Finite Group Databasebeta

← #13244159

Subgroups — #13244159

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#8112—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×D7D411yescharacteristic
30C22×C4×D7D411yescharacteristic
31C22×C4×D7D412yes—
32C22×C4×D7Q811yescharacteristic
33C22×C4×D7Q811yescharacteristic
34C2×C7×M16—22no—
35C2×C7×C22⋊C4C2×C412yes—
36C2×C22⋊Dic7C2×C412yes—
37C2×(C2×D7)⋊C4—24no—
38C2×Dic7⋊C4—22no—
39C2×C4×Dic7—22no—
40C2×#1136—22no—
41D7×M16—48no—
42C2×#1131—24no—
43C2×C8×D7—24no—
44#6440—44no—
45C7×C22⋊C8—22no—
46C7×C22⋊C8C2×C414yes—
47C7×#95—22no—
48C7⋊#95—22no—
49C22⋊(C7⋊C8)—22no—
50C22⋊(C7⋊C8)C2×C414yes—
51(C2×D7)⋊C8—24no—
52(C2×D7)⋊C8—24no—
53C2×#407—72no—
54#2280—71no—
55#1306—72no—
56#1306—71no—
57#1306D711yescharacteristic, complemented
58C24×C7C22×C411yescharacteristic
59C23×D7—44no—
60C23×D7—42no—
61C23×D7C4212yes—
62C23×D7C22⋊C412yes—
63C23×D7C4⋊C411yescharacteristic
64C23×D7C4⋊C411yescharacteristic
65C22×C4×C7—22no—
66C22×C4×C7—42no—
67C22×C4×C7C22×C412yes—
68C22×C4×C7C2×D412yes—
69C22×C4×C7C2×D411yescharacteristic
70C22×C4×C7C2×Q811yescharacteristic
71C22×Dic7—24no—
72C22×Dic7—22no—
73C22×Dic7C4211yescharacteristic
74C22×Dic7C4211yescharacteristic
75C22×Dic7C22⋊C411yescharacteristic
76C22×Dic7C22⋊C411yescharacteristic
77C22×Dic7C4⋊C411yescharacteristic
78C22×Dic7C4⋊C411yescharacteristic
79C2×C4×D7—28no—
80C2×C4×D7—44no—
81C2×C4×D7—44no—
82C2×C4×D7—42no—
83C2×C4×D7—42no—
84C2×C4×D7—42no—
85C2×C4×D7—42no—
86C2×C4×D7—44no—
87C2×C4×D7C22⋊C412yes—
88C2×C4×D7C22⋊C412yes—
89C2×C4×D7C22⋊C412yes—
90C2×C4×D7C22⋊C412yes—
91C2×C4×D7C4⋊C412yes—
92C2×C4×D7C4⋊C412yes—
93C2×C4×D7C4⋊C412yes—
94C2×C4×D7C4⋊C412yes—
95C7×M16—44no—
96C2×C7×C8—24no—
97C7×C22⋊C4—42no—
98C22⋊Dic7—42no—
99(C2×D7)⋊C4—48no—
100Dic7⋊C4—44no—
101C4×Dic7—44no—
102#1136—44no—
103C2×C7⋊C8—24no—
104#1131—48no—
105C8×D7—48no—
106C24×C4—71no—
107C22×M16—142no—
108C2×#117—72no—
109#407—72no—
110#407—74no—
111#407—142no—
112#407—142no—
113#407C2×D712yes—
114C2×C22⋊C8—74no—
115C2×C22⋊C8—142no—
116#381—71no—
117#381C2×D711yescharacteristic
118#380—142no—
119#379—72no—
120C4×C22⋊C4—142no—
121C23×C7—22no—
122C23×C7—42no—
123C23×C7C2×C4211yescharacteristic
124C23×C7C2×C22⋊C411yescharacteristic
125C23×C7C2×C4⋊C411yescharacteristic
126C22×D7—24no—
127C22×D7—24no—
128C22×D7—24no—
129C22×D7—24no—
130C22×D7—42no—
131C22×D7—41no—
132C22×D7—41no—
133C22×D7—42no—
134C22×D7—42no—
135C22×D7—42no—
136C22×D7—416no—
137C22×D7#9512yes—
138C22×D7#9512yes—
139C2×C4×C7—24no—
140C2×C4×C7—42no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—44no—
144C2×C4×C7—42no—
145C2×C4×C7C2×C22⋊C412yes—
146C2×C4×C7C2×C4⋊C412yes—
147C2×Dic7—22no—
148C2×Dic7—22no—
149C2×Dic7—22no—
150C2×Dic7—22no—
151C2×Dic7—22no—
152C2×Dic7—44no—
153C2×Dic7—42no—
154C2×Dic7#9511yescharacteristic
155C2×Dic7#9511yescharacteristic
156C2×Dic7#9511yescharacteristic
157C2×Dic7#9511yescharacteristic
158C2×Dic7#9914yes—
159C4×D7—24no—
160C4×D7—24no—
161C4×D7—24no—
162C4×D7—24no—
163C4×D7—42no—
164C4×D7—42no—
165C4×D7—42no—
166C4×D7—42no—
167C7×C8—44no—
168C7⋊C8—44no—
169C25—71no—
170C23×C4—71no—
171C23×C4—72no—
172C23×C4—72no—
173C23×C4—72no—
174C23×C4—72no—
175C23×C4—71no—
176C23×C4—71no—
177C23×C4—71no—
178C23×C4—71no—
179C23×C4—284no—
180C23×C4C22×D711yescharacteristic
181C2×M16—22no—
182C2×M16—144no—
183C2×M16—142no—
184C2×M16—288no—
185C22×C8—144no—
186#117—284no—
187C2×C4⋊C4—142no—
188C2×C22⋊C4—72no—
189C2×C22⋊C4—144no—
190C2×C22⋊C4C4×D712yes—
191C2×C42—142no—
192C22⋊C8—22no—
193C22⋊C8—74no—
194C22⋊C8—144no—
195C22⋊C8—144no—
196C22⋊C8—142no—
197C22⋊C8C4×D714yes—
198#95—22no—
199#95—142no—
200C22×C7—22no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—22no—
204C22×C7—22no—
205C22×C7—44no—
206C22×C7C2×#9511yescharacteristic
207C22×C7C2×#9912yes—
208C2×D7—22no—
209C2×D7—22no—
210C2×D7—22no—
211C2×D7—22no—
212C2×D7—22no—
213C2×D7—44no—
214C2×D7—48no—
215C2×D7—48no—
216C2×D7—44no—
217C2×D7#34414yes—
218C4×C7—22no—
219C4×C7—22no—
220C4×C7—82no—
221Dic7—21no—
222Dic7—21no—
223Dic7—82no—
224Dic7#34212yes—
225Dic7#34212yes—
226C24—72no—
227C24—71no—
228C24—72no—
229C24—71no—
230C24—282no—
231C24—284no—
232C24C2×C4×D711yescharacteristic
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—71no—
236C22×C4—71no—
237C22×C4—71no—
238C22×C4—71no—
239C22×C4—71no—
240C22×C4—71no—
241C22×C4—72no—
242C22×C4—72no—
243C22×C4—72no—
244C22×C4—72no—
245C22×C4—72no—
246C22×C4—72no—
247C22×C4—72no—
248C22×C4—72no—
249C22×C4—148no—
250C22×C4—144no—
251C22×C4—142no—
252C22×C4—284no—
253C22×C4—284no—
254C22×C4—284no—
255C22×C4—282no—
256C22×C4—282no—
257C22×C4—282no—
258C22×C4—282no—
259C22×C4C2×C4×D712yes—
260C22×C4D4×D712yes—
261C22×C4D4×D711yescharacteristic
262C22×C4Q8×D711yescharacteristic
263M16—44no—
264M16—288no—
265M16—284no—
266C2×C8—24no—
267C2×C8—144no—
268C2×C8—288no—
269C4⋊C4—284no—
270C22⋊C4—42no—
271C22⋊C4—288no—
272C22⋊C4—282no—
273C42—284no—
274C2×C7—22no—
275C2×C7—41no—
276C2×C7—41no—
277C2×C7C2×#34211yescharacteristic
278C2×C7C2×#34412yes—
279D7—24no—
280D7—42no—
281C23—22no—
282C23—42no—
283C23—72no—
284C23—72no—
285C23—144no—
286C23—144no—
287C23—144no—
288C23—144no—
289C23—282no—
290C23—281no—
291C23—281no—
292C23—282no—
293C23—282no—
294C23—282no—
295C23—2816no—
296C23C42×D711yescharacteristic
297C23D7×C22⋊C411yescharacteristic
298C23D7×C4⋊C411yescharacteristic
299C2×C4—24no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—71no—
306C2×C4—71no—
307C2×C4—74no—
308C2×C4—71no—
309C2×C4—71no—
310C2×C4—144no—
311C2×C4—144no—
312C2×C4—144no—
313C2×C4—144no—
314C2×C4—142no—
315C2×C4—142no—
316C2×C4—142no—
317C2×C4—142no—
318C2×C4—142no—
319C2×C4—282no—
320C2×C4—282no—
321C2×C4—282no—
322C2×C4—282no—
323C2×C4—284no—
324C2×C4—282no—
325C2×C4D7×C22⋊C412yes—
326C2×C4D7×C4⋊C412yes—
327C8—44no—
328C8—284no—
329C7#811211yescharacteristic, complemented, Sylow
330C22—22no—
331C22—21no—
332C22—21no—
333C22—22no—
334C22—22no—
335C22—44no—
336C22—74no—
337C22—142no—
338C22—142no—
339C22—142no—
340C22—142no—
341C22—142no—
342C22—288no—
343C22—284no—
344C22—288no—
345C22—284no—
346C22D7×#9511yescharacteristic
347C22D7×#9912yes—
348C4—22no—
349C4—22no—
350C4—72no—
351C4—72no—
352C4—82no—
353C4—141no—
354C4—141no—
355C4—562no—
356C2—22no—
357C2—41no—
358C2—41no—
359C2—144no—
360C2—282no—
361C2D7×#34211yescharacteristic
362C2D7×#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#8112
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
57. #1306D7semidirect
329. C7#8112semidirect

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
#81127