Finite Group Databasebeta

← #11734717

Subgroups — #11734717

Order 896.

472 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 133 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×#395C211yesmaximal, characteristic, complemented
2#90003C212yesmaximal, complemented
3#89766C212yesmaximal, complemented
4#89760C211yesmaximal, characteristic
5#89759C212yesmaximal, complemented
6#89758C211yesmaximal, characteristic, complemented
7#89440C212yesmaximal, complemented
8#89437C212yesmaximal
9#89427C212yesmaximal, complemented
10C22×C7×Q8C2211yescharacteristic
11C22×D28C2211yescharacteristic
12C22×C4×D7C2212yes—
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C4⋊C4C2212yes—
15C2×(C2×D7)⋊C4C2212yes—
16C2×(C2×D7)⋊C4C2212yes—
17C2×(C2×D7)⋊C4C2212yes—
18C2×(C2×D7)⋊C4C2212yes—
19C2×(C2×D7)⋊C4C2212yes—
20C2×C4⋊Dic7C2212yes—
21C2×C4⋊Dic7C2211yescharacteristic
22C2×Dic7⋊C4C2212yes—
23C2×C4×Dic7C2211yescharacteristic
24C7×#95C2212yes—
25C7×#95C2211yescharacteristic
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2211yescharacteristic
31#2784—71nomaximal, Sylow
32C23×D7C2312yes—
33C2×C7×Q8—42no—
34C2×C7×Q8—41no—
35C22×C4×C7C2312yes—
36C22×C4×C7C2311yescharacteristic
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2311yescharacteristic
39C22×C4×C7C2312yes—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C2×D28—41no—
44C2×D28—41no—
45C2×D28Q814yescomplemented
46C2×C4×D7—24no—
47C2×C4×D7—42no—
48C2×C4×D7—42no—
49C7×C4⋊C4—42no—
50C7×C4⋊C4—41no—
51(C2×D7)⋊C4—24no—
52(C2×D7)⋊C4—42no—
53(C2×D7)⋊C4—42no—
54(C2×D7)⋊C4—42no—
55(C2×D7)⋊C4—42no—
56C4⋊Dic7—42no—
57C4⋊Dic7—41no—
58Dic7⋊C4—24no—
59C4×Dic7—41no—
60#400—71no—
61#396—72no—
62#395D711yescharacteristic, complemented
63#393—72no—
64#390—71no—
65#387—72no—
66#386—72no—
67#385—72no—
68#380—72no—
69C23×C7C2411yescharacteristic
70C22×D7—22no—
71C22×D7—22no—
72C22×D7—22no—
73C22×D7—22no—
74C22×D7—22no—
75C22×D7—22no—
76C22×D7C2×Q814yes—
77C7×Q8—44no—
78C2×C4×C7—21no—
79C2×C4×C7—22no—
80C2×C4×C7—22no—
81C2×C4×C7—21no—
82C2×C4×C7—21no—
83C2×C4×C7—22no—
84C2×C4×C7—22no—
85C2×C4×C7—22no—
86C2×C4×C7—22no—
87C2×C4×C7—22no—
88C2×C4×C7—21no—
89C2×C4×C7C2×Q812yes—
90C2×C4×C7#4114yes—
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—22no—
100D28—44no—
101C4×D7—48no—
102C22×Q8C2×D711yescharacteristic
103C22×D4—71no—
104C23×C4—72no—
105C2×C4⋊C4—72no—
106C2×C4⋊C4—72no—
107C2×C4⋊C4—71no—
108C2×C4⋊C4C2×D712yes—
109C2×C4⋊C4C2×D711yescharacteristic
110C2×C22⋊C4—72no—
111C2×C22⋊C4—72no—
112C2×C22⋊C4—72no—
113C2×C22⋊C4—72no—
114C2×C22⋊C4—72no—
115C2×C42—71no—
116#95—71no—
117#95—72no—
118#95—72no—
119#95—72no—
120#95—72no—
121#95C2×D711yescharacteristic
122#95C2×D712yes—
123C22×C7C22×Q811yescharacteristic
124C22×C7C2×#4112yes—
125C22×C7H(5,𝔽2)12yes—
126C22×C7H(5,𝔽2)11yescharacteristic
127C22×C7H(5,𝔽2)11yescharacteristic
128C2×D7—24no—
129C2×D7—24no—
130C2×D7—24no—
131C2×D7—42no—
132C2×D7—42no—
133C2×D7—42no—
134C2×D7—42no—
135C4×C7—24no—
136C4×C7—22no—
137C4×C7—41no—
138C4×C7—41no—
139C4×C7—42no—
140Dic7—42no—
141Dic7—42no—
142Dic7—42no—
143C24—72no—
144C2×Q8—42no—
145C2×Q8—41no—
146C2×D4—74no—
147C2×D4—281no—
148C2×D4—281no—
149C22×C4—72no—
150C22×C4—72no—
151C22×C4—72no—
152C22×C4—144no—
153C22×C4—282no—
154C22×C4—282no—
155C22×C4C22×D712yes—
156C22×C4C22×D711yescharacteristic
157C22×C4C22×D711yescharacteristic
158C22×C4C22×D711yescharacteristic
159C22×C4C22×D712yes—
160C4⋊C4—42no—
161C4⋊C4—41no—
162C4⋊C4—144no—
163C4⋊C4—282no—
164C4⋊C4—281no—
165C22⋊C4—144no—
166C22⋊C4—282no—
167C22⋊C4—282no—
168C22⋊C4—282no—
169C22⋊C4—282no—
170C42—281no—
171C2×C7#53812yes—
172C2×C7#54311yescharacteristic
173C2×C7#55412yes—
174C2×C7#55511yescharacteristic
175C2×C7#56011yescharacteristic
176D7—44no—
177C23—74no—
178C23—142no—
179C23—142no—
180C23—142no—
181C23—142no—
182C23—142no—
183C23—142no—
184C23C23×D711yescharacteristic
185Q8—44no—
186D4—284no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—21no—
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—288no—
208C2×C4Q8×D712yes—
209C2×C4#116114yes—
210C7#278411yescharacteristic, complemented, Sylow
211C22—144no—
212C22—144no—
213C22—144no—
214C22—282no—
215C22—282no—
216C22—282no—
217C22—282no—
218C22#654812yes—
219C22#654811yescharacteristic
220C22C2×Q8×D711yescharacteristic
221C22C2×#116112yes—
222C22#655311yescharacteristic
223C4—22no—
224C4—24no—
225C4—41no—
226C4—42no—
227C4—41no—
228C4—282no—
229C4—282no—
230C4—282no—
231C2—284no—
232C2#9028611yescharacteristic
233C2#9032511yescharacteristic
234C2#9033512yes—
235C2#9042112yes—
236C2#9042511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#395C2
prime radicalC22×D28C22
hypercenter#395D7
hypocenterC7#2784
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C7×#395C2semidirect
2. #90003C2semidirect
3. #89766C2semidirect
5. #89759C2semidirect
6. #89758C2semidirect
7. #89440C2semidirect
9. #89427C2semidirect
45. C2×D28Q8semidirect
62. #395D7semidirect
210. C7#2784semidirect

Maximal subgroups

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

subgroupindex
C7×#3952
#900032
#897662
#897602
#897592
#897582
#894402
#894372
#894272
#27847