Finite Group Databasebeta

← #13407419

Subgroups — #13407419

Order 1792.

1052 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1052 conjugacy classes of subgroups — too many to draw. Only its 191 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90388C211yesmaximal, characteristic
2C2×C7×#491C211yesmaximal, characteristic, complemented
3#11740706C211yesmaximal, characteristic
4C2×#89864C211yesmaximal, characteristic, complemented
5#11729494C211yesmaximal, characteristic, complemented
6#11728035C214yesmaximal, complemented
7#11728027C214yesmaximal
8#11727584C211yesmaximal, characteristic
9#11727576C211yesmaximal, characteristic, complemented
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×C7×#125C2211yescharacteristic
12C2×#6509C2212yes—
13C22×#1144C2214yes—
14#90388—24no—
15#90388—24no—
16C2×#6461—22no—
17C2×#6460C2212yes—
18C2×#6453—22no—
19C2×#6450C2212yes—
20C2×#6435C2211yescharacteristic
21C2×#6432C2211yescharacteristic
22C7×#491—24no—
23C2×C7×#103C2214yes—
24C2×C7×#97C2211yescharacteristic
25#90002C2212yes—
26C2×(C2×D7)⋊C8C2212yes—
27C2×#6390C2214yes—
28#89864—24no—
29#89765—22no—
30#89752—22no—
31#89708C2211yescharacteristic
32C2×C4×C7⋊C8C2211yescharacteristic
33#89345C2212yes—
34#89324C2214yes—
35#25609—71nomaximal, Sylow
36C22×C7×Q8C2312yes—
37C22×C4×D7C2311yescharacteristic
38C22×Dic14C2312yes—
39C22×C7×C8C2312yes—
40C7×C4⋊Q8—22no—
41C7×C4⋊Q8—22no—
42C7×#125—22no—
43C7×#125D414yes—
44C2×C7×C4⋊C4—22no—
45C2×C7×C4⋊C4—22no—
46C2×C7×C4⋊C4C2312yes—
47C2×C42×C7C2311yescharacteristic
48#6509—44no—
49C2×#1144—28no—
50C2×#1144—28no—
51C2×#1144—44no—
52C2×(C2×D7)⋊C4—22no—
53C2×(C2×D7)⋊C4—22no—
54C2×(C2×D7)⋊C4D412yes—
55C2×C4⋊Dic7—22no—
56C2×C4⋊Dic7—22no—
57C2×C4⋊Dic7C2312yes—
58C2×Dic7⋊C4—22no—
59C2×Dic7⋊C4—22no—
60C2×Dic7⋊C4D412yes—
61C2×C4×Dic7—22no—
62C2×C4×Dic7C2311yescharacteristic
63C22×C7⋊C8C2312yes—
64#6461—216no—
65#6460—24no—
66#6460—24no—
67#6453—216no—
68#6450—24no—
69#6450—24no—
70#6435—22no—
71#6435—22no—
72#6432—22no—
73#6432D414yes—
74C7×#103—28no—
75C7×#97—22no—
76C7⋊#95—22no—
77C7⋊#95—24no—
78(C2×D7)⋊C8—24no—
79#6390—44no—
80C4×C7⋊C8—22no—
81C2×#541—71no—
82C2×#494—71no—
83C2×#491D711yescharacteristic, complemented
84#2621—71no—
85#2001—74no—
86#1991—71no—
87#1960—71no—
88#1878—74no—
89#1750—71no—
90C23×D7C2×D411yescharacteristic
91C2×C7×Q8—22no—
92C2×C7×Q8—22no—
93C2×C7×Q8—24no—
94C2×C7×Q8—24no—
95C22×C4×C7—22no—
96C22×C4×C7—22no—
97C22×C4×C7C2×D412yes—
98C22×C4×C7C2411yescharacteristic
99C22×Dic7—22no—
100C22×Dic7—22no—
101C22×Dic7C2×D412yes—
102C22×Dic7C2×D411yescharacteristic
103C2×C4×D7—21no—
104C2×C4×D7—21no—
105C2×C4×D7—22no—
106C2×C4×D7—22no—
107C2×Dic14—24no—
108C2×Dic14—24no—
109C2×Dic14—22no—
110C2×Dic14C2×D414yes—
111C2×C7×C8—22no—
112C2×C7×C8—22no—
113C2×C7×C8—22no—
114C7×C4⋊C4—28no—
115C7×C4⋊C4—22no—
116C7×C4⋊C4—44no—
117C7×C4⋊C4C2×D414yes—
118C42×C7C2×D412yes—
119C42×C7C2×D412yes—
120#1144—416no—
121(C2×D7)⋊C4—28no—
122(C2×D7)⋊C4—24no—
123(C2×D7)⋊C4—44no—
124C4⋊Dic7—28no—
125C4⋊Dic7—28no—
126C4⋊Dic7—24no—
127Dic7⋊C4—28no—
128Dic7⋊C4—24no—
129Dic7⋊C4—44no—
130C4×Dic7—28no—
131C4×Dic7—21no—
132C4×Dic7—21no—
133C2×C7⋊C8—22no—
134C2×C7⋊C8—22no—
135C2×C7⋊C8#4114yes—
136C22×Q16—74no—
137#541—144no—
138#541—144no—
139C2×C4⋊Q8—71no—
140C2×C4⋊Q8C2×D711yescharacteristic
141C2×#126—142no—
142C2×#125—142no—
143C2×#125C2×D711yescharacteristic
144C2×C22⋊Q8—72no—
145C2×C22⋊Q8—72no—
146C2×C4×Q8—72no—
147C2×#117—71no—
148#494—144no—
149#491—24no—
150C2×#103—74no—
151C2×#103C2×D714yes—
152C2×C22⋊C8—72no—
153C2×#97C2×D711yescharacteristic
154C2×C4×C8—71no—
155#398—142no—
156#397—142no—
157#391—72no—
158#382—71no—
159#340—74no—
160#336—72no—
161C23×C7C22×D411yescharacteristic
162C22×D7—21no—
163C22×D7—21no—
164C22×D7—21no—
165C22×D7—24no—
166C7×Q8—44no—
167C7×Q8—44no—
168C2×C4×C7—24no—
169C2×C4×C7—24no—
170C2×C4×C7—44no—
171C2×C4×C7—42no—
172C2×C4×C7—42no—
173C2×C4×C7—42no—
174C2×C4×C7#12014yes—
175C2×C4×C7C22×D411yescharacteristic
176C2×C4×C7C22×D411yescharacteristic
177C2×C4×C7C2×#4112yes—
178C2×C4×C7H(5,𝔽2)12yes—
179C2×Dic7—22no—
180C2×Dic7—24no—
181C2×Dic7—21no—
182C2×Dic7—21no—
183C2×Dic7—21no—
184C2×Dic7—24no—
185C2×Dic7—42no—
186C2×Dic7—42no—
187C2×Dic7—42no—
188C2×Dic7—44no—
189C4×D7—42no—
190C4×D7—42no—
191Dic14—28no—
192Dic14—44no—
193C7×C8—42no—
194C7⋊C8—24no—
195C22×Q8—72no—
196C22×Q8C22×D712yes—
197C23×C4—71no—
198C2×Q16—148no—
199C2×Q16—148no—
200C2×Q16—284no—
201C22×C8—72no—
202C22×C8C22×D712yes—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—74no—
206C4⋊Q8—142no—
207#126—1416no—
208#125—22no—
209#125—1416no—
210#125C22⋊D714yes—
211C22⋊Q8—144no—
212C22⋊Q8—144no—
213C22⋊Q8—284no—
214C4×Q8—144no—
215C4×Q8—144no—
216#117—142no—
217#117—142no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—72no—
221C2×C4⋊C4—72no—
222C2×C4⋊C4—142no—
223C2×C4⋊C4—142no—
224C2×C4⋊C4—142no—
225C2×C4⋊C4—142no—
226C2×C4⋊C4C22×D712yes—
227C2×C22⋊C4—72no—
228C2×C22⋊C4—142no—
229C2×C22⋊C4—142no—
230C2×C42—71no—
231C2×C42—142no—
232C2×C42C22×D711yescharacteristic
233#103—28no—
234#103—284no—
235C22⋊C8—144no—
236#97—22no—
237C4×C8—142no—
238#95—144no—
239#95—142no—
240C22×C7C2×#12011yescharacteristic
241C22×C7#53211yescharacteristic
242C22×C7#53511yescharacteristic
243C22×C7#57814yes—
244C2×D7—42no—
245C2×D7—41no—
246C2×D7—42no—
247C2×D7—42no—
248C4×C7—24no—
249C4×C7—44no—
250C4×C7—82no—
251C4×C7#54612yes—
252C4×C7#55012yes—
253Dic7—44no—
254Dic7—41no—
255Dic7—42no—
256Dic7—82no—
257C24—71no—
258C2×Q8—22no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—22no—
262C2×Q8—74no—
263C2×Q8—142no—
264C2×Q8—144no—
265C2×Q8—144no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—72no—
269C22×C4—71no—
270C22×C4—142no—
271C22×C4—141no—
272C22×C4—142no—
273C22×C4—142no—
274C22×C4—141no—
275C22×C4—142no—
276C22×C4D4×D712yes—
277C22×C4C23×D711yescharacteristic
278Q16—2816no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—74no—
283C2×C8—142no—
284C2×C8—142no—
285C4⋊C4—28no—
286C4⋊C4—22no—
287C4⋊C4—44no—
288C4⋊C4—148no—
289C4⋊C4—148no—
290C4⋊C4—144no—
291C4⋊C4—148no—
292C4⋊C4—144no—
293C4⋊C4—284no—
294C4⋊C4C2×C22⋊D714yes—
295C22⋊C4—148no—
296C22⋊C4—144no—
297C22⋊C4—284no—
298C42—148no—
299C42—141no—
300C42—141no—
301C42D4×D712yes—
302C42C2×C22⋊D712yes—
303C2×C7#243111yescharacteristic
304C2×C7#301112yes—
305C2×C7#317712yes—
306C2×C7#322312yes—
307D7—81no—
308C23—144no—
309C23—141no—
310C23—141no—
311C23—141no—
312C23C2×D4×D711yescharacteristic
313Q8—44no—
314Q8—44no—
315Q8—148no—
316Q8—284no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—144no—
324C2×C4—142no—
325C2×C4—144no—
326C2×C4—141no—
327C2×C4—141no—
328C2×C4—141no—
329C2×C4—282no—
330C2×C4—282no—
331C2×C4—282no—
332C2×C4—282no—
333C2×C4—284no—
334C2×C4—282no—
335C2×C4#650014yes—
336C2×C4C2×D4×D711yescharacteristic
337C2×C4D7×#4112yes—
338C2×C4#655312yes—
339C2×C4C22×C22⋊D711yescharacteristic
340C8—42no—
341C8—144no—
342C7#2560911yescharacteristic, complemented, Sylow
343C22—282no—
344C22—282no—
345C22—281no—
346C22—282no—
347C22#9035911yescharacteristic
348C22#9036511yescharacteristic
349C22#9047014yes—
350C22C2×#650011yescharacteristic
351C4—24no—
352C4—44no—
353C4—82no—
354C4—281no—
355C4—284no—
356C4—282no—
357C4—562no—
358C4#9036712yes—
359C4#9052512yes—
360C2—561no—
361C2#1173917512yes—
362C2#1173918912yes—
363C2#1174040311yescharacteristic
364C2#1174142812yes—
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#491C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
hypercenterC2×#491D7
hypocenterC7#25609
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
2. C2×C7×#491C2semidirect
4. C2×#89864C2semidirect
5. #11729494C2semidirect
6. #11728035C2semidirect
9. #11727576C2semidirect
83. C2×#491D7semidirect
342. C7#25609semidirect

Maximal subgroups

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

subgroupindex
C2×#903882
C2×C7×#4912
#117407062
C2×#898642
#117294942
#117280352
#117280272
#117275842
#117275762
#256097