Finite Group Databasebeta

← #13255123

Subgroups — #13255123

Order 1792.

542 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×C7×#489C211yesmaximal, characteristic
2C2×#89852C211yesmaximal, characteristic, complemented
3#11740437C211yesmaximal, characteristic
4#11729540C211yesmaximal, characteristic, complemented
5C2×#89335C211yesmaximal, characteristic
6C2×#89260C211yesmaximal, characteristic, complemented
7#11725465C211yesmaximal, characteristic, complemented
8C2×C7×C4⋊Q8C2211yescharacteristic
9C2×C7×#124C2211yescharacteristic
10C2×#6433C2211yescharacteristic
11C7×#489—22no—
12C7×#489—22no—
13C2×C7×#103—21no—
14C2×C7×#102—21no—
15C2×C7×#97C2211yescharacteristic
16#90001—21no—
17C2×#6409—22no—
18#89994—21no—
19#89852—22no—
20#89852—22no—
21C2×#6382—21no—
22C2×#6381—21no—
23#89725—21no—
24#89721—21no—
25#89720—22no—
26#89708C2211yescharacteristic
27C2×#6377C2211yescharacteristic
28C2×#6377C2211yescharacteristic
29#89335—22no—
30#89260—22no—
31#10086—71nomaximal, Sylow
32C22×C7×Q8—21no—
33C22×C7×Q8D411yescharacteristic
34C22×C7×D4D411yescharacteristic
35C22×C7×C8—21no—
36C7×C4⋊Q8—21no—
37C7×C4⋊Q8—21no—
38C7×C4⋊Q8—21no—
39C7×C4⋊Q8D412yes—
40C7×#124—21no—
41C7×#124—21no—
42C7×#124—21no—
43C7×#124—21no—
44C2×C7×C4⋊C4—21no—
45C2×C7×C4⋊C4—21no—
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4—21no—
48C2×C42×C7C2311yescharacteristic
49C2×C22⋊Dic7—22no—
50C2×C22⋊Dic7—22no—
51C2×C4⋊Dic7D411yescharacteristic
52C2×C4⋊Dic7D411yescharacteristic
53C2×C4⋊Dic7D412yes—
54C2×Dic7⋊C4—21no—
55C2×Dic7⋊C4—21no—
56C2×Dic7⋊C4—22no—
57C22×C7⋊C8—21no—
58C22×C7⋊C8—21no—
59#6433—21no—
60#6433—21no—
61#6433—21no—
62#6433—21no—
63C7×#103—24no—
64C7×#102—24no—
65C7×#97—21no—
66C7×#97D412yes—
67#6409—44no—
68C7⋊#95—22no—
69C7⋊#95—21no—
70C7⋊#95—21no—
71#6382—24no—
72#6381—24no—
73#6377—21no—
74#6377—21no—
75#6377—21no—
76#6377D412yes—
77C2×#501—71no—
78C2×#489D711yescharacteristic
79#2820—71no—
80#1960—71no—
81C2×#330—71no—
82C2×#329—71no—
83#1295—71no—
84C24×C7—21no—
85C2×C7×Q8—21no—
86C2×C7×Q8—22no—
87C2×C7×Q8—21no—
88C2×C7×Q8—42no—
89C2×C7×Q8—42no—
90C2×C7×Q8—42no—
91C2×C7×Q8—41no—
92C2×C7×D4—21no—
93C2×C7×D4—21no—
94C2×C7×D4—42no—
95C22×C4×C7—21no—
96C22×C4×C7—41no—
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C22×Dic7—22no—
101C22×Dic7—21no—
102C22×Dic7—21no—
103C2×C7×C8—22no—
104C2×C7×C8—42no—
105C7×C4⋊C4—22no—
106C7×C4⋊C4—22no—
107C7×C4⋊C4—41no—
108C7×C4⋊C4—41no—
109C7×C22⋊C4—41no—
110C7×C22⋊C4—41no—
111C7×C22⋊C4—41no—
112C7×C22⋊C4—41no—
113C42×C7C2×D411yescharacteristic
114C42×C7C2×D411yescharacteristic
115C42×C7C2×D411yescharacteristic
116C42×C7#4111yescharacteristic
117C22⋊Dic7—82no—
118C22⋊Dic7—82no—
119C4⋊Dic7—21no—
120C4⋊Dic7—21no—
121C4⋊Dic7—21no—
122C4⋊Dic7—21no—
123C4⋊Dic7—42no—
124Dic7⋊C4—41no—
125Dic7⋊C4—41no—
126Dic7⋊C4—41no—
127Dic7⋊C4—41no—
128Dic7⋊C4—82no—
129C2×C7⋊C8—24no—
130C2×C7⋊C8—22no—
131C2×C7⋊C8—22no—
132C2×C7⋊C8—41no—
133C2×C7⋊C8—41no—
134C2×C4⋊Q8C2×D711yescharacteristic
135C2×#125—71no—
136C2×#124C2×D711yescharacteristic
137#501—142no—
138#501—142no—
139#489—22no—
140#489—22no—
141C2×#107—141no—
142C2×#106—141no—
143C2×#103—21no—
144C2×#103—142no—
145C2×#102—21no—
146C2×#97—71no—
147C2×#97—71no—
148C2×#97C2×D711yescharacteristic
149#399—141no—
150#397—141no—
151#396—142no—
152#395—141no—
153#394—141no—
154#382—71no—
155#330—142no—
156#329—142no—
157C23×C7—41no—
158C23×C7—41no—
159C23×C7—41no—
160C23×C7—42no—
161C23×C7—42no—
162C23×C7#12011yescharacteristic
163C7×Q8—42no—
164C7×Q8—44no—
165C7×Q8—81no—
166C7×Q8—82no—
167C7×D4—42no—
168C7×D4—81no—
169C2×C4×C7—21no—
170C2×C4×C7—21no—
171C2×C4×C7—22no—
172C2×C4×C7—22no—
173C2×C4×C7—41no—
174C2×C4×C7—42no—
175C2×C4×C7—42no—
176C2×C4×C7—81no—
177C2×C4×C7—81no—
178C2×C4×C7#12011yescharacteristic
179C2×C4×C7#12011yescharacteristic
180C2×C4×C7#12111yescharacteristic
181C2×C4×C7#12111yescharacteristic
182C2×C4×C7#12111yescharacteristic
183C2×C4×C7#12711yescharacteristic
184C2×Dic7—41no—
185C2×Dic7—41no—
186C2×Dic7—42no—
187C2×Dic7—41no—
188C2×Dic7—41no—
189C2×Dic7—42no—
190C2×Dic7—42no—
191C2×Dic7—42no—
192C7×C8—42no—
193C7⋊C8—24no—
194C7⋊C8—42no—
195C22×Q8—21no—
196C22×Q8C22⋊D711yescharacteristic
197C22×D4C22⋊D711yescharacteristic
198C22×C8—21no—
199C22×C8—141no—
200C22×C8—141no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8C22⋊D712yes—
205#125—141no—
206#125—141no—
207#125—141no—
208#125—141no—
209#124—21no—
210#124—21no—
211#124—21no—
212#124—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—71no—
216C2×C4⋊C4—71no—
217C2×C4⋊C4—72no—
218C2×C4⋊C4—141no—
219C2×C4⋊C4—141no—
220C2×C4⋊C4—142no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—142no—
224C2×C22⋊C4—142no—
225C2×C42C22×D711yescharacteristic
226#107—144no—
227#106—144no—
228#103—24no—
229#103—284no—
230#102—24no—
231#97—21no—
232#97—72no—
233#97—141no—
234#97—141no—
235#97—141no—
236#97C22⋊D712yes—
237#95—142no—
238#95—141no—
239#95—141no—
240C22×C7—82no—
241C22×C7—82no—
242C22×C7—81no—
243C22×C7—81no—
244C22×C7—81no—
245C22×C7#39211yescharacteristic
246C22×C7#45311yescharacteristic
247C22×C7#45411yescharacteristic
248C22×C7#45512yes—
249C22×C7#45612yes—
250C4×C7—21no—
251C4×C7—21no—
252C4×C7—21no—
253C4×C7—21no—
254C4×C7—21no—
255C4×C7—21no—
256C4×C7—81no—
257C4×C7—82no—
258Dic7—82no—
259Dic7—81no—
260Dic7—81no—
261C24—21no—
262C2×Q8—22no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—41no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—42no—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—142no—
275C22×C4—141no—
276C22×C4—141no—
277C22×C4D4×D711yescharacteristic
278C22×C4D4×D711yescharacteristic
279C22×C4C2×C22⋊D711yescharacteristic
280C2×C8—22no—
281C2×C8—42no—
282C2×C8—144no—
283C2×C8—142no—
284C2×C8—142no—
285C2×C8—281no—
286C2×C8—281no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—141no—
292C4⋊C4—141no—
293C4⋊C4—141no—
294C4⋊C4—141no—
295C4⋊C4—282no—
296C4⋊C4—281no—
297C4⋊C4—281no—
298C4⋊C4—281no—
299C4⋊C4—281no—
300C4⋊C4—562no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—562no—
306C22⋊C4—562no—
307C42D4×D711yescharacteristic
308C42#115911yescharacteristic
309C42C2×C22⋊D711yescharacteristic
310C42C2×C22⋊D711yescharacteristic
311C2×C7—161no—
312C2×C7#166112yes—
313C2×C7#166412yes—
314C2×C7#200411yescharacteristic
315C2×C7#200511yescharacteristic
316C2×C7#200611yescharacteristic
317C23—41no—
318C23—41no—
319C23—42no—
320C23—41no—
321C23—42no—
322C23#650011yescharacteristic
323Q8—42no—
324Q8—44no—
325Q8—81no—
326Q8—82no—
327D4—42no—
328D4—81no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—281no—
339C2×C4—282no—
340C2×C4—281no—
341C2×C4—282no—
342C2×C4—281no—
343C2×C4—281no—
344C2×C4—282no—
345C2×C4—282no—
346C2×C4#650011yescharacteristic
347C2×C4#650111yescharacteristic
348C2×C4#650211yescharacteristic
349C2×C4#650211yescharacteristic
350C2×C4#650311yescharacteristic
351C2×C4#651611yescharacteristic
352C8—42no—
353C8—144no—
354C8—282no—
355C7#1008611yescharacteristic, complemented, Sylow
356C22—81no—
357C22—82no—
358C22—82no—
359C22—81no—
360C22—81no—
361C22#8983312yes—
362C22#8997011yescharacteristic
363C22#8997311yescharacteristic
364C22#8997712yes—
365C22#8999511yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—81no—
373C4—82no—
374C4—561no—
375C4—562no—
376C4—561no—
377C2—161no—
378C2#1172899912yes—
379C2#1172900112yes—
380C2#1172948511yescharacteristic
381C2#1172985211yescharacteristic
382C2#1172986211yescharacteristic
subgroupG/H
centerC23#6500
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC2×C7×#489C2
socleC23×C7#120
prime radicalC22×C7×D4D4
hypercenterC2×#489D7
hypocenterC7#10086
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#489

Complements

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

NH ≅ G/Ntype
2. C2×#89852C2semidirect
4. #11729540C2semidirect
6. C2×#89260C2semidirect
7. #11725465C2semidirect
355. C7#10086semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4892
C2×#898522
#117404372
#117295402
C2×#893352
C2×#892602
#117254652
#100867