Finite Group Databasebeta

← #13319971

Subgroups — #13319971

Order 1792.

848 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90209C211yesmaximal, characteristic
2#11736759C212yesmaximal, complemented
3C7×#1464C211yesmaximal, characteristic
4#11729326C211yesmaximal, characteristic
5#11728584C212yesmaximal, complemented
6#11725882C214yesmaximal, complemented
7#11725859C214yesmaximal
8C2×C7×#117C2211yescharacteristic
9C2×#6487C411yescharacteristic
10C2×#6487C411yescharacteristic
11#90209—24no—
12#90209—24no—
13C2×#6452C412yes—
14C2×#6452C2212yes—
15C2×#6442C412yes—
16C2×#6442C2212yes—
17C2×#6433C411yescharacteristic
18C2×#6432C411yescharacteristic
19C2×C4×Dic14C2211yescharacteristic
20C2×C4×Dic14C2211yescharacteristic
21C7×#432C2212yes—
22#89698C2212yes—
23#89600—24no—
24#89491C2214yes—
25#89490C2214yes—
26#89473—24no—
27C7×#324C2212yes—
28C7×#323C2212yes—
29#89323C2212yes—
30#89320C2212yes—
31#89264C2218yes—
32#17410—71nomaximal, Sylow
33C23×C4×C7C2×C411yescharacteristic
34C22×Dic14C2×C411yescharacteristic
35C22×Dic14C2×C411yescharacteristic
36C7×#117—22no—
37C7×#117—22no—
38C2×C7×C4⋊C4C2×C411yescharacteristic
39C2×C7×C4⋊C4C2311yescharacteristic
40C2×C7×C22⋊C4C2×C411yescharacteristic
41C2×C7×C22⋊C4C2311yescharacteristic
42C2×C42×C7C2×C411yescharacteristic
43C2×C42×C7C2311yescharacteristic
44C2×C22⋊Dic7C2×C412yes—
45C2×C22⋊Dic7C2×C412yes—
46C2×C4⋊Dic7C2×C411yescharacteristic
47C2×C4⋊Dic7C2×C412yes—
48C2×C4⋊Dic7C2×C412yes—
49C2×C4⋊Dic7C2×C411yescharacteristic
50#6487—22no—
51#6487—22no—
52#6487—22no—
53#6487—22no—
54C2×Dic7⋊C4C2×C412yes—
55C2×Dic7⋊C4C2×C412yes—
56C2×Dic7⋊C4C2×C412yes—
57C2×Dic7⋊C4C2×C412yes—
58C2×Dic7⋊C4C2×C412yes—
59C2×Dic7⋊C4C2312yes—
60C2×C4×Dic7C2×C412yes—
61C2×C4×Dic7C2312yes—
62#6452—28no—
63#6452—24no—
64#6452—24no—
65#6442—28no—
66#6442—24no—
67#6442—24no—
68#6433—22no—
69#6433—22no—
70#6432—22no—
71#6432—22no—
72C4×Dic14—22no—
73C4×Dic14—22no—
74C4×Dic14D414yes—
75C4×Dic14D414yes—
76C7×C4⋊C8—22no—
77C7×C22⋊C8C2314yes—
78C7×#97—22no—
79C22⋊(C7⋊C8)C2314yes—
80#6389—24no—
81Dic7⋊C8—24no—
82Dic7⋊C8—24no—
83C8×Dic7—24no—
84C4⋊(C7⋊C8)—22no—
85#6377—22no—
86C2×#541—71no—
87#2983—72no—
88#1473—74no—
89#1470—74no—
90#1467—72no—
91#1464—71no—
92#1464D711yescharacteristic
93C24×C7C22×C411yescharacteristic
94C22×C4×C7—21no—
95C22×C4×C7—21no—
96C22×C4×C7—42no—
97C22×C4×C7C22×C411yescharacteristic
98C22×C4×C7C22×C411yescharacteristic
99C22×C4×C7C22×C411yescharacteristic
100C22×C4×C7C22×C411yescharacteristic
101C22×C4×C7C22×C411yescharacteristic
102C22×C4×C7C2411yescharacteristic
103C22×Dic7C22×C412yes—
104C22×Dic7C22×C412yes—
105C22×Dic7C22×C412yes—
106C22×Dic7C22×C412yes—
107C2×Dic14—21no—
108C2×Dic14—21no—
109C2×Dic14—42no—
110C2×Dic14—42no—
111C2×Dic14C22⋊C412yes—
112C2×Dic14C22⋊C412yes—
113C2×C7×C8—24no—
114C7×C4⋊C4—22no—
115C7×C4⋊C4—22no—
116C7×C22⋊C4—22no—
117C7×C22⋊C4—22no—
118C42×C7—21no—
119C42×C7—21no—
120C42×C7C2×D412yes—
121C42×C7C2×D412yes—
122C22⋊Dic7—24no—
123C22⋊Dic7—24no—
124C4⋊Dic7—21no—
125C4⋊Dic7—21no—
126C4⋊Dic7—24no—
127C4⋊Dic7—24no—
128C4⋊Dic7C22⋊C412yes—
129C4⋊Dic7C22⋊C412yes—
130Dic7⋊C4—22no—
131Dic7⋊C4—22no—
132Dic7⋊C4—24no—
133Dic7⋊C4—24no—
134Dic7⋊C4—24no—
135Dic7⋊C4—24no—
136Dic7⋊C4C22⋊C418yes—
137C4×Dic7—24no—
138C4×Dic7C2×D414yes—
139C4×Dic7C2×D414yes—
140C2×C7⋊C8—24no—
141#541—144no—
142#541—144no—
143C2×C4⋊Q8—71no—
144C2×#126—72no—
145C2×#126—72no—
146C2×#125—72no—
147C2×#125—72no—
148C2×#125—71no—
149C2×C22⋊Q8—71no—
150C2×C22⋊Q8—71no—
151C2×C4×Q8—71no—
152C2×C4×Q8—71no—
153C2×#117C2×D711yescharacteristic
154#446—144no—
155#446—144no—
156#432—72no—
157#432—74no—
158#432C2×D712yes—
159#431—74no—
160#324—72no—
161#324—78no—
162#324C2×D712yes—
163#323—72no—
164#323C2×D712yes—
165C23×C7—21no—
166C23×C7—22no—
167C23×C7—41no—
168C23×C7—41no—
169C23×C7C23×C411yescharacteristic
170C2×C4×C7—22no—
171C2×C4×C7—21no—
172C2×C4×C7—21no—
173C2×C4×C7—21no—
174C2×C4×C7—22no—
175C2×C4×C7—41no—
176C2×C4×C7—41no—
177C2×C4×C7—42no—
178C2×C4×C7—42no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7C2×C22⋊C412yes—
182C2×C4×C7C2×C22⋊C411yescharacteristic
183C2×C4×C7C2×C22⋊C411yescharacteristic
184C2×C4×C7C22×D412yes—
185C2×Dic7—24no—
186C2×Dic7—22no—
187C2×Dic7—22no—
188C2×Dic7—24no—
189C2×Dic7—42no—
190C2×Dic7—42no—
191C2×Dic7C2×C22⋊C414yes—
192C2×Dic7C2×C22⋊C414yes—
193Dic14—42no—
194Dic14—42no—
195Dic14—42no—
196Dic14—42no—
197C7×C8—44no—
198C7⋊C8—44no—
199C22×Q8—71no—
200C22×Q8—71no—
201C23×C4C4×D711yescharacteristic
202C4⋊Q8—142no—
203C4⋊Q8—142no—
204#126—144no—
205#126—144no—
206#126—148no—
207#125—142no—
208#125—142no—
209#125—144no—
210#125—144no—
211#125—148no—
212C22⋊Q8—142no—
213C22⋊Q8—142no—
214C22⋊Q8—142no—
215C22⋊Q8—142no—
216C4×Q8—74no—
217C4×Q8—74no—
218C4×Q8—142no—
219C4×Q8—142no—
220#117—22no—
221#117—22no—
222C2×C4⋊C4—71no—
223C2×C4⋊C4—72no—
224C2×C4⋊C4—71no—
225C2×C4⋊C4—72no—
226C2×C4⋊C4—72no—
227C2×C4⋊C4—72no—
228C2×C4⋊C4—72no—
229C2×C4⋊C4—72no—
230C2×C4⋊C4—72no—
231C2×C4⋊C4—72no—
232C2×C4⋊C4C4×D711yescharacteristic
233C2×C4⋊C4C22×D711yescharacteristic
234C2×C22⋊C4—72no—
235C2×C22⋊C4—72no—
236C2×C22⋊C4C4×D711yescharacteristic
237C2×C22⋊C4C22×D711yescharacteristic
238C2×C42—72no—
239C2×C42—72no—
240C2×C42C4×D711yescharacteristic
241C2×C42C22×D711yescharacteristic
242C4⋊C8—22no—
243C4⋊C8—144no—
244C4⋊C8—144no—
245C4⋊C8—142no—
246C22⋊C8—74no—
247C22⋊C8C22×D714yes—
248#97—22no—
249#97—144no—
250#97—142no—
251C4×C8—144no—
252C22×C7—21no—
253C22×C7—21no—
254C22×C7—41no—
255C22×C7—41no—
256C22×C7—41no—
257C22×C7—82no—
258C22×C7C22×C22⋊C411yescharacteristic
259C22×C7#56812yes—
260C4×C7—21no—
261C4×C7—21no—
262C4×C7—22no—
263C4×C7—22no—
264C4×C7—41no—
265C4×C7—41no—
266C4×C7—41no—
267Dic7—24no—
268Dic7—24no—
269Dic7—42no—
270Dic7—42no—
271C24C2×C4×D711yescharacteristic
272C2×Q8—72no—
273C2×Q8—72no—
274C2×Q8—141no—
275C2×Q8—141no—
276C2×Q8—282no—
277C2×Q8—282no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—42no—
281C22×C4—72no—
282C22×C4—72no—
283C22×C4—72no—
284C22×C4—72no—
285C22×C4C2×C4×D711yescharacteristic
286C22×C4C2×C4×D711yescharacteristic
287C22×C4C2×C4×D711yescharacteristic
288C22×C4C2×C4×D711yescharacteristic
289C22×C4C2×C4×D711yescharacteristic
290C22×C4C23×D711yescharacteristic
291C2×C8—24no—
292C2×C8—144no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—78no—
296C4⋊C4—72no—
297C4⋊C4—72no—
298C4⋊C4—144no—
299C4⋊C4—144no—
300C4⋊C4—144no—
301C4⋊C4—144no—
302C4⋊C4—144no—
303C4⋊C4—144no—
304C4⋊C4—141no—
305C4⋊C4—141no—
306C4⋊C4—142no—
307C4⋊C4—142no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—144no—
311C22⋊C4—144no—
312C42—21no—
313C42—21no—
314C42—74no—
315C42—74no—
316C42—144no—
317C42D4×D712yes—
318C42D4×D712yes—
319C2×C7—22no—
320C2×C7—81no—
321C2×C7#287911yescharacteristic
322C2×C7#288411yescharacteristic
323C2×C7#288911yescharacteristic
324C23—21no—
325C23—22no—
326C23—41no—
327C23—41no—
328C23C22×C4×D711yescharacteristic
329Q8—282no—
330Q8—282no—
331Q8—282no—
332Q8—282no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—74no—
345C2×C4—74no—
346C2×C4—144no—
347C2×C4—142no—
348C2×C4—142no—
349C2×C4—144no—
350C2×C4—282no—
351C2×C4—282no—
352C2×C4D7×C22⋊C411yescharacteristic
353C2×C4D7×C22⋊C411yescharacteristic
354C2×C4D7×C22⋊C412yes—
355C2×C4C2×D4×D712yes—
356C8—44no—
357C8—284no—
358C7#1741011yescharacteristic, complemented, Sylow
359C22—21no—
360C22—21no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—82no—
365C22C2×D7×C22⋊C411yescharacteristic
366C22#9043612yes—
367C4—21no—
368C4—21no—
369C4—22no—
370C4—22no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—144no—
375C4—144no—
376C4—282no—
377C4—282no—
378C2—22no—
379C2—81no—
380C2#1173607511yescharacteristic
381C2#1173613411yescharacteristic
382C2#1173616011yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1464C2
socleC22×C7C22×C22⋊C4
prime radicalC24×C7C22×C4
cosocleC22×C4×C7C24
hypercenter#1464D7
hypocenterC7#17410
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #11736759C2semidirect
5. #11728584C2semidirect
6. #11725882C2semidirect
358. C7#17410semidirect

Maximal subgroups

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

subgroupindex
C2×#902092
#117367592
C7×#14642
#117293262
#117285842
#117258822
#117258592
#174107