Finite Group Databasebeta

← #13370354

Subgroups — #13370354

Order 1792.

1198 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic14C211yesmaximal, characteristic
2#11736491C212yesmaximal, complemented
3C7×#1800C211yesmaximal, characteristic
4C4×#6407C211yesmaximal, characteristic
5#11728278C212yesmaximal, complemented
6#11727817C212yesmaximal
7Dic7×#100C212yesmaximal
8#11726728C214yesmaximal, complemented
9C23×Dic14C412yes—
10C2×C4×C7×D4C2211yescharacteristic
11C2×D4×Dic7C2212yes—
12C2×#6487C412yes—
13D4×Dic14—24no—
14D4×Dic14—22no—
15D4×Dic14—22no—
16C2×#6451C2212yes—
17C2×#6441C414yes—
18C2×#6432C2211yescharacteristic
19C2×C4×Dic14C2211yescharacteristic
20C7×#423C2212yes—
21C2×C7×#100C2212yes—
22C2×#6407C2211yescharacteristic
23C2×#6407C2211yescharacteristic
24#89890C2214yes—
25M16×Dic7C2214yes—
26C4×#1136C2212yes—
27#89603—22no—
28#89602—22no—
29#89600—24no—
30C7×#343C2212yes—
31#89349C2214yes—
32#89349C2214yes—
33#89325C2212yes—
34Q8×#100—71nomaximal, Sylow
35C22×C7×D4C2311yescharacteristic
36C23×C4×C7C2×C412yes—
37C23×Dic7C2×C414yes—
38C22×Dic14—24no—
39C22×Dic14—22no—
40C22×Dic14—22no—
41C22×Dic14—48no—
42C22×Dic14C2×C411yescharacteristic
43C22×Dic14C2×C412yes—
44C22×Dic14C2×C414yes—
45C2×C7×M16C2314yes—
46C4×C7×D4—22no—
47C4×C7×D4—22no—
48C2×C7×C4⋊C4C2311yescharacteristic
49C2×C7×C22⋊C4C2×C412yes—
50C2×C42×C7C2311yescharacteristic
51C2×C22⋊Dic7C2×C414yes—
52D4×Dic7—24no—
53D4×Dic7—24no—
54C2×C4⋊Dic7C2×C412yes—
55C2×C4⋊Dic7C2×C411yescharacteristic
56C2×C4⋊Dic7C2312yes—
57#6487—24no—
58#6487—24no—
59C2×Dic7⋊C4C2×C414yes—
60C2×Dic7⋊C4C2×C412yes—
61C2×C4×Dic7C2312yes—
62C2×#1136C2312yes—
63C2×#1136C2312yes—
64#6451—24no—
65#6451—24no—
66#6441—28no—
67#6441—28no—
68#6432—22no—
69#6432—22no—
70C4×Dic14D414yes—
71C4×Dic14D412yes—
72C4×Dic14D412yes—
73C7×C4⋊C8—22no—
74C7×C4⋊C8—22no—
75C7×#100—24no—
76#6407—22no—
77#6407Q814yescomplemented
78#6389—24no—
79Dic7⋊C8—24no—
80Dic7⋊C8—24no—
81C8×Dic7—24no—
82#6377—22no—
83C4×C7⋊C8—22no—
84C2×D4×Q8—71no—
85Q8×M16—72no—
86#1997—72no—
87#1997—74no—
88#1800—72no—
89#1800D711yescharacteristic
90C4×#100—72no—
91C4×#100—71no—
92C24×C7C22×C412yes—
93C2×C7×D4—22no—
94C2×C7×D4—22no—
95C2×C7×D4C2×Q812yes—
96C2×C7×D4#4112yes—
97C22×C4×C7—22no—
98C22×C4×C7—22no—
99C22×C4×C7—44no—
100C22×C4×C7C22×C411yescharacteristic
101C22×C4×C7C22×C412yes—
102C22×C4×C7C22×C411yescharacteristic
103C22×C4×C7C2411yescharacteristic
104C22×Dic7—24no—
105C22×Dic7—24no—
106C22×Dic7—48no—
107C22×Dic7C22×C412yes—
108C22×Dic7C22×C414yes—
109C22×Dic7C22×C412yes—
110C2×Dic14—24no—
111C2×Dic14—22no—
112C2×Dic14—22no—
113C2×Dic14—24no—
114C2×Dic14—24no—
115C2×Dic14—48no—
116C2×Dic14—416no—
117C2×Dic14—48no—
118C2×Dic14—48no—
119C2×Dic14—44no—
120C2×Dic14C22⋊C411yescharacteristic
121C2×Dic14C22⋊C412yes—
122C2×Dic14C22⋊C411yescharacteristic
123C7×M16—24no—
124C7×M16—24no—
125C2×C7×C8—24no—
126C7×C4⋊C4—22no—
127C7×C22⋊C4—24no—
128C42×C7C2×D412yes—
129C42×C7C2×D412yes—
130C22⋊Dic7—28no—
131C4⋊Dic7—24no—
132C4⋊Dic7—24no—
133C4⋊Dic7C22⋊C412yes—
134C4⋊Dic7C22⋊C412yes—
135Dic7⋊C4—28no—
136Dic7⋊C4C22⋊C414yes—
137Dic7⋊C4C22⋊C414yes—
138C4×Dic7C2×D414yes—
139C4×Dic7C2×D414yes—
140#1136—22no—
141#1136—22no—
142#1136—22no—
143#1136C2×Q814yescomplemented
144C2×C7⋊C8—22no—
145C2×C7⋊C8—22no—
146C23×Q8—72no—
147D4×Q8—142no—
148D4×Q8—142no—
149D4×Q8—144no—
150C2×C4⋊Q8—72no—
151C2×C4⋊Q8—71no—
152C2×C22⋊Q8—72no—
153C2×C22⋊Q8—74no—
154C2×C4×Q8—71no—
155C2×C4×D4—72no—
156C2×C4×D4C2×D711yescharacteristic
157#446—144no—
158#446—142no—
159C8×Q8—142no—
160#423—74no—
161#423C2×D712yes—
162C2×#100—71no—
163C2×#100—71no—
164C2×#100C2×D712yes—
165C4×M16—74no—
166C4×M16—72no—
167#343—74no—
168#343—74no—
169#343—72no—
170#343C2×D712yes—
171C23×C7—22no—
172C23×C7—22no—
173C23×C7—42no—
174C23×C7—42no—
175C23×C7C4×Q814yes—
176C23×C7C23×C411yescharacteristic
177C7×D4—24no—
178C7×D4—24no—
179C2×C4×C7—22no—
180C2×C4×C7—22no—
181C2×C4×C7—41no—
182C2×C4×C7—44no—
183C2×C4×C7—44no—
184C2×C4×C7—42no—
185C2×C4×C7C2×C22⋊C412yes—
186C2×C4×C7C2×C22⋊C411yescharacteristic
187C2×C4×C7C2×C22⋊C411yescharacteristic
188C2×C4×C7C22×D412yes—
189C2×C4×C7C22×Q811yescharacteristic
190C2×C4×C7C2×#4111yescharacteristic
191C2×C4×C72−1+412yes—
192C2×Dic7—24no—
193C2×Dic7—24no—
194C2×Dic7—42no—
195C2×Dic7—48no—
196C2×Dic7—48no—
197C2×Dic7—44no—
198C2×Dic7C2×C22⋊C414yes—
199C2×Dic7C2×C22⋊C412yes—
200C2×Dic7C2×C22⋊C412yes—
201Dic14—22no—
202Dic14—24no—
203Dic14—48no—
204Dic14—44no—
205Dic14—44no—
206Dic14—48no—
207Dic14#10014yescomplemented
208C7×C8—44no—
209C7⋊C8—24no—
210C7⋊C8—42no—
211C22×Q8—71no—
212C22×Q8—74no—
213C22×Q8—72no—
214C22×Q8—142no—
215C22×Q8—144no—
216C22×Q8—142no—
217C22×Q8—288no—
218C22×D4C22×D711yescharacteristic
219C23×C4—74no—
220C23×C4C4×D712yes—
221C2×M16—72no—
222C2×M16—72no—
223C2×M16C22×D714yes—
224C4⋊Q8—144no—
225C4⋊Q8—144no—
226C4⋊Q8—142no—
227C4⋊Q8—142no—
228C22⋊Q8—148no—
229C22⋊Q8—148no—
230C22⋊Q8—144no—
231C22⋊Q8—144no—
232C4×Q8—72no—
233C4×Q8—74no—
234C4×Q8—72no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—144no—
238C4×D4—144no—
239C2×C4⋊C4—72no—
240C2×C4⋊C4—71no—
241C2×C4⋊C4—74no—
242C2×C4⋊C4—72no—
243C2×C4⋊C4—72no—
244C2×C4⋊C4C22×D711yescharacteristic
245C2×C22⋊C4—74no—
246C2×C22⋊C4C4×D712yes—
247C2×C42—72no—
248C2×C42C22×D711yescharacteristic
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—144no—
252C4⋊C8—144no—
253#100—24no—
254#100—74no—
255#100—142no—
256#97—144no—
257#97—142no—
258C4×C8—144no—
259C4×C8—142no—
260C22×C7—21no—
261C22×C7—21no—
262C22×C7—24no—
263C22×C7—44no—
264C22×C7—44no—
265C22×C7—42no—
266C22×C7—42no—
267C22×C7C22×C22⋊C411yescharacteristic
268C22×C7C2×C4×Q811yescharacteristic
269C22×C7#51911yescharacteristic
270C4×C7—21no—
271C4×C7—22no—
272C4×C7—42no—
273C4×C7C2×#10012yes—
274C4×C7#54812yes—
275C4×C7D4×Q812yes—
276Dic7—22no—
277Dic7—24no—
278Dic7—44no—
279Dic7C2×#10014yes—
280C24C2×C4×D712yes—
281C2×Q8—71no—
282C2×Q8—72no—
283C2×Q8—71no—
284C2×Q8—144no—
285C2×Q8—144no—
286C2×Q8—144no—
287C2×Q8—142no—
288C2×Q8—142no—
289C2×Q8—284no—
290C2×Q8—288no—
291C2×Q8—288no—
292C2×Q8—2816no—
293C2×Q8—288no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4#115912yes—
297C2×D4Q8×D712yes—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—44no—
301C22×C4—72no—
302C22×C4—72no—
303C22×C4—74no—
304C22×C4—144no—
305C22×C4—144no—
306C22×C4—288no—
307C22×C4C2×C4×D711yescharacteristic
308C22×C4C2×C4×D712yes—
309C22×C4C2×C4×D711yescharacteristic
310C22×C4C23×D711yescharacteristic
311M16—24no—
312M16—24no—
313M16—74no—
314M16—142no—
315M16—142no—
316M16—142no—
317C2×C8—24no—
318C2×C8—142no—
319C2×C8—142no—
320C4⋊C4—22no—
321C4⋊C4—74no—
322C4⋊C4—74no—
323C4⋊C4—72no—
324C4⋊C4—72no—
325C4⋊C4—148no—
326C4⋊C4—144no—
327C4⋊C4—144no—
328C22⋊C4—24no—
329C22⋊C4—148no—
330C42—74no—
331C42—74no—
332C42D4×D712yes—
333C42D4×D712yes—
334C2×C7—21no—
335C2×C7—21no—
336C2×C7—44no—
337C2×C7Q8×C22⋊C411yescharacteristic
338C2×C7C22×#10011yescharacteristic
339C2×C7#288911yescharacteristic
340C23—22no—
341C23—22no—
342C23—42no—
343C23—42no—
344C23#645014yes—
345C23C22×C4×D711yescharacteristic
346Q8—74no—
347Q8—144no—
348Q8—142no—
349Q8—284no—
350Q8—288no—
351Q8—288no—
352Q8—284no—
353D4—24no—
354D4—24no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—72no—
362C2×C4—74no—
363C2×C4—72no—
364C2×C4—144no—
365C2×C4—144no—
366C2×C4—288no—
367C2×C4—282no—
368C2×C4—288no—
369C2×C4—284no—
370C2×C4D7×C22⋊C412yes—
371C2×C4D7×C22⋊C411yescharacteristic
372C2×C4D7×C22⋊C411yescharacteristic
373C2×C4C2×D4×D712yes—
374C2×C4C2×#115911yescharacteristic
375C2×C4C2×Q8×D711yescharacteristic
376C2×C4#655412yes—
377C8—44no—
378C8—144no—
379C8—282no—
380C7Q8×#10011yescharacteristic, complemented, Sylow
381C22—21no—
382C22—21no—
383C22—24no—
384C22—44no—
385C22—44no—
386C22—42no—
387C22—42no—
388C22C2×D7×C22⋊C411yescharacteristic
389C22C2×#645011yescharacteristic
390C22#9020711yescharacteristic
391C4—21no—
392C4—22no—
393C4—42no—
394C4—74no—
395C4—142no—
396C4—144no—
397C4—284no—
398C4D7×#10012yes—
399C4#9028912yes—
400C4#9032312yes—
401C2—21no—
402C2—21no—
403C2—44no—
404C2#1173132111yescharacteristic
405C2C2×D7×#10011yescharacteristic
406C2#1173616011yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1800C2
socleC22×C7C22×C22⋊C4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenter#1800D7
hypocenterC7Q8×#100
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #11736491C2semidirect
5. #11728278C2semidirect
8. #11726728C2semidirect
77. #6407Q8semidirect
143. #1136C2×Q8semidirect
207. Dic14#100semidirect
380. C7Q8×#100semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic142
#117364912
C7×#18002
C4×#64072
#117282782
#117278172
Dic7×#1002
#117267282
Q8×#1007