Finite Group Databasebeta

← #13372261

Subgroups — #13372261

Order 1792.

870 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89667C212yesmaximal
2#11736494C212yesmaximal, complemented
3C4×#6467C211yesmaximal, characteristic
4C7×#1848C211yesmaximal, characteristic, complemented
5#11728476C214yesmaximal, complemented
6#11728458C212yesmaximal, complemented
7#11728283C212yesmaximal, complemented
8C8×C4⋊Dic7C211yesmaximal, characteristic, complemented
9C2×#6470C2214yes—
10C2×#6467C2211yescharacteristic
11C4×#1157C2212yes—
12C2×C7×#108C2212yes—
13C2×C7×#108C2212yes—
14C7×#423C2212yes—
15C2×C4×C7×C8C2211yescharacteristic
16C2×#6399C2214yes—
17#89891C2214yes—
18#89885C2212yes—
19#89878C2211yescharacteristic
20C2×C8×Dic7C2212yes—
21C4×C4⋊Dic7C2211yescharacteristic
22C2×C4⋊(C7⋊C8)C2211yescharacteristic
23#89667—28no—
24#89611—24no—
25#89609—22no—
26D7×C4⋊C8—22no—
27C4×Dic28C411yescharacteristic, complemented
28C4×Dic28C411yescharacteristic, complemented
29C4×D56C411yescharacteristic, complemented
30C4×D56C411yescharacteristic, complemented
31C4×#1132C412yescomplemented
32C4×#1132C412yescomplemented
33#89350C2214yes—
34#89345C2212yes—
35#22688—71nomaximal, Sylow
36C2×#1157C2312yes—
37C2×C7×M16C2314yes—
38C22×C7×C8C2311yescharacteristic
39C22×C7×C8C2311yescharacteristic
40C2×C42×C7C2311yescharacteristic
41C4×C22⋊D7—22no—
42C4×C22⋊D7—22no—
43#6489C2312yes—
44C2×C4⋊Dic7—21no—
45C2×C4×Dic7C2312yes—
46C22×C7⋊C8C2312yes—
47#6470—28no—
48#6470—44no—
49C2×Dic28C2×C411yescharacteristic
50#6467D414yes—
51#6467Q814yes—
52C2×D56C2×C411yescharacteristic
53C2×#1132C2×C412yes—
54C2×#1131—24no—
55C2×C8×D7—24no—
56C4×D28C2×C412yes—
57C4×D28C2×C412yes—
58#6435—22no—
59C42×D7—22no—
60C4×Dic14C2×C412yes—
61C4×Dic14C2×C412yes—
62C7×#108—24no—
63C7×#108—42no—
64C7×C4⋊C8—22no—
65C7×C4⋊C8—22no—
66C4×C7×C8C2×C411yescharacteristic
67C4×C7×C8C2×C411yescharacteristic
68C4×C7×C8C2×C411yescharacteristic
69C4×C7×C8C2×C411yescharacteristic
70C7⋊#95—21no—
71#6399—28no—
72#6395C2×C412yes—
73#6395C2×C412yes—
74#6392C2×C411yescharacteristic
75#6392C2×C411yescharacteristic
76#6391C2×C412yes—
77#6390C2×C412yes—
78#6390C2×C412yes—
79Dic7⋊C8—24no—
80Dic7⋊C8—24no—
81C8×Dic7—24no—
82C4⋊(C7⋊C8)D411yescharacteristic
83C4⋊(C7⋊C8)D411yescharacteristic
84C4⋊(C7⋊C8)D412yes—
85C2×#443—72no—
86C4×#135—71no—
87#2906—72no—
88#1945—72no—
89#1866—74no—
90#1848D711yescharacteristic, complemented
91#1806—72no—
92C8×C4⋊C4—71no—
93C22×C4×C7#4111yescharacteristic
94C22×C4×C7#4111yescharacteristic
95C22×C4×C7C2411yescharacteristic
96C2×C22⋊D7—22no—
97C22×Dic7—22no—
98#1157—22no—
99#1157—22no—
100#1157C2×D414yes—
101#1157C2×Q814yes—
102C2×D28C22×C412yes—
103C2×C4×D7—22no—
104C2×C4×D7—22no—
105C2×C4×D7—22no—
106C2×Dic14C22×C412yes—
107C7×M16—24no—
108C7×M16—24no—
109C2×C7×C8—21no—
110C2×C7×C8—21no—
111C2×C7×C8—24no—
112C2×C7×C8C22×C411yescharacteristic
113C2×C7×C8C22×C411yescharacteristic
114C2×C7×C8C22×C411yescharacteristic
115C2×C7×C8C22×C411yescharacteristic
116C2×C7×C8C2×D412yes—
117C2×C7×C8C2×Q812yes—
118C42×C7C22×C411yescharacteristic
119C42×C7C22×C411yescharacteristic
120C42×C7C2×D411yescharacteristic
121C42×C7C2×D411yescharacteristic
122C22⋊Dic7—22no—
123(C2×D7)⋊C4—22no—
124(C2×D7)⋊C4—22no—
125C4⋊Dic7—24no—
126C4⋊Dic7C22×C412yes—
127C4⋊Dic7C22×C412yes—
128Dic7⋊C4—22no—
129Dic7⋊C4—22no—
130C4×Dic7—22no—
131C4×Dic7—22no—
132C4×Dic7—22no—
133C4×Dic7—22no—
134C4×Dic7—22no—
135C2×C7⋊C8—22no—
136C2×C7⋊C8—22no—
137C2×C7⋊C8C2×D412yes—
138C2×C7⋊C8C2×D412yes—
139Dic28C4⋊C414yescomplemented
140D56C4⋊C414yescomplemented
141#1132C4⋊C418yescomplemented
142#1131—48no—
143C8×D7—48no—
144C2×#135—71no—
145C2×#131—74no—
146C4×#41—72no—
147#443—148no—
148C4×Q16—71no—
149C4×Q16—71no—
150C4×SD16—72no—
151C4×SD16—72no—
152C4×D8—71no—
153C4×D8—71no—
154C2×#108C2×D712yes—
155C2×#108C2×D712yes—
156#427—71no—
157#424—74no—
158#424—144no—
159#423—142no—
160#423C2×D712yes—
161C2×C4⋊C8—71no—
162C2×C4⋊C8—142no—
163C2×#104—74no—
164#416—72no—
165C2×C4×C8—72no—
166C2×C4×C8C2×D711yescharacteristic
167C4×C4⋊C4—71no—
168#339—74no—
169#336—72no—
170C23×C7C2×#4111yescharacteristic
171C22×D7—22no—
172C2×C4×C7—22no—
173C2×C4×C7—41no—
174C2×C4×C7C4×D411yescharacteristic
175C2×C4×C7C4×D411yescharacteristic
176C2×C4×C7C4×D411yescharacteristic
177C2×C4×C7C4×D411yescharacteristic
178C2×C4×C7C23×C411yescharacteristic
179C2×C4×C7C22×D411yescharacteristic
180C2×C4×C7C22×D411yescharacteristic
181C2×C4×C7C22×Q811yescharacteristic
182C2×C4×C7H(5,𝔽2)11yescharacteristic
183C2×C4×C72−1+411yescharacteristic
184C22⋊D7—24no—
185C22⋊D7—24no—
186C2×Dic7—22no—
187C2×Dic7—24no—
188C2×Dic7—22no—
189C2×Dic7—22no—
190C2×Dic7—42no—
191C2×Dic7—42no—
192D28—22no—
193D28C2×C4⋊C414yes—
194C4×D7—24no—
195C4×D7—24no—
196C4×D7—44no—
197C4×D7—82no—
198Dic14—22no—
199Dic14C2×C4⋊C414yes—
200C7×C8—21no—
201C7×C8—21no—
202C7×C8—44no—
203C7×C8C2×C4⋊C412yes—
204C7×C8C2×C4⋊C412yes—
205C7⋊C8—22no—
206C7⋊C8—22no—
207C2×#41—72no—
208#135—74no—
209#135—74no—
210C2×Q16—71no—
211C2×SD16—72no—
212C2×D8—71no—
213#131—148no—
214#131—284no—
215C2×M16—144no—
216C2×M16C22×D714yes—
217C22×C8—72no—
218C22×C8—144no—
219C22×C8C22×D711yescharacteristic
220C22×C8C22×D711yescharacteristic
221C4×Q8—72no—
222C4×Q8—72no—
223C4×D4—72no—
224C4×D4—72no—
225C4×D4—142no—
226C4×D4—142no—
227#117—72no—
228#117—142no—
229C2×C4⋊C4—141no—
230C2×C42—72no—
231C2×C42—142no—
232C2×C42C22×D711yescharacteristic
233#108—24no—
234#108—42no—
235#107—71no—
236#107—71no—
237#106—72no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—71no—
241C4⋊C8—71no—
242C4⋊C8—72no—
243C4⋊C8—144no—
244C4⋊C8—144no—
245#104—148no—
246#103—72no—
247#103—72no—
248#102—72no—
249#102—72no—
250C4×C8—144no—
251C4×C8C4×D711yescharacteristic
252C4×C8C4×D711yescharacteristic
253C4×C8C4×D711yescharacteristic
254C4×C8C4×D711yescharacteristic
255#95—141no—
256C22×C7—21no—
257C22×C7—21no—
258C22×C7C2×C4×D411yescharacteristic
259C22×C7#54711yescharacteristic
260C22×C7#55411yescharacteristic
261C2×D7—24no—
262C2×D7—42no—
263C2×D7—42no—
264C4×C7—21no—
265C4×C7—21no—
266C4×C7—21no—
267C4×C7#44312yescomplemented
268C4×C7C22×C4⋊C411yescharacteristic
269C4×C7#52011yescharacteristic
270C4×C7D4211yescharacteristic
271C4×C7D4×Q811yescharacteristic
272Dic7—24no—
273Dic7—44no—
274Dic7—42no—
275Dic7—42no—
276#41—74no—
277#41—74no—
278#41—142no—
279#41—142no—
280C2×Q8—72no—
281C2×D4—72no—
282C2×D4—142no—
283C22×C4—142no—
284C22×C4—142no—
285C22×C4—142no—
286C22×C4—142no—
287C22×C4#116111yescharacteristic
288C22×C4#116111yescharacteristic
289C22×C4C23×D711yescharacteristic
290Q16—74no—
291SD16—78no—
292D8—74no—
293M16—24no—
294M16—24no—
295M16—288no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—24no—
299C2×C8—72no—
300C2×C8—72no—
301C2×C8—142no—
302C2×C8—142no—
303C2×C8—288no—
304C2×C8C2×C4×D711yescharacteristic
305C2×C8C2×C4×D711yescharacteristic
306C2×C8C2×C4×D711yescharacteristic
307C2×C8C2×C4×D711yescharacteristic
308C2×C8D4×D712yes—
309C2×C8Q8×D712yes—
310C4⋊C4—72no—
311C4⋊C4—72no—
312C4⋊C4—144no—
313C4⋊C4—142no—
314C4⋊C4—142no—
315C22⋊C4—142no—
316C22⋊C4—142no—
317C22⋊C4—142no—
318C42—142no—
319C42—142no—
320C42—142no—
321C42—142no—
322C42—142no—
323C42C2×C4×D711yescharacteristic
324C42C2×C4×D711yescharacteristic
325C42D4×D711yescharacteristic
326C42D4×D711yescharacteristic
327C2×C7—21no—
328C2×C7—21no—
329C2×C7D4×C4⋊C411yescharacteristic
330C2×C7C2×#44311yescharacteristic
331C2×C7#295711yescharacteristic
332D7—44no—
333C23—142no—
334C23C2×#116111yescharacteristic
335Q8—74no—
336Q8—142no—
337D4—74no—
338D4—144no—
339D4—144no—
340D4—142no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—142no—
344C2×C4—144no—
345C2×C4—144no—
346C2×C4—144no—
347C2×C4—142no—
348C2×C4—142no—
349C2×C4—282no—
350C2×C4—282no—
351C2×C4—284no—
352C2×C4—562no—
353C2×C4#645611yescharacteristic
354C2×C4#645611yescharacteristic
355C2×C4#645611yescharacteristic
356C2×C4#645611yescharacteristic
357C2×C4C22×C4×D711yescharacteristic
358C2×C4C2×D4×D711yescharacteristic
359C2×C4C2×D4×D711yescharacteristic
360C2×C4#654811yescharacteristic
361C2×C4C2×Q8×D711yescharacteristic
362C2×C4#655111yescharacteristic
363C8—21no—
364C8—21no—
365C8—44no—
366C8—142no—
367C8—142no—
368C8D7×C4⋊C412yes—
369C8D7×C4⋊C412yes—
370C7#2268811yescharacteristic, complemented, Sylow
371C22—21no—
372C22—21no—
373C22—144no—
374C22—282no—
375C22—282no—
376C22C2×#645611yescharacteristic
377C22#9041711yescharacteristic
378C22#9042111yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—144no—
383C4—284no—
384C4—282no—
385C4—282no—
386C4#8966712yescomplemented
387C4C2×D7×C4⋊C411yescharacteristic
388C4#9019511yescharacteristic
389C4#9040811yescharacteristic
390C4#9041811yescharacteristic
391C2—21no—
392C2—21no—
393C2—284no—
394C2#1173180511yescharacteristic
395C2C2×#8966711yescharacteristic
396C2#1173669411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1848C2
socleC22×C7C2×C4×D4
prime radicalC2×#6467C22
cosocleC22×C4×C7C24
hypercenter#1848D7
hypocenterC7#22688
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1848

Complements

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

NH ≅ G/Ntype
2. #11736494C2semidirect
4. C7×#1848C2semidirect
5. #11728476C2semidirect
6. #11728458C2semidirect
7. #11728283C2semidirect
8. C8×C4⋊Dic7C2semidirect
27. C4×Dic28C4semidirect
28. C4×Dic28C4semidirect
29. C4×D56C4semidirect
30. C4×D56C4semidirect
31. C4×#1132C4semidirect
32. C4×#1132C4semidirect
90. #1848D7semidirect
139. Dic28C4⋊C4semidirect
140. D56C4⋊C4semidirect
141. #1132C4⋊C4semidirect
267. C4×C7#443semidirect
370. C7#22688semidirect
386. C4#89667semidirect

Maximal subgroups

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

subgroupindex
C2×#896672
#117364942
C4×#64672
C7×#18482
#117284762
#117284582
#117282832
C8×C4⋊Dic72
#226887