Finite Group Databasebeta

← #13257456

Subgroups — #13257456

Order 1792.

408 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×#89614C211yesmaximal, characteristic
2#11728534C211yesmaximal, characteristic, complemented
3#11728511C211yesmaximal, characteristic
4#11728281C211yesmaximal, characteristic
5#11728261C211yesmaximal, characteristic, complemented
6#11728176C211yesmaximal, characteristic, complemented
7C7×#1301C211yesmaximal, characteristic, complemented
8C2×C4×D28C2211yescharacteristic
9C2×#6432C2211yescharacteristic
10C7×#431C2211yescharacteristic
11C7×#423C2211yescharacteristic
12C2×C7×C4⋊C8C2211yescharacteristic
13C2×#6397—21no—
14C2×#6395—21no—
15C2×#6391—21no—
16#89697C2211yescharacteristic
17#89696C2211yescharacteristic
18#89614—22no—
19#89614—22no—
20#89349—21no—
21#89277—22no—
22#89276—22no—
23#89274—22no—
24#89254—22no—
25#10453—71nomaximal, Sylow
26C22×D28D411yescharacteristic
27C22×C4×D7—21no—
28C22×Dic14—21no—
29C2×C7×M16—21no—
30C22×C7×C8—21no—
31C2×C42×C7C2311yescharacteristic
32C2×(C2×D7)⋊C4—21no—
33C2×C4⋊Dic7—21no—
34C2×C4⋊Dic7—21no—
35C2×C4⋊Dic7D411yescharacteristic
36C2×#1136—21no—
37C4×D28—21no—
38C4×D28—21no—
39C4×D28—21no—
40C4×D28—21no—
41#6432—21no—
42#6432—21no—
43#6432—21no—
44#6432D412yes—
45C7×C4⋊C8—21no—
46C7×C4⋊C8—21no—
47C7×C4⋊C8—21no—
48C7×C4⋊C8—21no—
49C7×C4⋊C8D412yes—
50C7×C22⋊C8—21no—
51C4×C7×C8—21no—
52C22⋊(C7⋊C8)—21no—
53#6397—42no—
54#6395—24no—
55#6391—24no—
56C4⋊(C7⋊C8)—21no—
57C4⋊(C7⋊C8)—21no—
58C4⋊(C7⋊C8)D412yes—
59C4×C7⋊C8—21no—
60C2×#476—71no—
61#1800—71no—
62#1571—71no—
63#1549—71no—
64#1548—71no—
65#1478—71no—
66#1301D711yescharacteristic, complemented
67C23×D7—21no—
68C22×C4×C7C2×D411yescharacteristic
69C22×C4×C7#4111yescharacteristic
70C22×C4×C7#4111yescharacteristic
71C22×Dic7—21no—
72C22×Dic7—41no—
73C2×D28—21no—
74C2×D28—21no—
75C2×D28—21no—
76C2×D28—21no—
77C2×D28—21no—
78C2×D28Q1612yes—
79C2×C4×D7—41no—
80C2×C4×D7—41no—
81C2×C4×D7—82no—
82C2×Dic14—22no—
83C2×Dic14—41no—
84C2×Dic14—82no—
85C7×M16—41no—
86C7×M16—41no—
87C2×C7×C8—22no—
88C2×C7×C8—41no—
89C2×C7×C8—41no—
90C2×C7×C8—41no—
91C2×C7×C8—41no—
92C42×C7C2×D411yescharacteristic
93C42×C7C2×D411yescharacteristic
94C42×C7C2×D411yescharacteristic
95C42×C7#4111yescharacteristic
96(C2×D7)⋊C4—41no—
97(C2×D7)⋊C4—41no—
98C4⋊Dic7—22no—
99C4⋊Dic7—21no—
100C4⋊Dic7—21no—
101C4⋊Dic7—22no—
102C4⋊Dic7—41no—
103C4⋊Dic7—41no—
104#1136—41no—
105#1136—41no—
106C2×C7⋊C8—22no—
107C2×C7⋊C8—41no—
108C2×C4⋊Q8—71no—
109C2×C4×D4—71no—
110#476—142no—
111#476—142no—
112#431—71no—
113#431C2×D711yescharacteristic
114C2×#106—141no—
115#423—71no—
116#423C2×D711yescharacteristic
117C2×C4⋊C8C2×D711yescharacteristic
118C2×#102—141no—
119C2×#100—141no—
120#343—141no—
121#335—142no—
122#333—142no—
123#332—142no—
124D4⋊C8—142no—
125C23×C7#12311yescharacteristic
126C22×D7—22no—
127C22×D7—41no—
128C22×D7—41no—
129C22×D7—81no—
130C22×D7—81no—
131C2×C4×C7—22no—
132C2×C4×C7—41no—
133C2×C4×C7#12011yescharacteristic
134C2×C4×C7#12111yescharacteristic
135C2×C4×C7#12111yescharacteristic
136C2×C4×C7#12111yescharacteristic
137C2×C4×C7#12311yescharacteristic
138C2×C4×C7#12411yescharacteristic
139C2×C4×C7C2×Q1611yescharacteristic
140C2×C4×C7#13511yescharacteristic
141C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C7#13711yescharacteristic
143C2×Dic7—42no—
144C2×Dic7—41no—
145C2×Dic7—82no—
146C2×Dic7—81no—
147D28—22no—
148D28—22no—
149D28—41no—
150D28—41no—
151C4×D7—82no—
152C4×D7—82no—
153Dic14—82no—
154Dic14—82no—
155C7×C8—42no—
156C7×C8—81no—
157C7×C8—81no—
158C7⋊C8—42no—
159C7⋊C8—81no—
160C22×Q8—141no—
161C22×D4—71no—
162C23×C4—141no—
163C2×M16—21no—
164C2×M16—141no—
165C22×C8—21no—
166C4⋊Q8—72no—
167C4⋊Q8—141no—
168C4⋊Q8—141no—
169C4⋊Q8—141no—
170C4×D4—141no—
171C4×D4—141no—
172C4×D4—141no—
173C4×D4—141no—
174C2×C4⋊C4—71no—
175C2×C4⋊C4—141no—
176C2×C4⋊C4—141no—
177C2×C22⋊C4—141no—
178C2×C42C22×D711yescharacteristic
179#106—144no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—72no—
185C4⋊C8—141no—
186C4⋊C8—141no—
187C4⋊C8D2812yes—
188#102—144no—
189#100—282no—
190C22⋊C8—21no—
191C22⋊C8—141no—
192C4×C8—21no—
193C4×C8—141no—
194C22×C7—21no—
195C22×C7—21no—
196C22×C7#39411yescharacteristic
197C22×C7#48411yescharacteristic
198C22×C7#48511yescharacteristic
199C2×D7—42no—
200C2×D7—82no—
201C2×D7—81no—
202C2×D7—81no—
203C2×D7—82no—
204C4×C7—21no—
205C4×C7—21no—
206C4×C7—21no—
207C4×C7#44911yescharacteristic
208C4×C7#45111yescharacteristic
209C4×C7#46711yescharacteristic
210C4×C7#46811yescharacteristic
211C4×C7#47112yes—
212Dic7—82no—
213Dic7—81no—
214C24—141no—
215C2×Q8—142no—
216C2×Q8—281no—
217C2×Q8—562no—
218C2×D4—72no—
219C2×D4—141no—
220C2×D4—141no—
221C2×D4—141no—
222C2×D4—141no—
223C2×D4—141no—
224C22×C4—141no—
225C22×C4—281no—
226C22×C4—281no—
227C22×C4—281no—
228C22×C4—562no—
229C22×C4#115711yescharacteristic
230C22×C4D4×D711yescharacteristic
231C22×C4#116111yescharacteristic
232M16—41no—
233M16—41no—
234M16—281no—
235M16—281no—
236C2×C8—22no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—142no—
242C2×C8—281no—
243C4⋊C4—142no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—142no—
247C4⋊C4—281no—
248C4⋊C4—281no—
249C22⋊C4—281no—
250C22⋊C4—281no—
251C42C2×D2811yescharacteristic
252C42#115711yescharacteristic
253C42D4×D711yescharacteristic
254C42D4×D711yescharacteristic
255C2×C7—21no—
256C2×C7—21no—
257C2×C7#203611yescharacteristic
258C2×C7#204611yescharacteristic
259C2×C7#204811yescharacteristic
260D7—82no—
261C23—142no—
262C23—281no—
263C23—281no—
264C23—561no—
265C23—561no—
266C23#645711yescharacteristic
267Q8—562no—
268Q8—562no—
269D4—142no—
270D4—142no—
271D4—281no—
272D4—281no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—281no—
276C2×C4—282no—
277C2×C4—562no—
278C2×C4—562no—
279C2×C4—562no—
280C2×C4—561no—
281C2×C4#643811yescharacteristic
282C2×C4#644511yescharacteristic
283C2×C4#644711yescharacteristic
284C2×C4#644711yescharacteristic
285C2×C4#645711yescharacteristic
286C2×C4#645811yescharacteristic
287C2×C4#647411yescharacteristic
288C2×C4#647811yescharacteristic
289C2×C4#647911yescharacteristic
290C2×C4D7×Q1611yescharacteristic
291C8—42no—
292C8—81no—
293C8—81no—
294C8—282no—
295C8—561no—
296C7#1045311yescharacteristic, complemented, Sylow
297C22—21no—
298C22—21no—
299C22—282no—
300C22—562no—
301C22—562no—
302C22—561no—
303C22—561no—
304C22#8944411yescharacteristic
305C22#8963911yescharacteristic
306C22#8965611yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—561no—
311C4—562no—
312C4#8955211yescharacteristic
313C4#8955311yescharacteristic
314C4#8958011yescharacteristic
315C4#8958511yescharacteristic
316C4#8967012yes—
317C2—21no—
318C2—21no—
319C2—562no—
320C2#1172657911yescharacteristic
321C2#1172668711yescharacteristic
322C2#1172677211yescharacteristic
subgroupG/H
centerC22#89444
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1301C2
socleC22×C7#394
prime radicalC22×D28D4
hypercenter#1301D7
hypocenterC7#10453
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
2. #11728534C2semidirect
5. #11728261C2semidirect
6. #11728176C2semidirect
7. C7×#1301C2semidirect
66. #1301D7semidirect
296. C7#10453semidirect

Maximal subgroups

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

subgroupindex
C2×#896142
#117285342
#117285112
#117282812
#117282612
#117281762
C7×#13012
#104537