Finite Group Databasebeta

← #11731582

Subgroups — #11731582

Order 896.

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

normal subgroups only

This group has 620 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×#6493C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D7C211yesmaximal, characteristic, complemented
3C7×#381C211yesmaximal, characteristic, complemented
4#89733C212yesmaximal, complemented
5#89732C212yesmaximal, complemented
6#89449C211yesmaximal, characteristic, complemented
7#89444C211yesmaximal, characteristic, complemented
8#89443C211yesmaximal, characteristic, complemented
9#89442C212yesmaximal, complemented
10#89437C212yesmaximal, complemented
11#89431C211yesmaximal, characteristic, complemented
12C23×C4×C7C2211yescharacteristic
13C22×C22⋊D7C2211yescharacteristic
14C22×C22⋊D7C2212yes—
15C22×D28C2212yes—
16C22×D28C2211yescharacteristic
17C22×C4×D7C2211yescharacteristic
18C2×C7×C22⋊C4C2212yes—
19C2×C22⋊Dic7C2211yescharacteristic
20#6493—24no—
21C4×C22⋊D7—24no—
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×(C2×D7)⋊C4C2211yescharacteristic
25C2×(C2×D7)⋊C4C2212yes—
26C2×(C2×D7)⋊C4C2212yes—
27C2×(C2×D7)⋊C4C2212yes—
28C2×C4⋊Dic7C2211yescharacteristic
29C2×C4⋊Dic7C2212yes—
30C2×Dic7⋊C4C2211yescharacteristic
31C2×C4×Dic7C2211yescharacteristic
32C7×#95C2211yescharacteristic
33C7×#95C2212yes—
34C7×#95C2211yescharacteristic
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2635—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C23×D7C2312yes—
40C23×D7C2311yescharacteristic
41C22×C4×C7—42no—
42C22×C4×C7D414yescomplemented
43C22×C4×C7C2312yes—
44C22×C4×C7C2312yes—
45C22×C4×C7C2311yescharacteristic
46C22×C4×C7C2311yescharacteristic
47C2×C22⋊D7—22no—
48C2×C22⋊D7—22no—
49C2×C22⋊D7—24no—
50C2×C22⋊D7—22no—
51C2×C22⋊D7—42no—
52C2×C22⋊D7—42no—
53C22×Dic7C2311yescharacteristic
54C22×Dic7C2312yes—
55C22×Dic7C2311yescharacteristic
56C22×Dic7C2311yescharacteristic
57C2×D28—24no—
58C2×D28—22no—
59C2×D28—42no—
60C2×D28—42no—
61C2×D28—42no—
62C2×C4×D7—22no—
63C2×C4×D7—42no—
64C7×C22⋊C4—42no—
65C22⋊Dic7—22no—
66(C2×D7)⋊C4—22no—
67(C2×D7)⋊C4—24no—
68(C2×D7)⋊C4—24no—
69(C2×D7)⋊C4—42no—
70(C2×D7)⋊C4—42no—
71(C2×D7)⋊C4—42no—
72C4⋊Dic7—22no—
73C4⋊Dic7—42no—
74Dic7⋊C4—22no—
75C4×Dic7—22no—
76C2×#121—71no—
77C2×C4×D4—71no—
78#400—71no—
79#399—71no—
80#394—72no—
81#394—71no—
82#392—72no—
83#392—71no—
84#388—72no—
85#386—72no—
86#381D711yescharacteristic, complemented
87C23×C7—21no—
88C23×C7—21no—
89C23×C7—22no—
90C23×C7—22no—
91C23×C7C2×D412yes—
92C23×C7C2411yescharacteristic
93C22×D7—21no—
94C22×D7—21no—
95C22×D7—22no—
96C22×D7—22no—
97C22×D7—22no—
98C22×D7—22no—
99C22×D7—22no—
100C22×D7—22no—
101C22×D7—22no—
102C22×D7—22no—
103C22×D7—22no—
104C22×D7#4112yes—
105C2×C4×C7—22no—
106C2×C4×C7—22no—
107C2×C4×C7—22no—
108C2×C4×C7—22no—
109C2×C4×C7—22no—
110C2×C4×C7—22no—
111C2×C4×C7—22no—
112C2×C4×C7—22no—
113C2×C4×C7—44no—
114C2×C4×C7C2×D412yes—
115C2×C4×C7C2×D412yes—
116C22⋊D7—28no—
117C22⋊D7—48no—
118C2×Dic7—21no—
119C2×Dic7—21no—
120C2×Dic7—21no—
121C2×Dic7—21no—
122C2×Dic7—22no—
123C2×Dic7—22no—
124C2×Dic7—22no—
125C2×Dic7—22no—
126C2×Dic7—22no—
127C2×Dic7#4112yes—
128D28—44no—
129D28—48no—
130C4×D7—44no—
131C22×D4—71no—
132C22×D4—72no—
133C22×D4—71no—
134C22×D4—72no—
135C23×C4—71no—
136C23×C4C2×D711yescharacteristic
137#121—144no—
138C4×D4—144no—
139C2×C4⋊C4—71no—
140C2×C4⋊C4—71no—
141C2×C4⋊C4—72no—
142C2×C22⋊C4—71no—
143C2×C22⋊C4—71no—
144C2×C22⋊C4—72no—
145C2×C22⋊C4—72no—
146C2×C22⋊C4—72no—
147C2×C22⋊C4—72no—
148C2×C22⋊C4—72no—
149C2×C22⋊C4C2×D712yes—
150C2×C42—71no—
151#95—72no—
152#95—72no—
153#95C2×D712yes—
154#95C2×D711yescharacteristic
155#95C2×D711yescharacteristic
156C22×C7—22no—
157C22×C7—22no—
158C22×C7—22no—
159C22×C7—42no—
160C22×C7—42no—
161C22×C7C22×D411yescharacteristic
162C22×C7C2×#4111yescharacteristic
163C22×C7C2×#4111yescharacteristic
164C22×C7H(5,𝔽2)12yes—
165C22×C7H(5,𝔽2)12yes—
166C2×D7—22no—
167C2×D7—24no—
168C2×D7—41no—
169C2×D7—41no—
170C2×D7—42no—
171C2×D7—42no—
172C2×D7—42no—
173C2×D7—42no—
174C2×D7—42no—
175C2×D7—42no—
176C2×D7—42no—
177C2×D7—42no—
178C4×C7—22no—
179C4×C7—22no—
180C4×C7—42no—
181C4×C7—42no—
182Dic7—22no—
183Dic7—41no—
184Dic7—41no—
185Dic7—42no—
186C24—71no—
187C24—72no—
188C24C22×D711yescharacteristic
189C2×D4—144no—
190C2×D4—142no—
191C2×D4—142no—
192C2×D4—142no—
193C2×D4—144no—
194C2×D4—142no—
195C2×D4—282no—
196C2×D4—282no—
197C2×D4—282no—
198C2×D4—282no—
199C2×D4—282no—
200C22×C4—42no—
201C22×C4—71no—
202C22×C4—71no—
203C22×C4—72no—
204C22×C4—71no—
205C22×C4—142no—
206C22×C4—282no—
207C22×C4D2814yescomplemented
208C22×C4C22×D712yes—
209C22×C4C22×D712yes—
210C22×C4C22×D711yescharacteristic
211C22×C4C22×D711yescharacteristic
212C4⋊C4—142no—
213C4⋊C4—142no—
214C4⋊C4—282no—
215C22⋊C4—42no—
216C22⋊C4—144no—
217C22⋊C4—142no—
218C22⋊C4—142no—
219C22⋊C4—144no—
220C22⋊C4—282no—
221C22⋊C4—282no—
222C22⋊C4—282no—
223C42—142no—
224C2×C7—42no—
225C2×C7#52511yescharacteristic
226C2×C7#53411yescharacteristic
227C2×C7#53511yescharacteristic
228C2×C7#53812yes—
229C2×C7#54012yes—
230D7—42no—
231D7—82no—
232C23—22no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—72no—
237C23—141no—
238C23—142no—
239C23—141no—
240C23—142no—
241C23—142no—
242C23—142no—
243C23—142no—
244C23—142no—
245C23—142no—
246C23—142no—
247C23—142no—
248C23C2×D2812yes—
249C23C23×D711yescharacteristic
250D4—148no—
251D4—284no—
252D4—288no—
253D4—288no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—44no—
263C2×C4—72no—
264C2×C4—141no—
265C2×C4—141no—
266C2×C4—141no—
267C2×C4—141no—
268C2×C4—142no—
269C2×C4—142no—
270C2×C4—142no—
271C2×C4—142no—
272C2×C4—142no—
273C2×C4—284no—
274C2×C4C2×D2812yes—
275C2×C4C2×D2812yes—
276C7#263511yescharacteristic, complemented, Sylow
277C22—22no—
278C22—22no—
279C22—22no—
280C22—42no—
281C22—42no—
282C22—142no—
283C22—144no—
284C22—281no—
285C22—281no—
286C22—282no—
287C22—282no—
288C22—282no—
289C22—282no—
290C22—282no—
291C22—282no—
292C22—282no—
293C22—282no—
294C22C22×D2811yescharacteristic
295C22#654812yes—
296C22D7×#4111yescharacteristic
297C22D7×#4111yescharacteristic
298C22#655312yes—
299C4—22no—
300C4—22no—
301C4—42no—
302C4—42no—
303C4—142no—
304C4—281no—
305C4—282no—
306C4—281no—
307C2—42no—
308C2—282no—
309C2—562no—
310C2#9018411yescharacteristic
311C2#9021511yescharacteristic
312C2#9021611yescharacteristic
313C2#9034812yes—
314C2#9035412yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#381C2
hypercenter#381D7
hypocenterC7#2635
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C2×#6493C2semidirect
2. C2×C4×C22⋊D7C2semidirect
3. C7×#381C2semidirect
4. #89733C2semidirect
5. #89732C2semidirect
6. #89449C2semidirect
7. #89444C2semidirect
8. #89443C2semidirect
9. #89442C2semidirect
10. #89437C2semidirect
11. #89431C2semidirect
42. C22×C4×C7D4semidirect
86. #381D7semidirect
207. C22×C4D28semidirect
276. C7#2635semidirect

Maximal subgroups

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

subgroupindex
C2×#64932
C2×C4×C22⋊D72
C7×#3812
#897332
#897322
#894492
#894442
#894432
#894422
#894372
#894312
#26357