Finite Group Databasebeta

← #11732317

Subgroups — #11732317

Order 896.

688 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 165 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×#6502C211yesmaximal, characteristic
2C2×#6498C211yesmaximal, characteristic, complemented
3C22×C4×Dic7C211yesmaximal, characteristic, complemented
4C2×#6449C211yesmaximal, characteristic, complemented
5C7×#386C211yesmaximal, characteristic, complemented
6#89985C211yesmaximal, characteristic, complemented
7#89729C212yesmaximal, complemented
8#89723C212yesmaximal, complemented
9#89714C211yesmaximal, characteristic, complemented
10#89441C212yesmaximal, complemented
11#89419C212yesmaximal, complemented
12C22×C7×D4C2211yescharacteristic
13C23×C4×C7C2211yescharacteristic
14C22×C22⋊D7C2211yescharacteristic
15C22×C22⋊D7C2211yescharacteristic
16C23×Dic7C2211yescharacteristic
17C23×Dic7C2211yescharacteristic
18C2×C7×C22⋊C4C2211yescharacteristic
19C2×C7×C22⋊C4C2212yes—
20C2×C22⋊Dic7C2211yescharacteristic
21C2×C22⋊Dic7C2212yes—
22C2×C22⋊Dic7C2211yescharacteristic
23#6502—24no—
24#6498—24no—
25C2×(C2×D7)⋊C4C2212yes—
26C2×(C2×D7)⋊C4C2211yescharacteristic
27C2×(C2×D7)⋊C4C2212yes—
28C2×C4⋊Dic7C2212yes—
29C2×Dic7⋊C4C2211yescharacteristic
30C2×Dic7⋊C4C2212yes—
31C2×Dic7⋊C4C2212yes—
32C2×C4×Dic7—24no—
33C2×C4×Dic7C2212yes—
34C2×C4×Dic7C2212yes—
35#6449—24no—
36C7×#95C2212yes—
37C7⋊#95C2212yes—
38C7⋊#95C2212yes—
39#2854—71nomaximal, Sylow
40C24×C7C2311yescharacteristic
41C24×C7C2311yescharacteristic
42C23×D7C2311yescharacteristic
43C2×C7×D4—22no—
44C2×C7×D4—22no—
45C2×C7×D4—22no—
46C22×C4×C7—22no—
47C22×C4×C7—24no—
48C22×C4×C7C2312yes—
49C22×C4×C7C2311yescharacteristic
50C22×C4×C7C2312yes—
51C2×C22⋊D7—22no—
52C2×C22⋊D7—22no—
53C2×C22⋊D7—22no—
54C2×C22⋊D7—22no—
55C2×C22⋊D7—24no—
56C22×Dic7—22no—
57C22×Dic7—22no—
58C22×Dic7—24no—
59C22×Dic7D414yes—
60C22×Dic7D414yescomplemented
61C22×Dic7C2312yes—
62C22×Dic7C2311yescharacteristic
63C22×Dic7C2311yescharacteristic
64C22×Dic7C2312yes—
65C22×Dic7C2311yescharacteristic
66C7×C22⋊C4—22no—
67C7×C22⋊C4—24no—
68C22⋊Dic7—22no—
69C22⋊Dic7—24no—
70C22⋊Dic7—22no—
71(C2×D7)⋊C4—24no—
72(C2×D7)⋊C4—24no—
73(C2×D7)⋊C4—22no—
74C4⋊Dic7—24no—
75Dic7⋊C4—24no—
76Dic7⋊C4—22no—
77Dic7⋊C4—24no—
78C4×Dic7—24no—
79C4×Dic7—24no—
80C2×#123—71no—
81C2×#123—71no—
82C2×#121—71no—
83C22×C42—71no—
84#390—72no—
85#389—72no—
86#388—72no—
87#388—72no—
88#386—71no—
89#386D711yescharacteristic, complemented
90#380—71no—
91C23×C7—22no—
92C23×C7—22no—
93C23×C7—21no—
94C23×C7—21no—
95C23×C7—22no—
96C23×C7—22no—
97C23×C7—21no—
98C23×C7C2×D412yes—
99C23×C7C2×D412yes—
100C23×C7#4112yes—
101C23×C7C2411yescharacteristic
102C22×D7—21no—
103C22×D7—21no—
104C22×D7—22no—
105C22×D7—21no—
106C22×D7—22no—
107C7×D4—44no—
108C2×C4×C7—28no—
109C2×C4×C7—22no—
110C2×C4×C7—22no—
111C2×C4×C7—21no—
112C2×C4×C7—21no—
113C2×C4×C7—22no—
114C2×C4×C7—22no—
115C2×C4×C7—22no—
116C2×C4×C7—21no—
117C2×C4×C7#4114yes—
118C22⋊D7—44no—
119C22⋊D7—44no—
120C2×Dic7—28no—
121C2×Dic7—22no—
122C2×Dic7—22no—
123C2×Dic7—22no—
124C2×Dic7—28no—
125C2×Dic7—22no—
126C2×Dic7—21no—
127C2×Dic7—21no—
128C2×Dic7—22no—
129C2×Dic7—22no—
130C2×Dic7—21no—
131C2×Dic7C2×D412yes—
132C2×Dic7C2×D414yes—
133C2×Dic7C2×D412yes—
134C2×Dic7#4114yes—
135C22×D4—71no—
136C22×D4—71no—
137C22×D4C2×D711yescharacteristic
138C23×C4—71no—
139C23×C4—71no—
140C23×C4C2×D711yescharacteristic
141#123—144no—
142#123—144no—
143#121—144no—
144C2×C4⋊C4—71no—
145C2×C4⋊C4—72no—
146C2×C4⋊C4—72no—
147C2×C4⋊C4—72no—
148C2×C22⋊C4—71no—
149C2×C22⋊C4—71no—
150C2×C22⋊C4—72no—
151C2×C22⋊C4—71no—
152C2×C22⋊C4—72no—
153C2×C22⋊C4—72no—
154C2×C22⋊C4C2×D711yescharacteristic
155C2×C22⋊C4C2×D712yes—
156C2×C42—72no—
157C2×C42—72no—
158C2×C42—144no—
159#95—72no—
160#95—72no—
161#95C2×D712yes—
162C22×C7—22no—
163C22×C7—22no—
164C22×C7—24no—
165C22×C7—24no—
166C22×C7—41no—
167C22×C7—41no—
168C22×C7—41no—
169C22×C7—42no—
170C22×C7—42no—
171C22×C7#12114yes—
172C22×C7C22×D411yescharacteristic
173C22×C7C22×D411yescharacteristic
174C22×C7C2×#4111yescharacteristic
175C22×C7C2×#4112yes—
176C22×C7C2×#4112yes—
177C2×D7—41no—
178C2×D7—41no—
179C2×D7—41no—
180C2×D7—42no—
181C2×D7—42no—
182C4×C7—24no—
183C4×C7—41no—
184C4×C7—42no—
185Dic7—24no—
186Dic7—22no—
187Dic7—41no—
188Dic7—42no—
189Dic7#12314yescomplemented
190C24—71no—
191C24C22×D711yescharacteristic
192C24C22×D711yescharacteristic
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—142no—
197C2×D4—144no—
198C2×D4—142no—
199C2×D4—142no—
200C2×D4—142no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—71no—
204C22×C4—74no—
205C22×C4—74no—
206C22×C4—71no—
207C22×C4—72no—
208C22×C4—72no—
209C22×C4—71no—
210C22×C4—144no—
211C22×C4—142no—
212C22×C4—142no—
213C22×C4C22×D712yes—
214C22×C4C22×D711yescharacteristic
215C22×C4C22×D712yes—
216C4⋊C4—144no—
217C4⋊C4—142no—
218C4⋊C4—144no—
219C4⋊C4—144no—
220C22⋊C4—22no—
221C22⋊C4—24no—
222C22⋊C4—142no—
223C22⋊C4—144no—
224C22⋊C4—142no—
225C22⋊C4—144no—
226C22⋊C4—142no—
227C22⋊C4—144no—
228C42—144no—
229C42—144no—
230C2×C7—24no—
231C2×C7—81no—
232C2×C7C2×#12111yescharacteristic
233C2×C7C2×#12311yescharacteristic
234C2×C7#52511yescharacteristic
235C2×C7#52912yes—
236C2×C7#53212yes—
237D7—81no—
238C23—22no—
239C23—22no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—22no—
244C23—22no—
245C23—141no—
246C23—141no—
247C23—141no—
248C23—142no—
249C23—142no—
250C23#115712yes—
251C23D4×D712yes—
252C23D4×D712yes—
253C23C23×D711yescharacteristic
254D4—44no—
255D4—284no—
256D4—284no—
257C2×C4—28no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—72no—
267C2×C4—74no—
268C2×C4—72no—
269C2×C4—74no—
270C2×C4—148no—
271C2×C4—148no—
272C2×C4—142no—
273C2×C4—142no—
274C2×C4—142no—
275C2×C4—141no—
276C2×C4—142no—
277C2×C4—141no—
278C2×C4—141no—
279C2×C4—142no—
280C2×C4—142no—
281C2×C4#115914yes—
282C7#285411yescharacteristic, complemented, Sylow
283C22—24no—
284C22—22no—
285C22—24no—
286C22—22no—
287C22—41no—
288C22—42no—
289C22—41no—
290C22—42no—
291C22—41no—
292C22—281no—
293C22—282no—
294C22—281no—
295C22—282no—
296C22—281no—
297C22#644714yes—
298C22C2×#115711yescharacteristic
299C22C2×D4×D711yescharacteristic
300C22C2×D4×D711yescharacteristic
301C22C2×#115912yes—
302C22D7×#4112yes—
303C4—24no—
304C4—41no—
305C4—42no—
306C4—74no—
307C4—144no—
308C4—142no—
309C4—281no—
310C4—282no—
311C2—24no—
312C2—81no—
313C2—561no—
314C2C2×#644711yescharacteristic
315C2#9022912yes—
316C2#9027711yescharacteristic
317C2#9028712yes—
318C2D7×#12311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#386C2
prime radicalC2×#6502C2
hypercenter#386D7
hypocenterC7#2854
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#6498C2semidirect
3. C22×C4×Dic7C2semidirect
4. C2×#6449C2semidirect
5. C7×#386C2semidirect
6. #89985C2semidirect
7. #89729C2semidirect
8. #89723C2semidirect
9. #89714C2semidirect
10. #89441C2semidirect
11. #89419C2semidirect
60. C22×Dic7D4semidirect
89. #386D7semidirect
189. Dic7#123semidirect
282. C7#2854semidirect

Maximal subgroups

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

subgroupindex
C2×#65022
C2×#64982
C22×C4×Dic72
C2×#64492
C7×#3862
#899852
#897292
#897232
#897142
#894412
#894192
#28547