Finite Group Databasebeta

← #13272963

Subgroups — #13272963

Order 1792.

446 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89960C211yesmaximal, characteristic
2C2×C7×#358C211yesmaximal, characteristic, complemented
3C2×#89359C211yesmaximal, characteristic, complemented
4C2×(C2×D7)⋊C16C211yesmaximal, characteristic
5#11728384C211yesmaximal, characteristic
6C2×#89285C211yesmaximal, characteristic, complemented
7#11725612C211yesmaximal, characteristic, complemented
8C22×C7×Q16C2211yescharacteristic
9C2×#6509—21no—
10C22×C8×D7C2211yescharacteristic
11C22×C7×C16C2211yescharacteristic
12C2×C7×#107C2211yescharacteristic
13C2×#6409—21no—
14#89960—22no—
15#89960—22no—
16C2×#6392C2211yescharacteristic
17C2×#6392C2211yescharacteristic
18C22×C7⋊C16C2211yescharacteristic
19#89759—21no—
20C7×#358—22no—
21#89359—22no—
22#89324—21no—
23(C2×D7)⋊C16—22no—
24#89285—22no—
25#13382—71nomaximal, Sylow
26C22×C7×Q8—21no—
27C22×C4×D7D411yescharacteristic
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×C8C2311yescharacteristic
35C2×C7×C4⋊C4—21no—
36#6509—24no—
37#6509—42no—
38C2×(C2×D7)⋊C4—41no—
39C2×(C2×D7)⋊C4—41no—
40C2×C4⋊Dic7—21no—
41C2×C4⋊Dic7—21no—
42C2×Dic7⋊C4—41no—
43C22×C7⋊C8D411yescharacteristic
44C2×C8×D7—21no—
45C2×C8×D7—21no—
46C2×C8×D7—21no—
47C2×C8×D7—21no—
48C2×C8×D7—21no—
49C2×C8×D7—21no—
50C2×C7×C16—21no—
51C2×C7×C16—21no—
52C2×C7×C16—21no—
53C7×#107—21no—
54C7×#107—21no—
55#6409—24no—
56C7⋊#95—41no—
57#6392—21no—
58#6392—21no—
59#6392—21no—
60#6392D412yes—
61C2×C7⋊C16—21no—
62C2×C7⋊C16—21no—
63C2×C7⋊C16D412yes—
64C2×#467—71no—
65C2×#367—71no—
66C2×#358—71no—
67C2×#358D711yescharacteristic, complemented
68C2×C22⋊C16—71no—
69#1808—71no—
70#1380—71no—
71C23×D7D811yescharacteristic
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×C7C2×D411yescharacteristic
81C22×Dic7—41no—
82C22×Dic7—41no—
83C22×Dic7D811yescharacteristic
84C2×C4×D7—21no—
85C2×C4×D7—21no—
86C2×C4×D7—21no—
87C2×C4×D7—21no—
88C2×C4×D7—21no—
89C2×C4×D7—21no—
90C7×Q16—22no—
91C7×Q16—22no—
92C7×Q16—41no—
93C7×Q16—41no—
94C2×C7×C8C2×D411yescharacteristic
95C2×C7×C8C2×D411yescharacteristic
96C2×C7×C8C2×D411yescharacteristic
97C2×C7×C8#4111yescharacteristic
98C2×C7×C8#4111yescharacteristic
99C2×C7×C8#4111yescharacteristic
100C7×C4⋊C4—41no—
101C7×C4⋊C4—41no—
102(C2×D7)⋊C4—44no—
103(C2×D7)⋊C4—82no—
104C4⋊Dic7—22no—
105C4⋊Dic7—41no—
106C4⋊Dic7—41no—
107C4⋊Dic7—41no—
108Dic7⋊C4—44no—
109C2×C7⋊C8—21no—
110C2×C7⋊C8—21no—
111C2×C7⋊C8—21no—
112C8×D7—41no—
113C8×D7—41no—
114C8×D7—41no—
115C8×D7—41no—
116C7×C16—41no—
117C7⋊C16—22no—
118C22×Q16C2×D711yescharacteristic
119C23×C8—71no—
120C2×C22⋊Q8—141no—
121C22×C16—71no—
122C22×C16C2×D711yescharacteristic
123#467—142no—
124#467—142no—
125C2×#107—71no—
126C2×#107—71no—
127C2×#107C2×D711yescharacteristic
128C2×#103—141no—
129#380—141no—
130#367—142no—
131#358—22no—
132#358—142no—
133C22⋊C16—142no—
134#340—141no—
135C23×C7C2×D811yescharacteristic
136C22×D7—21no—
137C22×D7—21no—
138C22×D7—21no—
139C22×D7—21no—
140C22×D7—21no—
141C22×D7—21no—
142C22×D7SD3212yes—
143C7×Q8—42no—
144C7×Q8—42no—
145C7×Q8—81no—
146C7×Q8—81no—
147C2×C4×C7—42no—
148C2×C4×C7—81no—
149C2×C4×C7—81no—
150C2×C4×C7—81no—
151C2×C4×C7—81no—
152C2×C4×C7—81no—
153C2×C4×C7#12011yescharacteristic
154C2×C4×C7#12111yescharacteristic
155C2×C4×C7#12311yescharacteristic
156C2×C4×C7#13511yescharacteristic
157C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C7#13711yescharacteristic
159C2×Dic7—21no—
160C2×Dic7—21no—
161C2×Dic7—21no—
162C2×Dic7—42no—
163C2×Dic7—81no—
164C2×Dic7—81no—
165C2×Dic7—81no—
166C2×Dic7—81no—
167C2×Dic7—81no—
168C4×D7—41no—
169C4×D7—41no—
170C4×D7—41no—
171C4×D7—41no—
172C7×C8#12111yescharacteristic
173C7×C8#12111yescharacteristic
174C7×C8#12311yescharacteristic
175C7×C8#12411yescharacteristic
176C7⋊C8—41no—
177C22×Q8—21no—
178C23×C4—71no—
179C2×Q16—21no—
180C2×Q16—21no—
181C2×Q16—21no—
182C2×Q16—21no—
183C2×Q16—21no—
184C2×Q16C22⋊D712yes—
185C22×C8—71no—
186C22×C8—141no—
187C22×C8—141no—
188C22×C8—141no—
189C22×C8—141no—
190C22×C8—141no—
191C22×C8—141no—
192C22×C8C22×D711yescharacteristic
193C22⋊Q8—144no—
194C22⋊Q8—282no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—141no—
197C2×C4⋊C4—141no—
198C2×C4⋊C4—281no—
199C2×C22⋊C4—281no—
200C2×C22⋊C4—281no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—72no—
205C2×C16—141no—
206C2×C16—141no—
207#107—21no—
208#107—21no—
209#107—72no—
210#107—141no—
211#107—141no—
212#107—141no—
213#103—144no—
214#95—281no—
215C22×C7#44711yescharacteristic
216C22×C7#46611yescharacteristic
217C22×C7#48011yescharacteristic
218C22×C7C2×SD3211yescharacteristic
219C22×C7#50811yescharacteristic
220C22×C7#51011yescharacteristic
221C22×C7#51011yescharacteristic
222C2×D7—22no—
223C2×D7—22no—
224C2×D7—22no—
225C2×D7—41no—
226C2×D7—41no—
227C2×D7—41no—
228C2×D7—41no—
229C4×C7—82no—
230C4×C7—161no—
231C4×C7#39411yescharacteristic
232C4×C7#45211yescharacteristic
233C4×C7#46811yescharacteristic
234C4×C7#48211yescharacteristic
235Dic7—41no—
236Dic7—82no—
237Dic7—161no—
238C24—71no—
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—71no—
248C22×C4—141no—
249C22×C4—141no—
250C22×C4—141no—
251C22×C4—141no—
252C22×C4—141no—
253C22×C4—141no—
254C22×C4—281no—
255C22×C4—281no—
256C22×C4D4×D711yescharacteristic
257Q16—22no—
258Q16—22no—
259Q16—41no—
260Q16—41no—
261C2×C8—141no—
262C2×C8—141no—
263C2×C8—141no—
264C2×C8—281no—
265C2×C8—281no—
266C2×C8—281no—
267C2×C8—281no—
268C2×C8#115711yescharacteristic
269C2×C8#115711yescharacteristic
270C2×C8D4×D711yescharacteristic
271C2×C8D4×D711yescharacteristic
272C2×C8#115911yescharacteristic
273C2×C8C2×C22⋊D711yescharacteristic
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—142no—
277C4⋊C4—284no—
278C4⋊C4—281no—
279C4⋊C4—281no—
280C4⋊C4—281no—
281C22⋊C4—284no—
282C22⋊C4—562no—
283C16—41no—
284C16—142no—
285C2×C7#203311yescharacteristic
286C2×C7#218511yescharacteristic
287C2×C7#218911yescharacteristic
288C2×C7#221711yescharacteristic
289C2×C7#221911yescharacteristic
290C2×C7#223511yescharacteristic
291C2×C7#223711yescharacteristic
292D7—42no—
293C23—72no—
294C23—141no—
295C23—141no—
296C23—141no—
297C23—141no—
298C23—141no—
299C23—141no—
300C23D7×D811yescharacteristic
301Q8—42no—
302Q8—42no—
303Q8—81no—
304Q8—81no—
305C2×C4—42no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—141no—
312C2×C4—141no—
313C2×C4—141no—
314C2×C4—281no—
315C2×C4—281no—
316C2×C4—281no—
317C2×C4—281no—
318C2×C4—282no—
319C2×C4—561no—
320C2×C4—561no—
321C2×C4—561no—
322C2×C4—561no—
323C2×C4—561no—
324C2×C4#644611yescharacteristic
325C2×C4#644711yescharacteristic
326C2×C4#647411yescharacteristic
327C2×C4#647811yescharacteristic
328C2×C4#647911yescharacteristic
329C2×C4#650011yescharacteristic
330C8—281no—
331C8#644711yescharacteristic
332C8#644811yescharacteristic
333C8#649211yescharacteristic
334C8#650211yescharacteristic
335C7#1338211yescharacteristic, complemented, Sylow
336C22—142no—
337C22—142no—
338C22—142no—
339C22—281no—
340C22—281no—
341C22—281no—
342C22—281no—
343C22#8954611yescharacteristic
344C22#8954711yescharacteristic
345C22D7×SD3211yescharacteristic
346C22#8968711yescharacteristic
347C22#8969011yescharacteristic
348C22#8969111yescharacteristic
349C22#8992811yescharacteristic
350C4—82no—
351C4—161no—
352C4—281no—
353C4—562no—
354C4—1121no—
355C4#8955111yescharacteristic
356C4#8955211yescharacteristic
357C4#8973111yescharacteristic
358C4#8994511yescharacteristic
359C2—282no—
360C2#1172696811yescharacteristic
361C2#1172698211yescharacteristic
362C2#1172698311yescharacteristic
363C2#1172698511yescharacteristic
364C2#1172796011yescharacteristic
365C2#1172923711yescharacteristic
366C2#1172925711yescharacteristic
subgroupG/H
centerC23D7×D8
commutator, cosocleC22×C7×C8C23
FrattiniC22×C8C22×D7
FittingC2×C7×#358C2
socleC23×C7C2×D8
prime radicalC23×D7D8
hypercenterC2×#358D7
hypocenterC7#13382
LCSG ▷ C22×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
2. C2×C7×#358C2semidirect
3. C2×#89359C2semidirect
6. C2×#89285C2semidirect
7. #11725612C2semidirect
67. C2×#358D7semidirect
335. C7#13382semidirect

Maximal subgroups

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

subgroupindex
C2×#899602
C2×C7×#3582
C2×#893592
C2×(C2×D7)⋊C162
#117283842
C2×#892852
#117256122
#133827