Finite Group Databasebeta

← #13264323

Subgroups — #13264323

Order 1792.

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

normal subgroups only

This group has 420 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#11739706C211yesmaximal, characteristic
2C7×#1464C211yesmaximal, characteristic
3#11729341C211yesmaximal, characteristic
4#11725265C212yesmaximal, complemented
5#11725264C212yesmaximal, complemented
6C2×C7×#117C2211yescharacteristic
7C22×#1136—21no—
8C2×#6433C411yescharacteristic
9C7×#432—21no—
10#89982—21no—
11#89758C2211yescharacteristic
12#89755C2211yescharacteristic
13#89744—21no—
14#89725C411yescharacteristic
15#89721C411yescharacteristic
16#89717—21no—
17#89705—21no—
18#89704C411yescharacteristic
19C7×#324C2212yes—
20C7×#323—21no—
21#89325—21no—
22#89325C2212yes—
23#11767—71nomaximal, Sylow
24C23×C4×C7D411yescharacteristic
25C7×#117—41no—
26C7×#117—41no—
27C2×C7×C4⋊C4D411yescharacteristic
28C2×C7×C4⋊C4C2311yescharacteristic
29C2×C7×C22⋊C4C2×C411yescharacteristic
30C2×C7×C22⋊C4D411yescharacteristic
31C2×C42×C7C2×C411yescharacteristic
32C2×C42×C7D411yescharacteristic
33C2×C22⋊Dic7—21no—
34C2×C22⋊Dic7—21no—
35C2×C4⋊Dic7C2×C411yescharacteristic
36C2×C4⋊Dic7C2×C411yescharacteristic
37C2×Dic7⋊C4—21no—
38C2×Dic7⋊C4—21no—
39C2×Dic7⋊C4—21no—
40C2×Dic7⋊C4—21no—
41C2×C4×Dic7—21no—
42C2×#1136—24no—
43C2×#1136—22no—
44C2×#1136—41no—
45C2×#1136—42no—
46C22×C7⋊C8—41no—
47#6433—41no—
48#6433—41no—
49C7×C4⋊C8—41no—
50C7×C22⋊C8—22no—
51C7×#97—41no—
52C7⋊#95—21no—
53C7⋊#95—21no—
54C7⋊#95—21no—
55C7⋊#95—21no—
56C7⋊#95—21no—
57C7⋊#95C2×C411yescharacteristic
58C7⋊#95C2×C411yescharacteristic
59#2611—71no—
60#1742—71no—
61#1464D711yescharacteristic
62#1353—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×C7×C8—42no—
81C7×C4⋊C4—41no—
82C7×C4⋊C4—41no—
83C7×C22⋊C4—41no—
84C7×C22⋊C4—41no—
85C42×C7—21no—
86C42×C7—21no—
87C42×C7—21no—
88C42×C7—21no—
89C22⋊Dic7—42no—
90C22⋊Dic7—42no—
91C4⋊Dic7—41no—
92C4⋊Dic7—41no—
93Dic7⋊C4—42no—
94Dic7⋊C4—42no—
95Dic7⋊C4—41no—
96Dic7⋊C4—41no—
97Dic7⋊C4—41no—
98Dic7⋊C4—41no—
99C4×Dic7—42no—
100#1136—24no—
101#1136—44no—
102#1136—42no—
103C2×C7⋊C8—44no—
104C2×C7⋊C8—42no—
105C22×M16—141no—
106C2×#125—71no—
107C2×#117C2×D711yescharacteristic
108#432—21no—
109#400—71no—
110#399—71no—
111#398—71no—
112#397—71no—
113#389—141no—
114#388—141no—
115#387—141no—
116#385—141no—
117#376—71no—
118#343—72no—
119#343—141no—
120#324C2×D712yes—
121#323—21no—
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—
160C7×C8—82no—
161C7⋊C8—44no—
162C23×C4D2811yescharacteristic
163C2×M16—144no—
164C2×M16—142no—
165C2×M16—281no—
166C2×M16—282no—
167C22×C8—281no—
168#125—281no—
169#125—281no—
170#117—41no—
171#117—41no—
172C2×C4⋊C4—71no—
173C2×C4⋊C4—71no—
174C2×C4⋊C4—141no—
175C2×C4⋊C4—141no—
176C2×C4⋊C4—141no—
177C2×C4⋊C4—141no—
178C2×C4⋊C4D2811yescharacteristic
179C2×C4⋊C4C22×D711yescharacteristic
180C2×C22⋊C4—141no—
181C2×C22⋊C4—141no—
182C2×C22⋊C4C4×D711yescharacteristic
183C2×C22⋊C4C22⋊D711yescharacteristic
184C2×C42—141no—
185C2×C42C4×D711yescharacteristic
186C2×C42C22⋊D711yescharacteristic
187C4⋊C8—41no—
188C22⋊C8—22no—
189#97—41no—
190#95—71no—
191#95—71no—
192#95—141no—
193#95—141no—
194#95—141no—
195#95—141no—
196#95—141no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—22no—
200C22×C7—42no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7—81no—
204C22×C7C2×#9911yescharacteristic
205C22×C7#41711yescharacteristic
206C22×C7C2×#10411yescharacteristic
207C4×C7—21no—
208C4×C7—21no—
209C4×C7—21no—
210C4×C7—21no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7—41no—
215Dic7—42no—
216Dic7—81no—
217Dic7—81no—
218Dic7—81no—
219C24(C2×D7)⋊C411yescharacteristic
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—141no—
226C22×C4—141no—
227C22×C4—141no—
228C22×C4—141no—
229C22×C4(C2×D7)⋊C411yescharacteristic
230C22×C4(C2×D7)⋊C411yescharacteristic
231C22×C4(C2×D7)⋊C411yescharacteristic
232C22×C4C2×C4×D711yescharacteristic
233C22×C4C2×D2811yescharacteristic
234C22×C4C2×C22⋊D711yescharacteristic
235M16—144no—
236M16—284no—
237M16—282no—
238C2×C8—42no—
239C2×C8—284no—
240C2×C8—282no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—281no—
244C4⋊C4—281no—
245C4⋊C4—282no—
246C4⋊C4—282no—
247C4⋊C4—281no—
248C4⋊C4—281no—
249C4⋊C4—281no—
250C4⋊C4—281no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—282no—
254C22⋊C4—282no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—282no—
260C2×C7—21no—
261C2×C7—21no—
262C2×C7—42no—
263C2×C7#150411yescharacteristic
264C2×C7#212311yescharacteristic
265C2×C7#213211yescharacteristic
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23#639912yes—
271C23C2×(C2×D7)⋊C411yescharacteristic
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—142no—
289C2×C4—281no—
290C2×C4—281no—
291C2×C4—281no—
292C2×C4—281no—
293C2×C4—281no—
294C2×C4—281no—
295C2×C4—281no—
296C2×C4—281no—
297C2×C4—281no—
298C2×C4—281no—
299C2×C4—281no—
300C2×C4#638011yescharacteristic
301C2×C4#638011yescharacteristic
302C2×C4#647111yescharacteristic
303C2×C4#647211yescharacteristic
304C8—82no—
305C8—284no—
306C7#1176711yescharacteristic, complemented, Sylow
307C22—21no—
308C22—21no—
309C22—22no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—81no—
314C22C2×#638011yescharacteristic
315C22#8990511yescharacteristic
316C22C2×#639911yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—282no—
326C4—561no—
327C4—561no—
328C4—561no—
329C2—21no—
330C2—21no—
331C2—42no—
332C2#1172773011yescharacteristic
333C2#1172781211yescharacteristic
334C2#1172790211yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#1464C2
socleC22×C7C2×#99
prime radicalC24×C7C22⋊C4
cosocleC2×C7×C4⋊C4C23
hypercenter#1464D7
hypocenterC7#11767
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
4. #11725265C2semidirect
5. #11725264C2semidirect
306. C7#11767semidirect

Maximal subgroups

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

subgroupindex
#117397062
C7×#14642
#117293412
#117252652
#117252642
#117677