Finite Group Databasebeta

← #13246773

Subgroups — #13246773

Order 1792.

456 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2929C211yesmaximal, characteristic
2#11729731C211yesmaximal, characteristic, complemented
3#11729725C211yesmaximal, 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×#6392—21no—
21#89347—21no—
22#89346—21no—
23#89333—22no—
24#89332—22no—
25#8627—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#6392—42no—
65#2929D711yescharacteristic
66#1810—71no—
67#1808—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×#107—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—21no—
140C23×C7—21no—
141C23×C7—21no—
142C23×C7—41no—
143C23×C7—41no—
144C23×C7#11711yescharacteristic
145C23×C7#12111yescharacteristic
146C23×C7C22⋊Q811yescharacteristic
147C7×Q8—41no—
148C7×D4—41no—
149C7×D4—42no—
150C7×D4—42no—
151C7×D4—41no—
152C7×D4—41no—
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—21no—
161C2×C4×C7—21no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7C2×C22⋊C411yescharacteristic
168C2×C4×C7C2×C4⋊C411yescharacteristic
169C2×C4×C7#12111yescharacteristic
170C2×C4×C7C22⋊Q811yescharacteristic
171C2×Dic7—82no—
172C2×Dic7—82no—
173C2×Dic7—82no—
174C7×C8—21no—
175C7×C8—21no—
176C7×C8—21no—
177C7×C8—41no—
178C7×C8—41no—
179C7×C8#9912yes—
180C7⋊C8—82no—
181C2×#41C2×Dic711yescharacteristic
182C23×C4C22×D711yescharacteristic
183#131—42no—
184C2×M16—21no—
185C2×M16—142no—
186C2×M16C2×Dic711yescharacteristic
187C2×M16C2×Dic711yescharacteristic
188C22×C8—21no—
189C22×C8—42no—
190C22×C8C2×Dic711yescharacteristic
191C22×C8C2×Dic711yescharacteristic
192C22×C8C22⋊D711yescharacteristic
193C22×C8C22⋊D711yescharacteristic
194C22×C8C22⋊D711yescharacteristic
195C22×C8C22⋊D711yescharacteristic
196#123—21no—
197C22⋊Q8Dic1411yescharacteristic
198C22⋊Q8D2811yescharacteristic
199#121Dic1411yescharacteristic
200#121D2811yescharacteristic
201#120—21no—
202C4×D4—21no—
203C4×D4—21no—
204#117C2×Dic711yescharacteristic
205C2×C4⋊C4—142no—
206C2×C22⋊C4—142no—
207#108—282no—
208#107—282no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C22⋊C8—21no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214C22⋊C8—21no—
215C22⋊C8—142no—
216C22⋊C8—142no—
217#97—21no—
218C4×C8—21no—
219#95—142no—
220C22×C7—21no—
221C22×C7—21no—
222C22×C7—21no—
223C22×C7—21no—
224C22×C7—21no—
225C22×C7—21no—
226C22×C7—21no—
227C22×C7—21no—
228C22×C7—42no—
229C22×C7—42no—
230C22×C7—41no—
231C22×C7—41no—
232C22×C7—41no—
233C22×C7#38011yescharacteristic
234C22×C7#47211yescharacteristic
235C22×C7#47311yescharacteristic
236C4×C7—21no—
237C4×C7—21no—
238C4×C7—21no—
239C4×C7—41no—
240C4×C7—41no—
241C4×C7—41no—
242C4×C7C2×#9911yescharacteristic
243C4×C7#41011yescharacteristic
244Dic7—162no—
245C24#115711yescharacteristic
246#41—41no—
247#41—41no—
248C2×Q8C4⋊Dic711yescharacteristic
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4C4⋊Dic711yescharacteristic
253C22×C4—21no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—282no—
257C22×C4C2×Dic1411yescharacteristic
258C22×C4C2×D2811yescharacteristic
259C22×C4#115711yescharacteristic
260C22×C4C22×Dic711yescharacteristic
261C22×C4C2×C22⋊D711yescharacteristic
262C22×C4C2×C22⋊D711yescharacteristic
263M16—42no—
264M16—41no—
265M16—41no—
266M16—282no—
267M16—282no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—42no—
277C2×C8—282no—
278C2×C8C22⋊Dic711yescharacteristic
279C2×C8C22⋊Dic711yescharacteristic
280C2×C8C22⋊Dic711yescharacteristic
281C2×C8C22⋊Dic711yescharacteristic
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—282no—
285C4⋊C4—282no—
286C4⋊C4C4⋊Dic711yescharacteristic
287C4⋊C4C4⋊Dic711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—562no—
294C42—21no—
295C2×C7—21no—
296C2×C7—21no—
297C2×C7—41no—
298C2×C7—41no—
299C2×C7—81no—
300C2×C7#179911yescharacteristic
301C2×C7#181411yescharacteristic
302C2×C7#181611yescharacteristic
303C23—21no—
304C23—21no—
305C23—21no—
306C23—41no—
307C23—41no—
308C23#648711yescharacteristic
309C23#648911yescharacteristic
310C23#649311yescharacteristic
311Q8—41no—
312D4—41no—
313D4—42no—
314D4—42no—
315D4—41no—
316D4—41no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—562no—
332C2×C4—562no—
333C2×C4—562no—
334C2×C4#648711yescharacteristic
335C2×C4C2×C4⋊Dic711yescharacteristic
336C2×C4#649311yescharacteristic
337C2×C4C2×C22⋊Dic711yescharacteristic
338C8—21no—
339C8—21no—
340C8—21no—
341C8—41no—
342C8—41no—
343C8—562no—
344C8#640812yes—
345C7#862711yescharacteristic, complemented, Sylow
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—42no—
355C22—42no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22#8991411yescharacteristic
360C22#8991511yescharacteristic
361C22#8998411yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—1122no—
369C4C2×#640811yescharacteristic
370C4#9001211yescharacteristic
371C2—21no—
372C2—21no—
373C2—41no—
374C2—41no—
375C2—81no—
376C2#1172937311yescharacteristic
377C2#1172976711yescharacteristic
378C2#1172977011yescharacteristic
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#8627
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. #11729725C2semidirect
5. #11725490C2semidirect
6. C2×C7×#131C4semidirect
13. C7×#424C4semidirect
14. C7×#408C4semidirect
16. C7×#405C4semidirect
120. C2×#131Dic7semidirect
129. #424Dic7semidirect
130. #408Dic7semidirect
133. #405Dic7semidirect
345. C7#8627semidirect

Maximal subgroups

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

subgroupindex
C7×#29292
#117297312
#117297252
#117294632
#117254902
#86277