Finite Group Databasebeta

← #13264166

Subgroups — #13264166

Order 1792.

434 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 87 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#11739937C211yesmaximal, characteristic
2C7×#1461C211yesmaximal, characteristic, complemented
3C4×#6410C211yesmaximal, characteristic, complemented
4#11725265C212yesmaximal, complemented
5#11725262C212yesmaximal
6C2×C4×C7×Q8C2211yescharacteristic
7C2×C4×D28C411yescharacteristic
8C7×#431—21no—
9#90003—21no—
10C2×#6410—21no—
11#89765—21no—
12#89763C411yescharacteristic
13#89763C2211yescharacteristic
14#89758C411yescharacteristic
15#89758C2211yescharacteristic
16C4×(C2×D7)⋊C4—21no—
17C4×(C2×D7)⋊C4—21no—
18#89704C411yescharacteristic
19C4×#1136—21no—
20C7×#324—21no—
21C7×#324C2212yes—
22#89325C2212yes—
23#11734—71nomaximal, Sylow
24C22×C7×Q8D411yescharacteristic
25C22×D28C2×C411yescharacteristic
26C22×C4×D7—21no—
27C4×C7×Q8—41no—
28C4×C7×Q8—41no—
29C2×C7×C4⋊C4C2×C411yescharacteristic
30C2×C7×C4⋊C4D411yescharacteristic
31C2×C7×C4⋊C4C2311yescharacteristic
32C2×C42×C7C2×C411yescharacteristic
33C2×C42×C7D411yescharacteristic
34C2×C42×C7D411yescharacteristic
35C2×(C2×D7)⋊C4—21no—
36C2×(C2×D7)⋊C4—21no—
37C2×(C2×D7)⋊C4—21no—
38C2×(C2×D7)⋊C4—21no—
39C2×(C2×D7)⋊C4—21no—
40C2×C4⋊Dic7C2×C411yescharacteristic
41C2×C4×Dic7—21no—
42C2×#1136—22no—
43C4×D28—41no—
44C4×D28—41no—
45C7×C4⋊C8—41no—
46C7×C22⋊C8—22no—
47C4×C7×C8—41no—
48#6410—24no—
49C7⋊#95—21no—
50C7⋊#95—21no—
51C7⋊#95—21no—
52C7⋊#95—21no—
53C7⋊#95—21no—
54C7⋊#95C2×C411yescharacteristic
55C7⋊#95C2×C411yescharacteristic
56#6377—41no—
57C4×C7⋊C8—41no—
58#2398—71no—
59C4×#101—71no—
60#1461D711yescharacteristic, complemented
61#1353—72no—
62#1345—72no—
63C23×D7—21no—
64C2×C7×Q8—22no—
65C2×C7×Q8—22no—
66C2×C7×Q8—41no—
67C22×C4×C7C22⋊C411yescharacteristic
68C22×C4×C7C22⋊C411yescharacteristic
69C22×C4×C7C22⋊C411yescharacteristic
70C22×C4×C7C22⋊C411yescharacteristic
71C22×C4×C7C22×C411yescharacteristic
72C22×C4×C7C2×D411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C22×Dic7—21no—
75C22×Dic7—21no—
76C22×Dic7—21no—
77C2×D28—41no—
78C2×D28—42no—
79C2×C4×D7—42no—
80C2×C4×D7—42no—
81C2×C4×D7—41no—
82C2×C4×D7—41no—
83C2×C7×C8—42no—
84C7×C4⋊C4—41no—
85C7×C4⋊C4—41no—
86C7×C4⋊C4—41no—
87C42×C7—21no—
88C42×C7—21no—
89C42×C7—21no—
90C42×C7—21no—
91C42×C7—21no—
92C42×C7—21no—
93(C2×D7)⋊C4—42no—
94(C2×D7)⋊C4—42no—
95(C2×D7)⋊C4—42no—
96(C2×D7)⋊C4—42no—
97(C2×D7)⋊C4—42no—
98C4⋊Dic7—41no—
99C4×Dic7—42no—
100#1136—24no—
101#1136—42no—
102C2×C7⋊C8—42no—
103C2×C4×Q8C2×D711yescharacteristic
104C2×C4×D4—71no—
105#431—21no—
106C2×#101—141no—
107C4×M16—141no—
108#400—71no—
109#400—71no—
110#397—141no—
111#394—71no—
112#394—71no—
113#393—141no—
114C4×C22⋊C4—141no—
115C4×C22⋊C4—141no—
116#376—71no—
117#343—72no—
118#324—21no—
119#324C2×D712yes—
120C23×C7C2×C22⋊C411yescharacteristic
121C22×D7—22no—
122C22×D7—41no—
123C22×D7—41no—
124C22×D7—41no—
125C22×D7—41no—
126C22×D7—41no—
127C22×D7—41no—
128C7×Q8—44no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—22no—
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#9911yescharacteristic
144C2×C4×C7#9911yescharacteristic
145C2×C4×C7#10412yes—
146C2×C4×C7#13512yes—
147C2×Dic7—41no—
148C2×Dic7—41no—
149C2×Dic7—41no—
150C2×Dic7—41no—
151C2×Dic7—41no—
152C2×Dic7—41no—
153C2×Dic7—41no—
154C2×Dic7—41no—
155C2×Dic7—41no—
156D28—42no—
157D28—42no—
158C4×D7—82no—
159C4×D7—82no—
160C7×C8—82no—
161C7⋊C8—44no—
162C22×Q8D2811yescharacteristic
163C22×D4—71no—
164C23×C4—141no—
165C2×M16—142no—
166C4×Q8—41no—
167C4×Q8—41no—
168C4×D4—281no—
169C4×D4—281no—
170C2×C4⋊C4—71no—
171C2×C4⋊C4C4×D711yescharacteristic
172C2×C4⋊C4C22⋊D711yescharacteristic
173C2×C4⋊C4C22×D711yescharacteristic
174C2×C22⋊C4—141no—
175C2×C22⋊C4—141no—
176C2×C22⋊C4—141no—
177C2×C22⋊C4—141no—
178C2×C22⋊C4—141no—
179C2×C42—141no—
180C2×C42C4×D711yescharacteristic
181C2×C42D2811yescharacteristic
182C2×C42C22⋊D711yescharacteristic
183C4⋊C8—41no—
184#101—144no—
185C22⋊C8—22no—
186#97—281no—
187C4×C8—41no—
188C4×C8—281no—
189#95—71no—
190#95—71no—
191#95—141no—
192#95—141no—
193#95—141no—
194#95—141no—
195#95—141no—
196C22×C7—21no—
197C22×C7—21no—
198C22×C7C2×#9911yescharacteristic
199C22×C7#41611yescharacteristic
200C22×C7C2×#10411yescharacteristic
201C2×D7—42no—
202C2×D7—42no—
203C2×D7—42no—
204C2×D7—81no—
205C2×D7—81no—
206C2×D7—81no—
207C2×D7—81no—
208C4×C7—21no—
209C4×C7—22no—
210C4×C7—22no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7#35512yes—
216Dic7—81no—
217Dic7—81no—
218Dic7—81no—
219C24—141no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—41no—
223C2×D4—281no—
224C2×D4—282no—
225C22×C4—141no—
226C22×C4—141no—
227C22×C4—141no—
228C22×C4—282no—
229C22×C4—282no—
230C22×C4—281no—
231C22×C4—281no—
232C22×C4(C2×D7)⋊C411yescharacteristic
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—144no—
240M16—282no—
241C2×C8—42no—
242C2×C8—282no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—281no—
247C22⋊C4—282no—
248C22⋊C4—282no—
249C22⋊C4—282no—
250C22⋊C4—282no—
251C22⋊C4—282no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—282no—
259C2×C7—21no—
260C2×C7—21no—
261C2×C7#150111yescharacteristic
262C2×C7#212011yescharacteristic
263C2×C7C2×#35511yescharacteristic
264D7—82no—
265C23—142no—
266C23—281no—
267C23—281no—
268C23—281no—
269C23—281no—
270C23—281no—
271C23—281no—
272C23C2×(C2×D7)⋊C411yescharacteristic
273Q8—44no—
274D4—282no—
275D4—282no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
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—562no—
300C2×C4—562no—
301C2×C4#638011yescharacteristic
302C2×C4#638011yescharacteristic
303C2×C4#639912yes—
304C2×C4#646712yes—
305C8—82no—
306C8—284no—
307C7#1173411yescharacteristic, complemented, Sylow
308C22—21no—
309C22—21no—
310C22—282no—
311C22—282no—
312C22—282no—
313C22—561no—
314C22—561no—
315C22—561no—
316C22—561no—
317C22C2×#638011yescharacteristic
318C22#8988511yescharacteristic
319C22C2×#639911yescharacteristic
320C4—21no—
321C4—22no—
322C4—22no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—561no—
328C4—561no—
329C4—561no—
330C4#8927212yes—
331C2—21no—
332C2—21no—
333C2—562no—
334C2#1172769611yescharacteristic
335C2#1172781011yescharacteristic
336C2C2×#8927211yescharacteristic
subgroupG/H
centerC2×C4#6380
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#1461C2
socleC22×C7C2×#99
prime radicalC22×D28C2×C4
cosocleC2×C7×C4⋊C4C23
hypercenter#1461D7
hypocenterC7#11734
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. C7×#1461C2semidirect
3. C4×#6410C2semidirect
4. #11725265C2semidirect
60. #1461D7semidirect
307. C7#11734semidirect

Maximal subgroups

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

subgroupindex
#117399372
C7×#14612
C4×#64102
#117252652
#117252622
#117347