Finite Group Databasebeta

← #13252770

Subgroups — #13252770

Order 1792.

720 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6437C211yesmaximal, characteristic
2C7×#1767C211yesmaximal, characteristic, complemented
3C2×#89278C214yesmaximal, complemented
4C4×C4⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
5C2×#6437C411yescharacteristic
6C2×C4×D28—21no—
7C2×C4×D28—21no—
8C2×C4×D28C2212yes—
9C2×C7×C4⋊C8C2212yes—
10C2×C7×#97—21no—
11C43×C7C2211yescharacteristic
12#89711—21no—
13#89711—21no—
14#89711C412yes—
15#89707C411yescharacteristic
16C2×C4⋊(C7⋊C8)—21no—
17C2×C4×C7⋊C8C2212yes—
18C7×#336—21no—
19#89319—21no—
20#89278—28no—
21#9678—71nomaximal, Sylow
22C22×D28—21no—
23C22×D28—21no—
24C22×D28C2×C412yes—
25C22×C4×D7—22no—
26C22×C7×C8—22no—
27C2×C42×C7C2×C411yescharacteristic
28C2×C42×C7C2×C411yescharacteristic
29C2×C42×C7D411yescharacteristic
30C2×C42×C7D411yescharacteristic
31C2×C42×C7D411yescharacteristic
32C2×C42×C7D411yescharacteristic
33C2×C42×C7C2311yescharacteristic
34C2×(C2×D7)⋊C4—22no—
35C2×(C2×D7)⋊C4—22no—
36C2×(C2×D7)⋊C4—22no—
37C2×C4⋊Dic7—21no—
38C2×C4⋊Dic7—21no—
39C2×C4⋊Dic7C2×C412yes—
40C22×C7⋊C8—22no—
41#6437—24no—
42C4×D28—24no—
43C4×D28—24no—
44C4×D28—44no—
45C4×D28—42no—
46C4×D28—42no—
47C7×C4⋊C8—22no—
48C7×C4⋊C8D414yescomplemented
49C7×#97—42no—
50C4⋊(C7⋊C8)—24no—
51C4×C7⋊C8—24no—
52C4×#127—71no—
53#1767D711yescharacteristic, complemented
54C4×C4⋊C8—71no—
55C2×D4⋊C8—74no—
56C23×D7—22no—
57C22×C4×C7C22⋊C411yescharacteristic
58C22×C4×C7C22⋊C411yescharacteristic
59C22×C4×C7C22⋊C411yescharacteristic
60C22×C4×C7C22⋊C411yescharacteristic
61C22×C4×C7C22×C411yescharacteristic
62C22×C4×C7C2×D411yescharacteristic
63C22×C4×C7C2×D411yescharacteristic
64C22×Dic7—22no—
65C2×D28—24no—
66C2×D28—24no—
67C2×D28—24no—
68C2×D28—24no—
69C2×D28—22no—
70C2×D28—42no—
71C2×D28—42no—
72C2×D28—42no—
73C2×D28—42no—
74C2×D28M1614yes—
75C2×C4×D7—44no—
76C2×C4×D7—42no—
77C2×C4×D7—42no—
78C2×C4×D7—44no—
79C2×C7×C8—24no—
80C2×C7×C8—42no—
81C2×C7×C8—42no—
82C42×C7—22no—
83C42×C7—22no—
84C42×C7—22no—
85C42×C7—22no—
86C42×C7—41no—
87C42×C7D812yes—
88C42×C7SD1612yes—
89C42×C7C2×D412yes—
90C42×C7#4112yes—
91(C2×D7)⋊C4—44no—
92(C2×D7)⋊C4—44no—
93(C2×D7)⋊C4—42no—
94(C2×D7)⋊C4—42no—
95C4⋊Dic7—24no—
96C4⋊Dic7—42no—
97C4⋊Dic7—42no—
98C2×C7⋊C8—24no—
99C2×C7⋊C8—42no—
100C2×C7⋊C8—42no—
101C2×#127—71no—
102C2×C4×D4—72no—
103C2×C4×D4—141no—
104C2×C4×D4—141no—
105C2×C4⋊C8—141no—
106C2×C4⋊C8C2×D712yes—
107C2×#97—21no—
108C2×C4×C8—72no—
109#390—72no—
110#390—141no—
111#390—141no—
112#384—71no—
113C43C2×D711yescharacteristic
114#336—21no—
115#336—141no—
116D4⋊C8—148no—
117C23×C7C2×C22⋊C411yescharacteristic
118C22×D7—24no—
119C22×D7—42no—
120C22×D7—42no—
121C22×D7—42no—
122C22×D7—42no—
123C22×D7—42no—
124C22×D7—42no—
125C2×C4×C7—21no—
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—21no—
136C2×C4×C7#10212yes—
137C2×C4×C7#10212yes—
138C2×C4×C7#10412yes—
139C2×C4×C7C4×D412yes—
140C2×C4×C7#12112yes—
141C2×C4×C7#12411yescharacteristic
142C2×C4×C7#12711yescharacteristic
143C2×C4×C7C2×M1612yes—
144C2×C4×C7#13112yes—
145C2×C4×C7C2×D811yescharacteristic
146C2×C4×C7C2×SD1611yescharacteristic
147C2×C4×C7AGL(1,ℤ/8)12yes—
148C2×Dic7—42no—
149C2×Dic7—42no—
150C2×Dic7—42no—
151D28—28no—
152D28—48no—
153D28—48no—
154D28—44no—
155C4×D7—84no—
156C4×D7—84no—
157C7×C8—44no—
158C7⋊C8—44no—
159C22×D4—72no—
160C22×D4—141no—
161C22×D4—141no—
162C23×C4—142no—
163C22×C8—22no—
164C22×C8—142no—
165#127—144no—
166C4×D4—144no—
167C4×D4—144no—
168C4×D4—284no—
169C4×D4—282no—
170C4×D4—282no—
171C2×C4⋊C4—72no—
172C2×C4⋊C4—141no—
173C2×C4⋊C4—141no—
174C2×C22⋊C4—142no—
175C2×C22⋊C4—142no—
176C2×C22⋊C4—142no—
177C2×C42C4×D711yescharacteristic
178C2×C42C4×D711yescharacteristic
179C2×C42D2811yescharacteristic
180C2×C42D2811yescharacteristic
181C2×C42C22⋊D711yescharacteristic
182C2×C42C22⋊D711yescharacteristic
183C2×C42C22×D711yescharacteristic
184C4⋊C8—22no—
185C4⋊C8—144no—
186C4⋊C8D2814yescomplemented
187#97—42no—
188C4×C8—144no—
189C22×C7#39011yescharacteristic
190C22×C7#40711yescharacteristic
191C22×C7#40811yescharacteristic
192C22×C7C2×#10211yescharacteristic
193C22×C7#41811yescharacteristic
194C22×C7C2×#10411yescharacteristic
195C22×C7#42111yescharacteristic
196C2×D7—44no—
197C2×D7—44no—
198C2×D7—44no—
199C2×D7—82no—
200C2×D7—82no—
201C2×D7—82no—
202C2×D7—82no—
203C4×C7—21no—
204C4×C7—21no—
205C4×C7—22no—
206C4×C7—22no—
207C4×C7—22no—
208C4×C7—41no—
209C4×C7#43514yes—
210C4×C7#45912yes—
211C4×C7#46012yes—
212Dic7—82no—
213C24—142no—
214C2×D4—74no—
215C2×D4—144no—
216C2×D4—144no—
217C2×D4—144no—
218C2×D4—142no—
219C2×D4—144no—
220C2×D4—282no—
221C2×D4—282no—
222C2×D4—282no—
223C2×D4—282no—
224C22×C4—142no—
225C22×C4—284no—
226C22×C4—282no—
227C22×C4—282no—
228C22×C4—284no—
229C22×C4(C2×D7)⋊C411yescharacteristic
230C22×C4(C2×D7)⋊C411yescharacteristic
231C22×C4(C2×D7)⋊C411yescharacteristic
232C22×C4(C2×D7)⋊C411yescharacteristic
233C22×C4C2×C4×D711yescharacteristic
234C22×C4C2×D2811yescharacteristic
235C22×C4C2×C22⋊D711yescharacteristic
236C2×C8—24no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—144no—
240C2×C8—282no—
241C2×C8—282no—
242C4⋊C4—144no—
243C4⋊C4—282no—
244C4⋊C4—282no—
245C22⋊C4—284no—
246C22⋊C4—282no—
247C22⋊C4—282no—
248C22⋊C4—284no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42—22no—
253C42—41no—
254C42#114112yes—
255C42#114312yes—
256C42C2×D2812yes—
257C42#115712yes—
258C2×C7#148611yescharacteristic
259C2×C7#149011yescharacteristic
260C2×C7#149211yescharacteristic
261C2×C7#149611yescharacteristic
262C2×C7#195511yescharacteristic
263C2×C7#198011yescharacteristic
264C2×C7#198211yescharacteristic
265D7—84no—
266C23—144no—
267C23—282no—
268C23—282no—
269C23—282no—
270C23—282no—
271C23—282no—
272C23—282no—
273C23C2×(C2×D7)⋊C411yescharacteristic
274D4—148no—
275D4—288no—
276D4—288no—
277D4—284no—
278C2×C4—21no—
279C2×C4—21no—
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—21no—
289C2×C4—282no—
290C2×C4—282no—
291C2×C4—282no—
292C2×C4—564no—
293C2×C4—564no—
294C2×C4#638312yes—
295C2×C4#638312yes—
296C2×C4#639912yes—
297C2×C4C4×D2812yes—
298C2×C4#643711yescharacteristic
299C2×C4#643811yescharacteristic
300C2×C4D7×M1612yes—
301C2×C4#647012yes—
302C2×C4#649312yes—
303C2×C4C2×#114111yescharacteristic
304C2×C4C2×#114311yescharacteristic
305C2×C4#651212yes—
306C8—44no—
307C8—284no—
308C7#967811yescharacteristic, complemented, Sylow
309C22—284no—
310C22—284no—
311C22—284no—
312C22—562no—
313C22—562no—
314C22—562no—
315C22—562no—
316C22#8971111yescharacteristic
317C22C2×#638311yescharacteristic
318C22#8977311yescharacteristic
319C22#8989811yescharacteristic
320C22#8990111yescharacteristic
321C22C2×#639911yescharacteristic
322C22#8991111yescharacteristic
323C4—21no—
324C4—21no—
325C4—22no—
326C4—22no—
327C4—22no—
328C4—41no—
329C4—562no—
330C4#8947814yes—
331C4#8978712yes—
332C4#8979912yes—
333C2—564no—
334C2#1172724111yescharacteristic
335C2#1172759111yescharacteristic
336C2#1172762111yescharacteristic
337C2#1172825011yescharacteristic
338C2#1172825211yescharacteristic
339C2#1172855311yescharacteristic
340C2#1172855511yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D7
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1767C2
socleC23×C7C2×C22⋊C4
prime radicalC2×#6437C4
cosocleC2×C42×C7C23
hypercenter#1767D7
hypocenterC7#9678
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C43 ◁ #1767

Complements

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

NH ≅ G/Ntype
2. C7×#1767C2semidirect
3. C2×#89278C2semidirect
4. C4×C4⋊(C7⋊C8)C2semidirect
48. C7×C4⋊C8D4semidirect
53. #1767D7semidirect
186. C4⋊C8D28semidirect
308. C7#9678semidirect

Maximal subgroups

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

subgroupindex
C4×#64372
C7×#17672
C2×#892782
C4×C4⋊(C7⋊C8)2
#96787