Finite Group Databasebeta

← #13242023

Subgroups — #13242023

Order 1792.

594 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 127 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×#89700C211yesmaximal, characteristic
2C7×#1749C211yesmaximal, characteristic
3#11729283C211yesmaximal, characteristic
4#11725284C214yesmaximal, complemented
5C22×C42×C7C2211yescharacteristic
6C22×C4⋊Dic7C411yescharacteristic
7C22×C4⋊Dic7C411yescharacteristic
8C22×Dic7⋊C4—21no—
9C2×C7×C22⋊C8—21no—
10C2×C7×#97—21no—
11C2×C7⋊#95—21no—
12C2×C22⋊(C7⋊C8)—21no—
13#89700—21no—
14#89700—21no—
15#89700C412yes—
16#89700C2212yes—
17C2×#6377—21no—
18C7×#336C2212yes—
19#89319C2212yes—
20#7627—71nomaximal, Sylow
21C23×C4×C7C2×C411yescharacteristic
22C23×C4×C7D411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×Dic7—21no—
25C23×Dic7—21no—
26C22×C7×C8—22no—
27C2×C42×C7—42no—
28C2×C42×C7C2×C411yescharacteristic
29C2×C42×C7D411yescharacteristic
30C2×C42×C7D411yescharacteristic
31C2×C42×C7C2311yescharacteristic
32C2×C4⋊Dic7—21no—
33C2×C4⋊Dic7—24no—
34C2×C4⋊Dic7—21no—
35C2×C4⋊Dic7—42no—
36C2×C4⋊Dic7C2×C412yes—
37C2×C4⋊Dic7C2×C412yes—
38C2×Dic7⋊C4—22no—
39C2×Dic7⋊C4—22no—
40C2×Dic7⋊C4—44no—
41C22×C7⋊C8—22no—
42C7×C22⋊C8—24no—
43C7×#97—42no—
44C7⋊#95—22no—
45C7⋊#95—22no—
46C22⋊(C7⋊C8)—24no—
47#6377—42no—
48C2×#389—71no—
49#1749—71no—
50#1749D711yescharacteristic
51#1271—74no—
52C24×C7C22⋊C411yescharacteristic
53C22×C4×C7—22no—
54C22×C4×C7—22no—
55C22×C4×C7—22no—
56C22×C4×C7—22no—
57C22×C4×C7—22no—
58C22×C4×C7—22no—
59C22×C4×C7—22no—
60C22×C4×C7—41no—
61C22×C4×C7—41no—
62C22×C4×C7C22⋊C411yescharacteristic
63C22×C4×C7C22⋊C411yescharacteristic
64C22×C4×C7C22⋊C411yescharacteristic
65C22×C4×C7C22×C411yescharacteristic
66C22×C4×C7C2×D411yescharacteristic
67C22×C4×C7C2×D411yescharacteristic
68C22×Dic7—24no—
69C22×Dic7—22no—
70C22×Dic7—22no—
71C22×Dic7—42no—
72C22×Dic7—42no—
73C22×Dic7—42no—
74C22×Dic7—42no—
75C22×Dic7—42no—
76C2×C7×C8—24no—
77C2×C7×C8—42no—
78C2×C7×C8—42no—
79C42×C7—22no—
80C42×C7—22no—
81C42×C7—41no—
82C42×C7—41no—
83C4⋊Dic7—24no—
84C4⋊Dic7—22no—
85C4⋊Dic7—42no—
86C4⋊Dic7—42no—
87C4⋊Dic7M1614yes—
88Dic7⋊C4—44no—
89Dic7⋊C4—44no—
90C2×C7⋊C8—24no—
91C2×C7⋊C8—42no—
92C2×C7⋊C8—42no—
93C22×C4⋊C4—71no—
94C22×C4⋊C4—71no—
95C22×C4⋊C4—141no—
96C22×C42C2×D711yescharacteristic
97C2×C22⋊C8—21no—
98C2×C22⋊C8—141no—
99C2×#97—21no—
100C2×#97—141no—
101#389—72no—
102#389—72no—
103#389—141no—
104#389—141no—
105C2×#95—141no—
106#336—72no—
107#336C2×D712yes—
108C23×C7—21no—
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—21no—
112C23×C7#10112yes—
113C23×C7#10412yes—
114C23×C7#10412yes—
115C23×C7C2×C22⋊C411yescharacteristic
116C2×C4×C7—28no—
117C2×C4×C7—21no—
118C2×C4×C7—21no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
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—44no—
132C2×C4×C7—44no—
133C2×C4×C7#9912yes—
134C2×C4×C7C2×M1612yes—
135C2×C4×C7AGL(1,ℤ/8)12yes—
136C2×C4×C7#13712yes—
137C2×Dic7—24no—
138C2×Dic7—48no—
139C2×Dic7—42no—
140C2×Dic7—42no—
141C2×Dic7—42no—
142C2×Dic7—42no—
143C2×Dic7—42no—
144C2×Dic7—84no—
145C7×C8—44no—
146C7⋊C8—44no—
147C23×C4—141no—
148C23×C4—141no—
149C23×C4C4×D711yescharacteristic
150C23×C4D2811yescharacteristic
151C23×C4C22⋊D711yescharacteristic
152C22×C8—22no—
153C22×C8—142no—
154C2×C4⋊C4—72no—
155C2×C4⋊C4—72no—
156C2×C4⋊C4—144no—
157C2×C4⋊C4—141no—
158C2×C4⋊C4—141no—
159C2×C4⋊C4—142no—
160C2×C4⋊C4—142no—
161C2×C4⋊C4—284no—
162C2×C4⋊C4—282no—
163C2×C42—42no—
164C2×C42C4×D711yescharacteristic
165C2×C42D2811yescharacteristic
166C2×C42C22⋊D711yescharacteristic
167C2×C42C22×D711yescharacteristic
168C22⋊C8—24no—
169C22⋊C8—144no—
170#97—42no—
171#97—282no—
172#95—142no—
173#95—142no—
174C22×C7—22no—
175C22×C7—22no—
176C22×C7—22no—
177C22×C7—22no—
178C22×C7—22no—
179C22×C7—22no—
180C22×C7#33014yes—
181C22×C7#40711yescharacteristic
182C22×C7C2×#9911yescharacteristic
183C22×C7C2×#10111yescharacteristic
184C22×C7#41811yescharacteristic
185C22×C7#41911yescharacteristic
186C22×C7C2×#10411yescharacteristic
187C22×C7C2×#10411yescharacteristic
188C4×C7—22no—
189C4×C7—22no—
190C4×C7—22no—
191C4×C7—22no—
192C4×C7—41no—
193C4×C7—41no—
194Dic7—44no—
195Dic7—82no—
196C24(C2×D7)⋊C411yescharacteristic
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—144no—
207C22×C4—142no—
208C22×C4—142no—
209C22×C4—282no—
210C22×C4—282no—
211C22×C4—282no—
212C22×C4—282no—
213C22×C4—282no—
214C22×C4(C2×D7)⋊C411yescharacteristic
215C22×C4(C2×D7)⋊C411yescharacteristic
216C22×C4(C2×D7)⋊C411yescharacteristic
217C22×C4C2×C4×D711yescharacteristic
218C22×C4C2×D2811yescharacteristic
219C22×C4C2×C22⋊D711yescharacteristic
220C2×C8—24no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—144no—
224C2×C8—282no—
225C2×C8—282no—
226C4⋊C4—74no—
227C4⋊C4—144no—
228C4⋊C4—142no—
229C4⋊C4—282no—
230C4⋊C4—284no—
231C4⋊C4—284no—
232C4⋊C4—282no—
233C42—22no—
234C42—22no—
235C42—41no—
236C42—41no—
237C2×C7—24no—
238C2×C7#146311yescharacteristic
239C2×C7#149011yescharacteristic
240C2×C7#149111yescharacteristic
241C2×C7#150711yescharacteristic
242C2×C7#150811yescharacteristic
243C2×C7C2×#33011yescharacteristic
244C2×C7#155611yescharacteristic
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23#637912yes—
250C23#639812yes—
251C23#639912yes—
252C23C2×(C2×D7)⋊C411yescharacteristic
253C2×C4—28no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—144no—
271C2×C4—288no—
272C2×C4—282no—
273C2×C4—282no—
274C2×C4—282no—
275C2×C4—282no—
276C2×C4—282no—
277C2×C4—564no—
278C2×C4#638012yes—
279C2×C4D7×M1612yes—
280C2×C4#647212yes—
281C2×C4#651212yes—
282C8—44no—
283C8—284no—
284C7#762711yescharacteristic, complemented, Sylow
285C22—22no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22#8926014yes—
292C22C2×#637911yescharacteristic
293C22C2×#638011yescharacteristic
294C22#8977311yescharacteristic
295C22#8989211yescharacteristic
296C22#8989811yescharacteristic
297C22C2×#639811yescharacteristic
298C22C2×#639911yescharacteristic
299C4—22no—
300C4—22no—
301C4—22no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—284no—
306C4—562no—
307C2—24no—
308C2#1172719811yescharacteristic
309C2C2×#8926011yescharacteristic
310C2#1172761111yescharacteristic
311C2#1172772511yescharacteristic
312C2#1172773411yescharacteristic
313C2#1172855311yescharacteristic
314C2#1172857711yescharacteristic
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1749C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C22⋊C4
cosocleC2×C42×C7C23
hypercenter#1749D7
hypocenterC7#7627
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
4. #11725284C2semidirect
284. C7#7627semidirect

Maximal subgroups

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

subgroupindex
C2×#897002
C7×#17492
#117292832
#117252842
#76277