Finite Group Databasebeta

← #13244166

Subgroups — #13244166

Order 1792.

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

normal subgroups only

This group has 942 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
2D7×#381C211yesmaximal, characteristic
3C7×#1306C211yesmaximal, characteristic, complemented
4#11725492C212yesmaximal, complemented
5#11725435C211yesmaximal, characteristic, complemented
6C23×C4×D7C2211yescharacteristic
7C2×D7×M16—22no—
8C2×D7×C22⋊C4C412yes—
9C7×#407C2212yes—
10C7×#381C2211yescharacteristic
11#89983C2211yescharacteristic
12#89979C2212yes—
13#89898—22no—
14#89715C412yes—
15#89498—22no—
16#89498C414yescomplemented
17D7×C22⋊C8—22no—
18D7×C22⋊C8C414yescomplemented
19#89436—22no—
20D7×#95—22no—
21#8114—71nomaximal, Sylow
22C24×D7C2×C411yescharacteristic
23C23×C4×C7C2311yescharacteristic
24C23×Dic7C2×C411yescharacteristic
25C22×C4×D7—22no—
26C22×C4×D7—44no—
27C22×C4×D7C2×C412yes—
28C22×C4×D7C2×C412yes—
29C22×C4×D7D412yes—
30C22×C4×D7D412yes—
31C22×C4×D7D412yes—
32C22×C4×D7Q812yes—
33C2×C7×M16—22no—
34C2×C7×C22⋊C4C2×C412yes—
35C2×C22⋊Dic7C2×C412yes—
36C2×(C2×D7)⋊C4—22no—
37C2×#1136—22no—
38D7×M16—48no—
39C2×#1131—24no—
40C2×C8×D7—24no—
41D7×C22⋊C4—44no—
42C7×C22⋊C8—22no—
43C7×C22⋊C8C2×C414yescomplemented
44C7×#95—22no—
45C7⋊#95—22no—
46C7⋊#95—24no—
47C22⋊(C7⋊C8)—22no—
48C22⋊(C7⋊C8)C2×C414yescomplemented
49(C2×D7)⋊C8—24no—
50(C2×D7)⋊C8—24no—
51C2×#407—72no—
52C2×#381—71no—
53#1306—72no—
54#1306—71no—
55#1306D711yescharacteristic, complemented
56C24×C7C22×C411yescharacteristic
57C23×D7—24no—
58C23×D7—44no—
59C23×D7C4212yes—
60C23×D7C22⋊C411yescharacteristic
61C23×D7C22⋊C411yescharacteristic
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—22no—
70C22×Dic7—42no—
71C22×Dic7C4211yescharacteristic
72C22×Dic7C4211yescharacteristic
73C22×Dic7C22⋊C412yes—
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×D7—44no—
82C2×C4×D7—48no—
83C2×C4×D7C22⋊C412yes—
84C2×C4×D7C22⋊C412yes—
85C2×C4×D7C22⋊C412yes—
86C2×C4×D7C22⋊C412yes—
87C2×C4×D7C4⋊C412yes—
88C2×C4×D7C4⋊C412yes—
89C2×C4×D7C4⋊C412yes—
90C2×C4×D7C4⋊C412yes—
91C7×M16—44no—
92C2×C7×C8—24no—
93C7×C22⋊C4—42no—
94C22⋊Dic7—42no—
95(C2×D7)⋊C4—44no—
96#1136—44no—
97C2×C7⋊C8—24no—
98#1131—48no—
99C8×D7—48no—
100C24×C4—71no—
101C22×M16—142no—
102C22×C22⋊C4—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#381—72no—
112#381—142no—
113#381C2×D711yescharacteristic
114C2×#95—142no—
115C23×C7—22no—
116C23×C7—42no—
117C23×C7C2×C4211yescharacteristic
118C23×C7C2×C22⋊C411yescharacteristic
119C23×C7C2×C4⋊C411yescharacteristic
120C22×D7—22no—
121C22×D7—24no—
122C22×D7—24no—
123C22×D7—24no—
124C22×D7—24no—
125C22×D7—24no—
126C22×D7—42no—
127C22×D7—42no—
128C22×D7—42no—
129C22×D7—416no—
130C22×D7#9512yes—
131C22×D7#9512yes—
132C22×D7#9914yes—
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—24no—
142C2×Dic7—41no—
143C2×Dic7—41no—
144C2×Dic7—42no—
145C2×Dic7—44no—
146C2×Dic7—42no—
147C2×Dic7#9512yes—
148C2×Dic7#9512yes—
149C4×D7—24no—
150C4×D7—24no—
151C4×D7—24no—
152C4×D7—24no—
153C4×D7—42no—
154C4×D7—42no—
155C4×D7—42no—
156C4×D7—42no—
157C4×D7—88no—
158C7×C8—44no—
159C7⋊C8—44no—
160C25—71no—
161C23×C4—71no—
162C23×C4—72no—
163C23×C4—72no—
164C23×C4—72no—
165C23×C4—72no—
166C23×C4—72no—
167C23×C4—72no—
168C23×C4—142no—
169C23×C4—284no—
170C23×C4C22×D711yescharacteristic
171C2×M16—22no—
172C2×M16—144no—
173C2×M16—142no—
174C2×M16—288no—
175C22×C8—144no—
176C2×C22⋊C4—72no—
177C2×C22⋊C4—142no—
178C2×C22⋊C4—284no—
179C2×C22⋊C4C4×D712yes—
180C22⋊C8—22no—
181C22⋊C8—74no—
182C22⋊C8—144no—
183C22⋊C8—144no—
184C22⋊C8—142no—
185C22⋊C8C4×D714yescomplemented
186#95—22no—
187#95—144no—
188#95—142no—
189C22×C7—22no—
190C22×C7—21no—
191C22×C7—21no—
192C22×C7—22no—
193C22×C7—22no—
194C22×C7—44no—
195C22×C7C2×#9511yescharacteristic
196C22×C7C2×#9912yes—
197C2×D7—28no—
198C2×D7—22no—
199C2×D7—22no—
200C2×D7—22no—
201C2×D7—22no—
202C2×D7—44no—
203C2×D7—48no—
204C2×D7—48no—
205C2×D7#34214yes—
206C2×D7#34414yes—
207C4×C7—22no—
208C4×C7—22no—
209C4×C7—82no—
210Dic7—22no—
211Dic7—22no—
212Dic7—82no—
213C24—71no—
214C24—71no—
215C24—72no—
216C24—72no—
217C24—144no—
218C24—284no—
219C24C2×C4×D711yescharacteristic
220C22×C4—22no—
221C22×C4—42no—
222C22×C4—72no—
223C22×C4—71no—
224C22×C4—71no—
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—72no—
234C22×C4—148no—
235C22×C4—142no—
236C22×C4—284no—
237C22×C4—284no—
238C22×C4—284no—
239C22×C4—284no—
240C22×C4—284no—
241C22×C4—284no—
242C22×C4—288no—
243C22×C4—282no—
244C22×C4C2×C4×D712yes—
245C22×C4D4×D712yes—
246C22×C4D4×D711yescharacteristic
247C22×C4Q8×D711yescharacteristic
248M16—44no—
249M16—288no—
250M16—284no—
251C2×C8—24no—
252C2×C8—144no—
253C2×C8—288no—
254C22⋊C4—42no—
255C22⋊C4—284no—
256C22⋊C4—282no—
257C2×C7—22no—
258C2×C7—41no—
259C2×C7—41no—
260C2×C7C2×#34211yescharacteristic
261C2×C7C2×#34412yes—
262D7—24no—
263D7—24no—
264C23—22no—
265C23—42no—
266C23—72no—
267C23—72no—
268C23—74no—
269C23—144no—
270C23—144no—
271C23—144no—
272C23—144no—
273C23—144no—
274C23—142no—
275C23—282no—
276C23—282no—
277C23—282no—
278C23—2816no—
279C23C42×D711yescharacteristic
280C23D7×C22⋊C411yescharacteristic
281C23D7×C4⋊C411yescharacteristic
282C2×C4—24no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—72no—
289C2×C4—72no—
290C2×C4—144no—
291C2×C4—144no—
292C2×C4—144no—
293C2×C4—144no—
294C2×C4—144no—
295C2×C4—282no—
296C2×C4—282no—
297C2×C4—282no—
298C2×C4—282no—
299C2×C4—281no—
300C2×C4—281no—
301C2×C4—282no—
302C2×C4—284no—
303C2×C4—282no—
304C2×C4—568no—
305C2×C4D7×C22⋊C412yes—
306C2×C4D7×C4⋊C412yes—
307C8—44no—
308C8—284no—
309C7#811411yescharacteristic, complemented, Sylow
310C22—22no—
311C22—21no—
312C22—21no—
313C22—22no—
314C22—22no—
315C22—44no—
316C22—74no—
317C22—74no—
318C22—148no—
319C22—142no—
320C22—142no—
321C22—142no—
322C22—142no—
323C22—284no—
324C22—288no—
325C22—288no—
326C22D7×#9511yescharacteristic
327C22D7×#9912yes—
328C4—22no—
329C4—22no—
330C4—82no—
331C4—142no—
332C4—142no—
333C4—562no—
334C2—22no—
335C2—41no—
336C2—41no—
337C2—144no—
338C2—144no—
339C2D7×#34211yescharacteristic
340C2D7×#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#8114
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
43. C7×C22⋊C8C2×C4semidirect
48. C22⋊(C7⋊C8)C2×C4semidirect
55. #1306D7semidirect
185. C22⋊C8C4×D7semidirect
309. C7#8114semidirect

Maximal subgroups

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

subgroupindex
D7×#4072
D7×#3812
C7×#13062
#117254922
#117254352
#81147