Finite Group Databasebeta

← #13272458

Subgroups — #13272458

Order 1792.

616 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C7×#2409C211yesmaximal, characteristic
2C4×#6408C211yesmaximal, characteristic
3#11729371C211yesmaximal, characteristic
4#11725443C214yesmaximal
5C23×C7×Q8C411yescharacteristic, complemented
6C2×C7×C22⋊Q8—21no—
7C2×C4×C7×D4C2211yescharacteristic
8C7×#395—21no—
9C7×#393—21no—
10C7×#391—21no—
11C7×#386C2212yes—
12C7×#385C412yescomplemented
13C7×#380C411yescharacteristic, complemented
14C2×#6408C2212yes—
15C2×#6407C2212yes—
16C4×C22⋊Dic7—21no—
17#89696—21no—
18#89325—21no—
19#89321—21no—
20#13296—71nomaximal, Sylow
21C22×C7×Q8—21no—
22C22×C7×Q8—21no—
23C22×C7×Q8—41no—
24C22×C7×Q8—41no—
25C22×C7×Q8—44no—
26C22×C7×D4C2311yescharacteristic
27C23×C4×C7C2×C411yescharacteristic
28C23×C4×C7C2×C412yes—
29C23×C4×C7D411yescharacteristic
30C7×C22⋊Q8—44no—
31C4×C7×D4—41no—
32C4×C7×D4—41no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—22no—
35C2×C7×C4⋊C4D411yescharacteristic
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4C2×C412yes—
38C2×C7×C22⋊C4C2×C411yescharacteristic
39C2×C7×C22⋊C4D411yescharacteristic
40C2×C42×C7D411yescharacteristic
41C2×C22⋊Dic7—22no—
42C2×C4×Dic7—41no—
43C2×#1136—22no—
44C7×#95—21no—
45C7×#95—22no—
46C7×#95—22no—
47#6408—22no—
48#6408Q814yescomplemented
49#6407—24no—
50C7⋊#95—41no—
51C4⋊(C7⋊C8)—41no—
52C4⋊(C7⋊C8)—41no—
53#2409D711yescharacteristic
54#1800—71no—
55C4×#99—71no—
56#1390—74no—
57C24×C7C22×C411yescharacteristic
58C24×C7C2×D411yescharacteristic
59C2×C7×Q8—24no—
60C2×C7×Q8—42no—
61C2×C7×Q8—44no—
62C2×C7×Q8—42no—
63C2×C7×Q8—48no—
64C2×C7×Q8—44no—
65C2×C7×Q8—44no—
66C2×C7×Q8—41no—
67C2×C7×Q8—41no—
68C2×C7×D4—41no—
69C2×C7×D4—41no—
70C2×C7×D4C2×Q812yes—
71C2×C7×D4#4112yes—
72C22×C4×C7—22no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—22no—
78C22×C4×C7—42no—
79C22×C4×C7—44no—
80C22×C4×C7—42no—
81C22×C4×C7—42no—
82C22×C4×C7C22⋊C411yescharacteristic
83C22×C4×C7C22⋊C411yescharacteristic
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C2×D411yescharacteristic
87C22×Dic7—42no—
88C7×C4⋊C4—41no—
89C7×C4⋊C4—42no—
90C7×C4⋊C4—44no—
91C7×C22⋊C4—41no—
92C7×C22⋊C4—41no—
93C7×C22⋊C4—44no—
94C7×C22⋊C4—42no—
95C42×C7—21no—
96C42×C7—21no—
97C22⋊Dic7—24no—
98C22⋊Dic7—42no—
99C4×Dic7—82no—
100#1136—42no—
101#1136—42no—
102C2×C7⋊C8—42no—
103C23×Q8Dic711yescharacteristic, complemented
104C2×C22⋊Q8—21no—
105C2×C4×D4C2×D711yescharacteristic
106#423—141no—
107C2×#100—72no—
108C2×#99—72no—
109#395—21no—
110#393—21no—
111#391—21no—
112#386C2×D712yes—
113#385Dic712yescomplemented
114#380Dic711yescharacteristic, complemented
115C4×C22⋊C4—141no—
116#343—141no—
117#342—141no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—41no—
123C23×C7—41no—
124C23×C7—41no—
125C23×C7—41no—
126C23×C7C2×C22⋊C411yescharacteristic
127C23×C7C4×Q812yes—
128C23×C7C22⋊Q812yes—
129C7×Q8—44no—
130C7×Q8—48no—
131C7×Q8—44no—
132C7×D4—42no—
133C7×D4—42no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—22no—
137C2×C4×C7—24no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7—44no—
144C2×C4×C7—42no—
145C2×C4×C7—42no—
146C2×C4×C7—42no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—44no—
151C2×C4×C7—42no—
152C2×C4×C7—42no—
153C2×C4×C7—42no—
154C2×C4×C7—81no—
155C2×C4×C7#9911yescharacteristic
156C2×C4×C7#9911yescharacteristic
157C2×C4×C7#12411yescharacteristic
158C2×C4×C7C4⋊Q811yescharacteristic
159C2×Dic7—44no—
160C2×Dic7—82no—
161C2×Dic7—82no—
162C7⋊C8—82no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—41no—
166C22×Q8—41no—
167C22×Q8—44no—
168C22×D4C22×D711yescharacteristic
169C23×C4C2×Dic711yescharacteristic
170C23×C4C2×Dic712yes—
171C23×C4C22⋊D711yescharacteristic
172C2×M16—142no—
173C22⋊Q8—44no—
174C4×D4—41no—
175C4×D4—41no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4C22⋊D711yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—142no—
181C2×C22⋊C4C2×Dic712yes—
182C2×C22⋊C4C2×Dic711yescharacteristic
183C2×C22⋊C4C22⋊D711yescharacteristic
184C2×C42—281no—
185C2×C42C22⋊D711yescharacteristic
186C4⋊C8—281no—
187C4⋊C8—281no—
188#100—144no—
189#99—74no—
190#99—142no—
191#95—21no—
192#95—22no—
193#95—22no—
194#95—281no—
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—22no—
198C22×C7—22no—
199C22×C7—42no—
200C22×C7—42no—
201C22×C7—42no—
202C22×C7—41no—
203C22×C7—41no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7—81no—
207C22×C7#39111yescharacteristic
208C22×C7C2×#9911yescharacteristic
209C22×C7#41011yescharacteristic
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—44no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7—41no—
217C4×C7—82no—
218C4×C7AΣL(1,𝔽16)12yes—
219Dic7—84no—
220C24C22×Dic711yescharacteristic
221C24C2×C22⋊D711yescharacteristic
222C2×Q8—24no—
223C2×Q8—42no—
224C2×Q8—44no—
225C2×Q8—42no—
226C2×Q8—44no—
227C2×Q8—48no—
228C2×Q8—44no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4Q8×D712yes—
234C2×D4#116112yes—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—282no—
246C22×C4C22⋊Dic711yescharacteristic
247C22×C4C22⋊Dic711yescharacteristic
248C22×C4C22⋊Dic711yescharacteristic
249C22×C4C22⋊Dic711yescharacteristic
250C22×C4C2×C22⋊D711yescharacteristic
251M16—282no—
252M16—282no—
253C2×C8—282no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—44no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—44no—
260C22⋊C4—42no—
261C22⋊C4—144no—
262C22⋊C4—282no—
263C42—21no—
264C42—21no—
265C42—562no—
266C2×C7—21no—
267C2×C7—21no—
268C2×C7—42no—
269C2×C7—42no—
270C2×C7#199611yescharacteristic
271C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
272C2×C7#212311yescharacteristic
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23Q8×Dic712yes—
282C23#650912yes—
283C23C2×C22⋊Dic711yescharacteristic
284Q8—44no—
285Q8—48no—
286Q8—44no—
287D4—42no—
288D4—42no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—81no—
310C2×C4—284no—
311C2×C4—562no—
312C2×C4—562no—
313C2×C4#640811yescharacteristic
314C2×C4#640811yescharacteristic
315C2×C4#650711yescharacteristic
316C2×C4#651011yescharacteristic
317C8—562no—
318C7#1329611yescharacteristic, complemented, Sylow
319C22—21no—
320C22—21no—
321C22—22no—
322C22—22no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22C2×#640811yescharacteristic
332C22#9000211yescharacteristic
333C22#9001211yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—44no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—82no—
342C4—564no—
343C4#8933112yes—
344C2—21no—
345C2—21no—
346C2—42no—
347C2—42no—
348C2C2×#8933111yescharacteristic
349C2#1172945511yescharacteristic
350C2#1172962411yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC24×C7C22×C4
FrattiniC22×D4C22×D7
FittingC7×#2409C2
socleC22×C7C2×#99
prime radical, cosocleC22×C7×D4C23
hypercenter#2409D7
hypocenterC7#13296
LCSG ▷ C24×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
5. C23×C7×Q8C4semidirect
12. C7×#385C4semidirect
13. C7×#380C4semidirect
48. #6408Q8semidirect
103. C23×Q8Dic7semidirect
113. #385Dic7semidirect
114. #380Dic7semidirect
318. C7#13296semidirect

Maximal subgroups

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

subgroupindex
C7×#24092
C4×#64082
#117293712
#117254432
#132967