Finite Group Databasebeta

← #11734142

Subgroups — #11734142

Order 896.

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

normal subgroups only

This group has 932 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
1C22×#1159C211yesmaximal, characteristic, complemented
2C2×#6502C212yesmaximal, complemented
3C2×#6447C211yesmaximal, characteristic, complemented
4C7×#392C211yesmaximal, characteristic, complemented
5#89995C211yesmaximal, characteristic, complemented
6#89995C212yesmaximal, complemented
7#89732C212yesmaximal, complemented
8#89719C211yesmaximal, characteristic, complemented
9#89714C212yesmaximal, complemented
10#89445C211yesmaximal, characteristic, complemented
11#89439C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C22×C7×D4C2212yes—
14C22×C22⋊D7C2211yescharacteristic
15C22×C22⋊D7C2212yes—
16C22×C22⋊D7C2211yescharacteristic
17C22×C22⋊D7C2212yes—
18C23×Dic7C2212yes—
19C2×#1159—28no—
20C2×#1159—24no—
21C22×D28C2211yescharacteristic
22C22×C4×D7C2211yescharacteristic
23C22×Dic14C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2212yes—
26C2×C22⋊Dic7C2212yes—
27C2×C22⋊Dic7C2212yes—
28#6502—28no—
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2212yes—
31C2×(C2×D7)⋊C4C2212yes—
32C2×Dic7⋊C4C2211yescharacteristic
33C2×Dic7⋊C4C2212yes—
34#6447—24no—
35C7×#95C2211yescharacteristic
36C7⋊#95C2211yescharacteristic
37C7⋊#95C2212yes—
38C7⋊#95C2212yes—
39#2615—71nomaximal, Sylow
40C24×C7C2311yescharacteristic
41C24×C7C2312yes—
42C23×D7C2311yescharacteristic
43C23×D7C2311yescharacteristic
44C2×C7×D4—24no—
45C2×C7×D4—24no—
46C2×C7×D4—22no—
47C2×C7×D4—42no—
48C2×C7×D4—42no—
49C22×C4×C7C2311yescharacteristic
50C22×C4×C7C2312yes—
51C22×C4×C7C2311yescharacteristic
52C2×C22⋊D7—24no—
53C2×C22⋊D7—24no—
54C2×C22⋊D7—24no—
55C2×C22⋊D7—22no—
56C2×C22⋊D7—24no—
57C2×C22⋊D7—22no—
58C2×C22⋊D7—24no—
59C2×C22⋊D7—42no—
60C2×C22⋊D7—42no—
61C2×C22⋊D7—42no—
62C22×Dic7—24no—
63C22×Dic7—24no—
64C22×Dic7D418yescomplemented
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2312yes—
67C22×Dic7C2312yes—
68C22×Dic7C2311yescharacteristic
69#1159—232no—
70C2×D28—22no—
71C2×D28—42no—
72C2×C4×D7—24no—
73C2×C4×D7D414yescomplemented
74C2×Dic14—22no—
75C2×Dic14—24no—
76C7×C22⋊C4—22no—
77C7×C22⋊C4—42no—
78C22⋊Dic7—24no—
79C22⋊Dic7—42no—
80(C2×D7)⋊C4—22no—
81(C2×D7)⋊C4—24no—
82(C2×D7)⋊C4—42no—
83Dic7⋊C4—22no—
84Dic7⋊C4—24no—
85C22×#41—71no—
86C2×#121—72no—
87C2×#121—71no—
88#393—71no—
89#393—71no—
90#392—72no—
91#392—71no—
92#392—72no—
93#392D711yescharacteristic, complemented
94#380—72no—
95#380—71no—
96C23×C7—22no—
97C23×C7—22no—
98C23×C7—22no—
99C23×C7—22no—
100C23×C7—22no—
101C23×C7—22no—
102C23×C7—22no—
103C23×C7—21no—
104C23×C7—22no—
105C23×C7—21no—
106C23×C7—21no—
107C23×C7C2×D414yes—
108C23×C7C2411yescharacteristic
109C22×D7—22no—
110C22×D7—21no—
111C22×D7—22no—
112C22×D7—21no—
113C22×D7—21no—
114C22×D7—22no—
115C22×D7—22no—
116C22×D7—21no—
117C22×D7—21no—
118C22×D7C2×D412yes—
119C7×D4—28no—
120C7×D4—48no—
121C2×C4×C7—22no—
122C2×C4×C7—22no—
123C2×C4×C7—21no—
124C2×C4×C7—22no—
125C2×C4×C7—22no—
126C2×C4×C7—22no—
127C2×C4×C7C2×D412yes—
128C22⋊D7—216no—
129C22⋊D7—44no—
130C22⋊D7—48no—
131C22⋊D7—44no—
132C2×Dic7—22no—
133C2×Dic7—22no—
134C2×Dic7—216no—
135C2×Dic7—21no—
136C2×Dic7—22no—
137C2×Dic7—22no—
138C2×Dic7—22no—
139C2×Dic7—22no—
140C2×Dic7C2×D414yes—
141C2×Dic7C2×D414yes—
142C2×Dic7C2×D412yes—
143D28—44no—
144C4×D7—28no—
145Dic14—28no—
146C2×#41—148no—
147C2×#41—144no—
148C22×Q8—71no—
149C22×D4—71no—
150C22×D4—72no—
151C22×D4—71no—
152C22×D4—72no—
153C22×D4—71no—
154C22×D4C2×D711yescharacteristic
155C22×D4C2×D712yes—
156C23×C4—72no—
157C23×C4—71no—
158#121—144no—
159#121—148no—
160C2×C4⋊C4—71no—
161C2×C4⋊C4—72no—
162C2×C22⋊C4—72no—
163C2×C22⋊C4—72no—
164C2×C22⋊C4—71no—
165C2×C22⋊C4—72no—
166C2×C22⋊C4—72no—
167C2×C22⋊C4C2×D711yescharacteristic
168C2×C22⋊C4C2×D712yes—
169#95—72no—
170#95—72no—
171#95—71no—
172#95C2×D711yescharacteristic
173C22×C7—24no—
174C22×C7—24no—
175C22×C7—24no—
176C22×C7—42no—
177C22×C7—42no—
178C22×C7—42no—
179C22×C7—42no—
180C22×C7—41no—
181C22×C7—41no—
182C22×C7—42no—
183C22×C7—42no—
184C22×C7—41no—
185C22×C7C22×D411yescharacteristic
186C22×C7C22×D412yes—
187C22×C7H(5,𝔽2)12yes—
188C22×C7H(5,𝔽2)11yescharacteristic
189C22×C72−1+411yescharacteristic
190C2×D7—22no—
191C2×D7—22no—
192C2×D7—22no—
193C2×D7—42no—
194C2×D7—42no—
195C2×D7—41no—
196C2×D7—41no—
197C2×D7—42no—
198C2×D7—42no—
199C2×D7—41no—
200C4×C7—22no—
201C4×C7—41no—
202C4×C7—42no—
203Dic7—24no—
204Dic7—41no—
205Dic7—42no—
206Dic7#12014yes—
207C24—71no—
208C24—71no—
209C24C22×D711yescharacteristic
210C24C22×D712yes—
211#41—1432no—
212C2×Q8—144no—
213C2×Q8—142no—
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—22no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—142no—
220C2×D4—144no—
221C2×D4—142no—
222C2×D4—144no—
223C2×D4—144no—
224C2×D4—144no—
225C2×D4—142no—
226C2×D4—144no—
227C2×D4—282no—
228C2×D4—282no—
229C2×D4—282no—
230C2×D4—282no—
231C22×C4—71no—
232C22×C4—78no—
233C22×C4—72no—
234C22×C4—74no—
235C22×C4—72no—
236C22×C4—71no—
237C22×C4—144no—
238C22×C4—144no—
239C22×C4—144no—
240C22×C4C22×D711yescharacteristic
241C22×C4C22×D712yes—
242C22×C4C22×D711yescharacteristic
243C4⋊C4—142no—
244C4⋊C4—144no—
245C22⋊C4—22no—
246C22⋊C4—42no—
247C22⋊C4—144no—
248C22⋊C4—144no—
249C22⋊C4—142no—
250C22⋊C4—282no—
251C22⋊C4—282no—
252C2×C7—44no—
253C2×C7—81no—
254C2×C7C2×#12011yescharacteristic
255C2×C7#53511yescharacteristic
256C2×C7#53512yes—
257C2×C7#53712yes—
258C2×C7#53711yescharacteristic
259D7—42no—
260D7—81no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—21no—
268C23—22no—
269C23—21no—
270C23—22no—
271C23—21no—
272C23—72no—
273C23—142no—
274C23—141no—
275C23—142no—
276C23—141no—
277C23—141no—
278C23—142no—
279C23—142no—
280C23—141no—
281C23—141no—
282C23D4×D714yes—
283C23C23×D711yescharacteristic
284Q8—148no—
285D4—28no—
286D4—48no—
287D4—1416no—
288D4—284no—
289D4—284no—
290D4—288no—
291D4—284no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—72no—
299C2×C4—74no—
300C2×C4—74no—
301C2×C4—148no—
302C2×C4—1416no—
303C2×C4—142no—
304C2×C4—142no—
305C2×C4—141no—
306C2×C4—142no—
307C2×C4—142no—
308C2×C4—142no—
309C2×C4—142no—
310C2×C4D4×D712yes—
311C7#261511yescharacteristic, complemented, Sylow
312C22—24no—
313C22—24no—
314C22—24no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22—42no—
323C22—41no—
324C22—142no—
325C22—142no—
326C22—142no—
327C22—282no—
328C22—282no—
329C22—281no—
330C22—281no—
331C22—282no—
332C22—282no—
333C22—281no—
334C22C2×D4×D711yescharacteristic
335C22C2×D4×D712yes—
336C22#654812yes—
337C22#654811yescharacteristic
338C22#655411yescharacteristic
339C4—22no—
340C4—41no—
341C4—42no—
342C4—74no—
343C4—144no—
344C4—281no—
345C4—282no—
346C2—44no—
347C2—81no—
348C2—282no—
349C2—561no—
350C2D7×#12011yescharacteristic
351C2#9028312yes—
352C2#9028812yes—
353C2#9029911yescharacteristic
354C2#9040611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#392C2
hypercenter#392D7
hypocenterC7#2615
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#1159C2semidirect
2. C2×#6502C2semidirect
3. C2×#6447C2semidirect
4. C7×#392C2semidirect
5. #89995C2semidirect
6. #89995C2semidirect
7. #89732C2semidirect
8. #89719C2semidirect
9. #89714C2semidirect
10. #89445C2semidirect
11. #89439C2semidirect
64. C22×Dic7D4semidirect
73. C2×C4×D7D4semidirect
93. #392D7semidirect
311. C7#2615semidirect

Maximal subgroups

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

subgroupindex
C22×#11592
C2×#65022
C2×#64472
C7×#3922
#899952
#897322
#897192
#897142
#894452
#894392
#26157