Finite Group Databasebeta

← #11732136

Subgroups — #11732136

Order 896.

610 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 157 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×#6498C211yesmaximal, characteristic
2C2×#6498C211yesmaximal, characteristic
3C22×C4⋊Dic7C211yesmaximal, characteristic, complemented
4C2×#6440C211yesmaximal, characteristic, complemented
5C7×#386C211yesmaximal, characteristic
6#89982C211yesmaximal, characteristic, complemented
7#89723C212yesmaximal, complemented
8#89720C212yesmaximal, complemented
9#89715C211yesmaximal, characteristic, complemented
10#89429C212yesmaximal, complemented
11#89419C212yesmaximal, complemented
12C22×C7×D4C2211yescharacteristic
13C23×C4×C7C2211yescharacteristic
14C23×Dic7C2211yescharacteristic
15C23×Dic7C2211yescharacteristic
16C2×C7×C22⋊C4C2211yescharacteristic
17C2×C7×C22⋊C4C2212yes—
18C2×C22⋊Dic7C2211yescharacteristic
19C2×C22⋊Dic7C2211yescharacteristic
20C2×C22⋊Dic7C2212yes—
21C2×C22⋊Dic7C2211yescharacteristic
22C2×C22⋊Dic7C2211yescharacteristic
23#6498—24no—
24#6498—24no—
25C2×C4⋊Dic7—24no—
26C2×C4⋊Dic7C2212yes—
27C2×C4⋊Dic7C2212yes—
28C2×Dic7⋊C4C2212yes—
29C2×Dic7⋊C4C2211yescharacteristic
30C2×Dic7⋊C4C2212yes—
31C2×Dic7⋊C4C2212yes—
32C2×Dic7⋊C4C2211yescharacteristic
33C2×C4×Dic7C2212yes—
34#6440—24no—
35C7×#95C2212yes—
36C7⋊#95C2212yes—
37C7⋊#95C2212yes—
38C7⋊#95C2212yes—
39#2413—71nomaximal, Sylow
40C24×C7C2311yescharacteristic
41C24×C7C2311yescharacteristic
42C2×C7×D4—22no—
43C2×C7×D4—22no—
44C2×C7×D4—22no—
45C22×C4×C7—22no—
46C22×C4×C7—24no—
47C22×C4×C7C2312yes—
48C22×C4×C7C2311yescharacteristic
49C22×C4×C7C2312yes—
50C22×Dic7—22no—
51C22×Dic7—22no—
52C22×Dic7—22no—
53C22×Dic7—22no—
54C22×Dic7—22no—
55C22×Dic7D414yes—
56C22×Dic7C2311yescharacteristic
57C22×Dic7C2311yescharacteristic
58C22×Dic7C2312yes—
59C22×Dic7C2312yes—
60C22×Dic7C2311yescharacteristic
61C22×Dic7C2311yescharacteristic
62C7×C22⋊C4—22no—
63C7×C22⋊C4—24no—
64C22⋊Dic7—22no—
65C22⋊Dic7—22no—
66C22⋊Dic7—22no—
67C22⋊Dic7—24no—
68C22⋊Dic7D414yes—
69C4⋊Dic7—24no—
70C4⋊Dic7—24no—
71Dic7⋊C4—24no—
72Dic7⋊C4—24no—
73Dic7⋊C4—22no—
74Dic7⋊C4—22no—
75Dic7⋊C4—24no—
76C4×Dic7—24no—
77C2×#123—71no—
78C2×#123—71no—
79C2×#117—71no—
80C22×C4⋊C4—71no—
81#398—72no—
82#396—72no—
83#389—72no—
84#388—72no—
85#386D711yescharacteristic
86#385—71no—
87#381—71no—
88C23×C7—22no—
89C23×C7—22no—
90C23×C7—21no—
91C23×C7—21no—
92C23×C7—22no—
93C23×C7—21no—
94C23×C7—22no—
95C23×C7C2×D412yes—
96C23×C7C2×D412yes—
97C23×C7#4112yes—
98C23×C7C2411yescharacteristic
99C7×D4—44no—
100C2×C4×C7—28no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7—22no—
104C2×C4×C7—21no—
105C2×C4×C7—21no—
106C2×C4×C7—22no—
107C2×C4×C7—21no—
108C2×C4×C7—22no—
109C2×C4×C7#4114yes—
110C2×Dic7—28no—
111C2×Dic7—21no—
112C2×Dic7—21no—
113C2×Dic7—22no—
114C2×Dic7—21no—
115C2×Dic7—21no—
116C2×Dic7—22no—
117C2×Dic7—21no—
118C2×Dic7—21no—
119C2×Dic7—22no—
120C2×Dic7—22no—
121C2×Dic7—22no—
122C2×Dic7—22no—
123C2×Dic7—22no—
124C2×Dic7—44no—
125C2×Dic7C2×D414yes—
126C2×Dic7C2×D412yes—
127C2×Dic7C2×D412yes—
128C22×D4C2×D711yescharacteristic
129C23×C4—71no—
130C23×C4—71no—
131C23×C4C2×D711yescharacteristic
132#123—144no—
133#123—144no—
134#117—144no—
135C2×C4⋊C4—72no—
136C2×C4⋊C4—72no—
137C2×C4⋊C4—72no—
138C2×C4⋊C4—71no—
139C2×C4⋊C4—72no—
140C2×C4⋊C4—71no—
141C2×C4⋊C4—72no—
142C2×C4⋊C4—144no—
143C2×C22⋊C4—71no—
144C2×C22⋊C4—71no—
145C2×C22⋊C4—71no—
146C2×C22⋊C4—72no—
147C2×C22⋊C4—71no—
148C2×C22⋊C4C2×D711yescharacteristic
149C2×C22⋊C4C2×D712yes—
150C2×C42—72no—
151#95—72no—
152#95—72no—
153#95—72no—
154#95C2×D712yes—
155C22×C7—22no—
156C22×C7—22no—
157C22×C7—24no—
158C22×C7—24no—
159C22×C7—41no—
160C22×C7—41no—
161C22×C7—41no—
162C22×C7—42no—
163C22×C7—42no—
164C22×C7#12114yes—
165C22×C7C22×D411yescharacteristic
166C22×C7C22×D411yescharacteristic
167C22×C7C2×#4111yescharacteristic
168C22×C7C2×#4112yes—
169C22×C72−1+412yes—
170C4×C7—24no—
171C4×C7—41no—
172C4×C7—42no—
173Dic7—22no—
174Dic7—22no—
175Dic7—24no—
176Dic7—42no—
177Dic7—41no—
178Dic7—41no—
179C24C22×D711yescharacteristic
180C24C22×D711yescharacteristic
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—22no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—74no—
187C22×C4—72no—
188C22×C4—71no—
189C22×C4—71no—
190C22×C4—71no—
191C22×C4—71no—
192C22×C4—72no—
193C22×C4—142no—
194C22×C4—142no—
195C22×C4—142no—
196C22×C4—142no—
197C22×C4—142no—
198C22×C4C22×D712yes—
199C22×C4C22×D711yescharacteristic
200C22×C4C22×D712yes—
201C4⋊C4—144no—
202C4⋊C4—144no—
203C4⋊C4—142no—
204C4⋊C4—142no—
205C4⋊C4—144no—
206C4⋊C4—144no—
207C4⋊C4—144no—
208C22⋊C4—22no—
209C22⋊C4—24no—
210C22⋊C4—74no—
211C22⋊C4—144no—
212C22⋊C4—142no—
213C22⋊C4—142no—
214C22⋊C4—142no—
215C42—144no—
216C2×C7—24no—
217C2×C7—81no—
218C2×C7C2×#12111yescharacteristic
219C2×C7#52511yescharacteristic
220C2×C7#53611yescharacteristic
221C2×C7#54712yes—
222C2×C7#55212yes—
223C23—22no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—22no—
229C23—22no—
230C23#115712yes—
231C23D4×D712yes—
232C23D4×D712yes—
233C23C23×D711yescharacteristic
234D4—44no—
235C2×C4—28no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—74no—
245C2×C4—72no—
246C2×C4—72no—
247C2×C4—148no—
248C2×C4—141no—
249C2×C4—141no—
250C2×C4—141no—
251C2×C4—141no—
252C2×C4—142no—
253C2×C4—142no—
254C2×C4—142no—
255C2×C4—141no—
256C2×C4—141no—
257C2×C4—142no—
258C2×C4—142no—
259C2×C4—142no—
260C2×C4—142no—
261C2×C4—284no—
262C2×C4#115914yes—
263C7#241311yescharacteristic, complemented, Sylow
264C22—24no—
265C22—22no—
266C22—24no—
267C22—22no—
268C22—41no—
269C22—42no—
270C22—41no—
271C22—42no—
272C22—41no—
273C22#644714yes—
274C22C2×#115711yescharacteristic
275C22C2×D4×D711yescharacteristic
276C22C2×D4×D711yescharacteristic
277C22C2×#115912yes—
278C22#655412yes—
279C4—24no—
280C4—41no—
281C4—42no—
282C4—142no—
283C4—142no—
284C4—144no—
285C4—282no—
286C4—281no—
287C4—281no—
288C2—24no—
289C2—81no—
290C2C2×#644711yescharacteristic
291C2#9023212yes—
292C2#9027711yescharacteristic
293C2#9030212yes—
294C2#9033911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#386C2
prime radicalC22×C7×D4C22
hypercenter#386D7
hypocenterC7#2413
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
3. C22×C4⋊Dic7C2semidirect
4. C2×#6440C2semidirect
6. #89982C2semidirect
7. #89723C2semidirect
8. #89720C2semidirect
9. #89715C2semidirect
10. #89429C2semidirect
11. #89419C2semidirect
263. C7#2413semidirect

Maximal subgroups

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

subgroupindex
C2×#64982
C22×C4⋊Dic72
C2×#64402
C7×#3862
#899822
#897232
#897202
#897152
#894292
#894192
#24137