Finite Group Databasebeta

← #13272026

Subgroups — #13272026

Order 1792.

590 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89923C211yesmaximal, characteristic
2C2×C7×#357C211yesmaximal, characteristic, complemented
3C2×#89357C211yesmaximal, characteristic, complemented
4C2×#89304C211yesmaximal, characteristic, complemented
5C2×#89297C211yesmaximal, characteristic, complemented
6#11728361C211yesmaximal, characteristic, complemented
7C2×#89286C211yesmaximal, characteristic, complemented
8C22×C7×D8C2211yescharacteristic
9C2×#6503—21no—
10C22×#1141—21no—
11C22×D56C2211yescharacteristic
12C22×C7×C16C2211yescharacteristic
13C2×C7×#107C2211yescharacteristic
14#89923—22no—
15#89923C414yescomplemented
16C2×#6392C2211yescharacteristic
17C2×C8×Dic7C2211yescharacteristic
18C22×C7⋊C16C2211yescharacteristic
19#89749—21no—
20C2×#6382—21no—
21C7×#357—22no—
22#89357—22no—
23#89304—22no—
24#89297—22no—
25#89286—22no—
26#13206—71nomaximal, Sylow
27C22×C7×D4—21no—
28C22×C22⋊D7—42no—
29C22×D28—21no—
30C2×C7×D8—21no—
31C2×C7×D8—21no—
32C2×C7×D8—21no—
33C2×C7×D8—21no—
34C2×C7×D8—21no—
35C2×C7×D8C2×C412yes—
36C22×C7×C8C2311yescharacteristic
37C2×C7×C4⋊C4—21no—
38#6503—24no—
39#6503—42no—
40C2×#1141—24no—
41C2×#1141—42no—
42C2×#1141—42no—
43C2×C4⋊Dic7—21no—
44C2×C4×Dic7D411yescharacteristic
45C22×C7⋊C8D411yescharacteristic
46C2×D56—21no—
47C2×D56—21no—
48C2×D56—21no—
49C2×D56—21no—
50C2×D56—21no—
51C2×D56C2×C412yes—
52C2×C7×C16—21no—
53C2×C7×C16—21no—
54C2×C7×C16D412yescomplemented
55C7×#107—21no—
56C7×#107—21no—
57C7⋊#95—42no—
58#6392—21no—
59#6392—21no—
60C8×Dic7C2×C412yes—
61C8×Dic7D412yes—
62C2×C7⋊C16—21no—
63C2×C7⋊C16—21no—
64C2×C7⋊C16D412yescomplemented
65#6382—42no—
66C2×#493—71no—
67C2×#357—71no—
68C2×#357—71no—
69C2×#357—71no—
70C2×#357D711yescharacteristic, complemented
71C2×#346—71no—
72#1846—71no—
73C24×C7—41no—
74C23×D7—41no—
75C2×C7×D4—22no—
76C2×C7×D4—41no—
77C2×C7×D4—41no—
78C2×C7×D4—41no—
79C2×C7×D4—41no—
80C2×C7×D4—41no—
81C22×C4×C7—41no—
82C22×C4×C7C2×D411yescharacteristic
83C2×C22⋊D7—48no—
84C2×C22⋊D7—84no—
85C2×C22⋊D7—84no—
86C22×Dic7—41no—
87C22×Dic7D812yes—
88C2×D28—22no—
89C2×D28—41no—
90C2×D28—41no—
91C2×D28—41no—
92C2×D28—41no—
93C2×D28—41no—
94C7×D8—22no—
95C7×D8—22no—
96C7×D8—41no—
97C7×D8—41no—
98C2×C7×C8C22×C411yescharacteristic
99C2×C7×C8C2×D411yescharacteristic
100C2×C7×C8C2×D411yescharacteristic
101C2×C7×C8C2×D411yescharacteristic
102C2×C7×C8#4111yescharacteristic
103C2×C7×C8#4111yescharacteristic
104C7×C4⋊C4—41no—
105C7×C4⋊C4—41no—
106#1141—48no—
107C4⋊Dic7—41no—
108C4⋊Dic7—41no—
109C4×Dic7C22⋊C412yes—
110C4×Dic7SD1612yes—
111C2×C7⋊C8C22⋊C412yes—
112C2×C7⋊C8D812yes—
113C2×C7⋊C8SD1612yes—
114D56—22no—
115D56—22no—
116D56—41no—
117D56—41no—
118C7×C16—22no—
119C7⋊C16—22no—
120C22×D8—71no—
121C22×D8—141no—
122C22×D8C2×D711yescharacteristic
123C2×#127—141no—
124C22×C16—71no—
125C22×C16C2×D711yescharacteristic
126#493—74no—
127#493—142no—
128C2×#107—71no—
129C2×#107C2×D711yescharacteristic
130C2×#106—141no—
131C2×C4×C8—71no—
132#382—141no—
133#357—22no—
134#357—142no—
135#357—142no—
136#357—142no—
137#346—142no—
138C23×C7—42no—
139C23×C7—81no—
140C23×C7—81no—
141C23×C7—82no—
142C23×C7—82no—
143C23×C7C2×D811yescharacteristic
144C22×D7—42no—
145C22×D7—81no—
146C22×D7—81no—
147C22×D7—82no—
148C22×D7—82no—
149C7×D4—42no—
150C7×D4—42no—
151C7×D4—81no—
152C7×D4—81no—
153C2×C4×C7—81no—
154C2×C4×C7—82no—
155C2×C4×C7C2×C22⋊C411yescharacteristic
156C2×C4×C7#12411yescharacteristic
157C2×C4×C7#12711yescharacteristic
158C2×C4×C7C2×D811yescharacteristic
159C2×C4×C7C2×SD1611yescharacteristic
160C2×C4×C7C2×SD1611yescharacteristic
161C22⋊D7—816no—
162C2×Dic7—22no—
163C2×Dic7—22no—
164C2×Dic7—81no—
165C2×Dic7—82no—
166C2×Dic7#10214yes—
167D28—42no—
168D28—42no—
169D28—81no—
170D28—81no—
171C7×C8C4×D411yescharacteristic
172C7×C8C4×D411yescharacteristic
173C7×C8#12111yescharacteristic
174C7×C8#12111yescharacteristic
175C7⋊C8#10214yes—
176C22×D4—21no—
177C22×D4—141no—
178C22×D4—282no—
179C2×D8—21no—
180C2×D8—21no—
181C2×D8—21no—
182C2×D8—21no—
183C2×D8—21no—
184C2×D8—72no—
185C2×D8—144no—
186C2×D8—141no—
187C2×D8—141no—
188C2×D8—141no—
189C2×D8—141no—
190C2×D8—141no—
191C2×D8—282no—
192C2×D8—282no—
193C2×D8C4×D712yes—
194C22×C8—71no—
195C22×C8C22×D711yescharacteristic
196#127—144no—
197#127—282no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—141no—
200C2×C42—71no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—72no—
204C2×C16—141no—
205C2×C16—141no—
206C2×C16C22⋊D712yescomplemented
207#107—21no—
208#107—21no—
209#107—141no—
210#107—141no—
211#106—282no—
212C4×C8—72no—
213C4×C8—72no—
214#95—282no—
215C22×C7—84no—
216C22×C7—82no—
217C22×C7—162no—
218C22×C7—161no—
219C22×C7—161no—
220C22×C7C2×#10211yescharacteristic
221C22×C7#48611yescharacteristic
222C22×C7#49311yescharacteristic
223C22×C7#50912yes—
224C22×C7#50912yes—
225C2×D7—84no—
226C2×D7—82no—
227C2×D7—162no—
228C2×D7—161no—
229C2×D7—161no—
230C4×C7—161no—
231C4×C7#39011yescharacteristic
232C4×C7C2×#10211yescharacteristic
233C4×C7#48611yescharacteristic
234C4×C7#49211yescharacteristic
235Dic7—24no—
236Dic7—161no—
237C24—41no—
238C24—281no—
239C2×D4—22no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—142no—
246C2×D4—288no—
247C2×D4—281no—
248C2×D4—281no—
249C2×D4—281no—
250C2×D4—281no—
251C2×D4—281no—
252C2×D4—564no—
253C2×D4—564no—
254C22×C4—41no—
255C22×C4—72no—
256C22×C4—281no—
257C22×C4D4×D711yescharacteristic
258D8—22no—
259D8—22no—
260D8—41no—
261D8—41no—
262D8—142no—
263D8—142no—
264D8—288no—
265D8—281no—
266D8—281no—
267C2×C8—72no—
268C2×C8—72no—
269C2×C8—72no—
270C2×C8C2×C4×D711yescharacteristic
271C2×C8#115711yescharacteristic
272C2×C8D4×D711yescharacteristic
273C2×C8D4×D711yescharacteristic
274C2×C8#115911yescharacteristic
275C2×C8C2×C22⋊D711yescharacteristic
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—281no—
279C4⋊C4—281no—
280C42—72no—
281C42—72no—
282C16—22no—
283C16—142no—
284C2×C7—162no—
285C2×C7#195711yescharacteristic
286C2×C7#214012yes—
287C2×C7#224412yes—
288C2×C7#225011yescharacteristic
289C2×C7#225011yescharacteristic
290D7—162no—
291C23—42no—
292C23—81no—
293C23—82no—
294C23—82no—
295C23—81no—
296C23—282no—
297C23—561no—
298C23—562no—
299C23—562no—
300C23—561no—
301C23D7×D811yescharacteristic
302D4—42no—
303D4—42no—
304D4—81no—
305D4—81no—
306D4—282no—
307D4—282no—
308D4—5616no—
309D4—561no—
310D4—561no—
311C2×C4—74no—
312C2×C4—81no—
313C2×C4—82no—
314C2×C4—142no—
315C2×C4—142no—
316C2×C4—561no—
317C2×C4—562no—
318C2×C4D7×C22⋊C411yescharacteristic
319C2×C4#644811yescharacteristic
320C2×C4D7×D811yescharacteristic
321C2×C4D7×SD1611yescharacteristic
322C2×C4D7×SD1611yescharacteristic
323C2×C4#650311yescharacteristic
324C8—74no—
325C8(C22⋊D7)⋊C411yescharacteristic
326C8#644711yescharacteristic
327C8C4×C22⋊D711yescharacteristic
328C8#650211yescharacteristic
329C7#1320611yescharacteristic, complemented, Sylow
330C22—82no—
331C22—84no—
332C22—161no—
333C22—161no—
334C22—162no—
335C22—562no—
336C22—564no—
337C22—1121no—
338C22—1121no—
339C22—1122no—
340C22#8953211yescharacteristic
341C22D7×#10211yescharacteristic
342C22#8968312yes—
343C22#8968612yes—
344C22#8992311yescharacteristic
345C4—144no—
346C4—161no—
347C4—1121no—
348C4#8953211yescharacteristic
349C4D7×#10211yescharacteristic
350C4#8972911yescharacteristic
351C4#8994711yescharacteristic
352C2—162no—
353C2—1122no—
354C2#1172694012yes—
355C2#1172694412yes—
356C2#1172791111yescharacteristic
357C2#1172922011yescharacteristic
358C2#1172923511yescharacteristic
subgroupG/H
centerC23D7×D8
commutatorC2×C7×C8C22×C4
FrattiniC22×C8C22×D7
FittingC2×C7×#357C2
socleC23×C7C2×D8
prime radicalC2×#89923C2
cosocleC22×C7×C8C23
hypercenterC2×#357D7
hypocenterC7#13206
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
2. C2×C7×#357C2semidirect
3. C2×#89357C2semidirect
4. C2×#89304C2semidirect
5. C2×#89297C2semidirect
6. #11728361C2semidirect
7. C2×#89286C2semidirect
15. #89923C4semidirect
54. C2×C7×C16D4semidirect
64. C2×C7⋊C16D4semidirect
70. C2×#357D7semidirect
206. C2×C16C22⋊D7semidirect
329. C7#13206semidirect

Maximal subgroups

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

subgroupindex
C2×#899232
C2×C7×#3572
C2×#893572
C2×#893042
C2×#892972
#117283612
C2×#892862
#132067