Finite Group Databasebeta

← #11732215

Subgroups — #11732215

Order 896.

584 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 161 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×D4×Dic7C211yesmaximal, characteristic
2C22×Dic7⋊C4C211yesmaximal, characteristic, complemented
3C7×#386C211yesmaximal, characteristic
4#89992C211yesmaximal, characteristic
5#89982C211yesmaximal, characteristic, complemented
6#89722C211yesmaximal, characteristic, complemented
7#89720C212yesmaximal, complemented
8#89718C212yesmaximal, complemented
9#89714C211yesmaximal, characteristic, complemented
10#89429C212yesmaximal, complemented
11#89426C212yesmaximal, complemented
12C22×C7×D4C2211yescharacteristic
13C23×C4×C7C2211yescharacteristic
14C23×Dic7C2211yescharacteristic
15C23×Dic7C2211yescharacteristic
16C23×Dic7C2211yescharacteristic
17C2×C7×C22⋊C4C2211yescharacteristic
18C2×C7×C22⋊C4C2212yes—
19C2×C22⋊Dic7C2211yescharacteristic
20C2×C22⋊Dic7C2212yes—
21C2×C22⋊Dic7C2211yescharacteristic
22C2×C22⋊Dic7C2211yescharacteristic
23D4×Dic7—24no—
24C2×C4⋊Dic7C2212yes—
25C2×C4⋊Dic7C2211yescharacteristic
26C2×Dic7⋊C4—24no—
27C2×Dic7⋊C4C2212yes—
28C2×Dic7⋊C4C2212yes—
29C2×Dic7⋊C4C2212yes—
30C2×Dic7⋊C4C2212yes—
31C2×C4×Dic7C2211yescharacteristic
32C7×#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2579—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C24×C7C2311yescharacteristic
40C2×C7×D4—22no—
41C2×C7×D4—22no—
42C2×C7×D4Q814yes—
43C22×C4×C7—22no—
44C22×C4×C7—42no—
45C22×C4×C7C2312yes—
46C22×C4×C7C2312yes—
47C22×C4×C7C2311yescharacteristic
48C22×Dic7—22no—
49C22×Dic7—24no—
50C22×Dic7—22no—
51C22×Dic7—42no—
52C22×Dic7—42no—
53C22×Dic7D414yes—
54C22×Dic7C2312yes—
55C22×Dic7C2311yescharacteristic
56C22×Dic7C2311yescharacteristic
57C22×Dic7C2311yescharacteristic
58C22×Dic7C2311yescharacteristic
59C22×Dic7C2312yes—
60C7×C22⋊C4—22no—
61C7×C22⋊C4—42no—
62C22⋊Dic7—22no—
63C22⋊Dic7—22no—
64C22⋊Dic7—22no—
65C22⋊Dic7—24no—
66C4⋊Dic7—22no—
67C4⋊Dic7—24no—
68Dic7⋊C4—24no—
69Dic7⋊C4—24no—
70Dic7⋊C4—42no—
71Dic7⋊C4—42no—
72C4×Dic7—22no—
73C2×C4×D4—71no—
74C22×C4⋊C4—71no—
75#398—72no—
76#396—72no—
77#387—72no—
78#386—71no—
79#386D711yescharacteristic
80#385—71no—
81#385—72no—
82#385—71no—
83#380—71no—
84C23×C7—22no—
85C23×C7—22no—
86C23×C7—21no—
87C23×C7—22no—
88C23×C7—21no—
89C23×C7—22no—
90C23×C7C2×D412yes—
91C23×C7C2×Q812yes—
92C23×C7C2×Q812yes—
93C23×C7#4112yes—
94C23×C7C2411yescharacteristic
95C7×D4—28no—
96C2×C4×C7—21no—
97C2×C4×C7—21no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7—22no—
104C2×C4×C7—44no—
105C2×C4×C7C2×Q812yes—
106C2×Dic7—28no—
107C2×Dic7—22no—
108C2×Dic7—22no—
109C2×Dic7—22no—
110C2×Dic7—22no—
111C2×Dic7—21no—
112C2×Dic7—21no—
113C2×Dic7—22no—
114C2×Dic7—22no—
115C2×Dic7—22no—
116C2×Dic7—22no—
117C2×Dic7—22no—
118C2×Dic7—44no—
119C2×Dic7—44no—
120C2×Dic7C2×D412yes—
121C2×Dic7C2×D412yes—
122C2×Dic7#4114yes—
123C22×D4C2×D711yescharacteristic
124C23×C4—71no—
125C23×C4—71no—
126C23×C4—71no—
127C23×C4C2×D711yescharacteristic
128C4×D4—144no—
129C2×C4⋊C4—72no—
130C2×C4⋊C4—72no—
131C2×C4⋊C4—71no—
132C2×C4⋊C4—72no—
133C2×C4⋊C4—72no—
134C2×C4⋊C4—72no—
135C2×C4⋊C4—144no—
136C2×C22⋊C4—71no—
137C2×C22⋊C4—71no—
138C2×C22⋊C4—71no—
139C2×C22⋊C4—72no—
140C2×C22⋊C4C2×D711yescharacteristic
141C2×C22⋊C4C2×D712yes—
142C2×C42—71no—
143#95—72no—
144#95—72no—
145#95—72no—
146#95—72no—
147#95C2×D712yes—
148C22×C7—22no—
149C22×C7—22no—
150C22×C7—24no—
151C22×C7—24no—
152C22×C7—22no—
153C22×C7—24no—
154C22×C7—41no—
155C22×C7—41no—
156C22×C7—42no—
157C22×C7C22⋊Q814yes—
158C22×C7C22×D411yescharacteristic
159C22×C7C22×Q811yescharacteristic
160C22×C7C2×#4111yescharacteristic
161C22×C7C2×#4112yes—
162C22×C7H(5,𝔽2)12yes—
163C4×C7—22no—
164C4×C7—42no—
165C4×C7—42no—
166Dic7—24no—
167Dic7—22no—
168Dic7—22no—
169Dic7—41no—
170Dic7—41no—
171Dic7—42no—
172C24C22×D711yescharacteristic
173C24C22×D711yescharacteristic
174C2×D4—22no—
175C2×D4—22no—
176C2×D4Dic1414yes—
177C22×C4—22no—
178C22×C4—42no—
179C22×C4—71no—
180C22×C4—72no—
181C22×C4—71no—
182C22×C4—74no—
183C22×C4—72no—
184C22×C4—71no—
185C22×C4—71no—
186C22×C4—144no—
187C22×C4—142no—
188C22×C4—142no—
189C22×C4—282no—
190C22×C4—282no—
191C22×C4C22×D712yes—
192C22×C4C22×D712yes—
193C22×C4C22×D711yescharacteristic
194C4⋊C4—144no—
195C4⋊C4—144no—
196C4⋊C4—144no—
197C4⋊C4—142no—
198C4⋊C4—282no—
199C4⋊C4—282no—
200C22⋊C4—22no—
201C22⋊C4—42no—
202C22⋊C4—142no—
203C22⋊C4—144no—
204C22⋊C4—142no—
205C22⋊C4—142no—
206C42—142no—
207C2×C7—24no—
208C2×C7—42no—
209C2×C7C2×C22⋊Q811yescharacteristic
210C2×C7#52511yescharacteristic
211C2×C7#53411yescharacteristic
212C2×C7#55312yes—
213C2×C7#55412yes—
214C23—22no—
215C23—22no—
216C23—21no—
217C23—21no—
218C23—22no—
219C23—22no—
220C23C2×Dic1412yes—
221C23C2×Dic1412yes—
222C23D4×D712yes—
223C23#115912yes—
224C23C23×D711yescharacteristic
225D4—28no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—44no—
235C2×C4—72no—
236C2×C4—72no—
237C2×C4—74no—
238C2×C4—148no—
239C2×C4—142no—
240C2×C4—142no—
241C2×C4—142no—
242C2×C4—142no—
243C2×C4—141no—
244C2×C4—141no—
245C2×C4—142no—
246C2×C4—142no—
247C2×C4—142no—
248C2×C4—142no—
249C2×C4—142no—
250C2×C4—284no—
251C2×C4—284no—
252C2×C4C2×Dic1412yes—
253C7#257911yescharacteristic, complemented, Sylow
254C22—24no—
255C22—22no—
256C22—24no—
257C22—22no—
258C22—22no—
259C22—24no—
260C22—41no—
261C22—42no—
262C22—41no—
263C22#644114yes—
264C22C22×Dic1411yescharacteristic
265C22C2×D4×D711yescharacteristic
266C22C2×#115911yescharacteristic
267C22#654812yes—
268C22D7×#4112yes—
269C4—22no—
270C4—42no—
271C4—42no—
272C4—142no—
273C4—142no—
274C4—144no—
275C4—281no—
276C4—282no—
277C4—281no—
278C2—24no—
279C2—42no—
280C2C2×#644111yescharacteristic
281C2#9023012yes—
282C2#9028011yescharacteristic
283C2#9029212yes—
284C2#9034711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#386C2
prime radicalC22×C7×D4C22
hypercenter#386D7
hypocenterC7#2579
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×Dic7⋊C4C2semidirect
5. #89982C2semidirect
6. #89722C2semidirect
7. #89720C2semidirect
8. #89718C2semidirect
9. #89714C2semidirect
10. #89429C2semidirect
11. #89426C2semidirect
253. C7#2579semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic72
C22×Dic7⋊C42
C7×#3862
#899922
#899822
#897222
#897202
#897182
#897142
#894292
#894262
#25797