Finite Group Databasebeta

← #13272481

Subgroups — #13272481

Order 1792.

420 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 93 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
1#11733988C211yesmaximal, characteristic
2#11727851C211yesmaximal, characteristic
3Dic7×#101C211yesmaximal, characteristic
4C7×#1394C211yesmaximal, characteristic
5#11725505C211yesmaximal, characteristic
6#11725505C211yesmaximal, characteristic
7#11725450C211yesmaximal, characteristic
8C2×Q8×Dic7C2211yescharacteristic
9C2×#6451C411yescharacteristic
10C2×C7×#101C2211yescharacteristic
11C2×C7×#101C2211yescharacteristic
12C7×#391C2211yescharacteristic
13#90002C2211yescharacteristic
14C2×#6410C2211yescharacteristic
15C2×#6410C2211yescharacteristic
16#89890—21no—
17M16×Dic7—21no—
18#89748C411yescharacteristic
19#89707C411yescharacteristic
20C4×Dic7⋊C4C411yescharacteristic
21#89428—21no—
22#89428—21no—
23#89419—21no—
24#89419—21no—
25#89349—21no—
26#89349—21no—
27#13302—71nomaximal, Sylow
28C22×C7×Q8C2311yescharacteristic
29C22×Dic14—21no—
30C2×C7×M16—21no—
31C2×C7×M16—21no—
32C2×C7×C4⋊C4C2×C411yescharacteristic
33C2×C42×C7C2×C411yescharacteristic
34Q8×Dic7—41no—
35Q8×Dic7—41no—
36C2×C4⋊Dic7—21no—
37C2×C4⋊Dic7—21no—
38C2×C4⋊Dic7C2×C411yescharacteristic
39C2×C4⋊Dic7C2×C411yescharacteristic
40C2×C4⋊Dic7D411yescharacteristic
41C2×C4⋊Dic7D411yescharacteristic
42C2×Dic7⋊C4—21no—
43C2×Dic7⋊C4—21no—
44C2×Dic7⋊C4—21no—
45C2×Dic7⋊C4—21no—
46C2×C4×Dic7C2×C411yescharacteristic
47C2×C4×Dic7C2×C411yescharacteristic
48C2×C4×Dic7D411yescharacteristic
49C2×C4×Dic7D411yescharacteristic
50C2×#1136—21no—
51C2×#1136—21no—
52#6451—41no—
53#6451—41no—
54C7×#101—21no—
55C7×#101—21no—
56C7×#101—21no—
57C7×#101Q812yescomplemented
58C7×#95—21no—
59C7×#95—21no—
60#6410—21no—
61#6410—21no—
62#6410—21no—
63#6410Q812yescomplemented
64C7⋊#95—21no—
65C7⋊#95—21no—
66C7⋊#95—21no—
67#6389—41no—
68Dic7⋊C8—41no—
69Dic7⋊C8—41no—
70C8×Dic7—41no—
71#2567—71no—
72#1801—71no—
73C4×#101—71no—
74#1394—71no—
75#1394—71no—
76#1394—71no—
77#1394D711yescharacteristic
78C2×C7×Q8—41no—
79C2×C7×Q8—41no—
80C2×C7×Q8C2×Q811yescharacteristic
81C2×C7×Q8C2×Q811yescharacteristic
82C2×C7×Q8#4111yescharacteristic
83C2×C7×Q8#4111yescharacteristic
84C22×C4×C7—21no—
85C22×C4×C7—21no—
86C22×C4×C7C22×C411yescharacteristic
87C22×C4×C7C2×D411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C22×Dic7—21no—
90C22×Dic7—21no—
91C22×Dic7C22⋊C411yescharacteristic
92C22×Dic7C22⋊C411yescharacteristic
93C22×Dic7C22⋊C411yescharacteristic
94C22×Dic7C22⋊C411yescharacteristic
95C2×Dic14—42no—
96C2×Dic14—81no—
97C2×Dic14—81no—
98C7×M16—22no—
99C7×M16—41no—
100C7×M16—41no—
101C7×M16—41no—
102C2×C7×C8—41no—
103C2×C7×C8—41no—
104C7×C4⋊C4—41no—
105C42×C7—41no—
106C4⋊Dic7—41no—
107C4⋊Dic7—41no—
108C4⋊Dic7—41no—
109C4⋊Dic7—41no—
110C4⋊Dic7—81no—
111C4⋊Dic7—81no—
112Dic7⋊C4—24no—
113Dic7⋊C4—41no—
114Dic7⋊C4—41no—
115Dic7⋊C4—81no—
116Dic7⋊C4—81no—
117C4×Dic7—21no—
118C4×Dic7—21no—
119C4×Dic7—21no—
120C4×Dic7—21no—
121C4×Dic7—21no—
122C4×Dic7—21no—
123C4×Dic7—41no—
124#1136—22no—
125#1136—41no—
126#1136—41no—
127#1136—41no—
128C2×C7⋊C8—41no—
129C2×C7⋊C8—41no—
130C2×C4⋊Q8—71no—
131C2×C4×Q8—71no—
132#423—141no—
133C2×#101—71no—
134C2×#101—71no—
135C2×#101C2×D711yescharacteristic
136C2×#101C2×D711yescharacteristic
137C4×M16—141no—
138#395—141no—
139#395—141no—
140#391—71no—
141#391C2×D711yescharacteristic
142#389—141no—
143#389—141no—
144#384—71no—
145#384—71no—
146C4×C4⋊C4—71no—
147#343—141no—
148#343—141no—
149C23×C7C2×C22⋊C411yescharacteristic
150C7×Q8—41no—
151C7×Q8—41no—
152C7×Q8—41no—
153C7×Q8—41no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—21no—
160C2×C4×C7—22no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7C4×Q811yescharacteristic
167C2×C4×C7C4×Q811yescharacteristic
168C2×C4×C7C22⋊Q811yescharacteristic
169C2×C4×C7C22⋊Q811yescharacteristic
170C2×C4×C7#12411yescharacteristic
171C2×C4×C7C4⋊Q811yescharacteristic
172C2×Dic7—22no—
173C2×Dic7—22no—
174C2×Dic7—21no—
175C2×Dic7—21no—
176C2×Dic7—21no—
177C2×Dic7—41no—
178C2×Dic7—41no—
179C2×Dic7—41no—
180C2×Dic7—41no—
181C2×Dic7—41no—
182C2×Dic7—41no—
183C2×Dic7—41no—
184C2×Dic7—41no—
185C2×Dic7#9911yescharacteristic
186C2×Dic7#9911yescharacteristic
187Dic14—84no—
188C7×C8—42no—
189C7×C8—81no—
190C7⋊C8—42no—
191C7⋊C8—81no—
192C22×Q8—141no—
193C22×Q8C22×D711yescharacteristic
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—141no—
197C2×M16—141no—
198C4⋊Q8—281no—
199C4⋊Q8—281no—
200C4×Q8—281no—
201C4×Q8—281no—
202C2×C4⋊C4—71no—
203C2×C4⋊C4—71no—
204C2×C4⋊C4—71no—
205C2×C4⋊C4—71no—
206C2×C4⋊C4—141no—
207C2×C4⋊C4—141no—
208C2×C4⋊C4—141no—
209C2×C4⋊C4—141no—
210C2×C4⋊C4—141no—
211C2×C4⋊C4—141no—
212C2×C4⋊C4C4×D711yescharacteristic
213C2×C42—71no—
214C2×C42—71no—
215C2×C42—71no—
216C2×C42—71no—
217C2×C42C4×D711yescharacteristic
218C4⋊C8—281no—
219C4⋊C8—281no—
220#101—21no—
221#101—21no—
222#101—21no—
223#101—72no—
224#101—141no—
225#101—141no—
226#101—141no—
227#101Dic1412yescomplemented
228#97—281no—
229C4×C8—281no—
230#95—21no—
231#95—21no—
232#95—141no—
233#95—141no—
234#95—141no—
235C22×C7—21no—
236C22×C7—21no—
237C22×C7#39111yescharacteristic
238C22×C7C2×#9911yescharacteristic
239C22×C7#41011yescharacteristic
240C4×C7—21no—
241C4×C7—21no—
242C4×C7—21no—
243C4×C7—21no—
244C4×C7—21no—
245C4×C7—21no—
246C4×C7—42no—
247C4×C7—81no—
248Dic7—21no—
249Dic7—42no—
250Dic7—41no—
251Dic7—41no—
252Dic7—41no—
253Dic7—81no—
254Dic7#35612yes—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—282no—
258C2×Q8—561no—
259C2×Q8—561no—
260C2×Q8C2×Dic1411yescharacteristic
261C2×Q8#115711yescharacteristic
262C2×Q8#115911yescharacteristic
263C2×Q8Q8×D711yescharacteristic
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—71no—
267C22×C4—71no—
268C22×C4—71no—
269C22×C4—71no—
270C22×C4—141no—
271C22×C4—141no—
272C22×C4C2×C4×D711yescharacteristic
273C22×C4D4×D711yescharacteristic
274C22×C4D4×D711yescharacteristic
275M16—22no—
276M16—41no—
277M16—41no—
278M16—41no—
279M16—142no—
280M16—281no—
281M16—281no—
282M16—281no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—281no—
286C2×C8—281no—
287C4⋊C4—41no—
288C4⋊C4—144no—
289C4⋊C4—281no—
290C4⋊C4—281no—
291C4⋊C4—281no—
292C4⋊C4—281no—
293C4⋊C4—281no—
294C4⋊C4—281no—
295C4⋊C4—561no—
296C4⋊C4—561no—
297C4⋊C4—561no—
298C4⋊C4—561no—
299C42—41no—
300C42—141no—
301C42—141no—
302C42—141no—
303C42—141no—
304C42—141no—
305C42—141no—
306C42—281no—
307C2×C7—21no—
308C2×C7—21no—
309C2×C7#199611yescharacteristic
310C2×C7C2×#35611yescharacteristic
311C2×C7#213411yescharacteristic
312C23D7×C22⋊C411yescharacteristic
313Q8—41no—
314Q8—41no—
315Q8—41no—
316Q8—41no—
317Q8—564no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—71no—
331C2×C4—71no—
332C2×C4—142no—
333C2×C4—142no—
334C2×C4—141no—
335C2×C4—141no—
336C2×C4—141no—
337C2×C4—281no—
338C2×C4—281no—
339C2×C4—281no—
340C2×C4—281no—
341C2×C4—281no—
342C2×C4—281no—
343C2×C4—281no—
344C2×C4—281no—
345C2×C4C4×Dic1411yescharacteristic
346C2×C4#644111yescharacteristic
347C2×C4#644811yescharacteristic
348C2×C4#645011yescharacteristic
349C2×C4#645111yescharacteristic
350C2×C4#645911yescharacteristic
351C8—42no—
352C8—81no—
353C8—282no—
354C8—561no—
355C7#1330211yescharacteristic, complemented, Sylow
356C22—21no—
357C22—21no—
358C22#8941811yescharacteristic
359C22#8951211yescharacteristic
360C22D7×#9911yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—42no—
368C4—72no—
369C4—81no—
370C4—141no—
371C4—282no—
372C4—281no—
373C4—281no—
374C4—281no—
375C4—561no—
376C2—21no—
377C2—21no—
378C2#1172670911yescharacteristic
379C2#1172692011yescharacteristic
380C2D7×#35611yescharacteristic
subgroupG/H
centerC22D7×#99
commutatorC22×C4×C7C22×C4
FrattiniC22×Q8C22×D7
FittingC7×#1394C2
socleC22×C7C2×#99
prime radicalC23×C7C2×C22⋊C4
cosocleC22×C7×Q8C23
hypercenter#1394D7
hypocenterC7#13302
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
57. C7×#101Q8semidirect
63. #6410Q8semidirect
227. #101Dic14semidirect
355. C7#13302semidirect

Maximal subgroups

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

subgroupindex
#117339882
#117278512
Dic7×#1012
C7×#13942
#117255052
#117254502
#133027