Finite Group Databasebeta

← #13244169

Subgroups — #13244169

Order 1792.

914 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 155 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
1D7×#407C212yesmaximal
2#11731495C211yesmaximal, characteristic
3C7×#1306C211yesmaximal, characteristic, complemented
4#11725492C212yesmaximal, complemented
5#11725435C211yesmaximal, characteristic, complemented
6C23×C4×D7C2211yescharacteristic
7C2×D7×M16—22no—
8C2×#6440C412yes—
9C7×#407C2212yes—
10C7×#381C2211yescharacteristic
11#89983C2211yescharacteristic
12#89979C2212yes—
13#89898—22no—
14#89727C412yes—
15#89498—22no—
16#89498C414yescomplemented
17D7×C22⋊C8—22no—
18D7×C22⋊C8C414yescomplemented
19#89439—22no—
20#89438—22no—
21#8115—71nomaximal, Sylow
22C24×D7C2×C411yescharacteristic
23C23×C4×C7C2311yescharacteristic
24C23×Dic7C2×C411yescharacteristic
25C22×C4×D7—44no—
26C22×C4×D7C2×C412yes—
27C22×C4×D7C2×C412yes—
28C22×C4×D7D412yes—
29C22×C4×D7D412yes—
30C22×C4×D7D412yes—
31C22×C4×D7Q812yes—
32C2×C7×M16—22no—
33C2×C7×C22⋊C4C2×C412yes—
34C2×C22⋊Dic7C2×C412yes—
35C2×(C2×D7)⋊C4—24no—
36C2×Dic7⋊C4—22no—
37C2×C4×Dic7—22no—
38C2×#1136—22no—
39D7×M16—48no—
40C2×#1131—24no—
41C2×C8×D7—24no—
42#6440—44no—
43C7×C22⋊C8—22no—
44C7×C22⋊C8C2×C414yes—
45C7×#95—22no—
46C7⋊#95—22no—
47C22⋊(C7⋊C8)—22no—
48C22⋊(C7⋊C8)C2×C414yes—
49(C2×D7)⋊C8—24no—
50(C2×D7)⋊C8—24no—
51C2×#407—72no—
52#2280—71no—
53#1306—72no—
54#1306—71no—
55#1306D711yescharacteristic, complemented
56C24×C7C22×C411yescharacteristic
57C23×D7—44no—
58C23×D7—42no—
59C23×D7C4211yescharacteristic
60C23×D7C4211yescharacteristic
61C23×D7C22⋊C412yes—
62C23×D7C4⋊C412yes—
63C22×C4×C7—22no—
64C22×C4×C7—42no—
65C22×C4×C7C22×C412yes—
66C22×C4×C7C2×D412yes—
67C22×C4×C7C2×D411yescharacteristic
68C22×C4×C7C2×Q811yescharacteristic
69C22×Dic7—24no—
70C22×Dic7—22no—
71C22×Dic7C4212yes—
72C22×Dic7C22⋊C411yescharacteristic
73C22×Dic7C22⋊C411yescharacteristic
74C22×Dic7C4⋊C412yes—
75C2×C4×D7—28no—
76C2×C4×D7—44no—
77C2×C4×D7—44no—
78C2×C4×D7—44no—
79C2×C4×D7—44no—
80C2×C4×D7—44no—
81C2×C4×D7C22⋊C412yes—
82C2×C4×D7C22⋊C412yes—
83C2×C4×D7C22⋊C412yes—
84C2×C4×D7C22⋊C412yes—
85C2×C4×D7C4⋊C412yes—
86C2×C4×D7C4⋊C412yes—
87C2×C4×D7C4⋊C412yes—
88C2×C4×D7C4⋊C412yes—
89C7×M16—44no—
90C2×C7×C8—24no—
91C7×C22⋊C4—42no—
92C22⋊Dic7—42no—
93(C2×D7)⋊C4—48no—
94Dic7⋊C4—44no—
95C4×Dic7—44no—
96#1136—44no—
97C2×C7⋊C8—24no—
98#1131—48no—
99C8×D7—48no—
100C24×C4—71no—
101C22×M16—142no—
102C2×#117—72no—
103#407—72no—
104#407—74no—
105#407—142no—
106#407—142no—
107#407C2×D712yes—
108C2×C22⋊C8—74no—
109C2×C22⋊C8—142no—
110#381—71no—
111#381C2×D711yescharacteristic
112#380—142no—
113#379—72no—
114C4×C22⋊C4—142no—
115C23×C7—22no—
116C23×C7—42no—
117C23×C7C2×C4211yescharacteristic
118C23×C7C2×C22⋊C411yescharacteristic
119C23×C7C2×C4⋊C411yescharacteristic
120C22×D7—24no—
121C22×D7—24no—
122C22×D7—24no—
123C22×D7—24no—
124C22×D7—41no—
125C22×D7—41no—
126C22×D7—42no—
127C22×D7—42no—
128C22×D7—42no—
129C22×D7—42no—
130C22×D7—416no—
131C22×D7#9512yes—
132C22×D7#9512yes—
133C2×C4×C7—24no—
134C2×C4×C7—42no—
135C2×C4×C7—41no—
136C2×C4×C7—41no—
137C2×C4×C7—44no—
138C2×C4×C7—42no—
139C2×C4×C7C2×C22⋊C412yes—
140C2×C4×C7C2×C4⋊C412yes—
141C2×Dic7—22no—
142C2×Dic7—24no—
143C2×Dic7—24no—
144C2×Dic7—44no—
145C2×Dic7—42no—
146C2×Dic7#9512yes—
147C2×Dic7#9512yes—
148C2×Dic7#9914yes—
149C4×D7—24no—
150C4×D7—24no—
151C4×D7—24no—
152C4×D7—24no—
153C4×D7—42no—
154C4×D7—42no—
155C4×D7—42no—
156C4×D7—42no—
157C7×C8—44no—
158C7⋊C8—44no—
159C25—71no—
160C23×C4—71no—
161C23×C4—72no—
162C23×C4—72no—
163C23×C4—72no—
164C23×C4—72no—
165C23×C4—72no—
166C23×C4—72no—
167C23×C4—284no—
168C23×C4C22×D711yescharacteristic
169C2×M16—22no—
170C2×M16—144no—
171C2×M16—142no—
172C2×M16—288no—
173C22×C8—144no—
174#117—284no—
175C2×C4⋊C4—142no—
176C2×C22⋊C4—72no—
177C2×C22⋊C4—144no—
178C2×C22⋊C4C4×D712yes—
179C2×C42—142no—
180C22⋊C8—22no—
181C22⋊C8—74no—
182C22⋊C8—144no—
183C22⋊C8—144no—
184C22⋊C8—142no—
185C22⋊C8C4×D714yes—
186#95—22no—
187#95—142no—
188C22×C7—22no—
189C22×C7—21no—
190C22×C7—21no—
191C22×C7—22no—
192C22×C7—22no—
193C22×C7—44no—
194C22×C7C2×#9511yescharacteristic
195C22×C7C2×#9912yes—
196C2×D7—22no—
197C2×D7—22no—
198C2×D7—22no—
199C2×D7—22no—
200C2×D7—22no—
201C2×D7—44no—
202C2×D7—48no—
203C2×D7—48no—
204C2×D7—44no—
205C2×D7#34414yes—
206C4×C7—22no—
207C4×C7—22no—
208C4×C7—82no—
209Dic7—22no—
210Dic7—82no—
211Dic7#34214yes—
212C24—71no—
213C24—72no—
214C24—72no—
215C24—71no—
216C24—282no—
217C24—284no—
218C24C2×C4×D711yescharacteristic
219C22×C4—22no—
220C22×C4—42no—
221C22×C4—71no—
222C22×C4—72no—
223C22×C4—71no—
224C22×C4—72no—
225C22×C4—72no—
226C22×C4—72no—
227C22×C4—72no—
228C22×C4—72no—
229C22×C4—72no—
230C22×C4—72no—
231C22×C4—72no—
232C22×C4—72no—
233C22×C4—148no—
234C22×C4—144no—
235C22×C4—142no—
236C22×C4—284no—
237C22×C4—284no—
238C22×C4—284no—
239C22×C4—284no—
240C22×C4—284no—
241C22×C4C2×C4×D712yes—
242C22×C4D4×D712yes—
243C22×C4D4×D711yescharacteristic
244C22×C4Q8×D711yescharacteristic
245M16—44no—
246M16—288no—
247M16—284no—
248C2×C8—24no—
249C2×C8—144no—
250C2×C8—288no—
251C4⋊C4—284no—
252C22⋊C4—42no—
253C22⋊C4—288no—
254C22⋊C4—282no—
255C42—284no—
256C2×C7—22no—
257C2×C7—41no—
258C2×C7—41no—
259C2×C7C2×#34211yescharacteristic
260C2×C7C2×#34412yes—
261D7—24no—
262D7—42no—
263C23—22no—
264C23—42no—
265C23—72no—
266C23—72no—
267C23—144no—
268C23—144no—
269C23—144no—
270C23—144no—
271C23—281no—
272C23—281no—
273C23—282no—
274C23—282no—
275C23—282no—
276C23—282no—
277C23—2816no—
278C23C42×D711yescharacteristic
279C23D7×C22⋊C411yescharacteristic
280C23D7×C4⋊C411yescharacteristic
281C2×C4—24no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—72no—
288C2×C4—74no—
289C2×C4—72no—
290C2×C4—144no—
291C2×C4—144no—
292C2×C4—144no—
293C2×C4—144no—
294C2×C4—144no—
295C2×C4—144no—
296C2×C4—142no—
297C2×C4—282no—
298C2×C4—282no—
299C2×C4—282no—
300C2×C4—282no—
301C2×C4—284no—
302C2×C4—282no—
303C2×C4D7×C22⋊C412yes—
304C2×C4D7×C4⋊C412yes—
305C8—44no—
306C8—284no—
307C7#811511yescharacteristic, complemented, Sylow
308C22—22no—
309C22—21no—
310C22—21no—
311C22—22no—
312C22—22no—
313C22—44no—
314C22—74no—
315C22—142no—
316C22—142no—
317C22—142no—
318C22—142no—
319C22—142no—
320C22—288no—
321C22—284no—
322C22—288no—
323C22—284no—
324C22D7×#9511yescharacteristic
325C22D7×#9912yes—
326C4—22no—
327C4—22no—
328C4—74no—
329C4—82no—
330C4—142no—
331C4—562no—
332C2—22no—
333C2—41no—
334C2—41no—
335C2—144no—
336C2—282no—
337C2D7×#34211yescharacteristic
338C2D7×#34412yes—
subgroupG/H
centerC22D7×#95
commutatorC23×C7C2×C42
FrattiniC23×C4C22×D7
FittingC7×#1306C2
socleC22×C7C2×#95
prime radicalC24×D7C2×C4
cosocleC23×C4×C7C23
hypercenter#1306D7
hypocenterC7#8115
LCSG ▷ C23×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1306

Complements

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

NH ≅ G/Ntype
3. C7×#1306C2semidirect
4. #11725492C2semidirect
5. #11725435C2semidirect
16. #89498C4semidirect
18. D7×C22⋊C8C4semidirect
55. #1306D7semidirect
307. C7#8115semidirect

Maximal subgroups

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

subgroupindex
D7×#4072
#117314952
C7×#13062
#117254922
#117254352
#81157