Finite Group Databasebeta

← #13264542

Subgroups — #13264542

Order 1792.

386 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 79 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
1#11739628C211yesmaximal, characteristic
2C7×#1821C211yesmaximal, characteristic
3#11729306C211yesmaximal, characteristic
4#11725302C212yesmaximal, complemented
5#11725300C212yesmaximal, complemented
6C22×C7×C4⋊C4C2211yescharacteristic
7C2×#6487C411yescharacteristic
8C7×#407—21no—
9#89984C411yescharacteristic
10#89983—21no—
11#89982—21no—
12#89979—21no—
13#89758C411yescharacteristic
14#89758C2211yescharacteristic
15#89756—21no—
16#89756—21no—
17#89751C411yescharacteristic
18#89751C2211yescharacteristic
19C7×#343—21no—
20C7×#343C2212yes—
21#89323—21no—
22#89323C2212yes—
23#11817—71nomaximal, Sylow
24C23×C4×C7C2×C411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C22×Dic14C2×C411yescharacteristic
28C2×C7×M16—22no—
29C2×C7×C4⋊C4—41no—
30C2×C7×C4⋊C4—41no—
31C2×C7×C4⋊C4C2×C411yescharacteristic
32C2×C7×C4⋊C4D411yescharacteristic
33C2×C7×C4⋊C4D411yescharacteristic
34C2×C7×C4⋊C4C2311yescharacteristic
35C2×C22⋊Dic7—21no—
36C2×C22⋊Dic7—21no—
37C2×C4⋊Dic7—21no—
38C2×C4⋊Dic7C2×C411yescharacteristic
39#6487—41no—
40#6487—41no—
41C2×Dic7⋊C4—21no—
42C2×Dic7⋊C4—21no—
43C2×Dic7⋊C4—21no—
44C2×#1136—41no—
45C7×C22⋊C8—41no—
46C7×C22⋊C8—41no—
47C7⋊#95—21no—
48C7⋊#95—21no—
49C7⋊#95—21no—
50C7⋊#95—21no—
51C7⋊#95—21no—
52C7⋊#95—21no—
53C7⋊#95C2×C411yescharacteristic
54C7⋊#95C2×C411yescharacteristic
55C22⋊(C7⋊C8)—22no—
56C22⋊(C7⋊C8)—41no—
57#2627—71no—
58#1821D711yescharacteristic
59#1463—71no—
60#1354—72no—
61#1353—72no—
62C24×C7C22⋊C411yescharacteristic
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—22no—
70C22×C4×C7—41no—
71C22×C4×C7—81no—
72C22×C4×C7—81no—
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22×C411yescharacteristic
77C22×C4×C7C2×D411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C22×Dic7—21no—
80C22×Dic7—21no—
81C22×Dic7—21no—
82C22×Dic7—21no—
83C2×Dic14—41no—
84C2×Dic14—81no—
85C7×M16—42no—
86C7×M16—42no—
87C2×C7×C8—42no—
88C7×C4⋊C4—41no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—41no—
92C22⋊Dic7—42no—
93C22⋊Dic7—41no—
94C22⋊Dic7—41no—
95C4⋊Dic7—41no—
96C4⋊Dic7—81no—
97Dic7⋊C4—42no—
98Dic7⋊C4—81no—
99Dic7⋊C4—81no—
100#1136—82no—
101C2×C7⋊C8—42no—
102C2×C22⋊Q8—71no—
103C22×C4⋊C4C2×D711yescharacteristic
104#407—21no—
105#407—141no—
106#400—71no—
107#400—71no—
108#398—141no—
109#398—141no—
110#395—71no—
111#395—71no—
112#385—141no—
113#381—141no—
114#380—71no—
115#343—21no—
116#343C2×D712yes—
117#324—72no—
118#324—141no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—41no—
122C23×C7—41no—
123C23×C7#9911yescharacteristic
124C23×C7#9911yescharacteristic
125C23×C7C2×C22⋊C411yescharacteristic
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—41no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—81no—
143C2×C4×C7—81no—
144C2×C4×C7—81no—
145C2×C4×C7#13712yes—
146C2×Dic7—41no—
147C2×Dic7—41no—
148C2×Dic7—41no—
149C2×Dic7—41no—
150C2×Dic7—41no—
151C2×Dic7—41no—
152C2×Dic7—42no—
153C2×Dic7—41no—
154C2×Dic7—41no—
155C2×Dic7—41no—
156C2×Dic7—41no—
157Dic14—82no—
158C7×C8—82no—
159C7⋊C8—82no—
160C22×Q8—71no—
161C23×C4C4×D711yescharacteristic
162C23×C4D2811yescharacteristic
163C23×C4C22⋊D711yescharacteristic
164C2×M16—22no—
165C2×M16—281no—
166C22⋊Q8—281no—
167C22⋊Q8—281no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—71no—
171C2×C4⋊C4—141no—
172C2×C4⋊C4—141no—
173C2×C4⋊C4—141no—
174C2×C4⋊C4—141no—
175C2×C4⋊C4C4×D711yescharacteristic
176C2×C4⋊C4D2811yescharacteristic
177C2×C4⋊C4C22⋊D711yescharacteristic
178C2×C4⋊C4C22×D711yescharacteristic
179C2×C22⋊C4—141no—
180C2×C22⋊C4—141no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183C22⋊C8—142no—
184C22⋊C8—281no—
185#95—71no—
186#95—71no—
187#95—141no—
188#95—141no—
189#95—141no—
190#95—141no—
191#95—141no—
192#95—141no—
193C22×C7—21no—
194C22×C7—21no—
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—42no—
202C22×C7—42no—
203C22×C7C2×#9911yescharacteristic
204C22×C7#41911yescharacteristic
205C22×C7#42111yescharacteristic
206C4×C7—22no—
207C4×C7—41no—
208C4×C7—41no—
209C4×C7—41no—
210C4×C7—41no—
211C4×C7—41no—
212Dic7—81no—
213Dic7—81no—
214Dic7—81no—
215Dic7—81no—
216C24(C2×D7)⋊C411yescharacteristic
217C2×Q8—281no—
218C2×Q8—561no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—41no—
227C22×C4—81no—
228C22×C4—81no—
229C22×C4—141no—
230C22×C4—141no—
231C22×C4—141no—
232C22×C4—141no—
233C22×C4(C2×D7)⋊C411yescharacteristic
234C22×C4(C2×D7)⋊C411yescharacteristic
235C22×C4(C2×D7)⋊C411yescharacteristic
236C22×C4C2×C4×D711yescharacteristic
237C22×C4C2×D2811yescharacteristic
238C22×C4C2×C22⋊D711yescharacteristic
239M16—42no—
240M16—42no—
241M16—562no—
242C2×C8—42no—
243C2×C8—282no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—281no—
249C4⋊C4—282no—
250C4⋊C4—561no—
251C4⋊C4—561no—
252C4⋊C4—561no—
253C22⋊C4—282no—
254C22⋊C4—281no—
255C22⋊C4—281no—
256C2×C7—21no—
257C2×C7—21no—
258C2×C7—41no—
259C2×C7—41no—
260C2×C7#151111yescharacteristic
261C2×C7#212111yescharacteristic
262C2×C7#213311yescharacteristic
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23#638011yescharacteristic
268C23#638011yescharacteristic
269C23C2×(C2×D7)⋊C411yescharacteristic
270Q8—562no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—281no—
291C2×C4—281no—
292C2×C4—281no—
293C2×C4—281no—
294C2×C4—281no—
295C2×C4—281no—
296C2×C4—282no—
297C2×C4—281no—
298C2×C4—281no—
299C2×C4—281no—
300C2×C4—281no—
301C2×C4#651412yes—
302C8—82no—
303C8—562no—
304C7#1181711yescharacteristic, complemented, Sylow
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—42no—
314C22—42no—
315C22C2×#638011yescharacteristic
316C22#8977611yescharacteristic
317C22#8977711yescharacteristic
318C4—22no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—561no—
325C4—561no—
326C4—561no—
327C4—561no—
328C2—21no—
329C2—21no—
330C2—41no—
331C2—41no—
332C2#1172728711yescharacteristic
333C2#1172784811yescharacteristic
334C2#1172788211yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#1821C2
socleC22×C7C2×#99
prime radicalC24×C7C22⋊C4
cosocleC2×C7×C4⋊C4C23
hypercenter#1821D7
hypocenterC7#11817
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
4. #11725302C2semidirect
5. #11725300C2semidirect
304. C7#11817semidirect

Maximal subgroups

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

subgroupindex
#117396282
C7×#18212
#117293062
#117253022
#117253002
#118177