Finite Group Databasebeta

← #13254517

Subgroups — #13254517

Order 1792.

510 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C7×#3195C211yesmaximal, characteristic
2C2×#89816C211yesmaximal, characteristic
3C2×#89815C211yesmaximal, characteristic
4#11729445C211yesmaximal, characteristic
5#11729445C211yesmaximal, characteristic
6#11727727C211yesmaximal, characteristic
7#11725494C211yesmaximal, characteristic
8C2×C7×C22⋊Q8C2211yescharacteristic
9C2×C7×C22⋊Q8C2211yescharacteristic
10C22×C4⋊Dic7C2211yescharacteristic
11C7×#483—21no—
12C7×#483—21no—
13C7×#470—22no—
14C7×#451—21no—
15C7×#451—21no—
16C7×#407C2211yescharacteristic
17C2×#6409—21no—
18C2×#6409—21no—
19#89979C2211yescharacteristic
20C2×C22⋊(C7⋊C8)C2211yescharacteristic
21C2×C22⋊(C7⋊C8)C2211yescharacteristic
22#89816—22no—
23#89816—22no—
24#89815—22no—
25#89815—22no—
26C2×#6382—21no—
27C2×#6381—21no—
28#89349—22no—
29#89323—22no—
30#89323—22no—
31#9995—71nomaximal, Sylow
32C22×C7×Q8D411yescharacteristic
33C22×C7×Q8D411yescharacteristic
34C23×C4×C7C2311yescharacteristic
35C23×Dic7—21no—
36C2×C7×Q16—22no—
37C2×C7×M16D412yes—
38C7×C22⋊Q8—21no—
39C7×C22⋊Q8—21no—
40C7×C22⋊Q8—21no—
41C7×C22⋊Q8—21no—
42C7×C22⋊Q8—21no—
43C7×C22⋊Q8—21no—
44C7×C22⋊Q8—21no—
45C7×C22⋊Q8—21no—
46C2×C7×C4⋊C4—21no—
47C2×C7×C4⋊C4—21no—
48C2×C7×C4⋊C4D411yescharacteristic
49C2×C7×C4⋊C4D411yescharacteristic
50C2×C7×C22⋊C4—21no—
51C2×C7×C22⋊C4—21no—
52C2×C4⋊Dic7—21no—
53C2×C4⋊Dic7—21no—
54C2×C4⋊Dic7—21no—
55C2×C4⋊Dic7—21no—
56C2×C4⋊Dic7—21no—
57C2×C4⋊Dic7D412yes—
58C2×#1136—21no—
59C22×C7⋊C8—21no—
60C22×C7⋊C8—21no—
61C7×#106—22no—
62C7×#103—22no—
63C7×#103—22no—
64C7×C22⋊C8—21no—
65C7×C22⋊C8—21no—
66#6409—24no—
67#6409—24no—
68C22⋊(C7⋊C8)—21no—
69C22⋊(C7⋊C8)—21no—
70C22⋊(C7⋊C8)—21no—
71C22⋊(C7⋊C8)—21no—
72C22⋊(C7⋊C8)D412yes—
73C22⋊(C7⋊C8)D412yes—
74#6382—24no—
75#6381—24no—
76#3195D711yescharacteristic
77C2×#484—71no—
78C2×#483—71no—
79#1821—71no—
80#1472—71no—
81#1472—71no—
82#1298—71no—
83C24×C7#4111yescharacteristic
84C2×C7×Q8—21no—
85C2×C7×Q8—21no—
86C2×C7×Q8—21no—
87C2×C7×Q8—21no—
88C2×C7×Q8—42no—
89C2×C7×Q8—42no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7—42no—
95C22×C4×C7C2×D411yescharacteristic
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C22×C4×C7C2×D411yescharacteristic
101C22×Dic7—22no—
102C22×Dic7—41no—
103C22×Dic7—41no—
104C22×Dic7—82no—
105C7×Q16—82no—
106C7×M16—42no—
107C2×C7×C8—22no—
108C7×C4⋊C4—21no—
109C7×C4⋊C4—21no—
110C7×C4⋊C4—21no—
111C7×C4⋊C4—21no—
112C7×C4⋊C4—41no—
113C7×C4⋊C4—41no—
114C7×C4⋊C4—41no—
115C7×C4⋊C4—41no—
116C7×C22⋊C4—41no—
117C7×C22⋊C4—41no—
118C7×C22⋊C4—41no—
119C7×C22⋊C4—41no—
120C4⋊Dic7—22no—
121C4⋊Dic7—22no—
122C4⋊Dic7—41no—
123C4⋊Dic7—41no—
124#1136—42no—
125C2×C7⋊C8—22no—
126C2×C7⋊C8—22no—
127C2×C7⋊C8—42no—
128C2×C7⋊C8—42no—
129C2×C7⋊C8—41no—
130C2×C22⋊Q8C2×D711yescharacteristic
131C2×C22⋊Q8C2×D711yescharacteristic
132C22×C4⋊C4—71no—
133#484—142no—
134#484—142no—
135#483—21no—
136#483—21no—
137#483—142no—
138#483—142no—
139#470—22no—
140#451—21no—
141#451—21no—
142C2×#107—141no—
143C2×#106—141no—
144C2×#103—141no—
145C2×#103—141no—
146#407—71no—
147#407C2×D711yescharacteristic
148C2×C22⋊C8—71no—
149C2×C22⋊C8—71no—
150#343—142no—
151#324—142no—
152#324—142no—
153C23×C7—22no—
154C23×C7—22no—
155C23×C7—41no—
156C23×C7#12111yescharacteristic
157C23×C7#12111yescharacteristic
158C23×C7#12111yescharacteristic
159C7×Q8—42no—
160C7×Q8—42no—
161C7×Q8—42no—
162C7×Q8—42no—
163C2×C4×C7—22no—
164C2×C4×C7—22no—
165C2×C4×C7—22no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—42no—
174C2×C4×C7—42no—
175C2×C4×C7—81no—
176C2×C4×C7—81no—
177C2×C4×C7#12011yescharacteristic
178C2×C4×C7#12011yescharacteristic
179C2×C4×C7#12011yescharacteristic
180C2×C4×C7#12711yescharacteristic
181C2×Dic7—42no—
182C2×Dic7—82no—
183C2×Dic7—82no—
184C2×Dic7—82no—
185C7×C8—42no—
186C7⋊C8—42no—
187C7⋊C8—42no—
188C7⋊C8—81no—
189C22×Q8C22⋊D711yescharacteristic
190C22×Q8C22⋊D711yescharacteristic
191C23×C4—141no—
192C23×C4C22×D711yescharacteristic
193C2×Q16—22no—
194C2×M16—141no—
195C2×M16C22⋊D712yes—
196C22×C8—141no—
197C22×C8—141no—
198C22⋊Q8—21no—
199C22⋊Q8—21no—
200C22⋊Q8—21no—
201C22⋊Q8—21no—
202C22⋊Q8—21no—
203C22⋊Q8—21no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—72no—
209C2×C4⋊C4—141no—
210C2×C4⋊C4—141no—
211C2×C4⋊C4—141no—
212C2×C4⋊C4—141no—
213C2×C4⋊C4—141no—
214C2×C4⋊C4C22⋊D711yescharacteristic
215C2×C4⋊C4C22⋊D711yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218#107—144no—
219#106—22no—
220#106—144no—
221#103—22no—
222#103—22no—
223#103—144no—
224#103—144no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227C22⋊C8—72no—
228C22⋊C8—72no—
229C22⋊C8—141no—
230C22⋊C8—141no—
231C22⋊C8—141no—
232C22⋊C8—141no—
233C22×C7—21no—
234C22×C7—21no—
235C22×C7—21no—
236C22×C7—21no—
237C22×C7—21no—
238C22×C7—21no—
239C22×C7—21no—
240C22×C7—44no—
241C22×C7#39211yescharacteristic
242C22×C7#47112yes—
243C22×C7#47312yes—
244C4×C7—21no—
245C4×C7—21no—
246C4×C7—21no—
247C4×C7—81no—
248C4×C7—81no—
249C4×C7—81no—
250C4×C7—81no—
251C4×C7#45812yes—
252Dic7—82no—
253C24#115911yescharacteristic
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—42no—
259C2×Q8—42no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—42no—
265C22×C4—142no—
266C22×C4—281no—
267C22×C4—281no—
268C22×C4—562no—
269C22×C4D4×D711yescharacteristic
270C22×C4D4×D711yescharacteristic
271C22×C4D4×D711yescharacteristic
272C22×C4C2×C22⋊D711yescharacteristic
273C22×C4C2×C22⋊D711yescharacteristic
274C22×C4C2×C22⋊D711yescharacteristic
275Q16—82no—
276M16—42no—
277M16—282no—
278C2×C8—22no—
279C2×C8—142no—
280C2×C8—142no—
281C2×C8—282no—
282C2×C8—282no—
283C2×C8—281no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—142no—
293C4⋊C4—142no—
294C4⋊C4—281no—
295C4⋊C4—281no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C2×C7—22no—
301C2×C7—22no—
302C2×C7—41no—
303C2×C7#201911yescharacteristic
304C2×C7#202011yescharacteristic
305C2×C7#202211yescharacteristic
306C23—22no—
307C23—22no—
308C23—41no—
309C23#650111yescharacteristic
310C23#650211yescharacteristic
311C23#650211yescharacteristic
312Q8—42no—
313Q8—42no—
314Q8—42no—
315Q8—42no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—282no—
331C2×C4—562no—
332C2×C4—562no—
333C2×C4—562no—
334C2×C4#650011yescharacteristic
335C2×C4#650011yescharacteristic
336C2×C4#650311yescharacteristic
337C2×C4#651611yescharacteristic
338C8—42no—
339C8—282no—
340C8—282no—
341C8—561no—
342C7#999511yescharacteristic, complemented, Sylow
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—44no—
351C22#8997512yes—
352C22#8997612yes—
353C22#8999511yescharacteristic
354C4—21no—
355C4—21no—
356C4—21no—
357C4—81no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—562no—
362C4#9001512yes—
363C2—22no—
364C2—22no—
365C2—41no—
366C2#1172960311yescharacteristic
367C2#1172986111yescharacteristic
368C2#1172986311yescharacteristic
subgroupG/H
centerC22#89995
commutator, cosocleC23×C4×C7C23
FrattiniC23×C4C22×D7
FittingC7×#3195C2
socleC22×C7#392
prime radicalC24×C7#41
hypercenter#3195D7
hypocenterC7#9995
LCSG ▷ C23×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3195

Complements

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

NH ≅ G/Ntype
342. C7#9995semidirect

Maximal subgroups

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

subgroupindex
C7×#31952
C2×#898162
C2×#898152
#117294452
#117277272
#117254942
#99957