Finite Group Databasebeta

← #11732316

Subgroups — #11732316

Order 896.

582 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 153 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
2C22×C4×Dic7C211yesmaximal, characteristic, complemented
3C7×#386C211yesmaximal, characteristic
4#89992C211yesmaximal, characteristic
5#89982C211yesmaximal, characteristic, complemented
6#89723C212yesmaximal, complemented
7#89722C211yesmaximal, characteristic, complemented
8#89717C212yesmaximal, complemented
9#89715C211yesmaximal, characteristic, complemented
10#89426C212yesmaximal, complemented
11#89424C212yesmaximal, 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—
24C2×C4⋊Dic7C2212yes—
25C2×Dic7⋊C4C2211yescharacteristic
26C2×Dic7⋊C4C2212yes—
27C2×Dic7⋊C4C2211yescharacteristic
28C2×C4×Dic7—24no—
29C2×C4×Dic7C2212yes—
30C2×C4×Dic7C2212yes—
31C7×#95C2212yes—
32C7⋊#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2853—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C24×C7C2311yescharacteristic
40C2×C7×D4—22no—
41C2×C7×D4—22no—
42C2×C7×D4—22no—
43C22×C4×C7—22no—
44C22×C4×C7—24no—
45C22×C4×C7C2312yes—
46C22×C4×C7C2311yescharacteristic
47C22×C4×C7C2312yes—
48C22×Dic7—22no—
49C22×Dic7—22no—
50C22×Dic7—22no—
51C22×Dic7—24no—
52C22×Dic7D414yes—
53C22×Dic7C2312yes—
54C22×Dic7C2311yescharacteristic
55C22×Dic7C2311yescharacteristic
56C22×Dic7C2311yescharacteristic
57C22×Dic7C2311yescharacteristic
58C22×Dic7C2312yes—
59C7×C22⋊C4—22no—
60C7×C22⋊C4—24no—
61C22⋊Dic7—22no—
62C22⋊Dic7—22no—
63C22⋊Dic7—22no—
64C22⋊Dic7—22no—
65C22⋊Dic7—24no—
66C4⋊Dic7—24no—
67Dic7⋊C4—24no—
68Dic7⋊C4—22no—
69Dic7⋊C4—22no—
70C4×Dic7—24no—
71C4×Dic7—24no—
72C2×#123—71no—
73C22×C42—71no—
74#388—72no—
75#388—72no—
76#387—72no—
77#387—72no—
78#386—71no—
79#386D711yescharacteristic
80#385—71no—
81#385—71no—
82#381—71no—
83C23×C7—22no—
84C23×C7—22no—
85C23×C7—21no—
86C23×C7—22no—
87C23×C7—22no—
88C23×C7—21no—
89C23×C7—21no—
90C23×C7C2×D412yes—
91C23×C7#4112yes—
92C23×C7#4112yes—
93C23×C7C2411yescharacteristic
94C7×D4—44no—
95C2×C4×C7—28no—
96C2×C4×C7—22no—
97C2×C4×C7—22no—
98C2×C4×C7—21no—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—21no—
103C2×C4×C7—21no—
104C2×C4×C7#4114yes—
105C2×Dic7—28no—
106C2×Dic7—28no—
107C2×Dic7—22no—
108C2×Dic7—21no—
109C2×Dic7—21no—
110C2×Dic7—21no—
111C2×Dic7—21no—
112C2×Dic7—22no—
113C2×Dic7—21no—
114C2×Dic7—21no—
115C2×Dic7—22no—
116C2×Dic7—22no—
117C2×Dic7—22no—
118C2×Dic7—22no—
119C2×Dic7—22no—
120C2×Dic7C2×D412yes—
121C2×Dic7C2×D412yes—
122C2×Dic7#4114yes—
123C22×D4C2×D711yescharacteristic
124C23×C4—71no—
125C23×C4—71no—
126C23×C4C2×D711yescharacteristic
127#123—144no—
128C2×C4⋊C4—71no—
129C2×C4⋊C4—71no—
130C2×C4⋊C4—72no—
131C2×C4⋊C4—72no—
132C2×C22⋊C4—71no—
133C2×C22⋊C4—71no—
134C2×C22⋊C4—71no—
135C2×C22⋊C4—71no—
136C2×C22⋊C4—72no—
137C2×C22⋊C4C2×D711yescharacteristic
138C2×C22⋊C4C2×D712yes—
139C2×C42—72no—
140C2×C42—72no—
141C2×C42—144no—
142#95—72no—
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—41no—
153C22×C7—41no—
154C22×C7—41no—
155C22×C7—42no—
156C22×C7—42no—
157C22×C7#12314yes—
158C22×C7C22×D411yescharacteristic
159C22×C7C2×#4111yescharacteristic
160C22×C7C2×#4111yescharacteristic
161C22×C7C2×#4112yes—
162C22×C7C2×#4112yes—
163C4×C7—24no—
164C4×C7—41no—
165C4×C7—42no—
166Dic7—22no—
167Dic7—22no—
168Dic7—24no—
169Dic7—41no—
170Dic7—41no—
171Dic7—42no—
172C24C22×D711yescharacteristic
173C24C22×D711yescharacteristic
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—74no—
180C22×C4—71no—
181C22×C4—71no—
182C22×C4—72no—
183C22×C4—72no—
184C22×C4—71no—
185C22×C4—71no—
186C22×C4—144no—
187C22×C4—142no—
188C22×C4—142no—
189C22×C4—142no—
190C22×C4C22×D712yes—
191C22×C4C22×D711yescharacteristic
192C22×C4C22×D712yes—
193C4⋊C4—142no—
194C4⋊C4—142no—
195C4⋊C4—144no—
196C4⋊C4—144no—
197C22⋊C4—22no—
198C22⋊C4—24no—
199C22⋊C4—144no—
200C22⋊C4—142no—
201C22⋊C4—142no—
202C22⋊C4—142no—
203C22⋊C4—142no—
204C42—144no—
205C42—144no—
206C2×C7—24no—
207C2×C7—81no—
208C2×C7C2×#12311yescharacteristic
209C2×C7#52511yescharacteristic
210C2×C7#52511yescharacteristic
211C2×C7#52912yes—
212C2×C7#52912yes—
213C23—22no—
214C23—22no—
215C23—21no—
216C23—21no—
217C23—22no—
218C23—21no—
219C23—22no—
220C23#115712yes—
221C23D4×D712yes—
222C23#115912yes—
223C23C23×D711yescharacteristic
224D4—44no—
225C2×C4—28no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—72no—
235C2×C4—72no—
236C2×C4—74no—
237C2×C4—148no—
238C2×C4—148no—
239C2×C4—141no—
240C2×C4—141no—
241C2×C4—142no—
242C2×C4—141no—
243C2×C4—141no—
244C2×C4—142no—
245C2×C4—142no—
246C2×C4—141no—
247C2×C4—141no—
248C2×C4—142no—
249C2×C4—142no—
250C2×C4—142no—
251C2×C4—142no—
252C2×C4#115914yes—
253C7#285311yescharacteristic, complemented, Sylow
254C22—24no—
255C22—22no—
256C22—24no—
257C22—22no—
258C22—41no—
259C22—42no—
260C22—41no—
261C22—42no—
262C22—41no—
263C22#644614yes—
264C22C2×#115711yescharacteristic
265C22C2×D4×D711yescharacteristic
266C22C2×#115911yescharacteristic
267C22C2×#115912yes—
268C22D7×#4112yes—
269C4—24no—
270C4—41no—
271C4—42no—
272C4—142no—
273C4—142no—
274C4—144no—
275C4—282no—
276C4—281no—
277C4—281no—
278C2—24no—
279C2—81no—
280C2C2×#644611yescharacteristic
281C2#9022912yes—
282C2#9027711yescharacteristic
283C2#9029112yes—
284C2#9034711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#386C2
prime radicalC22×C7×D4C22
hypercenter#386D7
hypocenterC7#2853
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic7C2semidirect
5. #89982C2semidirect
6. #89723C2semidirect
7. #89722C2semidirect
8. #89717C2semidirect
9. #89715C2semidirect
10. #89426C2semidirect
11. #89424C2semidirect
253. C7#2853semidirect

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
C7×#3862
#899922
#899822
#897232
#897222
#897172
#897152
#894262
#894242
#28537