Finite Group Databasebeta

← #11735796

Subgroups — #11735796

Order 896.

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

normal subgroups only

This group has 546 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
1C2×#6457C211yesmaximal, characteristic
2C2×#6455C211yesmaximal, characteristic, complemented
3C7×#400C211yesmaximal, characteristic, complemented
4#89764C211yesmaximal, characteristic, complemented
5#89755C211yesmaximal, characteristic, complemented
6#89446C211yesmaximal, characteristic, complemented
7#89441C212yesmaximal, complemented
8#89439C212yesmaximal, complemented
9#89437C212yesmaximal
10D7×#95C211yesmaximal, characteristic, complemented
11#89424C212yesmaximal, complemented
12C22×D28C2211yescharacteristic
13C22×C4×D7C2212yes—
14C22×C4×D7C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×(C2×D7)⋊C4C2212yes—
18C2×(C2×D7)⋊C4C2211yescharacteristic
19C2×(C2×D7)⋊C4C2211yescharacteristic
20C2×(C2×D7)⋊C4C2211yescharacteristic
21C2×(C2×D7)⋊C4C2212yes—
22C2×(C2×D7)⋊C4C2212yes—
23C2×C4⋊Dic7C2211yescharacteristic
24C2×Dic7⋊C4C2211yescharacteristic
25C2×Dic7⋊C4C2212yes—
26C2×Dic7⋊C4C2212yes—
27C2×C4×Dic7C2212yes—
28#6457—24no—
29#6455—24no—
30C7×#95C2212yes—
31C7×#95C2211yescharacteristic
32C7×#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2211yescharacteristic
37#2498—71nomaximal, Sylow
38C23×D7C2311yescharacteristic
39C23×D7C2311yescharacteristic
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2312yes—
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C22×Dic7C2312yes—
46C22×Dic7C2311yescharacteristic
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2312yes—
49C2×D28—22no—
50C2×D28—24no—
51C2×C4×D7—24no—
52C2×C4×D7—24no—
53C2×C4×D7—24no—
54C2×C4×D7—42no—
55C2×C4×D7D414yes—
56C7×C4⋊C4—22no—
57C7×C4⋊C4—22no—
58(C2×D7)⋊C4—24no—
59(C2×D7)⋊C4—22no—
60(C2×D7)⋊C4—22no—
61(C2×D7)⋊C4—22no—
62(C2×D7)⋊C4—24no—
63(C2×D7)⋊C4—42no—
64C4⋊Dic7—22no—
65Dic7⋊C4—22no—
66Dic7⋊C4—24no—
67Dic7⋊C4—42no—
68C4×Dic7—24no—
69C2×#123—71no—
70C2×#117—71no—
71#400D711yescharacteristic, complemented
72#399—71no—
73#398—71no—
74#397—71no—
75#388—72no—
76#387—72no—
77#386—72no—
78#380—72no—
79C2×#95—71no—
80C23×C7C2411yescharacteristic
81C22×D7—22no—
82C22×D7—21no—
83C22×D7—21no—
84C22×D7—21no—
85C22×D7—22no—
86C22×D7—22no—
87C22×D7—21no—
88C22×D7—21no—
89C22×D7C2×D412yes—
90C22×D7#4114yes—
91C2×C4×C7—22no—
92C2×C4×C7—22no—
93C2×C4×C7—21no—
94C2×C4×C7—21no—
95C2×C4×C7—22no—
96C2×C4×C7—22no—
97C2×C4×C7—22no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—21no—
101C2×C4×C7—22no—
102C2×C4×C7—21no—
103C2×C4×C7C2×D412yes—
104C2×Dic7—22no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7—22no—
108C2×Dic7—21no—
109C2×Dic7—22no—
110C2×Dic7—22no—
111C2×Dic7—22no—
112C2×Dic7C2×D412yes—
113C2×Dic7#4114yes—
114D28—44no—
115C4×D7—28no—
116C4×D7—48no—
117C22×D4—71no—
118C23×C4—71no—
119C23×C4—72no—
120#123—144no—
121#117—144no—
122C2×C4⋊C4—71no—
123C2×C4⋊C4—71no—
124C2×C4⋊C4—72no—
125C2×C4⋊C4—72no—
126C2×C4⋊C4C2×D711yescharacteristic
127C2×C4⋊C4C2×D711yescharacteristic
128C2×C22⋊C4—72no—
129C2×C22⋊C4—71no—
130C2×C22⋊C4—71no—
131C2×C22⋊C4—71no—
132C2×C22⋊C4—72no—
133C2×C22⋊C4—72no—
134C2×C42—72no—
135#95—72no—
136#95—71no—
137#95—72no—
138#95—72no—
139#95C2×D711yescharacteristic
140#95C2×D712yes—
141#95C2×D712yes—
142C22×C7C22×D411yescharacteristic
143C22×C7C2×#4112yes—
144C22×C7C2×#4112yes—
145C22×C7H(5,𝔽2)11yescharacteristic
146C22×C72−1+411yescharacteristic
147C2×D7—22no—
148C2×D7—22no—
149C2×D7—24no—
150C2×D7—24no—
151C2×D7—41no—
152C2×D7—41no—
153C2×D7—41no—
154C2×D7—42no—
155C2×D7—42no—
156C2×D7#12314yes—
157C4×C7—22no—
158C4×C7—41no—
159C4×C7—42no—
160C4×C7—41no—
161C4×C7—42no—
162Dic7—22no—
163Dic7—24no—
164Dic7—41no—
165Dic7—42no—
166C24—71no—
167C24—71no—
168C2×D4—142no—
169C2×D4—144no—
170C22×C4—72no—
171C22×C4—71no—
172C22×C4—74no—
173C22×C4—72no—
174C22×C4—71no—
175C22×C4—144no—
176C22×C4—144no—
177C22×C4—144no—
178C22×C4—282no—
179C22×C4C22×D712yes—
180C22×C4C22×D711yescharacteristic
181C22×C4C22×D712yes—
182C22×C4C22×D711yescharacteristic
183C22×C4C22×D711yescharacteristic
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—144no—
187C4⋊C4—142no—
188C4⋊C4—142no—
189C4⋊C4—282no—
190C22⋊C4—144no—
191C22⋊C4—142no—
192C22⋊C4—144no—
193C22⋊C4—142no—
194C22⋊C4—142no—
195C22⋊C4—282no—
196C42—144no—
197C2×C7C2×#12311yescharacteristic
198C2×C7#53211yescharacteristic
199C2×C7#53711yescharacteristic
200C2×C7#55112yes—
201C2×C7#55212yes—
202D7—24no—
203D7—81no—
204C23—72no—
205C23—74no—
206C23—141no—
207C23—142no—
208C23—141no—
209C23—141no—
210C23—141no—
211C23—142no—
212C23—142no—
213C23—141no—
214C23C23×D711yescharacteristic
215D4—284no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—72no—
229C2×C4—74no—
230C2×C4—148no—
231C2×C4—142no—
232C2×C4—142no—
233C2×C4—142no—
234C2×C4—142no—
235C2×C4—141no—
236C2×C4—142no—
237C2×C4—142no—
238C2×C4—142no—
239C2×C4—288no—
240C2×C4D4×D712yes—
241C7#249811yescharacteristic, complemented, Sylow
242C22—74no—
243C22—144no—
244C22—144no—
245C22—142no—
246C22—142no—
247C22—282no—
248C22—282no—
249C22—281no—
250C22—281no—
251C22—281no—
252C22C2×D4×D711yescharacteristic
253C22#655111yescharacteristic
254C22D7×#4112yes—
255C22D7×#4112yes—
256C22#655311yescharacteristic
257C4—22no—
258C4—42no—
259C4—41no—
260C4—41no—
261C4—42no—
262C4—144no—
263C4—142no—
264C4—282no—
265C4—281no—
266C2—144no—
267C2—561no—
268C2#9032011yescharacteristic
269C2D7×#12311yescharacteristic
270C2#9035911yescharacteristic
271C2#9038412yes—
272C2#9039612yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#400C2
prime radicalC22×D28C22
hypercenter#400D7
hypocenterC7#2498
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×#6455C2semidirect
3. C7×#400C2semidirect
4. #89764C2semidirect
5. #89755C2semidirect
6. #89446C2semidirect
7. #89441C2semidirect
8. #89439C2semidirect
10. D7×#95C2semidirect
11. #89424C2semidirect
71. #400D7semidirect
241. C7#2498semidirect

Maximal subgroups

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

subgroupindex
C2×#64572
C2×#64552
C7×#4002
#897642
#897552
#894462
#894412
#894392
#894372
D7×#952
#894242
#24987