Finite Group Databasebeta

← #13254575

Subgroups — #13254575

Order 1792.

500 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 101 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×#2581C211yesmaximal, characteristic
2#11739892C211yesmaximal, characteristic
3#11739876C211yesmaximal, characteristic
4#11729376C211yesmaximal, characteristic
5#11729375C211yesmaximal, characteristic
6#11727300C211yesmaximal, characteristic
7#11725437C211yesmaximal, characteristic
8C2×C4×C7×Q8C2211yescharacteristic
9C2×C4×C7×Q8C2211yescharacteristic
10#90161C2211yescharacteristic
11C7×#400—21no—
12C7×#395—21no—
13C7×#391—21no—
14C7×#389—21no—
15C7×#387—22no—
16C7×#382C2211yescharacteristic
17#90000—21no—
18#89997—21no—
19#89803—21no—
20C4×#1144—22no—
21#89794—22no—
22#89793—21no—
23#89793—21no—
24#89792—21no—
25#89742—21no—
26#89736—21no—
27#89697C2211yescharacteristic
28#89697C2211yescharacteristic
29#89696C2211yescharacteristic
30Q8⋊(C7⋊C8)—22no—
31Q8⋊(C7⋊C8)—22no—
32#89247—22no—
33#10003—71nomaximal, Sylow
34C22×C7×Q8D411yescharacteristic
35C22×C7×Q8D411yescharacteristic
36C4×C7×Q8—21no—
37C4×C7×Q8—21no—
38C4×C7×Q8—21no—
39C4×C7×Q8—21no—
40C4×C7×Q8—21no—
41C4×C7×Q8—21no—
42C4×C7×Q8—21no—
43C4×C7×Q8—21no—
44C2×C7×C4⋊C4—22no—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4—21no—
47C2×C7×C4⋊C4—21no—
48C2×C7×C4⋊C4D411yescharacteristic
49C2×C7×C4⋊C4D411yescharacteristic
50C2×C42×C7—21no—
51C2×C42×C7—21no—
52C2×C42×C7C2311yescharacteristic
53C2×#1144—22no—
54#6489—21no—
55#6487—21no—
56#6487—21no—
57C2×#1136—21no—
58#6433—21no—
59#6432D411yescharacteristic
60#6432D411yescharacteristic
61C4×Dic14—21no—
62C4×Dic14—21no—
63C7×#95—22no—
64C7×#95—22no—
65C7×#95—22no—
66C7×#95—21no—
67C7×#95D412yes—
68#6409—22no—
69#6409—22no—
70#6409—22no—
71C22⋊(C7⋊C8)—21no—
72C22⋊(C7⋊C8)—21no—
73#6384—22no—
74#6382—22no—
75#6381—22no—
76#6381—22no—
77C4⋊(C7⋊C8)—21no—
78C4⋊(C7⋊C8)—21no—
79C4⋊(C7⋊C8)—21no—
80C4⋊(C7⋊C8)D412yes—
81C4×C7⋊C8—21no—
82C4×C7⋊C8D412yes—
83#3104—71no—
84#3078—71no—
85#2581D711yescharacteristic
86#1518—71no—
87#1487—71no—
88#1479—71no—
89#1286—71no—
90C2×C7×Q8—21no—
91C2×C7×Q8—21no—
92C2×C7×Q8—21no—
93C2×C7×Q8—21no—
94C2×C7×Q8—21no—
95C2×C7×Q8—21no—
96C2×C7×Q8—21no—
97C2×C7×Q8—42no—
98C2×C7×Q8Q1612yescomplemented
99C22×C4×C7—21no—
100C22×C4×C7—21no—
101C22×C4×C7—22no—
102C22×C4×C7—21no—
103C22×C4×C7—21no—
104C22×C4×C7C2×D411yescharacteristic
105C22×C4×C7C2×D411yescharacteristic
106C22×C4×C7C2×D411yescharacteristic
107C2×Dic14—21no—
108C2×Dic14—21no—
109C7×C4⋊C4—21no—
110C7×C4⋊C4—21no—
111C7×C4⋊C4—21no—
112C7×C4⋊C4—21no—
113C7×C4⋊C4—44no—
114C7×C4⋊C4—41no—
115C7×C4⋊C4—41no—
116C7×C4⋊C4—41no—
117C7×C4⋊C4—41no—
118C7×C4⋊C4—41no—
119C7×C4⋊C4—41no—
120C42×C7—41no—
121C42×C7—41no—
122C42×C7—41no—
123C42×C7—41no—
124C42×C7C2×D411yescharacteristic
125C42×C7C2×D411yescharacteristic
126C42×C7C2×D411yescharacteristic
127C42×C7#4111yescharacteristic
128C22⋊Dic7—41no—
129#1144—44no—
130C4⋊Dic7—21no—
131C4⋊Dic7—21no—
132C4⋊Dic7—21no—
133C4⋊Dic7—21no—
134Dic7⋊C4—41no—
135Dic7⋊C4—41no—
136C4×Dic7—41no—
137#1136—41no—
138#1136—41no—
139C2×C7⋊C8—22no—
140C2×C7⋊C8—22no—
141C2×C7⋊C8—41no—
142#533—71no—
143C2×C4×Q8C2×D711yescharacteristic
144C2×C4×Q8C2×D711yescharacteristic
145#484—141no—
146#479—142no—
147#477—141no—
148#477—141no—
149#475—141no—
150#462—141no—
151#451—141no—
152C4×Q16—142no—
153#431—71no—
154#431—71no—
155#428—141no—
156#423—71no—
157#419—141no—
158#400—21no—
159#395—21no—
160#391—21no—
161#389—21no—
162#387—22no—
163#382C2×D711yescharacteristic
164#328—142no—
165Q8⋊C8—142no—
166Q8⋊C8—142no—
167C23×C7#12011yescharacteristic
168C7×Q8—22no—
169C7×Q8—22no—
170C7×Q8—22no—
171C7×Q8—22no—
172C7×Q8—41no—
173C7×Q8—41no—
174C7×Q8—41no—
175C7×Q8—41no—
176C2×C4×C7—22no—
177C2×C4×C7—22no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—42no—
185C2×C4×C7—42no—
186C2×C4×C7—42no—
187C2×C4×C7—42no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—42no—
191C2×C4×C7#12011yescharacteristic
192C2×C4×C7#12011yescharacteristic
193C2×C4×C7#12111yescharacteristic
194C2×C4×C7#12111yescharacteristic
195C2×C4×C7#12111yescharacteristic
196C2×C4×C7#12711yescharacteristic
197C2×C4×C7C2×Q1611yescharacteristic
198C2×C4×C7#13511yescharacteristic
199C2×C4×C7#13711yescharacteristic
200C2×C4×C7#13711yescharacteristic
201C2×Dic7—41no—
202C2×Dic7—41no—
203Dic14—42no—
204Dic14—81no—
205C7⋊C8—42no—
206C7⋊C8—42no—
207C7⋊C8—81no—
208C22×Q8C22⋊D711yescharacteristic
209C22×Q8C22⋊D711yescharacteristic
210C2×Q16—142no—
211C2×M16—141no—
212C4⋊Q8—71no—
213C4⋊Q8—71no—
214#125—141no—
215C22⋊Q8—141no—
216C22⋊Q8—141no—
217C4×Q8—21no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—141no—
226C4×Q8—141no—
227#117—141no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4C22⋊D711yescharacteristic
233C2×C4⋊C4C22⋊D711yescharacteristic
234C2×C42—21no—
235C2×C42—21no—
236C2×C42C22×D711yescharacteristic
237#107—142no—
238#107—142no—
239#106—142no—
240C4⋊C8—72no—
241C4⋊C8—141no—
242C4⋊C8—141no—
243C4⋊C8—141no—
244#103—142no—
245#103—142no—
246#103—142no—
247#103—142no—
248C22⋊C8—141no—
249C22⋊C8—141no—
250C4×C8—72no—
251C4×C8—141no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—21no—
256#95C22⋊D712yes—
257C22×C7—21no—
258C22×C7—21no—
259C22×C7#39211yescharacteristic
260C22×C7#45111yescharacteristic
261C22×C7#45211yescharacteristic
262C4×C7—21no—
263C4×C7—21no—
264C4×C7—21no—
265C4×C7—21no—
266C4×C7—42no—
267C4×C7—42no—
268C4×C7—82no—
269C4×C7—81no—
270C4×C7—81no—
271C4×C7#45111yescharacteristic
272C4×C7#45211yescharacteristic
273C4×C7#46711yescharacteristic
274C4×C7#47011yescharacteristic
275Dic7—81no—
276Dic7—81no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—42no—
285C2×Q8—141no—
286C2×Q8—141no—
287C2×Q8#114412yescomplemented
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4C2×C22⋊D711yescharacteristic
294C22×C4C2×C22⋊D711yescharacteristic
295C22×C4C2×C22⋊D711yescharacteristic
296Q16—284no—
297M16—281no—
298M16—281no—
299C2×C8—142no—
300C2×C8—142no—
301C2×C8—281no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—44no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—141no—
314C4⋊C4—141no—
315C4⋊C4—141no—
316C4⋊C4—141no—
317C4⋊C4—281no—
318C4⋊C4—281no—
319C22⋊C4—281no—
320C42—41no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42—281no—
325C42D4×D711yescharacteristic
326C42D4×D711yescharacteristic
327C42D4×D711yescharacteristic
328C42#115911yescharacteristic
329C2×C7—21no—
330C2×C7—21no—
331C2×C7#200111yescharacteristic
332C2×C7#200911yescharacteristic
333C2×C7#201011yescharacteristic
334C23#651611yescharacteristic
335Q8—22no—
336Q8—22no—
337Q8—22no—
338Q8—22no—
339Q8—41no—
340Q8—41no—
341Q8—41no—
342Q8—41no—
343Q8—282no—
344Q8—561no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—281no—
361C2×C4—281no—
362C2×C4#650011yescharacteristic
363C2×C4#650011yescharacteristic
364C2×C4#650111yescharacteristic
365C2×C4#650211yescharacteristic
366C2×C4#650211yescharacteristic
367C2×C4#650311yescharacteristic
368C2×C4#650511yescharacteristic
369C2×C4C2×#114411yescharacteristic
370C2×C4#651311yescharacteristic
371C2×C4#651411yescharacteristic
372C8—282no—
373C8—282no—
374C8—561no—
375C7#1000311yescharacteristic, complemented, Sylow
376C22—21no—
377C22—21no—
378C22#8999511yescharacteristic
379C22#9000011yescharacteristic
380C22#9001111yescharacteristic
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—42no—
386C4—42no—
387C4—82no—
388C4—81no—
389C4—81no—
390C4—561no—
391C4—561no—
392C4#8981911yescharacteristic
393C4#8982211yescharacteristic
394C4#8982311yescharacteristic
395C4#8982411yescharacteristic
396C2—21no—
397C2—21no—
398C2#1172945411yescharacteristic
399C2#1172946911yescharacteristic
400C2#1172962311yescharacteristic
subgroupG/H
centerC22#89995
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2581C2
socleC22×C7#392
prime radicalC23×C7#120
hypercenter#2581D7
hypocenterC7#10003
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2581

Complements

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

NH ≅ G/Ntype
98. C2×C7×Q8Q16semidirect
287. C2×Q8#1144semidirect
375. C7#10003semidirect

Maximal subgroups

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

subgroupindex
C7×#25812
#117398922
#117398762
#117293762
#117293752
#117273002
#117254372
#100037