Finite Group Databasebeta

← #13266807

Subgroups — #13266807

Order 1792.

386 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 63 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×#2823C211yesmaximal, characteristic
2#11727467C211yesmaximal, characteristic, complemented
3#11727458C211yesmaximal, characteristic
4#11727327C211yesmaximal, characteristic, complemented
5#11725456C211yesmaximal, characteristic, complemented
6#11725453C211yesmaximal, characteristic
7#11725441C211yesmaximal, characteristic, complemented
8C2×C7×#126—21no—
9#90191C2211yescharacteristic
10C7×#398—21no—
11C7×#398C2211yescharacteristic
12C7×#396—21no—
13C7×#396—21no—
14C7×#396—21no—
15C7×#394—21no—
16C7×#394C2211yescharacteristic
17C7×#384C2211yescharacteristic
18#89990—21no—
19#89743—21no—
20#89736—21no—
21#89325C2211yescharacteristic
22#89323C2211yescharacteristic
23#89323C2211yescharacteristic
24#89318—21no—
25#89247—21no—
26#89247—21no—
27#12249—71nomaximal, Sylow
28C22×C7×D4D411yescharacteristic
29C7×#126—42no—
30C7×#126—42no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4D411yescharacteristic
38C2×C7×C4⋊C4D411yescharacteristic
39C2×C7×C4⋊C4C2311yescharacteristic
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C42×C7D411yescharacteristic
47C2×#1142—41no—
48#6489—21no—
49#6487—21no—
50#6487—21no—
51C2×#1136—21no—
52#6453—41no—
53#6451—41no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95D411yescharacteristic
57C7×#95D411yescharacteristic
58#6406—41no—
59C22⋊(C7⋊C8)—21no—
60C22⋊(C7⋊C8)—21no—
61#6384—41no—
62#6382—41no—
63#6382—41no—
64#6381—41no—
65#2823D711yescharacteristic
66#2060—71no—
67#1611—71no—
68#1599—71no—
69#1346—71no—
70#1345—71no—
71#1325—71no—
72C24×C7—21no—
73C2×C7×D4—41no—
74C2×C7×D4—81no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7C2×D411yescharacteristic
81C22×C4×C7C2×D411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C2×Dic14—21no—
84C2×Dic14—21no—
85C7×C4⋊C4—41no—
86C7×C4⋊C4—41no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—42no—
89C7×C4⋊C4—42no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—41no—
92C7×C4⋊C4—81no—
93C7×C4⋊C4—81no—
94C7×C4⋊C4—81no—
95C7×C22⋊C4—42no—
96C7×C22⋊C4—42no—
97C7×C22⋊C4—41no—
98C7×C22⋊C4—41no—
99C7×C22⋊C4—81no—
100C7×C22⋊C4—81no—
101C7×C22⋊C4—81no—
102C42×C7—21no—
103C42×C7SD1612yescomplemented
104C22⋊Dic7—41no—
105#1142—82no—
106C4⋊Dic7—21no—
107C4⋊Dic7—21no—
108C4⋊Dic7—21no—
109C4⋊Dic7—21no—
110Dic7⋊C4—41no—
111Dic7⋊C4—41no—
112C4×Dic7—41no—
113#1136—22no—
114#1136—41no—
115C2×C7⋊C8—41no—
116C2×C7⋊C8—41no—
117C2×C7⋊C8—41no—
118#544—71no—
119C2×#126—21no—
120#483—141no—
121#450—141no—
122#428—141no—
123#398—21no—
124#398C2×D711yescharacteristic
125#396—21no—
126#396—21no—
127#396—21no—
128#394—21no—
129#394C2×D711yescharacteristic
130#384C2×D711yescharacteristic
131#343—71no—
132#339—141no—
133#328—141no—
134#328—141no—
135#324—71no—
136#324—71no—
137C23×C7—41no—
138C23×C7—41no—
139C23×C7—41no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7—41no—
144C23×C7#12011yescharacteristic
145C7×D4—42no—
146C7×D4—81no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—22no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
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×C7C2×SD1611yescharacteristic
169C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
170C2×Dic7—41no—
171C2×Dic7—41no—
172Dic14—81no—
173Dic14—81no—
174C7⋊C8—42no—
175C7⋊C8—81no—
176C7⋊C8—81no—
177C22×D4C22⋊D711yescharacteristic
178C2×SD16—281no—
179C2×M16—141no—
180C4⋊Q8—281no—
181#126—42no—
182#126—42no—
183#125—281no—
184C22⋊Q8—141no—
185C22⋊Q8—141no—
186#117—141no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C22⋊D711yescharacteristic
194C2×C4⋊C4C22⋊D711yescharacteristic
195C2×C4⋊C4C22×D711yescharacteristic
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C42C22⋊D711yescharacteristic
203#107—281no—
204#106—281no—
205#106—281no—
206#103—281no—
207#102—281no—
208C22⋊C8—141no—
209C22⋊C8—141no—
210#95—21no—
211#95—21no—
212#95C22⋊D711yescharacteristic
213#95C22⋊D711yescharacteristic
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—81no—
217C22×C7—81no—
218C22×C7—81no—
219C22×C7—81no—
220C22×C7—81no—
221C22×C7—81no—
222C22×C7—81no—
223C22×C7#45011yescharacteristic
224C22×C7#45611yescharacteristic
225C22×C7B(4,𝔽2)11yescharacteristic
226C4×C7—21no—
227C4×C7—21no—
228C4×C7—42no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7—81no—
232C4×C7—81no—
233C4×C7—81no—
234C4×C7—81no—
235Dic7—81no—
236Dic7—81no—
237C24—21no—
238C2×Q8—141no—
239C2×Q8—141no—
240C2×D4—41no—
241C2×D4—81no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4C2×C22⋊D711yescharacteristic
248C22×C4C2×C22⋊D711yescharacteristic
249C22×C4C2×C22⋊D711yescharacteristic
250SD16—562no—
251M16—142no—
252M16—281no—
253C2×C8—281no—
254C2×C8—281no—
255C2×C8—281no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—81no—
264C4⋊C4—81no—
265C4⋊C4—81no—
266C4⋊C4—141no—
267C4⋊C4—141no—
268C4⋊C4—141no—
269C4⋊C4—141no—
270C4⋊C4—281no—
271C4⋊C4—281no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—81no—
277C22⋊C4—81no—
278C22⋊C4—81no—
279C22⋊C4—281no—
280C42—21no—
281C42—281no—
282C42#114212yescomplemented
283C2×C7—21no—
284C2×C7—21no—
285C2×C7—161no—
286C2×C7#160411yescharacteristic
287C2×C7#220011yescharacteristic
288C2×C7#220311yescharacteristic
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23#651611yescharacteristic
297Q8—561no—
298Q8—561no—
299D4—42no—
300D4—81no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—281no—
323C2×C4—281no—
324C2×C4#649511yescharacteristic
325C2×C4C2×#114211yescharacteristic
326C8—282no—
327C8—561no—
328C8—561no—
329C7#1224911yescharacteristic, complemented, Sylow
330C22—21no—
331C22—21no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22#8999011yescharacteristic
340C22#9001611yescharacteristic
341C22#9001811yescharacteristic
342C4—21no—
343C4—21no—
344C4—42no—
345C4—41no—
346C4—41no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—561no—
352C4—561no—
353C2—21no—
354C2—21no—
355C2—161no—
356C2#1172963411yescharacteristic
357C2#1172970111yescharacteristic
358C2#1172971311yescharacteristic
subgroupG/H
centerC22#90016
commutator, cosocleC2×C7×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2823C2
socleC22×C7B(4,𝔽2)
prime radicalC22×C7×D4D4
hypercenter#2823D7
hypocenterC7#12249
LCSG ▷ C2×C7×C4⋊C4 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2823

Complements

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

NH ≅ G/Ntype
2. #11727467C2semidirect
4. #11727327C2semidirect
5. #11725456C2semidirect
7. #11725441C2semidirect
103. C42×C7SD16semidirect
282. C42#1142semidirect
329. C7#12249semidirect

Maximal subgroups

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

subgroupindex
C7×#28232
#117274672
#117274582
#117273272
#117254562
#117254532
#117254412
#122497