Finite Group Databasebeta

← #11735489

Subgroups — #11735489

Order 896.

604 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 135 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×#6456C211yesmaximal, characteristic, complemented
2C2×#6440C211yesmaximal, characteristic, complemented
3C7×#399C211yesmaximal, characteristic, complemented
4#89762C212yesmaximal, complemented
5#89761C212yesmaximal, complemented
6#89754C211yesmaximal, characteristic, complemented
7#89732C211yesmaximal, characteristic
8#89730C212yesmaximal, complemented
9#89728C212yesmaximal, complemented
10#89444C211yesmaximal, characteristic, complemented
11#89423C211yesmaximal, characteristic, complemented
12C22×C22⋊D7C2212yes—
13C23×Dic7C2211yescharacteristic
14C22×D28C2211yescharacteristic
15C22×C4×D7C2212yes—
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C4⋊C4C2212yes—
18C2×C7×C22⋊C4C2211yescharacteristic
19C2×C7×C22⋊C4C2212yes—
20C2×C22⋊Dic7C2211yescharacteristic
21C2×(C2×D7)⋊C4C2212yes—
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×(C2×D7)⋊C4C2212yes—
25C2×Dic7⋊C4C2211yescharacteristic
26C2×Dic7⋊C4C2211yescharacteristic
27C2×C4×Dic7C2211yescharacteristic
28C2×C4×Dic7C2211yescharacteristic
29#6456—24no—
30#6440—24no—
31C7×#95C2211yescharacteristic
32C7⋊#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2211yescharacteristic
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2598—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C23×D7C2312yes—
40C22×C4×C7C2312yes—
41C22×C4×C7C2312yes—
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2311yescharacteristic
44C2×C22⋊D7—24no—
45C2×C22⋊D7—24no—
46C2×C22⋊D7—42no—
47C22×Dic7—22no—
48C22×Dic7—22no—
49C22×Dic7—22no—
50C22×Dic7C2312yes—
51C22×Dic7C2311yescharacteristic
52C22×Dic7C2311yescharacteristic
53C22×Dic7C2311yescharacteristic
54C22×Dic7C2311yescharacteristic
55C2×D28—24no—
56C2×D28—22no—
57C2×C4×D7—24no—
58C2×C4×D7—42no—
59C2×C4×D7—42no—
60C7×C4⋊C4—22no—
61C7×C4⋊C4—42no—
62C7×C22⋊C4—22no—
63C7×C22⋊C4—42no—
64C22⋊Dic7—22no—
65(C2×D7)⋊C4—24no—
66(C2×D7)⋊C4—24no—
67(C2×D7)⋊C4—24no—
68(C2×D7)⋊C4—42no—
69Dic7⋊C4—22no—
70Dic7⋊C4—22no—
71C4×Dic7—22no—
72C4×Dic7D414yescomplemented
73C2×C4×D4—71no—
74C2×#117—71no—
75#400—71no—
76#399D711yescharacteristic, complemented
77#394—71no—
78#392—71no—
79#388—72no—
80#386—72no—
81#386—72no—
82#385—72no—
83#383—71no—
84C23×C7—21no—
85C23×C7—21no—
86C23×C7—22no—
87C23×C7—22no—
88C23×C7#4112yes—
89C23×C7C2411yescharacteristic
90C22×D7—22no—
91C22×D7—22no—
92C22×D7—22no—
93C22×D7—22no—
94C22×D7—22no—
95C22×D7—22no—
96C22×D7#4114yes—
97C2×C4×C7—22no—
98C2×C4×C7—21no—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7—22no—
104C2×C4×C7—22no—
105C2×C4×C7—22no—
106C2×C4×C7C2×D412yes—
107C22⋊D7—48no—
108C2×Dic7—28no—
109C2×Dic7—21no—
110C2×Dic7—21no—
111C2×Dic7—21no—
112C2×Dic7—21no—
113C2×Dic7—22no—
114C2×Dic7—21no—
115C2×Dic7—22no—
116C2×Dic7—22no—
117C2×Dic7—22no—
118C2×Dic7—22no—
119C2×Dic7C2×D412yes—
120C2×Dic7C2×D412yes—
121C2×Dic7#4112yes—
122D28—28no—
123C4×D7—48no—
124C22×D4—72no—
125C22×D4—71no—
126C23×C4—71no—
127C23×C4—72no—
128C4×D4—144no—
129#117—144no—
130C2×C4⋊C4—71no—
131C2×C4⋊C4—71no—
132C2×C4⋊C4C2×D711yescharacteristic
133C2×C4⋊C4C2×D712yes—
134C2×C22⋊C4—71no—
135C2×C22⋊C4—72no—
136C2×C22⋊C4—72no—
137C2×C22⋊C4—72no—
138C2×C22⋊C4—72no—
139C2×C22⋊C4C2×D711yescharacteristic
140C2×C22⋊C4C2×D712yes—
141C2×C42—71no—
142C2×C42—71no—
143#95—72no—
144#95—72no—
145#95—71no—
146#95—72no—
147#95—72no—
148#95C2×D711yescharacteristic
149C22×C7—22no—
150C22×C7—24no—
151C22×C7—41no—
152C22×C7—41no—
153C22×C7—42no—
154C22×C7C22×D411yescharacteristic
155C22×C7C2×#4111yescharacteristic
156C22×C7C2×#4112yes—
157C22×C7C2×#4111yescharacteristic
158C22×C7H(5,𝔽2)12yes—
159C2×D7—24no—
160C2×D7—24no—
161C2×D7—24no—
162C2×D7—42no—
163C2×D7—42no—
164C2×D7—42no—
165C2×D7—42no—
166C4×C7—22no—
167C4×C7—42no—
168C4×C7—41no—
169C4×C7—42no—
170Dic7—22no—
171Dic7—22no—
172Dic7—22no—
173Dic7—41no—
174Dic7—42no—
175C24—72no—
176C24C22×D711yescharacteristic
177C2×D4—144no—
178C2×D4—144no—
179C2×D4—144no—
180C2×D4—142no—
181C2×D4—282no—
182C22×C4—71no—
183C22×C4—71no—
184C22×C4—71no—
185C22×C4—72no—
186C22×C4—71no—
187C22×C4—144no—
188C22×C4—142no—
189C22×C4—142no—
190C22×C4—142no—
191C22×C4—282no—
192C22×C4—282no—
193C22×C4C22×D712yes—
194C22×C4C22×D712yes—
195C22×C4C22×D711yescharacteristic
196C22×C4C22×D711yescharacteristic
197C4⋊C4—22no—
198C4⋊C4—42no—
199C4⋊C4—142no—
200C4⋊C4—142no—
201C22⋊C4—22no—
202C22⋊C4—42no—
203C22⋊C4—144no—
204C22⋊C4—144no—
205C22⋊C4—144no—
206C22⋊C4—142no—
207C22⋊C4—282no—
208C42—74no—
209C42—142no—
210C2×C7—42no—
211C2×C7#52511yescharacteristic
212C2×C7#52511yescharacteristic
213C2×C7#53511yescharacteristic
214C2×C7#55112yes—
215C2×C7#55312yes—
216D7—44no—
217C23—21no—
218C23—22no—
219C23—21no—
220C23—22no—
221C23—74no—
222C23—142no—
223C23—142no—
224C23—142no—
225C23—142no—
226C23—142no—
227C23—142no—
228C23#116112yes—
229C23C23×D711yescharacteristic
230D4—148no—
231D4—288no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—72no—
242C2×C4—72no—
243C2×C4—72no—
244C2×C4—148no—
245C2×C4—141no—
246C2×C4—142no—
247C2×C4—141no—
248C2×C4—141no—
249C2×C4—141no—
250C2×C4—141no—
251C2×C4—142no—
252C2×C4—142no—
253C2×C4—142no—
254C2×C4—142no—
255C2×C4—288no—
256C2×C4D4×D712yes—
257C7#259811yescharacteristic, complemented, Sylow
258C22—22no—
259C22—24no—
260C22—41no—
261C22—41no—
262C22—42no—
263C22—144no—
264C22—144no—
265C22—144no—
266C22—282no—
267C22—282no—
268C22—282no—
269C22—282no—
270C22C2×D4×D711yescharacteristic
271C22#654812yes—
272C22C2×#116111yescharacteristic
273C22D7×#4111yescharacteristic
274C22D7×#4112yes—
275C4—22no—
276C4—42no—
277C4—41no—
278C4—42no—
279C4—142no—
280C4—142no—
281C4—142no—
282C4—282no—
283C4—281no—
284C2—42no—
285C2—284no—
286C2#9031811yescharacteristic
287C2#9032812yes—
288C2#9034711yescharacteristic
289C2#9035212yes—
290C2#9040611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#399C2
prime radical#89732C2
hypercenter#399D7
hypocenterC7#2598
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#6456C2semidirect
2. C2×#6440C2semidirect
3. C7×#399C2semidirect
4. #89762C2semidirect
5. #89761C2semidirect
6. #89754C2semidirect
8. #89730C2semidirect
9. #89728C2semidirect
10. #89444C2semidirect
11. #89423C2semidirect
72. C4×Dic7D4semidirect
76. #399D7semidirect
257. C7#2598semidirect

Maximal subgroups

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

subgroupindex
C2×#64562
C2×#64402
C7×#3992
#897622
#897612
#897542
#897322
#897302
#897282
#894442
#894232
#25987