Finite Group Databasebeta

← #13273390

Subgroups — #13273390

Order 1792.

482 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6400C211yesmaximal, characteristic
2C7×#1970C211yesmaximal, characteristic, complemented
3C2×#89357C211yesmaximal, characteristic, complemented
4C2×#89286C211yesmaximal, characteristic, complemented
5#11727530C211yesmaximal, characteristic, complemented
6C2×C4⋊(C7⋊C16)C211yesmaximal, characteristic, complemented
7#11725528C211yesmaximal, characteristic, complemented
8C22×C7×D8C2211yescharacteristic
9C22×D56C2211yescharacteristic
10C2×C7×#107C2211yescharacteristic
11C2×C7×#102—21no—
12C2×C4×C7×C8C2211yescharacteristic
13C7×#390—21no—
14C2×#6400—22no—
15C2×#6400—22no—
16C2×#6400—22no—
17C2×#6395—21no—
18C2×#6392C2211yescharacteristic
19C22×C7⋊C16C2211yescharacteristic
20C22×C7⋊C16C2211yescharacteristic
21#89711—21no—
22#89357—22no—
23#89286—22no—
24C4⋊(C7⋊C16)—22no—
25#13448—71nomaximal, Sylow
26C22×C7×D4—21no—
27C22×D28—21no—
28C2×C7×D8—21no—
29C2×C7×D8—21no—
30C2×C7×D8—21no—
31C2×C7×D8—21no—
32C2×C7×D8—21no—
33C2×C7×D8D412yes—
34C22×C7×C8D411yescharacteristic
35C22×C7×C8C2311yescharacteristic
36C2×C7×C4⋊C4—21no—
37C2×C7×C22⋊C4—41no—
38C2×C7×C22⋊C4—41no—
39C2×C42×C7D411yescharacteristic
40C2×(C2×D7)⋊C4—41no—
41C2×(C2×D7)⋊C4—41no—
42C2×C4⋊Dic7—21no—
43C2×D56—21no—
44C2×D56—21no—
45C2×D56—21no—
46C2×D56—21no—
47C2×D56—21no—
48C2×D56D412yes—
49C7×#107—21no—
50C7×#107—21no—
51C7×#102—42no—
52C4×C7×C8—21no—
53C4×C7×C8—21no—
54#6400—28no—
55#6395—42no—
56#6392—21no—
57#6392—21no—
58C2×C7⋊C16—21no—
59C2×C7⋊C16—21no—
60C2×C7⋊C16—21no—
61C2×C7⋊C16—21no—
62C2×C7⋊C16—21no—
63C2×C7⋊C16D412yes—
64C22×D16—71no—
65C2×C4⋊C16—71no—
66C2×#357—71no—
67C2×#357—71no—
68#1970—71no—
69#1970D711yescharacteristic, complemented
70#1380—71no—
71C24×C7—41no—
72C23×D7—41no—
73C2×C7×D4—22no—
74C2×C7×D4—41no—
75C2×C7×D4—41no—
76C2×C7×D4—41no—
77C2×C7×D4—41no—
78C2×C7×D4—41no—
79C22×C4×C7—41no—
80C22×C4×C7D811yescharacteristic
81C22×C4×C7D811yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C22×Dic7—41no—
84C2×D28—22no—
85C2×D28—41no—
86C2×D28—41no—
87C2×D28—41no—
88C2×D28—41no—
89C2×D28—41no—
90C7×D8—22no—
91C7×D8—22no—
92C7×D8—41no—
93C7×D8—41no—
94C2×C7×C8—21no—
95C2×C7×C8—21no—
96C2×C7×C8—21no—
97C2×C7×C8C2×D411yescharacteristic
98C2×C7×C8C2×D411yescharacteristic
99C2×C7×C8C2×D411yescharacteristic
100C2×C7×C8#4111yescharacteristic
101C2×C7×C8#4111yescharacteristic
102C2×C7×C8#4111yescharacteristic
103C7×C4⋊C4—41no—
104C7×C4⋊C4—41no—
105C7×C22⋊C4—82no—
106C7×C22⋊C4—82no—
107C42×C7—21no—
108C42×C7—21no—
109(C2×D7)⋊C4—82no—
110(C2×D7)⋊C4—82no—
111C4⋊Dic7—41no—
112C4⋊Dic7—41no—
113D56—22no—
114D56—22no—
115D56—41no—
116D56—41no—
117C7⋊C16—22no—
118C7⋊C16—41no—
119C22×D8—71no—
120C22×D8C2×D711yescharacteristic
121C2×D16—142no—
122C2×D16—142no—
123C2×D16—142no—
124C22×C16—71no—
125C22×C16—71no—
126C2×#107—71no—
127C2×#107C2×D711yescharacteristic
128C2×#102—21no—
129C2×#102—141no—
130C2×C4×C8C2×D711yescharacteristic
131#390—21no—
132#390—141no—
133C4⋊C16—142no—
134#357—142no—
135#357—142no—
136C23×C7—42no—
137C23×C7—81no—
138C23×C7—81no—
139C23×C7—81no—
140C23×C7—81no—
141C23×C7—81no—
142C23×C7—81no—
143C23×C7C2×D811yescharacteristic
144C22×D7—42no—
145C22×D7—81no—
146C22×D7—81no—
147C22×D7—81no—
148C22×D7—81no—
149C22×D7—81no—
150C22×D7—81no—
151C7×D4—42no—
152C7×D4—42no—
153C7×D4—81no—
154C7×D4—81no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—21no—
160C2×C4×C7—81no—
161C2×C4×C7—81no—
162C2×C4×C7—81no—
163C2×C4×C7D1612yescomplemented
164C2×C4×C7#12111yescharacteristic
165C2×C4×C7#12111yescharacteristic
166C2×C4×C7#12111yescharacteristic
167C2×C4×C7#13511yescharacteristic
168C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
170C2×Dic7—81no—
171C2×Dic7—81no—
172C2×Dic7—81no—
173D28—42no—
174D28—42no—
175D28—81no—
176D28—81no—
177C7×C8—41no—
178C7×C8#12011yescharacteristic
179C7×C8#12311yescharacteristic
180C7×C8#12311yescharacteristic
181C7×C8#12411yescharacteristic
182C22×D4—21no—
183C22×D4—141no—
184C2×D8—21no—
185C2×D8—21no—
186C2×D8—21no—
187C2×D8—21no—
188C2×D8—21no—
189C2×D8—72no—
190C2×D8—141no—
191C2×D8—141no—
192C2×D8—141no—
193C2×D8—141no—
194C2×D8—141no—
195C2×D8D2812yes—
196C22×C8C22⋊D711yescharacteristic
197C22×C8C22×D711yescharacteristic
198C2×C4⋊C4—21no—
199C2×C4⋊C4—141no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—281no—
203C2×C22⋊C4—281no—
204C2×C42C22⋊D711yescharacteristic
205D16—148no—
206C2×C16—72no—
207C2×C16—141no—
208C2×C16—141no—
209C2×C16—141no—
210C2×C16—141no—
211C2×C16—141no—
212#107—21no—
213#107—21no—
214#107—141no—
215#107—141no—
216#102—42no—
217#102—282no—
218C4×C8—21no—
219C4×C8—21no—
220C22×C7—82no—
221C22×C7—82no—
222C22×C7—82no—
223C22×C7—161no—
224C22×C7—161no—
225C22×C7—161no—
226C22×C7—161no—
227C22×C7#45911yescharacteristic
228C22×C7#45911yescharacteristic
229C22×C7#46611yescharacteristic
230C22×C7C2×D1611yescharacteristic
231C22×C7#50811yescharacteristic
232C22×C7#50911yescharacteristic
233C22×C7#50911yescharacteristic
234C2×D7—82no—
235C2×D7—82no—
236C2×D7—82no—
237C2×D7—161no—
238C2×D7—161no—
239C2×D7—161no—
240C2×D7—161no—
241C4×C7—22no—
242C4×C7—41no—
243C4×C7—161no—
244C4×C7#39411yescharacteristic
245C4×C7#46311yescharacteristic
246C4×C7#46311yescharacteristic
247C4×C7#46911yescharacteristic
248Dic7—161no—
249C24—41no—
250C24—281no—
251C2×D4—22no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—142no—
258C2×D4—281no—
259C2×D4—281no—
260C2×D4—281no—
261C2×D4—281no—
262C2×D4—281no—
263C22×C4—41no—
264C22×C4—281no—
265C22×C4#114111yescharacteristic
266C22×C4#114111yescharacteristic
267C22×C4C2×C22⋊D711yescharacteristic
268D8—22no—
269D8—22no—
270D8—41no—
271D8—41no—
272D8—142no—
273D8—142no—
274D8—281no—
275D8—281no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8C2×D2811yescharacteristic
280C2×C8#115711yescharacteristic
281C2×C8D4×D711yescharacteristic
282C2×C8D4×D711yescharacteristic
283C2×C8#115911yescharacteristic
284C2×C8#115911yescharacteristic
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—281no—
288C4⋊C4—281no—
289C22⋊C4—82no—
290C22⋊C4—82no—
291C22⋊C4—562no—
292C22⋊C4—562no—
293C42—21no—
294C42—21no—
295C16—142no—
296C16—281no—
297C2×C7—162no—
298C2×C7#205411yescharacteristic
299C2×C7#220611yescharacteristic
300C2×C7#220611yescharacteristic
301C2×C7#221011yescharacteristic
302C2×C7#221011yescharacteristic
303C2×C7#221511yescharacteristic
304C2×C7#221811yescharacteristic
305D7—162no—
306C23—42no—
307C23—81no—
308C23—81no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23—81no—
313C23—282no—
314C23—561no—
315C23—561no—
316C23—561no—
317C23—561no—
318C23—561no—
319C23—561no—
320C23C2×#114111yescharacteristic
321D4—42no—
322D4—42no—
323D4—81no—
324D4—81no—
325D4—282no—
326D4—282no—
327D4—561no—
328D4—561no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—561no—
338C2×C4—561no—
339C2×C4—561no—
340C2×C4#640012yescomplemented
341C2×C4#649311yescharacteristic
342C2×C4#649511yescharacteristic
343C2×C4#650111yescharacteristic
344C2×C4#650211yescharacteristic
345C2×C4#651211yescharacteristic
346C2×C4#651311yescharacteristic
347C8—41no—
348C8#644511yescharacteristic
349C8#644611yescharacteristic
350C8#644811yescharacteristic
351C8#644911yescharacteristic
352C7#1344811yescharacteristic, complemented, Sylow
353C22—82no—
354C22—82no—
355C22—82no—
356C22—161no—
357C22—161no—
358C22—161no—
359C22—161no—
360C22—562no—
361C22—562no—
362C22—562no—
363C22—1121no—
364C22—1121no—
365C22—1121no—
366C22—1121no—
367C22#8978711yescharacteristic
368C22#8981011yescharacteristic
369C22#8984411yescharacteristic
370C22C2×#640011yescharacteristic
371C22#8991911yescharacteristic
372C22#8996511yescharacteristic
373C22#8996611yescharacteristic
374C4—22no—
375C4—41no—
376C4—161no—
377C4—1121no—
378C4#8973311yescharacteristic
379C4#8978811yescharacteristic
380C4#8981111yescharacteristic
381C4#8982911yescharacteristic
382C2—162no—
383C2—1122no—
384C2#1172740411yescharacteristic
385C2#1172864311yescharacteristic
386C2#1172864411yescharacteristic
387C2#1172882511yescharacteristic
388C2#1172882711yescharacteristic
389C2#1172904711yescharacteristic
390C2#1172907711yescharacteristic
subgroupG/H
centerC23C2×#1141
commutator, cosocleC22×C7×C8C23
FrattiniC22×C8C22×D7
FittingC7×#1970C2
socleC23×C7C2×D8
prime radicalC22×#6400C2
hypercenter#1970D7
hypocenterC7#13448
LCSG ▷ C22×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ #1970

Complements

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

NH ≅ G/Ntype
2. C7×#1970C2semidirect
3. C2×#89357C2semidirect
4. C2×#89286C2semidirect
5. #11727530C2semidirect
6. C2×C4⋊(C7⋊C16)C2semidirect
7. #11725528C2semidirect
69. #1970D7semidirect
163. C2×C4×C7D16semidirect
340. C2×C4#6400semidirect
352. C7#13448semidirect

Maximal subgroups

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

subgroupindex
C22×#64002
C7×#19702
C2×#893572
C2×#892862
#117275302
C2×C4⋊(C7⋊C16)2
#117255282
#134487