Finite Group Databasebeta

← #11734860

Subgroups — #11734860

Order 896.

610 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6455C211yesmaximal, characteristic, complemented
2C2×D7×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#6446C211yesmaximal, characteristic
4C2×#6446C211yesmaximal, characteristic
5C7×#396C211yesmaximal, characteristic, complemented
6#89761C211yesmaximal, characteristic, complemented
7#89757C211yesmaximal, characteristic, complemented
8#89756C211yesmaximal, characteristic, complemented
9#89755C211yesmaximal, characteristic, complemented
10#89728C211yesmaximal, characteristic
11#89723C211yesmaximal, characteristic, complemented
12#89720C211yesmaximal, characteristic, complemented
13#89716C211yesmaximal, characteristic, complemented
14#89436C211yesmaximal, characteristic, complemented
15#89419C211yesmaximal, characteristic, complemented
16C22×C22⋊D7C2211yescharacteristic
17C22×C4×D7C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
19C22×C4×D7C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C7×C4⋊C4C2211yescharacteristic
22C2×C7×C4⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×C4⋊Dic7C2211yescharacteristic
33C2×C4⋊Dic7C2211yescharacteristic
34C2×C4⋊Dic7C2211yescharacteristic
35C2×C4⋊Dic7C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×Dic7⋊C4C2211yescharacteristic
40C2×Dic7⋊C4C2211yescharacteristic
41C2×C4×Dic7C2211yescharacteristic
42C2×C4×Dic7C2211yescharacteristic
43#6455—24no—
44D7×C4⋊C4—24no—
45#6446—24no—
46#6446—24no—
47C7×#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53C7⋊#95C2211yescharacteristic
54C7⋊#95C2211yescharacteristic
55#2413—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C23×D7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C2×C22⋊D7—22no—
65C2×C22⋊D7—22no—
66C2×C22⋊D7—22no—
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C2×C4×D7—22no—
75C2×C4×D7—22no—
76C2×C4×D7—22no—
77C2×C4×D7—22no—
78C2×C4×D7—22no—
79C2×C4×D7—22no—
80C2×C4×D7—22no—
81C2×C4×D7—22no—
82C2×C4×D7D414yes—
83C7×C4⋊C4—22no—
84C7×C4⋊C4—22no—
85C7×C4⋊C4—22no—
86C7×C22⋊C4—22no—
87C7×C22⋊C4—22no—
88C7×C22⋊C4—22no—
89C22⋊Dic7—22no—
90C22⋊Dic7—22no—
91C22⋊Dic7—22no—
92(C2×D7)⋊C4—22no—
93(C2×D7)⋊C4—22no—
94(C2×D7)⋊C4D414yes—
95C4⋊Dic7—22no—
96C4⋊Dic7—22no—
97C4⋊Dic7—22no—
98C4⋊Dic7—22no—
99Dic7⋊C4—22no—
100Dic7⋊C4—22no—
101Dic7⋊C4—22no—
102Dic7⋊C4—22no—
103Dic7⋊C4—22no—
104C4×Dic7—22no—
105C4×Dic7—22no—
106C2×#123—71no—
107C2×#123—71no—
108C2×#117—71no—
109C22×C4⋊C4—71no—
110#398—71no—
111#398—71no—
112#396—71no—
113#396D711yescharacteristic, complemented
114#389—71no—
115#389—71no—
116#388—71no—
117#388—71no—
118#386—71no—
119#385—71no—
120#381—71no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7C2411yescharacteristic
129C22×D7—21no—
130C22×D7—21no—
131C22×D7—21no—
132C22×D7—21no—
133C22×D7C2×D412yes—
134C22×D7C2×D412yes—
135C22×D7#4112yes—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7C2×D412yes—
152C2×C4×C7C2×D412yes—
153C2×C4×C7#4112yes—
154C22⋊D7—44no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—21no—
158C2×Dic7—21no—
159C2×Dic7—21no—
160C2×Dic7—21no—
161C2×Dic7—21no—
162C2×Dic7—21no—
163C2×Dic7—21no—
164C2×Dic7—21no—
165C2×Dic7—21no—
166C2×Dic7—21no—
167C2×Dic7—21no—
168C2×Dic7—21no—
169C2×Dic7—21no—
170C2×Dic7—21no—
171C2×Dic7—21no—
172C2×Dic7—21no—
173C2×Dic7C2×D412yes—
174C2×Dic7C2×D412yes—
175C2×Dic7#4112yes—
176C4×D7—28no—
177C4×D7—28no—
178C4×D7—44no—
179C22×D4—71no—
180C23×C4—71no—
181C23×C4—71no—
182C23×C4—71no—
183#123—144no—
184#123—144no—
185#117—144no—
186C2×C4⋊C4—71no—
187C2×C4⋊C4—71no—
188C2×C4⋊C4—71no—
189C2×C4⋊C4—71no—
190C2×C4⋊C4—71no—
191C2×C4⋊C4—71no—
192C2×C4⋊C4—71no—
193C2×C4⋊C4—71no—
194C2×C4⋊C4—71no—
195C2×C4⋊C4—144no—
196C2×C4⋊C4C2×D711yescharacteristic
197C2×C4⋊C4C2×D711yescharacteristic
198C2×C4⋊C4C2×D711yescharacteristic
199C2×C22⋊C4—71no—
200C2×C22⋊C4—71no—
201C2×C22⋊C4—71no—
202C2×C22⋊C4—71no—
203C2×C22⋊C4—71no—
204C2×C22⋊C4—71no—
205C2×C22⋊C4C2×D711yescharacteristic
206C2×C22⋊C4C2×D711yescharacteristic
207C2×C22⋊C4C2×D711yescharacteristic
208C2×C42—71no—
209C2×C42—71no—
210#95—71no—
211#95—71no—
212#95—71no—
213#95—71no—
214#95—71no—
215#95—71no—
216#95—71no—
217#95C2×D711yescharacteristic
218C22×C7—41no—
219C22×C7—41no—
220C22×C7—41no—
221C22×C7—41no—
222C22×C7—41no—
223C22×C7—41no—
224C22×C7—41no—
225C22×C7C22×D411yescharacteristic
226C22×C7C22×D411yescharacteristic
227C22×C7C2×#4111yescharacteristic
228C22×C7C2×#4111yescharacteristic
229C22×C7C2×#4111yescharacteristic
230C22×C72−1+411yescharacteristic
231C22×C72−1+411yescharacteristic
232C2×D7—22no—
233C2×D7—22no—
234C2×D7—22no—
235C2×D7—22no—
236C2×D7—22no—
237C2×D7—22no—
238C2×D7#12114yes—
239C4×C7—22no—
240C4×C7—22no—
241C4×C7—22no—
242C4×C7—41no—
243C4×C7—41no—
244C4×C7—41no—
245Dic7—22no—
246Dic7—22no—
247Dic7—22no—
248Dic7—41no—
249Dic7—41no—
250Dic7—41no—
251Dic7—41no—
252C24—71no—
253C24C22×D711yescharacteristic
254C2×D4—142no—
255C2×D4—142no—
256C2×D4—142no—
257C22×C4—71no—
258C22×C4—71no—
259C22×C4—71no—
260C22×C4—74no—
261C22×C4—71no—
262C22×C4—71no—
263C22×C4—71no—
264C22×C4—71no—
265C22×C4—142no—
266C22×C4—142no—
267C22×C4—142no—
268C22×C4—142no—
269C22×C4—142no—
270C22×C4—142no—
271C22×C4—142no—
272C22×C4—142no—
273C22×C4C22×D711yescharacteristic
274C22×C4C22×D711yescharacteristic
275C22×C4C22×D711yescharacteristic
276C22×C4C22×D711yescharacteristic
277C22×C4C22×D711yescharacteristic
278C22×C4C22×D711yescharacteristic
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—142no—
283C4⋊C4—142no—
284C4⋊C4—142no—
285C4⋊C4—142no—
286C4⋊C4—142no—
287C4⋊C4—142no—
288C4⋊C4—142no—
289C4⋊C4—142no—
290C4⋊C4—142no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—74no—
295C22⋊C4—142no—
296C22⋊C4—142no—
297C22⋊C4—142no—
298C22⋊C4—142no—
299C22⋊C4—142no—
300C42—142no—
301C42—142no—
302C2×C7—81no—
303C2×C7C2×#12111yescharacteristic
304C2×C7#52511yescharacteristic
305C2×C7#53611yescharacteristic
306C2×C7#54711yescharacteristic
307C2×C7#54711yescharacteristic
308C2×C7#55211yescharacteristic
309C2×C7#55211yescharacteristic
310D7—24no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—72no—
319C23—72no—
320C23—72no—
321C23—141no—
322C23—141no—
323C23—141no—
324C23—141no—
325C23C23×D711yescharacteristic
326D4—284no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—72no—
343C2×C4—72no—
344C2×C4—72no—
345C2×C4—148no—
346C2×C4—148no—
347C2×C4—141no—
348C2×C4—141no—
349C2×C4—141no—
350C2×C4—141no—
351C2×C4—141no—
352C2×C4—141no—
353C2×C4—141no—
354C2×C4—141no—
355C2×C4—141no—
356C2×C4—141no—
357C2×C4—141no—
358C2×C4—141no—
359C2×C4—141no—
360C2×C4—141no—
361C2×C4—141no—
362C2×C4—141no—
363C2×C4—141no—
364C2×C4—141no—
365C2×C4—284no—
366C2×C4D4×D712yes—
367C2×C4D4×D712yes—
368C2×C4#115912yes—
369C7#241311yescharacteristic, complemented, Sylow
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—74no—
378C22—142no—
379C22—142no—
380C22—142no—
381C22—142no—
382C22—142no—
383C22—142no—
384C22C2×D4×D711yescharacteristic
385C22C2×D4×D711yescharacteristic
386C22C2×#115911yescharacteristic
387C22#655111yescharacteristic
388C22D7×#4111yescharacteristic
389C22D7×#4111yescharacteristic
390C22#655411yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—142no—
398C4—142no—
399C4—142no—
400C4—281no—
401C4—281no—
402C4—281no—
403C4—281no—
404C2—81no—
405C2—144no—
406C2D7×#12111yescharacteristic
407C2#9029711yescharacteristic
408C2#9030211yescharacteristic
409C2#9031911yescharacteristic
410C2#9032211yescharacteristic
411C2#9033011yescharacteristic
412C2#9039911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#396C2
prime radicalC22×C22⋊D7C22
hypercenter#396D7
hypocenterC7#2413
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#6455C2semidirect
2. C2×D7×C4⋊C4C2semidirect
5. C7×#396C2semidirect
6. #89761C2semidirect
7. #89757C2semidirect
8. #89756C2semidirect
9. #89755C2semidirect
11. #89723C2semidirect
12. #89720C2semidirect
13. #89716C2semidirect
14. #89436C2semidirect
15. #89419C2semidirect
113. #396D7semidirect
369. C7#2413semidirect

Maximal subgroups

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

subgroupindex
C2×#64552
C2×D7×C4⋊C42
C2×#64462
C7×#3962
#897612
#897572
#897562
#897552
#897282
#897232
#897202
#897162
#894362
#894192
#24137