Finite Group Databasebeta

← #13251647

Subgroups — #13251647

Order 1792.

542 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 113 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#11739820C211yesmaximal, characteristic
2#11736708C211yesmaximal, characteristic, complemented
3C7×#1486C211yesmaximal, characteristic
4#11729352C211yesmaximal, characteristic
5#11728282C211yesmaximal, characteristic, complemented
6#11725544C211yesmaximal, characteristic, complemented
7#11725519C211yesmaximal, characteristic, complemented
8C2×C4×C7×D4C2211yescharacteristic
9C2×#6487—21no—
10C2×C4×Dic14C2211yescharacteristic
11C7×#431C2211yescharacteristic
12C7×#423C2211yescharacteristic
13#89994C412yes—
14#89984—21no—
15C2×#6398—21no—
16#89752C412yes—
17#89716—22no—
18#89708C2211yescharacteristic
19#89697C2211yescharacteristic
20#89696C2211yescharacteristic
21#89602—22no—
22#89490—22no—
23C8×Dic14—22no—
24C7×D4⋊C8—22no—
25#89349—21no—
26D4⋊(C7⋊C8)—22no—
27#9486—71nomaximal, Sylow
28C22×C7×D4C2×C411yescharacteristic
29C23×C4×C7—21no—
30C22×Dic14D411yescharacteristic
31C2×C7×M16—21no—
32C4×C7×D4—21no—
33C4×C7×D4—21no—
34C4×C7×D4—21no—
35C4×C7×D4—21no—
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C22⋊C4—21no—
38C2×C42×C7C2311yescharacteristic
39C2×C22⋊Dic7—22no—
40C2×C22⋊Dic7—22no—
41C2×C4⋊Dic7—21no—
42C2×C4⋊Dic7D411yescharacteristic
43#6487—42no—
44#6487—42no—
45C2×Dic7⋊C4—22no—
46C2×Dic7⋊C4C2×C412yes—
47C2×C4×Dic7D412yes—
48C2×#1136—21no—
49C4×Dic14—21no—
50C4×Dic14—21no—
51C4×Dic14—21no—
52C4×Dic14—21no—
53C7×C4⋊C8—21no—
54C7×C4⋊C8—21no—
55C7×C4⋊C8—21no—
56C7×C22⋊C8—21no—
57C4×C7×C8D412yes—
58C7⋊#95—21no—
59C7⋊#95—22no—
60C7⋊#95C2×C412yes—
61C22⋊(C7⋊C8)—21no—
62#6398—42no—
63Dic7⋊C8—24no—
64C8×Dic7—24no—
65C4⋊(C7⋊C8)—21no—
66C4⋊(C7⋊C8)—21no—
67C4⋊(C7⋊C8)—21no—
68C4×C7⋊C8D412yes—
69#2982—71no—
70#2536—71no—
71#1801—71no—
72#1486—71no—
73#1486D711yescharacteristic
74#1286—71no—
75#1286—71no—
76C24×C7—21no—
77C2×C7×D4—21no—
78C2×C7×D4—21no—
79C2×C7×D4—42no—
80C22×C4×C7—21no—
81C22×C4×C7—41no—
82C22×C4×C7—41no—
83C22×C4×C7—41no—
84C22×C4×C7—41no—
85C22×C4×C7—41no—
86C22×C4×C7—41no—
87C22×C4×C7C22×C411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C22×Dic7—22no—
91C22×Dic7C22⋊C412yes—
92C22×Dic7C22⋊C412yes—
93C2×Dic14—21no—
94C2×Dic14—21no—
95C2×Dic14—21no—
96C2×Dic14—21no—
97C2×Dic14D812yes—
98C2×Dic14SD1612yes—
99C7×M16—41no—
100C7×M16—41no—
101C2×C7×C8—22no—
102C2×C7×C8—41no—
103C7×C4⋊C4—21no—
104C7×C4⋊C4—21no—
105C7×C22⋊C4—41no—
106C7×C22⋊C4—41no—
107C42×C7C2×D411yescharacteristic
108C42×C7C2×D411yescharacteristic
109C42×C7#4111yescharacteristic
110C42×C7#4111yescharacteristic
111C22⋊Dic7—44no—
112C22⋊Dic7—44no—
113C4⋊Dic7—21no—
114C4⋊Dic7—21no—
115C4⋊Dic7—41no—
116C4⋊Dic7—41no—
117Dic7⋊C4—22no—
118Dic7⋊C4—22no—
119Dic7⋊C4—44no—
120C4×Dic7—22no—
121C4×Dic7—22no—
122#1136—41no—
123#1136—41no—
124C2×C7⋊C8—22no—
125C2×C7⋊C8—41no—
126C2×C22⋊Q8—141no—
127C2×C4×Q8—71no—
128C2×C4×D4C2×D711yescharacteristic
129C8×Q8—142no—
130C8×Q8—142no—
131#431—71no—
132#431—142no—
133#431C2×D711yescharacteristic
134#423—71no—
135#423C2×D711yescharacteristic
136C2×#101—141no—
137#398—72no—
138#394—72no—
139#388—142no—
140#382—71no—
141#380—141no—
142#343—141no—
143D4⋊C8—22no—
144D4⋊C8—142no—
145C23×C7—22no—
146C23×C7—41no—
147C23×C7—41no—
148C23×C7—41no—
149C23×C7—41no—
150C23×C7—41no—
151C23×C7—41no—
152C23×C7C2×C22⋊C411yescharacteristic
153C7×D4—22no—
154C7×D4—22no—
155C7×D4—41no—
156C7×D4—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—81no—
163C2×C4×C7—81no—
164C2×C4×C7—81no—
165C2×C4×C7—81no—
166C2×C4×C7C4×D411yescharacteristic
167C2×C4×C7C4×D411yescharacteristic
168C2×C4×C7#12011yescharacteristic
169C2×C4×C7#12111yescharacteristic
170C2×C4×C7#12111yescharacteristic
171C2×C4×C7#12311yescharacteristic
172C2×C4×C7C2×D811yescharacteristic
173C2×C4×C7C2×SD1611yescharacteristic
174C2×C4×C7#13511yescharacteristic
175C2×C4×C7#13511yescharacteristic
176C2×Dic7—22no—
177C2×Dic7—22no—
178C2×Dic7—22no—
179C2×Dic7—42no—
180C2×Dic7—42no—
181C2×Dic7—42no—
182C2×Dic7—42no—
183C2×Dic7#10214yes—
184Dic14—22no—
185Dic14—22no—
186Dic14—22no—
187Dic14—22no—
188C7×C8—42no—
189C7×C8—81no—
190C7⋊C8—42no—
191C7⋊C8—81no—
192C22×Q8—71no—
193C22×D4C4×D711yescharacteristic
194C23×C4—21no—
195C2×M16—21no—
196C2×M16—141no—
197C22⋊Q8—282no—
198C22⋊Q8—282no—
199C4×Q8—141no—
200C4×Q8—141no—
201C4×Q8—141no—
202C4×Q8—141no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C2×C4⋊C4—72no—
208C2×C4⋊C4—71no—
209C2×C4⋊C4—142no—
210C2×C4⋊C4—141no—
211C2×C4⋊C4C4×D711yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—142no—
214C2×C22⋊C4—142no—
215C2×C42—72no—
216C2×C42C22×D711yescharacteristic
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—144no—
221C4⋊C8—141no—
222C4⋊C8—141no—
223C4⋊C8—141no—
224#101—282no—
225C22⋊C8—21no—
226C22⋊C8—141no—
227C4×C8—72no—
228C4×C8—144no—
229C4×C8C22⋊D712yes—
230#95—72no—
231#95—142no—
232#95—141no—
233C22×C7—21no—
234C22×C7—21no—
235C22×C7—42no—
236C22×C7—42no—
237C22×C7—42no—
238C22×C7—81no—
239C22×C7—81no—
240C22×C7—81no—
241C22×C7—81no—
242C22×C7#38611yescharacteristic
243C22×C7C2×#10211yescharacteristic
244C22×C7#41611yescharacteristic
245C4×C7—21no—
246C4×C7—21no—
247C4×C7—21no—
248C4×C7—21no—
249C4×C7—81no—
250C4×C7#46511yescharacteristic
251C4×C7#46511yescharacteristic
252C4×C7#46611yescharacteristic
253C4×C7#46611yescharacteristic
254Dic7—24no—
255Dic7—42no—
256Dic7—82no—
257C24—21no—
258C2×Q8—72no—
259C2×Q8—72no—
260C2×Q8—141no—
261C2×Q8—141no—
262C2×Q8—141no—
263C2×Q8—141no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—42no—
267C22×C4—21no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—72no—
275C22×C4—72no—
276C22×C4—142no—
277C22×C4C2×C4×D711yescharacteristic
278C22×C4D4×D711yescharacteristic
279C22×C4D4×D711yescharacteristic
280M16—41no—
281M16—41no—
282M16—281no—
283M16—281no—
284C2×C8—22no—
285C2×C8—41no—
286C2×C8—142no—
287C2×C8—281no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—142no—
291C4⋊C4—141no—
292C4⋊C4—142no—
293C4⋊C4—141no—
294C4⋊C4—284no—
295C4⋊C4—281no—
296C4⋊C4—281no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—284no—
300C22⋊C4—284no—
301C42—142no—
302C42—142no—
303C42#115711yescharacteristic
304C42D4×D711yescharacteristic
305C42#115911yescharacteristic
306C42C2×C22⋊D711yescharacteristic
307C2×C7—21no—
308C2×C7—21no—
309C2×C7—82no—
310C2×C7#189111yescharacteristic
311C2×C7#189311yescharacteristic
312C2×C7#191211yescharacteristic
313C23—22no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23D7×C22⋊C411yescharacteristic
321Q8—142no—
322Q8—142no—
323Q8—142no—
324Q8—142no—
325D4—22no—
326D4—22no—
327D4—41no—
328D4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—74no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—142no—
340C2×C4—142no—
341C2×C4—142no—
342C2×C4—282no—
343C2×C4—282no—
344C2×C4—282no—
345C2×C4—282no—
346C2×C4(C22⋊D7)⋊C411yescharacteristic
347C2×C4#644611yescharacteristic
348C2×C4#644711yescharacteristic
349C2×C4D7×D811yescharacteristic
350C2×C4#647511yescharacteristic
351C2×C4D7×SD1611yescharacteristic
352C2×C4#647911yescharacteristic
353C2×C4C4×C22⋊D711yescharacteristic
354C2×C4#650011yescharacteristic
355C2×C4#650111yescharacteristic
356C8—42no—
357C8—81no—
358C8—282no—
359C8—561no—
360C7#948611yescharacteristic, complemented, Sylow
361C22—21no—
362C22—21no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22D7×#10211yescharacteristic
371C22#8954411yescharacteristic
372C22#8972811yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—81no—
378C4—144no—
379C4—282no—
380C4—562no—
381C4#8954711yescharacteristic
382C4#8955011yescharacteristic
383C4#8992911yescharacteristic
384C4#8994311yescharacteristic
385C2—21no—
386C2—21no—
387C2—82no—
388C2#1172774611yescharacteristic
389C2#1172795611yescharacteristic
390C2#1172816511yescharacteristic
subgroupG/H
centerC22#89728
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1486C2
socleC22×C7#386
prime radicalC22×C7×D4C2×C4
cosocleC2×C42×C7C23
hypercenter#1486D7
hypocenterC7#9486
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
2. #11736708C2semidirect
5. #11728282C2semidirect
6. #11725544C2semidirect
7. #11725519C2semidirect
360. C7#9486semidirect

Maximal subgroups

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

subgroupindex
#117398202
#117367082
C7×#14862
#117293522
#117282822
#117255442
#117255192
#94867