Finite Group Databasebeta

← #13264300

Subgroups — #13264300

Order 1792.

428 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 83 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#11739703C211yesmaximal, characteristic
2C7×#1742C211yesmaximal, characteristic
3#11729326C211yesmaximal, characteristic
4#11725301C212yesmaximal, complemented
5#11725300C212yesmaximal, complemented
6C22×C7×M16—21no—
7C2×C7×#117C2211yescharacteristic
8C2×C4×Dic14C411yescharacteristic
9#89982—21no—
10#89755C2211yescharacteristic
11#89751C2211yescharacteristic
12#89750—21no—
13#89725C411yescharacteristic
14#89719C411yescharacteristic
15#89716—21no—
16#89709C411yescharacteristic
17#89705—21no—
18#89698—21no—
19C7×#343—21no—
20C7×#343C2212yes—
21#89323C2212yes—
22#89320—21no—
23#11763—71nomaximal, Sylow
24C23×C4×C7D411yescharacteristic
25C22×Dic14C2×C411yescharacteristic
26C2×C7×M16—24no—
27C2×C7×M16—22no—
28C2×C7×M16—41no—
29C2×C7×M16—42no—
30C22×C7×C8—41no—
31C7×#117—41no—
32C7×#117—41no—
33C2×C7×C4⋊C4D411yescharacteristic
34C2×C7×C4⋊C4C2311yescharacteristic
35C2×C7×C22⋊C4C2×C411yescharacteristic
36C2×C7×C22⋊C4D411yescharacteristic
37C2×C42×C7C2×C411yescharacteristic
38C2×C42×C7D411yescharacteristic
39C2×C22⋊Dic7—21no—
40C2×C22⋊Dic7—21no—
41C2×C4⋊Dic7C2×C411yescharacteristic
42C2×Dic7⋊C4—21no—
43C2×Dic7⋊C4—21no—
44C2×Dic7⋊C4—21no—
45C2×C4×Dic7—21no—
46C4×Dic14—41no—
47C4×Dic14—41no—
48C7⋊#95—21no—
49C7⋊#95—21no—
50C7⋊#95—21no—
51C7⋊#95—21no—
52C7⋊#95—21no—
53C7⋊#95—21no—
54C7⋊#95C2×C411yescharacteristic
55C7⋊#95C2×C411yescharacteristic
56C22⋊(C7⋊C8)—22no—
57C4⋊(C7⋊C8)—41no—
58#6377—41no—
59#2599—71no—
60#1742D711yescharacteristic
61#1464—71no—
62#1354—72no—
63#1346—72no—
64C24×C7C22⋊C411yescharacteristic
65C22×C4×C7—22no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7C22⋊C411yescharacteristic
71C22×C4×C7C22⋊C411yescharacteristic
72C22×C4×C7C22⋊C411yescharacteristic
73C22×C4×C7C22×C411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C22×Dic7—21no—
77C22×Dic7—21no—
78C22×Dic7—21no—
79C22×Dic7—21no—
80C2×Dic14—41no—
81C2×Dic14—42no—
82C7×M16—24no—
83C7×M16—44no—
84C7×M16—42no—
85C2×C7×C8—44no—
86C2×C7×C8—42no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—41no—
89C7×C22⋊C4—41no—
90C7×C22⋊C4—41no—
91C42×C7—21no—
92C42×C7—21no—
93C42×C7—21no—
94C42×C7—21no—
95C22⋊Dic7—42no—
96C22⋊Dic7—42no—
97C4⋊Dic7—41no—
98Dic7⋊C4—42no—
99Dic7⋊C4—41no—
100Dic7⋊C4—41no—
101Dic7⋊C4—41no—
102Dic7⋊C4—41no—
103C4×Dic7—42no—
104C2×C7⋊C8—42no—
105C22×M16—21no—
106C2×C4×Q8—71no—
107C2×#117C2×D711yescharacteristic
108#432—141no—
109#398—71no—
110#397—71no—
111#395—71no—
112#393—71no—
113#388—141no—
114#387—141no—
115#387—141no—
116#385—141no—
117#383—71no—
118#343—21no—
119#343C2×D712yes—
120#324—72no—
121#323—141no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—41no—
125C23×C7—41no—
126C23×C7#10412yes—
127C23×C7C2×C22⋊C411yescharacteristic
128C2×C4×C7—22no—
129C2×C4×C7—22no—
130C2×C4×C7—22no—
131C2×C4×C7—22no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7#9911yescharacteristic
145C2×C4×C7#9911yescharacteristic
146C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C7#13711yescharacteristic
148C2×Dic7—22no—
149C2×Dic7—41no—
150C2×Dic7—41no—
151C2×Dic7—41no—
152C2×Dic7—41no—
153C2×Dic7—41no—
154C2×Dic7—41no—
155C2×Dic7—41no—
156C2×Dic7—41no—
157C2×Dic7—41no—
158C2×Dic7—41no—
159C2×Dic7—41no—
160Dic14—42no—
161Dic14—42no—
162C7×C8—44no—
163C7⋊C8—82no—
164C22×Q8—71no—
165C23×C4C22⋊D711yescharacteristic
166C2×M16—24no—
167C2×M16—22no—
168C2×M16—41no—
169C2×M16—42no—
170C22×C8—41no—
171C4×Q8—281no—
172C4×Q8—281no—
173#117—41no—
174#117—41no—
175C2×C4⋊C4—71no—
176C2×C4⋊C4—141no—
177C2×C4⋊C4—141no—
178C2×C4⋊C4—141no—
179C2×C4⋊C4C22⋊D711yescharacteristic
180C2×C4⋊C4C22×D711yescharacteristic
181C2×C22⋊C4—141no—
182C2×C22⋊C4—141no—
183C2×C22⋊C4C4×D711yescharacteristic
184C2×C22⋊C4D2811yescharacteristic
185C2×C42—141no—
186C2×C42C4×D711yescharacteristic
187C2×C42D2811yescharacteristic
188C4⋊C8—281no—
189C22⋊C8—142no—
190#97—281no—
191#95—71no—
192#95—71no—
193#95—141no—
194#95—141no—
195#95—141no—
196#95—141no—
197#95—141no—
198#95—141no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—22no—
202C22×C7—42no—
203C22×C7—41no—
204C22×C7—41no—
205C22×C7—81no—
206C22×C7C2×#9911yescharacteristic
207C22×C7#41711yescharacteristic
208C22×C7C2×#10411yescharacteristic
209C4×C7—21no—
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7—41no—
217Dic7—42no—
218Dic7—81no—
219Dic7—81no—
220Dic7—81no—
221C24(C2×D7)⋊C411yescharacteristic
222C2×Q8—281no—
223C2×Q8—282no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
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—24no—
240M16—44no—
241M16—42no—
242C2×C8—44no—
243C2×C8—42no—
244C2×C8—282no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—281no—
248C4⋊C4—282no—
249C4⋊C4—281no—
250C4⋊C4—281no—
251C4⋊C4—281no—
252C4⋊C4—281no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—282no—
256C22⋊C4—282no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—282no—
262C2×C7—21no—
263C2×C7—21no—
264C2×C7—42no—
265C2×C7#150411yescharacteristic
266C2×C7#212011yescharacteristic
267C2×C7#213511yescharacteristic
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23#637912yes—
273C23C2×(C2×D7)⋊C411yescharacteristic
274Q8—282no—
275Q8—282no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—142no—
293C2×C4—281no—
294C2×C4—281no—
295C2×C4—281no—
296C2×C4—281no—
297C2×C4—281no—
298C2×C4—281no—
299C2×C4—281no—
300C2×C4—281no—
301C2×C4—281no—
302C2×C4—281no—
303C2×C4—281no—
304C2×C4#638011yescharacteristic
305C2×C4#638011yescharacteristic
306C2×C4#649511yescharacteristic
307C2×C4#650511yescharacteristic
308C8—44no—
309C8—562no—
310C7#1176311yescharacteristic, complemented, Sylow
311C22—21no—
312C22—21no—
313C22—22no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22C2×#637911yescharacteristic
319C22C2×#638011yescharacteristic
320C22#8973811yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—282no—
330C4—561no—
331C4—561no—
332C4—561no—
333C2—21no—
334C2—21no—
335C2—42no—
336C2#1172715211yescharacteristic
337C2#1172789911yescharacteristic
338C2#1172790411yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#1742C2
socleC22×C7C2×#99
prime radicalC24×C7C22⋊C4
cosocleC2×C7×C4⋊C4C23
hypercenter#1742D7
hypocenterC7#11763
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1742

Complements

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

NH ≅ G/Ntype
4. #11725301C2semidirect
5. #11725300C2semidirect
310. C7#11763semidirect

Maximal subgroups

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

subgroupindex
#117397032
C7×#17422
#117293262
#117253012
#117253002
#117637