Finite Group Databasebeta

← #13273395

Subgroups — #13273395

Order 1792.

446 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6402C211yesmaximal, characteristic
2C7×#1969C211yesmaximal, characteristic, complemented
3C2×#89359C211yesmaximal, characteristic, complemented
4C2×#89286C211yesmaximal, characteristic
5#11727530C211yesmaximal, characteristic
6C2×C4⋊(C7⋊C16)C211yesmaximal, characteristic, complemented
7#11725528C211yesmaximal, characteristic, complemented
8C22×C7×Q16C2211yescharacteristic
9C22×D56C2211yescharacteristic
10C2×C7×#107C2211yescharacteristic
11C2×C7×#103—21no—
12C2×C4×C7×C8C2211yescharacteristic
13C7×#391—21no—
14C2×#6402—22no—
15C2×#6402—22no—
16C2×#6402—22no—
17C2×#6395—21no—
18C2×#6392C2211yescharacteristic
19C22×C7⋊C16C2211yescharacteristic
20C22×C7⋊C16C2211yescharacteristic
21#89711—21no—
22#89359—22no—
23#89286—22no—
24C4⋊(C7⋊C16)—22no—
25#13449—71nomaximal, Sylow
26C22×C7×Q8—21no—
27C22×D28—21no—
28C2×C7×Q16—21no—
29C2×C7×Q16—21no—
30C2×C7×Q16—21no—
31C2×C7×Q16—21no—
32C2×C7×Q16—21no—
33C2×C7×Q16D412yes—
34C22×C7×C8D411yescharacteristic
35C22×C7×C8C2311yescharacteristic
36C2×C7×C4⋊C4—21no—
37C2×C42×C7D411yescharacteristic
38C2×(C2×D7)⋊C4—41no—
39C2×(C2×D7)⋊C4—41no—
40C2×C4⋊Dic7—21no—
41C2×D56—21no—
42C2×D56—21no—
43C2×D56—21no—
44C2×D56—21no—
45C2×D56—21no—
46C2×D56D412yes—
47C7×#107—21no—
48C7×#107—21no—
49C7×#103—42no—
50C4×C7×C8—21no—
51C4×C7×C8—21no—
52C7×#95—41no—
53C7×#95—41no—
54#6402—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×SD32—71no—
65C2×C4⋊C16—71no—
66C2×#358—71no—
67C2×#357—71no—
68#1970—71no—
69#1969D711yescharacteristic, complemented
70#1380—71no—
71C23×D7—41no—
72C2×C7×Q8—22no—
73C2×C7×Q8—41no—
74C2×C7×Q8—41no—
75C2×C7×Q8—41no—
76C2×C7×Q8—41no—
77C2×C7×Q8—41no—
78C22×C4×C7—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×Q16—22no—
91C7×Q16—22no—
92C7×Q16—41no—
93C7×Q16—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—
105C42×C7—21no—
106C42×C7—21no—
107(C2×D7)⋊C4—82no—
108(C2×D7)⋊C4—82no—
109C4⋊Dic7—41no—
110C4⋊Dic7—41no—
111D56—22no—
112D56—22no—
113D56—41no—
114D56—41no—
115C7⋊C16—22no—
116C7⋊C16—41no—
117C22×Q16C2×D711yescharacteristic
118C22×D8—71no—
119C2×SD32—142no—
120C2×SD32—142no—
121C2×SD32—142no—
122C22×C16—71no—
123C22×C16—71no—
124C2×#107—71no—
125C2×#107C2×D711yescharacteristic
126C2×#103—21no—
127C2×#102—141no—
128C2×C4×C8C2×D711yescharacteristic
129#391—21no—
130#390—141no—
131C4⋊C16—142no—
132#358—142no—
133#357—142no—
134C23×C7C2×D811yescharacteristic
135C22×D7—42no—
136C22×D7—81no—
137C22×D7—81no—
138C22×D7—81no—
139C22×D7—81no—
140C22×D7—81no—
141C22×D7—81no—
142C7×Q8—42no—
143C7×Q8—42no—
144C7×Q8—81no—
145C7×Q8—81no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—42no—
152C2×C4×C7—81no—
153C2×C4×C7—81no—
154C2×C4×C7—81no—
155C2×C4×C7—81no—
156C2×C4×C7—81no—
157C2×C4×C7SD3212yescomplemented
158C2×C4×C7#12111yescharacteristic
159C2×C4×C7#12111yescharacteristic
160C2×C4×C7#12111yescharacteristic
161C2×C4×C7#13511yescharacteristic
162C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
164C2×Dic7—81no—
165C2×Dic7—81no—
166C2×Dic7—81no—
167D28—42no—
168D28—42no—
169D28—81no—
170D28—81no—
171C7×C8—41no—
172C7×C8#12011yescharacteristic
173C7×C8#12311yescharacteristic
174C7×C8#12311yescharacteristic
175C7×C8#12411yescharacteristic
176C22×Q8—21no—
177C22×D4—141no—
178C2×Q16—21no—
179C2×Q16—21no—
180C2×Q16—21no—
181C2×Q16—21no—
182C2×Q16—21no—
183C2×Q16D2812yes—
184C2×D8—72no—
185C2×D8—141no—
186C2×D8—141no—
187C2×D8—141no—
188C2×D8—141no—
189C2×D8—141no—
190C22×C8C22⋊D711yescharacteristic
191C22×C8C22×D711yescharacteristic
192C2×C4⋊C4—21no—
193C2×C4⋊C4—141no—
194C2×C22⋊C4—281no—
195C2×C22⋊C4—281no—
196C2×C42C22⋊D711yescharacteristic
197SD32—148no—
198C2×C16—72no—
199C2×C16—141no—
200C2×C16—141no—
201C2×C16—141no—
202C2×C16—141no—
203C2×C16—141no—
204#107—21no—
205#107—21no—
206#107—141no—
207#107—141no—
208#103—42no—
209#102—282no—
210C4×C8—21no—
211C4×C8—21no—
212#95—41no—
213#95—41no—
214C22×C7#45911yescharacteristic
215C22×C7#45911yescharacteristic
216C22×C7#46611yescharacteristic
217C22×C7C2×SD3211yescharacteristic
218C22×C7#50811yescharacteristic
219C22×C7#50911yescharacteristic
220C22×C7#51011yescharacteristic
221C2×D7—82no—
222C2×D7—82no—
223C2×D7—82no—
224C2×D7—161no—
225C2×D7—161no—
226C2×D7—161no—
227C2×D7—161no—
228C4×C7—22no—
229C4×C7—41no—
230C4×C7—82no—
231C4×C7—161no—
232C4×C7#39411yescharacteristic
233C4×C7#46311yescharacteristic
234C4×C7#46311yescharacteristic
235C4×C7#46911yescharacteristic
236Dic7—161no—
237C24—281no—
238C2×Q8—22no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×D4—142no—
245C2×D4—281no—
246C2×D4—281no—
247C2×D4—281no—
248C2×D4—281no—
249C2×D4—281no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—281no—
253C22×C4#114111yescharacteristic
254C22×C4#114111yescharacteristic
255C22×C4C2×C22⋊D711yescharacteristic
256Q16—22no—
257Q16—22no—
258Q16—41no—
259Q16—41no—
260D8—142no—
261D8—142no—
262D8—281no—
263D8—281no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8C2×D2811yescharacteristic
268C2×C8#115711yescharacteristic
269C2×C8D4×D711yescharacteristic
270C2×C8D4×D711yescharacteristic
271C2×C8#115911yescharacteristic
272C2×C8#115911yescharacteristic
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—281no—
276C4⋊C4—281no—
277C22⋊C4—562no—
278C22⋊C4—562no—
279C42—21no—
280C42—21no—
281C16—142no—
282C16—281no—
283C2×C7#205411yescharacteristic
284C2×C7#220711yescharacteristic
285C2×C7#220811yescharacteristic
286C2×C7#221011yescharacteristic
287C2×C7#221111yescharacteristic
288C2×C7#221711yescharacteristic
289C2×C7#222011yescharacteristic
290D7—162no—
291C23—282no—
292C23—561no—
293C23—561no—
294C23—561no—
295C23—561no—
296C23—561no—
297C23—561no—
298C23C2×#114111yescharacteristic
299Q8—42no—
300Q8—42no—
301Q8—81no—
302Q8—81no—
303D4—282no—
304D4—282no—
305D4—561no—
306D4—561no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—561no—
319C2×C4—561no—
320C2×C4—561no—
321C2×C4#640212yescomplemented
322C2×C4#649311yescharacteristic
323C2×C4#649511yescharacteristic
324C2×C4#650111yescharacteristic
325C2×C4#650211yescharacteristic
326C2×C4#651211yescharacteristic
327C2×C4#651311yescharacteristic
328C8—41no—
329C8#644511yescharacteristic
330C8#644611yescharacteristic
331C8#644811yescharacteristic
332C8#644911yescharacteristic
333C7#1344911yescharacteristic, complemented, Sylow
334C22—562no—
335C22—562no—
336C22—562no—
337C22—1121no—
338C22—1121no—
339C22—1121no—
340C22—1121no—
341C22#8978711yescharacteristic
342C22#8981011yescharacteristic
343C22#8984411yescharacteristic
344C22C2×#640211yescharacteristic
345C22#8995111yescharacteristic
346C22#8996511yescharacteristic
347C22#8996611yescharacteristic
348C4—22no—
349C4—41no—
350C4—82no—
351C4—161no—
352C4—1121no—
353C4#8973311yescharacteristic
354C4#8978811yescharacteristic
355C4#8981111yescharacteristic
356C4#8982911yescharacteristic
357C2—1122no—
358C2#1172740411yescharacteristic
359C2#1172866811yescharacteristic
360C2#1172866911yescharacteristic
361C2#1172883611yescharacteristic
362C2#1172883811yescharacteristic
363C2#1172906911yescharacteristic
364C2#1172907711yescharacteristic
subgroupG/H
centerC23C2×#1141
commutator, cosocleC22×C7×C8C23
FrattiniC22×C8C22×D7
FittingC7×#1969C2
socleC23×C7C2×D8
prime radicalC22×D56C22
hypercenter#1969D7
hypocenterC7#13449
LCSG ▷ C22×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ #1969

Complements

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

NH ≅ G/Ntype
2. C7×#1969C2semidirect
3. C2×#89359C2semidirect
6. C2×C4⋊(C7⋊C16)C2semidirect
7. #11725528C2semidirect
69. #1969D7semidirect
157. C2×C4×C7SD32semidirect
321. C2×C4#6402semidirect
333. C7#13449semidirect

Maximal subgroups

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

subgroupindex
C22×#64022
C7×#19692
C2×#893592
C2×#892862
#117275302
C2×C4⋊(C7⋊C16)2
#117255282
#134497