26C2×Dic7⋊C4C2212yes—
27C2×C4×Dic7C2211yescharacteristic
28#6459—28no—
29C7×#95C2212yes—
30C7×#95C2211yescharacteristic
31C7⋊#95C2212yes—
32C7⋊#95C2211yescharacteristic
33#2683—71nomaximal, Sylow
34C23×D7C2312yes—
35C2×C7×Q8—24no—
36C2×C7×Q8—41no—
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2312yes—
39C22×C4×C7C2311yescharacteristic
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×Dic7C2312yes—
43C22×Dic7C2312yes—
44C22×Dic7C2312yes—
45C2×D28—41no—
46C2×D28—41no—
47C2×D28Q814yescomplemented
48C2×C4×D7—24no—
49C2×C4×D7—24no—
50C2×C4×D7—42no—
51C2×Dic14—24no—
52C2×Dic14—41no—
53C7×C4⋊C4—24no—
54C7×C4⋊C4—41no—
55(C2×D7)⋊C4—24no—
56(C2×D7)⋊C4—24no—
57(C2×D7)⋊C4—24no—
58(C2×D7)⋊C4—24no—
59(C2×D7)⋊C4—42no—
60C4⋊Dic7—24no—
61Dic7⋊C4—24no—
62Dic7⋊C4—24no—
63Dic7⋊C4—42no—
64Dic7⋊C4D418yescomplemented
65C4×Dic7—41no—
66C2×C22⋊Q8—72no—
67C2×C22⋊Q8—72no—
68#396—72no—
69#396—72no—
70#395—71no—
71#395D711yescharacteristic, complemented
72#390—71no—
73#389—72no—
74#386—72no—
75C23×C7C2411yescharacteristic
76C22×D7—22no—
77C22×D7—22no—
78C22×D7—22no—
79C22×D7—22no—
80C22×D7—22no—
81C22×D7—22no—
82C22×D7C2×Q814yes—
83C7×Q8—44no—
84C2×C4×C7—22no—
85C2×C4×C7—21no—
86C2×C4×C7—21no—
87C2×C4×C7—22no—
88C2×C4×C7—22no—
89C2×C4×C7—22no—
90C2×C4×C7—22no—
91C2×C4×C7—22no—
92C2×C4×C7—21no—
93C2×C4×C7—21no—
94C2×C4×C7—22no—
95C2×C4×C7C2×D414yes—
96C2×C4×C7C2×Q812yes—
97C2×Dic7—22no—
98C2×Dic7—22no—
99C2×Dic7—22no—
100C2×Dic7—22no—
101C2×Dic7—22no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7C2×D414yes—
105C2×Dic7C2×D414yes—
106D28—44no—
107C4×D7—48no—
108Dic14—44no—
109C22×Q8—71no—
110C22×Q8C2×D711yescharacteristic
111C22×D4—71no—
112C23×C4—72no—
113C22⋊Q8—148no—
114C22⋊Q8—148no—
115C2×C4⋊C4—72no—
116C2×C4⋊C4—72no—
117C2×C4⋊C4—72no—
118C2×C4⋊C4—72no—
119C2×C4⋊C4—72no—
120C2×C4⋊C4C2×D712yes—
121C2×C4⋊C4C2×D711yescharacteristic
122C2×C22⋊C4—72no—
123C2×C22⋊C4—72no—
124C2×C22⋊C4—72no—
125C2×C22⋊C4—72no—
126C2×C22⋊C4—72no—
127C2×C42—71no—
128#95—72no—
129#95—71no—
130#95C2×D712yes—
131#95C2×D711yescharacteristic
132C22×C7C22×D412yes—
133C22×C7C22×Q811yescharacteristic
134C22×C7H(5,𝔽2)11yescharacteristic
135C22×C7H(5,𝔽2)11yescharacteristic
136C22×C72−1+412yes—
137C2×D7—24no—
138C2×D7—24no—
139C2×D7—24no—
140C2×D7—42no—
141C2×D7—42no—
142C2×D7—42no—
143C2×D7—42no—
144C4×C7—22no—
145C4×C7—24no—
146C4×C7—42no—
147C4×C7—41no—
148C4×C7—41no—
149Dic7—24no—
150Dic7—24no—
151Dic7—42no—
152C24—72no—
153C2×Q8—24no—
154C2×Q8—41no—
155C2×Q8—144no—
156C2×Q8—281no—
157C2×D4—74no—
158C2×D4—281no—
159C2×D4—281no—
160C22×C4—72no—
161C22×C4—72no—
162C22×C4—72no—
163C22×C4—144no—
164C22×C4—144no—
165C22×C4—282no—
166C22×C4C22×D711yescharacteristic
167C22×C4C22×D712yes—
168C22×C4C22×D711yescharacteristic
169C22×C4C22×D712yes—
170C22×C4C22×D711yescharacteristic
171C4⋊C4—24no—
172C4⋊C4—41no—
173C4⋊C4—78no—
174C4⋊C4—144no—
175C4⋊C4—144no—
176C4⋊C4—144no—
177C4⋊C4—282no—
178C22⋊C4—144no—
179C22⋊C4—144no—
180C22⋊C4—144no—
181C22⋊C4—144no—
182C22⋊C4—282no—
183C42—281no—
184C2×C7#53712yes—
185C2×C7#54311yescharacteristic
186C2×C7D4211yescharacteristic
187C2×C7D4×Q812yes—
188C2×C7#56311yescharacteristic
189D7—44no—
190C23—74no—
191C23—142no—
192C23—142no—
193C23—142no—
194C23—142no—
195C23—142no—
196C23—142no—
197C23C23×D711yescharacteristic
198Q8—44no—
199Q8—284no—
200D4—284no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—74no—
213C2×C4—74no—
214C2×C4—142no—
215C2×C4—142no—
216C2×C4—142no—
217C2×C4—142no—
218C2×C4—142no—
219C2×C4—142no—
220C2×C4—142no—
221C2×C4—288no—
222C2×C4D4×D714yes—
223C2×C4Q8×D712yes—
224C7#268311yescharacteristic, complemented, Sylow
225C22—144no—
226C22—144no—
227C22—144no—
228C22—282no—
229C22—282no—
230C22—282no—
231C22—282no—
232C22C2×D4×D712yes—
233C22#654811yescharacteristic
234C22C2×Q8×D711yescharacteristic
235C22#655112yes—
236C22#655311yescharacteristic
237C4—24no—
238C4—22no—
239C4—41no—
240C4—41no—
241C4—42no—
242C4—144no—
243C4—144no—
244C4—282no—
245C2—284no—
246C2#9028111yescharacteristic
247C2#9032012yes—
248C2#9032511yescharacteristic
249C2#9041812yes—
250C2#9042611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#395C2
prime radicalC22×D28C22
hypercenter#395D7
hypocenterC7#2683
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×#6509C2semidirect
2. C2×#6459C2semidirect
3. C7×#395C2semidirect
4. #89766C2semidirect
6. #89751C2semidirect
7. #89448C2semidirect
9. #89419C2semidirect
47. C2×D28Q8semidirect
64. Dic7⋊C4D4semidirect
71. #395D7semidirect
224. C7#2683semidirect

Maximal subgroups

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

subgroupindex
C2×#65092
C2×#64592
C7×#3952
#897662
#897602
#897512
#894482
#894372
#894192
#26837