Finite Group Databasebeta

← #13246783

Subgroups — #13246783

Order 1792.

456 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2929C211yesmaximal, characteristic
2#11729731C211yesmaximal, characteristic, complemented
3#11729724C211yesmaximal, characteristic, complemented
4#11729463C212yesmaximal
5#11725490C212yesmaximal, complemented
6C2×C7×#131C411yescharacteristic, complemented
7C23×C7×C8C2211yescharacteristic
8C7×#525C2211yescharacteristic
9C7×#435—21no—
10C7×#435—21no—
11C7×C8×D4—21no—
12C7×C8×D4—21no—
13C7×#424C411yescharacteristic, complemented
14C7×#408C411yescharacteristic, complemented
15C7×#407C2211yescharacteristic
16C7×#405C411yescharacteristic, complemented
17#89984C2212yes—
18#89979C2212yes—
19C2×#6393—21no—
20C2×#6391—21no—
21#89347—21no—
22#89346—21no—
23#89333—22no—
24#89332—22no—
25#8629—71nomaximal, Sylow
26C2×C7×#41C2×C411yescharacteristic
27C23×C4×C7C2311yescharacteristic
28C7×#131—42no—
29C2×C7×M16—21no—
30C2×C7×M16C2×C411yescharacteristic
31C2×C7×M16C2×C411yescharacteristic
32C22×C7×C8—21no—
33C22×C7×C8—42no—
34C22×C7×C8C2×C411yescharacteristic
35C22×C7×C8C2×C411yescharacteristic
36C22×C7×C8D411yescharacteristic
37C22×C7×C8D411yescharacteristic
38C22×C7×C8D411yescharacteristic
39C22×C7×C8D411yescharacteristic
40C7×#123—21no—
41C7×C22⋊Q8D411yescharacteristic
42C7×C22⋊Q8Q811yescharacteristic
43C7×#121D411yescharacteristic
44C7×#121Q811yescharacteristic
45C7×#120—21no—
46C4×C7×D4—21no—
47C4×C7×D4—21no—
48C7×#117C2×C411yescharacteristic
49C2×C22⋊Dic7—22no—
50C2×C4⋊Dic7—22no—
51C2×#1136—22no—
52C7×C4⋊C8—21no—
53C7×C4⋊C8—21no—
54C7×C22⋊C8—21no—
55C7×C22⋊C8—21no—
56C7×C22⋊C8—21no—
57C7×C22⋊C8—21no—
58C7×#97—21no—
59C4×C7×C8—21no—
60C7⋊#95—22no—
61C22⋊(C7⋊C8)—22no—
62C22⋊(C7⋊C8)—22no—
63#6393—42no—
64#6391—42no—
65#2929D711yescharacteristic
66#1810—71no—
67#1807—71no—
68#1513—72no—
69#1305—72no—
70C24×C7#4111yescharacteristic
71C7×#41—41no—
72C7×#41—41no—
73C2×C7×Q8C4⋊C411yescharacteristic
74C2×C7×D4—21no—
75C2×C7×D4—21no—
76C2×C7×D4—21no—
77C2×C7×D4C4⋊C411yescharacteristic
78C22×C4×C7—21no—
79C22×C4×C7—41no—
80C22×C4×C7—41no—
81C22×C4×C7C22×C411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C22×C4×C7C2×D411yescharacteristic
84C22×C4×C7C2×D411yescharacteristic
85C22×C4×C7C2×Q811yescharacteristic
86C22×C4×C7#4111yescharacteristic
87C22×Dic7—42no—
88C7×M16—42no—
89C7×M16—41no—
90C7×M16—41no—
91C2×C7×C8—22no—
92C2×C7×C8—22no—
93C2×C7×C8—22no—
94C2×C7×C8—21no—
95C2×C7×C8—21no—
96C2×C7×C8—41no—
97C2×C7×C8—41no—
98C2×C7×C8—41no—
99C2×C7×C8—42no—
100C2×C7×C8C22⋊C411yescharacteristic
101C2×C7×C8C22⋊C411yescharacteristic
102C2×C7×C8C22⋊C411yescharacteristic
103C2×C7×C8C22⋊C411yescharacteristic
104C7×C4⋊C4—21no—
105C7×C4⋊C4—21no—
106C7×C4⋊C4C4⋊C411yescharacteristic
107C7×C4⋊C4C4⋊C411yescharacteristic
108C7×C22⋊C4—21no—
109C7×C22⋊C4—21no—
110C7×C22⋊C4—21no—
111C7×C22⋊C4—21no—
112C7×C22⋊C4—21no—
113C42×C7—21no—
114C22⋊Dic7—82no—
115C4⋊Dic7—42no—
116C4⋊Dic7—42no—
117#1136—42no—
118#1136—42no—
119C2×C7⋊C8—42no—
120C2×#131Dic711yescharacteristic, complemented
121C23×C8C2×D711yescharacteristic
122#525C2×D711yescharacteristic
123#435—21no—
124#435—21no—
125C8×D4—21no—
126C8×D4—21no—
127C2×#108—141no—
128C2×#106—141no—
129#424Dic711yescharacteristic, complemented
130#408Dic711yescharacteristic, complemented
131#407—72no—
132#407C2×D711yescharacteristic
133#405Dic711yescharacteristic, complemented
134#380—72no—
135#341—141no—
136#340—141no—
137#328—142no—
138#327—142no—
139C23×C7—22no—
140C23×C7—21no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7#11711yescharacteristic
144C23×C7#12111yescharacteristic
145C23×C7C22⋊Q811yescharacteristic
146C7×Q8—41no—
147C7×D4—41no—
148C7×D4—42no—
149C7×D4—42no—
150C7×D4—42no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—21no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7C2×C22⋊C411yescharacteristic
166C2×C4×C7C2×C4⋊C411yescharacteristic
167C2×C4×C7#12111yescharacteristic
168C2×C4×C7C22⋊Q811yescharacteristic
169C2×Dic7—82no—
170C2×Dic7—82no—
171C2×Dic7—82no—
172C7×C8—21no—
173C7×C8—21no—
174C7×C8—21no—
175C7×C8—41no—
176C7×C8—41no—
177C7×C8#9912yes—
178C7⋊C8—82no—
179C2×#41C2×Dic711yescharacteristic
180C23×C4C22×D711yescharacteristic
181#131—42no—
182C2×M16—21no—
183C2×M16—142no—
184C2×M16C2×Dic711yescharacteristic
185C2×M16C2×Dic711yescharacteristic
186C22×C8—21no—
187C22×C8—42no—
188C22×C8C2×Dic711yescharacteristic
189C22×C8C2×Dic711yescharacteristic
190C22×C8C22⋊D711yescharacteristic
191C22×C8C22⋊D711yescharacteristic
192C22×C8C22⋊D711yescharacteristic
193C22×C8C22⋊D711yescharacteristic
194#123—21no—
195C22⋊Q8Dic1411yescharacteristic
196C22⋊Q8D2811yescharacteristic
197#121Dic1411yescharacteristic
198#121D2811yescharacteristic
199#120—21no—
200C4×D4—21no—
201C4×D4—21no—
202#117C2×Dic711yescharacteristic
203C2×C4⋊C4—142no—
204C2×C22⋊C4—142no—
205#108—282no—
206#106—282no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8—21no—
212C22⋊C8—21no—
213C22⋊C8—142no—
214C22⋊C8—142no—
215#97—21no—
216C4×C8—21no—
217#95—142no—
218C22×C7—21no—
219C22×C7—21no—
220C22×C7—21no—
221C22×C7—21no—
222C22×C7—21no—
223C22×C7—21no—
224C22×C7—21no—
225C22×C7—21no—
226C22×C7—44no—
227C22×C7—41no—
228C22×C7—41no—
229C22×C7—41no—
230C22×C7#38011yescharacteristic
231C22×C7#47112yes—
232C4×C7—21no—
233C4×C7—21no—
234C4×C7—21no—
235C4×C7—41no—
236C4×C7—41no—
237C4×C7—41no—
238C4×C7C2×#9911yescharacteristic
239C4×C7#41011yescharacteristic
240Dic7—162no—
241C24#115711yescharacteristic
242#41—41no—
243#41—41no—
244C2×Q8C4⋊Dic711yescharacteristic
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4C4⋊Dic711yescharacteristic
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—282no—
253C22×C4C2×Dic1411yescharacteristic
254C22×C4C2×D2811yescharacteristic
255C22×C4#115711yescharacteristic
256C22×C4C22×Dic711yescharacteristic
257C22×C4C2×C22⋊D711yescharacteristic
258C22×C4C2×C22⋊D711yescharacteristic
259M16—42no—
260M16—41no—
261M16—41no—
262M16—282no—
263M16—282no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—42no—
273C2×C8—282no—
274C2×C8C22⋊Dic711yescharacteristic
275C2×C8C22⋊Dic711yescharacteristic
276C2×C8C22⋊Dic711yescharacteristic
277C2×C8C22⋊Dic711yescharacteristic
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—282no—
281C4⋊C4—282no—
282C4⋊C4C4⋊Dic711yescharacteristic
283C4⋊C4C4⋊Dic711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—562no—
290C42—21no—
291C2×C7—22no—
292C2×C7—41no—
293C2×C7—41no—
294C2×C7—81no—
295C2×C7#179911yescharacteristic
296C2×C7#181411yescharacteristic
297C2×C7#181511yescharacteristic
298C23—22no—
299C23—21no—
300C23—41no—
301C23—41no—
302C23#648711yescharacteristic
303C23#648911yescharacteristic
304C23#649311yescharacteristic
305Q8—41no—
306D4—41no—
307D4—42no—
308D4—42no—
309D4—42no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—562no—
325C2×C4—562no—
326C2×C4—562no—
327C2×C4#648711yescharacteristic
328C2×C4C2×C4⋊Dic711yescharacteristic
329C2×C4#649311yescharacteristic
330C2×C4C2×C22⋊Dic711yescharacteristic
331C8—21no—
332C8—21no—
333C8—21no—
334C8—41no—
335C8—41no—
336C8—562no—
337C8#640812yes—
338C7#862911yescharacteristic, complemented, Sylow
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—44no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22#8991312yes—
352C22#8998411yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—1122no—
360C4C2×#640811yescharacteristic
361C4#9001211yescharacteristic
362C2—22no—
363C2—41no—
364C2—41no—
365C2—81no—
366C2#1172937311yescharacteristic
367C2#1172976711yescharacteristic
368C2#1172976911yescharacteristic
subgroupG/H
centerC22#89984
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2929C2
socleC22×C7#380
prime radicalC7×#525C22
cosocleC23×C4×C7C23
hypercenter#2929D7
hypocenterC7#8629
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2929

Complements

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

NH ≅ G/Ntype
2. #11729731C2semidirect
3. #11729724C2semidirect
5. #11725490C2semidirect
6. C2×C7×#131C4semidirect
13. C7×#424C4semidirect
14. C7×#408C4semidirect
16. C7×#405C4semidirect
120. C2×#131Dic7semidirect
129. #424Dic7semidirect
130. #408Dic7semidirect
133. #405Dic7semidirect
338. C7#8629semidirect

Maximal subgroups

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

subgroupindex
C7×#29292
#117297312
#117297242
#117294632
#117254902
#86297