Finite Group Databasebeta

← #13245159

Subgroups — #13245159

Order 1792.

874 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C22×#6461C211yesmaximal, characteristic
2C2×C7×#324C211yesmaximal, characteristic, complemented
3C2×#89323C211yesmaximal, characteristic, complemented
4C2×#89263C212yesmaximal
5#11727664C212yesmaximal, complemented
6C22×C7×C4⋊C4C2211yescharacteristic
7C22×(C2×D7)⋊C4C411yescharacteristic
8C22×(C2×D7)⋊C4C411yescharacteristic
9C22×(C2×D7)⋊C4C2211yescharacteristic
10C22×C4⋊Dic7C411yescharacteristic
11C22×Dic7⋊C4C411yescharacteristic
12C22×C4×Dic7C2211yescharacteristic
13C2×#6461—22no—
14C2×#6461—22no—
15C2×#6461—24no—
16C2×#6461—24no—
17C2×C7×C22⋊C8C2212yes—
18C2×C22⋊(C7⋊C8)C2212yes—
19C7×#324—22no—
20#89345—24no—
21#89323—22no—
22#89263—24no—
23#8309—71nomaximal, Sylow
24C24×D7C2×C411yescharacteristic
25C23×C4×C7C2×C411yescharacteristic
26C23×C4×C7C2×C411yescharacteristic
27C23×C4×C7C2311yescharacteristic
28C23×Dic7C2×C411yescharacteristic
29C23×Dic7C2×C411yescharacteristic
30C23×Dic7C2×C411yescharacteristic
31C22×C7×C8—22no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4—22no—
34C2×C7×C4⋊C4D412yes—
35C2×C7×C4⋊C4D412yes—
36C2×(C2×D7)⋊C4—22no—
37C2×(C2×D7)⋊C4—22no—
38C2×(C2×D7)⋊C4—22no—
39C2×(C2×D7)⋊C4—22no—
40C2×(C2×D7)⋊C4—21no—
41C2×(C2×D7)⋊C4—22no—
42C2×(C2×D7)⋊C4—21no—
43C2×(C2×D7)⋊C4—22no—
44C2×(C2×D7)⋊C4—22no—
45C2×(C2×D7)⋊C4D414yes—
46C2×C4⋊Dic7—22no—
47C2×C4⋊Dic7—22no—
48C2×C4⋊Dic7—22no—
49C2×Dic7⋊C4—22no—
50C2×Dic7⋊C4—22no—
51C2×Dic7⋊C4—22no—
52C2×C4×Dic7—21no—
53C2×C4×Dic7—22no—
54C2×C4×Dic7—21no—
55C2×C4×Dic7D414yes—
56C22×C7⋊C8—22no—
57#6461—216no—
58#6461—216no—
59C7×C22⋊C8—22no—
60C7×C22⋊C8—22no—
61C22⋊(C7⋊C8)—22no—
62C22⋊(C7⋊C8)—22no—
63C22×#126—71no—
64#1817—72no—
65C2×#324—71no—
66C2×#324D711yescharacteristic, complemented
67C2×#323—72no—
68C24×C7C22×C411yescharacteristic
69C23×D7—21no—
70C23×D7—21no—
71C23×D7—22no—
72C23×D7—22no—
73C23×D7—42no—
74C23×D7—42no—
75C23×D7C22⋊C412yes—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—22no—
81C22×C4×C7—22no—
82C22×C4×C7—42no—
83C22×C4×C7—42no—
84C22×C4×C7C22⋊C412yes—
85C22×C4×C7C22⋊C412yes—
86C22×C4×C7C2×D412yes—
87C22×C4×C7C2×D411yescharacteristic
88C22×C4×C7C2×D412yes—
89C22×C4×C7C2×D411yescharacteristic
90C22×Dic7—21no—
91C22×Dic7—21no—
92C22×Dic7—21no—
93C22×Dic7—21no—
94C22×Dic7—21no—
95C22×Dic7—21no—
96C22×Dic7—22no—
97C22×Dic7—22no—
98C22×Dic7—42no—
99C22×Dic7—42no—
100C22×Dic7C22⋊C412yes—
101C22×Dic7C22⋊C412yes—
102C22×Dic7C22⋊C412yes—
103C2×C7×C8—44no—
104C2×C7×C8—42no—
105C7×C4⋊C4—24no—
106C7×C4⋊C4—24no—
107(C2×D7)⋊C4—28no—
108(C2×D7)⋊C4—28no—
109(C2×D7)⋊C4—24no—
110(C2×D7)⋊C4—24no—
111C4⋊Dic7—28no—
112Dic7⋊C4—28no—
113C4×Dic7—24no—
114C4×Dic7—24no—
115C2×C7⋊C8—44no—
116C2×C7⋊C8—42no—
117C2×#126—144no—
118C2×#126—144no—
119C2×#126—142no—
120C2×#126—142no—
121C22×C4⋊C4—71no—
122C22×C4⋊C4—71no—
123C22×C4⋊C4C2×D711yescharacteristic
124C22×C22⋊C4—71no—
125C22×C22⋊C4—71no—
126C22×C22⋊C4—71no—
127C22×C42—71no—
128C2×C22⋊C8—72no—
129C2×C22⋊C8C2×D712yes—
130#336—144no—
131#324—22no—
132#324—142no—
133#323—144no—
134C23×C7—22no—
135C23×C7—22no—
136C23×C7C2×C22⋊C411yescharacteristic
137C23×C7C2×C22⋊C411yescharacteristic
138C23×C7C2×C22⋊C411yescharacteristic
139C23×C7#13112yes—
140C23×C7#13112yes—
141C22×D7—22no—
142C22×D7—24no—
143C22×D7—42no—
144C22×D7—42no—
145C22×D7—41no—
146C22×D7—41no—
147C22×D7—42no—
148C22×D7—42no—
149C22×D7—42no—
150C22×D7—42no—
151C22×D7—42no—
152C22×D7—84no—
153C22×D7—84no—
154C2×C4×C7—24no—
155C2×C4×C7—22no—
156C2×C4×C7—22no—
157C2×C4×C7—22no—
158C2×C4×C7—22no—
159C2×C4×C7—41no—
160C2×C4×C7—42no—
161C2×C4×C7—41no—
162C2×C4×C7—42no—
163C2×C4×C7—42no—
164C2×C4×C7—42no—
165C2×C4×C7—42no—
166C2×C4×C7—42no—
167C2×C4×C7#12012yes—
168C2×C4×C7#12012yes—
169C2×Dic7—22no—
170C2×Dic7—22no—
171C2×Dic7—22no—
172C2×Dic7—24no—
173C2×Dic7—24no—
174C2×Dic7—42no—
175C2×Dic7—42no—
176C2×Dic7—42no—
177C2×Dic7—41no—
178C2×Dic7—41no—
179C2×Dic7—42no—
180C2×Dic7—42no—
181C2×Dic7—42no—
182C7×C8—82no—
183C7⋊C8—82no—
184C25—71no—
185C23×C4—71no—
186C23×C4—71no—
187C23×C4—71no—
188C23×C4C4×D711yescharacteristic
189C23×C4C4×D711yescharacteristic
190C23×C4C22×D711yescharacteristic
191C22×C8—22no—
192C22×C8—142no—
193#126—1416no—
194#126—1416no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—142no—
198C2×C4⋊C4—142no—
199C2×C4⋊C4—142no—
200C2×C4⋊C4—142no—
201C2×C4⋊C4—142no—
202C2×C4⋊C4—142no—
203C2×C4⋊C4D2812yes—
204C2×C4⋊C4C22⋊D712yes—
205C2×C22⋊C4—74no—
206C2×C22⋊C4—142no—
207C2×C22⋊C4—142no—
208C2×C22⋊C4—142no—
209C2×C22⋊C4—142no—
210C2×C22⋊C4—142no—
211C2×C22⋊C4—142no—
212C2×C22⋊C4—141no—
213C2×C22⋊C4—142no—
214C2×C22⋊C4—141no—
215C2×C42—74no—
216C2×C42—141no—
217C2×C42—142no—
218C2×C42—141no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—142no—
222C22⋊C8—142no—
223C22×C7—24no—
224C22×C7—24no—
225C22×C7—21no—
226C22×C7—21no—
227C22×C7—21no—
228C22×C7—21no—
229C22×C7#37911yescharacteristic
230C22×C7#40811yescharacteristic
231C22×C7#40811yescharacteristic
232C22×C7#40812yes—
233C22×C7#40812yes—
234C22×C7#41012yes—
235C22×C7#41312yes—
236C2×D7—42no—
237C2×D7—42no—
238C2×D7—42no—
239C2×D7—82no—
240C2×D7—82no—
241C2×D7—84no—
242C2×D7—84no—
243C4×C7—22no—
244C4×C7—22no—
245C4×C7—42no—
246C4×C7—42no—
247Dic7—42no—
248Dic7—42no—
249Dic7—42no—
250C24—72no—
251C24—141no—
252C24—141no—
253C24—142no—
254C24—142no—
255C24—282no—
256C24—282no—
257C24C2×C4×D711yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—72no—
267C22×C4—72no—
268C22×C4—72no—
269C22×C4—141no—
270C22×C4—141no—
271C22×C4—141no—
272C22×C4—142no—
273C22×C4—142no—
274C22×C4—141no—
275C22×C4—141no—
276C22×C4—141no—
277C22×C4—282no—
278C22×C4—282no—
279C22×C4(C2×D7)⋊C412yes—
280C22×C4(C2×D7)⋊C412yes—
281C22×C4C2×D2811yescharacteristic
282C22×C4D4×D712yes—
283C22×C4D4×D712yes—
284C22×C4C2×C22⋊D711yescharacteristic
285C2×C8—44no—
286C2×C8—42no—
287C2×C8—284no—
288C2×C8—282no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—148no—
292C4⋊C4—148no—
293C22⋊C4—148no—
294C22⋊C4—148no—
295C22⋊C4—144no—
296C22⋊C4—144no—
297C42—144no—
298C42—144no—
299C2×C7—21no—
300C2×C7—21no—
301C2×C7—22no—
302C2×C7#147314yes—
303C2×C7#178011yescharacteristic
304C2×C7#178211yescharacteristic
305C2×C7#178511yescharacteristic
306D7—82no—
307C23—22no—
308C23—22no—
309C23—142no—
310C23—144no—
311C23—282no—
312C23—282no—
313C23—282no—
314C23—282no—
315C23—281no—
316C23—281no—
317C23—282no—
318C23—282no—
319C23—282no—
320C23—564no—
321C23—564no—
322C23D7×C22⋊C411yescharacteristic
323C23D7×C22⋊C411yescharacteristic
324C23#646412yes—
325C23#647012yes—
326C23C2×(C2×D7)⋊C411yescharacteristic
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—142no—
341C2×C4—142no—
342C2×C4—142no—
343C2×C4—144no—
344C2×C4—144no—
345C2×C4—282no—
346C2×C4—282no—
347C2×C4—282no—
348C2×C4—281no—
349C2×C4—281no—
350C2×C4—282no—
351C2×C4—282no—
352C2×C4—282no—
353C2×C4#644512yes—
354C2×C4#650012yes—
355C8—82no—
356C8—562no—
357C7#830911yescharacteristic, complemented, Sylow
358C22—24no—
359C22—24no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—282no—
365C22—282no—
366C22—282no—
367C22—564no—
368C22—562no—
369C22—562no—
370C22—564no—
371C22#8949912yes—
372C22#8949912yes—
373C22#8951212yes—
374C22#8952312yes—
375C22#8972711yescharacteristic
376C22#8988211yescharacteristic
377C22#8990111yescharacteristic
378C4—22no—
379C4—22no—
380C4—42no—
381C4—42no—
382C4—282no—
383C4—282no—
384C4—282no—
385C2—21no—
386C2—21no—
387C2—22no—
388C2—562no—
389C2#1172588314yes—
390C2#1172771011yescharacteristic
391C2#1172778111yescharacteristic
392C2#1172786811yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC24×C7C22×C4
FrattiniC23×C4C22×D7
FittingC2×C7×#324C2
socleC23×C7C2×C22⋊C4
prime radicalC24×D7C2×C4
cosocleC23×C4×C7C23
hypercenterC2×#324D7
hypocenterC7#8309
LCSG ▷ C24×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. C2×C7×#324C2semidirect
3. C2×#89323C2semidirect
5. #11727664C2semidirect
66. C2×#324D7semidirect
357. C7#8309semidirect

Maximal subgroups

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

subgroupindex
C22×#64612
C2×C7×#3242
C2×#893232
C2×#892632
#117276642
#83097