Finite Group Databasebeta

← #13256951

Subgroups — #13256951

Order 1792.

466 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 99 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#11730979C211yesmaximal, characteristic
2C7×#1996C211yesmaximal, characteristic
3#11729444C211yesmaximal, characteristic
4C2×#89333C211yesmaximal, characteristic
5#11725431C211yesmaximal, characteristic
6#11725430C211yesmaximal, characteristic
7#11725423C211yesmaximal, characteristic
8C2×C7×C22⋊Q8C2211yescharacteristic
9C2×#6441—21no—
10C7×#380C2211yescharacteristic
11C4×C7×C22⋊C4C2211yescharacteristic
12#89984C2211yescharacteristic
13C2×C7⋊#95C2211yescharacteristic
14C2×C22⋊(C7⋊C8)C2211yescharacteristic
15C2×C22⋊(C7⋊C8)C2211yescharacteristic
16#89722—21no—
17#89701—21no—
18#89700—21no—
19#89433—21no—
20#89428—21no—
21C7×#342—21no—
22C7×#342—21no—
23#89333—21no—
24#89333—21no—
25#89333—21no—
26#89333—21no—
27#10375—71nomaximal, Sylow
28C22×C7×Q8Q811yescharacteristic, complemented
29C23×C4×C7C2311yescharacteristic
30C23×Dic7—21no—
31C22×Dic14—21no—
32C7×C22⋊Q8—21no—
33C7×C22⋊Q8—21no—
34C7×C22⋊Q8—21no—
35C7×C22⋊Q8—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4Q811yescharacteristic, complemented
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4D411yescharacteristic
42C2×C7×C22⋊C4D411yescharacteristic
43C2×C42×C7—21no—
44C2×C22⋊Dic7—21no—
45C2×C4⋊Dic7—21no—
46C2×C4⋊Dic7—21no—
47C2×C4⋊Dic7—21no—
48C2×C4⋊Dic7—21no—
49C2×Dic7⋊C4—21no—
50C2×Dic7⋊C4—21no—
51C22×C7⋊C8—21no—
52C22×C7⋊C8—21no—
53#6441—42no—
54#6441—42no—
55C7×#95—21no—
56C7×#95—21no—
57C7⋊#95—21no—
58C7⋊#95—21no—
59C7⋊#95—21no—
60C7⋊#95D411yescharacteristic
61C7⋊#95D411yescharacteristic
62C7⋊#95D411yescharacteristic
63C7⋊#95D411yescharacteristic
64C22⋊(C7⋊C8)—21no—
65C22⋊(C7⋊C8)—21no—
66C22⋊(C7⋊C8)—21no—
67C22⋊(C7⋊C8)—21no—
68#2549—71no—
69#1996D711yescharacteristic
70#1520—71no—
71C2×#328—71no—
72#1294—71no—
73#1289—71no—
74#1282—71no—
75C24×C7C2×D411yescharacteristic
76C2×C7×Q8—21no—
77C2×C7×Q8—21no—
78C2×C7×Q8—81no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—41no—
84C22×C4×C7—81no—
85C22×C4×C7C2×D411yescharacteristic
86C22×C4×C7C2×D411yescharacteristic
87C22×C4×C7C2×Q811yescharacteristic
88C22×C4×C7#4111yescharacteristic
89C22×C4×C7#4111yescharacteristic
90C22×C4×C7#4111yescharacteristic
91C22×Dic7—21no—
92C22×Dic7—21no—
93C22×Dic7—21no—
94C22×Dic7—21no—
95C22×Dic7—42no—
96C22×Dic7—41no—
97C22×Dic7—41no—
98C22×Dic7—81no—
99C2×Dic14—42no—
100C2×Dic14—42no—
101C2×Dic14—81no—
102C7×C4⋊C4—21no—
103C7×C4⋊C4—21no—
104C7×C4⋊C4—41no—
105C7×C4⋊C4—41no—
106C7×C4⋊C4—81no—
107C7×C22⋊C4—21no—
108C7×C22⋊C4—21no—
109C7×C22⋊C4—41no—
110C7×C22⋊C4—41no—
111C7×C22⋊C4—41no—
112C7×C22⋊C4—41no—
113C7×C22⋊C4SD1612yes—
114C7×C22⋊C4Q1612yes—
115C42×C7—81no—
116C22⋊Dic7—41no—
117C22⋊Dic7—41no—
118C4⋊Dic7—42no—
119C4⋊Dic7—42no—
120C4⋊Dic7—81no—
121C4⋊Dic7—81no—
122Dic7⋊C4—42no—
123Dic7⋊C4—42no—
124C2×C7⋊C8—41no—
125C2×C7⋊C8—41no—
126C2×C7⋊C8—81no—
127C2×C7⋊C8—81no—
128C2×C22⋊Q8—141no—
129C2×C22⋊Q8C2×D711yescharacteristic
130C2×C22⋊C8—71no—
131C2×C22⋊C8—71no—
132#398—141no—
133#395—141no—
134#391—141no—
135#389—141no—
136#385—141no—
137#380—71no—
138#380C2×D711yescharacteristic
139C4×C22⋊C4C2×D711yescharacteristic
140C2×#95—71no—
141#342—21no—
142#342—21no—
143#328—141no—
144#328—141no—
145#328—141no—
146#328—141no—
147C23×C7—41no—
148C23×C7—41no—
149C23×C7#12011yescharacteristic
150C23×C7C22⋊Q811yescharacteristic
151C23×C7#12411yescharacteristic
152C23×C7C2×SD1611yescharacteristic
153C23×C7C2×Q1611yescharacteristic
154C23×C7AGL(1,ℤ/8)11yescharacteristic
155C23×C7#13711yescharacteristic
156C7×Q8—81no—
157C7×Q8—81no—
158C2×C4×C7—22no—
159C2×C4×C7—22no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
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—81no—
173C2×C4×C7—81no—
174C2×C4×C7—81no—
175C2×C4×C7—81no—
176C2×C4×C7—81no—
177C2×C4×C7C22⋊Q811yescharacteristic
178C2×C4×C7C22⋊Q811yescharacteristic
179C2×C4×C7#12411yescharacteristic
180C2×C4×C7#12411yescharacteristic
181C2×Dic7—22no—
182C2×Dic7—41no—
183C2×Dic7—41no—
184C2×Dic7—41no—
185C2×Dic7—41no—
186C2×Dic7—41no—
187C2×Dic7—41no—
188C2×Dic7—41no—
189C2×Dic7—41no—
190C2×Dic7—41no—
191C2×Dic7—41no—
192C2×Dic7—41no—
193C2×Dic7—82no—
194C2×Dic7—82no—
195Dic14—84no—
196C7⋊C8—81no—
197C7⋊C8—81no—
198C22×Q8—141no—
199C22×Q8Dic1411yescharacteristic, complemented
200C23×C4—141no—
201C23×C4C22×D711yescharacteristic
202C22×C8—141no—
203C22×C8—141no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206C22⋊Q8—21no—
207C22⋊Q8—21no—
208C22⋊Q8—282no—
209C22⋊Q8—282no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—141no—
213C2×C4⋊C4—141no—
214C2×C4⋊C4—141no—
215C2×C4⋊C4—141no—
216C2×C4⋊C4—141no—
217C2×C4⋊C4—141no—
218C2×C4⋊C4Dic1411yescharacteristic, complemented
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—141no—
222C2×C22⋊C4C22⋊D711yescharacteristic
223C2×C22⋊C4C22⋊D711yescharacteristic
224C2×C42—21no—
225C22⋊C8—141no—
226C22⋊C8—141no—
227C22⋊C8—141no—
228C22⋊C8—141no—
229#95—21no—
230#95—21no—
231#95—71no—
232#95—71no—
233#95—71no—
234#95—71no—
235#95—141no—
236#95—141no—
237#95—141no—
238C22×C7—21no—
239C22×C7—21no—
240C22×C7—21no—
241C22×C7—21no—
242C22×C7—21no—
243C22×C7—21no—
244C22×C7—41no—
245C22×C7—41no—
246C22×C7—41no—
247C22×C7—41no—
248C22×C7#39311yescharacteristic
249C22×C7#45011yescharacteristic
250C22×C7#45111yescharacteristic
251C22×C7#47511yescharacteristic
252C22×C7#47711yescharacteristic
253C22×C7#48711yescharacteristic
254C22×C7#48911yescharacteristic
255C4×C7—21no—
256C4×C7—21no—
257C4×C7—21no—
258C4×C7—21no—
259C4×C7—42no—
260C4×C7—42no—
261C4×C7—81no—
262C4×C7—81no—
263C4×C7—161no—
264Dic7—42no—
265Dic7—81no—
266Dic7—81no—
267Dic7—81no—
268C24C2×C22⋊D711yescharacteristic
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—81no—
272C2×Q8—282no—
273C2×Q8—282no—
274C2×Q8—561no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—81no—
281C22×C4—141no—
282C22×C4—141no—
283C22×C4—141no—
284C22×C4—141no—
285C22×C4—282no—
286C22×C4—281no—
287C22×C4—281no—
288C22×C4—561no—
289C22×C4C2×Dic1411yescharacteristic
290C22×C4#115711yescharacteristic
291C22×C4D4×D711yescharacteristic
292C22×C4D4×D711yescharacteristic
293C22×C4#115911yescharacteristic
294C22×C4#115911yescharacteristic
295C2×C8—281no—
296C2×C8—281no—
297C2×C8—561no—
298C2×C8—561no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—81no—
304C4⋊C4—282no—
305C4⋊C4—282no—
306C4⋊C4—282no—
307C4⋊C4—282no—
308C4⋊C4—561no—
309C4⋊C4—561no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—281no—
317C22⋊C4—281no—
318C22⋊C4#114312yes—
319C22⋊C4#114412yes—
320C42—81no—
321C2×C7—21no—
322C2×C7—21no—
323C2×C7—21no—
324C2×C7—21no—
325C2×C7—21no—
326C2×C7—21no—
327C2×C7#202411yescharacteristic
328C2×C7#203111yescharacteristic
329C2×C7#203211yescharacteristic
330C23—41no—
331C23—41no—
332C23#648711yescharacteristic
333C23#649911yescharacteristic
334C23#650011yescharacteristic
335C23C2×#114311yescharacteristic
336C23C2×#114411yescharacteristic
337C23#651211yescharacteristic
338C23#651411yescharacteristic
339Q8—81no—
340Q8—81no—
341Q8—564no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—142no—
362C2×C4—281no—
363C2×C4—281no—
364C2×C4—281no—
365C2×C4—281no—
366C2×C4—281no—
367C2×C4—281no—
368C2×C4—281no—
369C2×C4—281no—
370C2×C4—281no—
371C2×C4—281no—
372C2×C4—281no—
373C2×C4—562no—
374C2×C4—562no—
375C2×C4#644111yescharacteristic
376C2×C4#644111yescharacteristic
377C2×C4#644811yescharacteristic
378C2×C4#644811yescharacteristic
379C8—561no—
380C8—561no—
381C7#1037511yescharacteristic, complemented, Sylow
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22#8971911yescharacteristic
393C22#8979211yescharacteristic
394C22#8979311yescharacteristic
395C22#8981811yescharacteristic
396C22#8982211yescharacteristic
397C22#8982711yescharacteristic
398C22#8985311yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—42no—
405C4—81no—
406C4—81no—
407C4—161no—
408C4—282no—
409C4—561no—
410C4—561no—
411C4—561no—
412C2—21no—
413C2—21no—
414C2—21no—
415C2—21no—
416C2—21no—
417C2—21no—
418C2#1172742211yescharacteristic
419C2#1172777911yescharacteristic
420C2#1172779211yescharacteristic
subgroupG/H
centerC22#89719
commutator, cosocleC23×C4×C7C23
FrattiniC23×C4C22×D7
FittingC7×#1996C2
socleC22×C7#393
prime radicalC24×C7C2×D4
hypercenter#1996D7
hypocenterC7#10375
LCSG ▷ C23×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
28. C22×C7×Q8Q8semidirect
38. C2×C7×C4⋊C4Q8semidirect
199. C22×Q8Dic14semidirect
218. C2×C4⋊C4Dic14semidirect
381. C7#10375semidirect

Maximal subgroups

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

subgroupindex
#117309792
C7×#19962
#117294442
C2×#893332
#117254312
#117254302
#117254232
#103757