Finite Group Databasebeta

← #13247403

Subgroups — #13247403

Order 1792.

446 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 95 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#11739781C211yesmaximal, characteristic
2C4×C7×#100C211yesmaximal, characteristic
3#11729357C211yesmaximal, characteristic
4#11725472C212yesmaximal, complemented
5#11725471C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C2×C7×#100—21no—
8C4×C7×M16C2212yes—
9#89992—21no—
10#89984C411yescharacteristic
11#89983C411yescharacteristic
12C4×C22⋊Dic7C411yescharacteristic
13C2×C7⋊#95C411yescharacteristic
14#89753—21no—
15#89723—21no—
16C22⋊C4×Dic7—21no—
17#89709C2211yescharacteristic
18#89704C2211yescharacteristic
19#89698C2212yes—
20C7×#343—21no—
21#89320—21no—
22#89320—21no—
23#8772—71nomaximal, Sylow
24C22×C7×D4D411yescharacteristic
25C23×C4×C7C2×C411yescharacteristic
26C23×C4×C7C2×C411yescharacteristic
27C23×Dic7—21no—
28C2×C7×M16—22no—
29C4×C7×D4—22no—
30C4×C7×D4—22no—
31C2×C7×C4⋊C4D411yescharacteristic
32C2×C7×C22⋊C4D411yescharacteristic
33C2×C7×C22⋊C4D411yescharacteristic
34C2×C42×C7C2311yescharacteristic
35C2×C22⋊Dic7—21no—
36C2×C22⋊Dic7—21no—
37C2×C22⋊Dic7—21no—
38C2×C4⋊Dic7—21no—
39C2×C4×Dic7—21no—
40C2×C4×Dic7—21no—
41C7×#100—24no—
42C7×#97—22no—
43C4×C7×C8—22no—
44C7⋊#95—21no—
45C7⋊#95—21no—
46C7⋊#95—21no—
47C7⋊#95—21no—
48C7⋊#95—21no—
49C7⋊#95C2×C411yescharacteristic
50C7⋊#95C2×C411yescharacteristic
51C7⋊#95C2×C411yescharacteristic
52C7⋊#95C2×C411yescharacteristic
53C22⋊(C7⋊C8)—22no—
54C4⋊(C7⋊C8)—22no—
55#6377—22no—
56#2353—71no—
57C4×#100D711yescharacteristic
58#1466—71no—
59#1300—72no—
60#1288—72no—
61C24×C7C22⋊C411yescharacteristic
62C24×C7C22⋊C411yescharacteristic
63C2×C7×D4—22no—
64C2×C7×D4—22no—
65C2×C7×D4—41no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—81no—
71C22×C4×C7—81no—
72C22×C4×C7C22⋊C411yescharacteristic
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22×C411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×Dic7—21no—
78C22×Dic7—21no—
79C22×Dic7—21no—
80C22×Dic7—21no—
81C22×Dic7—42no—
82C22×Dic7—42no—
83C22×Dic7—81no—
84C7×M16—24no—
85C7×M16—42no—
86C2×C7×C8—42no—
87C7×C4⋊C4—22no—
88C7×C22⋊C4—22no—
89C7×C22⋊C4—22no—
90C42×C7#4112yes—
91C42×C7#4112yes—
92C22⋊Dic7—42no—
93C22⋊Dic7—42no—
94C22⋊Dic7—81no—
95C4⋊Dic7—81no—
96C4×Dic7—42no—
97C4×Dic7—81no—
98C2×C7⋊C8—42no—
99C2×C4×D4C2×D711yescharacteristic
100#432—72no—
101C2×#100—21no—
102C4×M16C2×D712yes—
103#388—141no—
104#387—141no—
105#386—141no—
106#383—71no—
107#381—71no—
108#380—71no—
109C4×C22⋊C4—71no—
110C4×C22⋊C4—141no—
111#376—71no—
112C2×#95—71no—
113#343—21no—
114#323—141no—
115#323—141no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—41no—
122C23×C7—41no—
123C23×C7—41no—
124C23×C7—41no—
125C23×C7#10412yes—
126C23×C7C2×C22⋊C411yescharacteristic
127C7×D4—44no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—41no—
133C2×C4×C7—41no—
134C2×C4×C7—41no—
135C2×C4×C7—41no—
136C2×C4×C7—81no—
137C2×C4×C7—81no—
138C2×C4×C7—81no—
139C2×C4×C7—81no—
140C2×C4×C7#11711yescharacteristic
141C2×C4×C7#11711yescharacteristic
142C2×C4×C7#12312yes—
143C2×C4×C7#12312yes—
144C2×Dic7—22no—
145C2×Dic7—41no—
146C2×Dic7—41no—
147C2×Dic7—41no—
148C2×Dic7—41no—
149C2×Dic7—41no—
150C2×Dic7—41no—
151C2×Dic7—41no—
152C2×Dic7—41no—
153C2×Dic7—41no—
154C2×Dic7—41no—
155C2×Dic7—41no—
156C2×Dic7—84no—
157C7×C8—44no—
158C7⋊C8—82no—
159C22×D4C22⋊D711yescharacteristic
160C23×C4—141no—
161C23×C4C4×D711yescharacteristic
162C23×C4C4×D711yescharacteristic
163C2×M16—22no—
164C4×D4—22no—
165C4×D4—22no—
166C2×C4⋊C4—141no—
167C2×C4⋊C4C22⋊D711yescharacteristic
168C2×C22⋊C4—141no—
169C2×C22⋊C4—141no—
170C2×C22⋊C4—141no—
171C2×C22⋊C4D2811yescharacteristic
172C2×C22⋊C4D2811yescharacteristic
173C2×C42—141no—
174C2×C42—141no—
175C2×C42C22×D711yescharacteristic
176C4⋊C8—142no—
177#100—24no—
178C22⋊C8—142no—
179#97—22no—
180#97—142no—
181C4×C8—22no—
182#95—71no—
183#95—71no—
184#95—71no—
185#95—71no—
186#95—141no—
187#95—141no—
188#95—141no—
189#95—141no—
190#95—141no—
191C22×C7—21no—
192C22×C7—21no—
193C22×C7—22no—
194C22×C7—42no—
195C22×C7—42no—
196C22×C7—41no—
197C22×C7—41no—
198C22×C7—41no—
199C22×C7—41no—
200C22×C7—41no—
201C22×C7—81no—
202C22×C7—81no—
203C22×C7#38111yescharacteristic
204C22×C7C2×#10411yescharacteristic
205C22×C7#42111yescharacteristic
206C4×C7—22no—
207C4×C7—22no—
208C4×C7—21no—
209C4×C7—21no—
210C4×C7—41no—
211C4×C7—41no—
212Dic7—42no—
213Dic7—81no—
214Dic7—81no—
215Dic7—81no—
216C24(C2×D7)⋊C411yescharacteristic
217C24(C2×D7)⋊C411yescharacteristic
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—41no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4—141no—
228C22×C4—141no—
229C22×C4—141no—
230C22×C4—141no—
231C22×C4—282no—
232C22×C4—282no—
233C22×C4—561no—
234C22×C4(C2×D7)⋊C411yescharacteristic
235C22×C4(C2×D7)⋊C411yescharacteristic
236C22×C4C2×C4×D711yescharacteristic
237C22×C4C2×D2811yescharacteristic
238C22×C4C2×C22⋊D711yescharacteristic
239M16—24no—
240M16—42no—
241C2×C8—42no—
242C2×C8—282no—
243C4⋊C4—22no—
244C4⋊C4—561no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—282no—
248C22⋊C4—282no—
249C22⋊C4—561no—
250C42—282no—
251C42—561no—
252C42#115912yes—
253C42#115912yes—
254C2×C7—21no—
255C2×C7—21no—
256C2×C7—42no—
257C2×C7—81no—
258C2×C7#182611yescharacteristic
259C2×C7#182711yescharacteristic
260C2×C7#182911yescharacteristic
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23#637912yes—
271C23C2×(C2×D7)⋊C411yescharacteristic
272D4—44no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—142no—
286C2×C4—281no—
287C2×C4—281no—
288C2×C4—281no—
289C2×C4—281no—
290C2×C4—281no—
291C2×C4—281no—
292C2×C4—281no—
293C2×C4—281no—
294C2×C4—281no—
295C2×C4—281no—
296C2×C4—281no—
297C2×C4—564no—
298C2×C4#644011yescharacteristic
299C2×C4#644011yescharacteristic
300C2×C4#644912yes—
301C2×C4#649812yes—
302C8—44no—
303C8—562no—
304C7#877211yescharacteristic, complemented, Sylow
305C22—21no—
306C22—21no—
307C22—22no—
308C22—42no—
309C22—42no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22—81no—
317C22C2×#637911yescharacteristic
318C22#8971511yescharacteristic
319C22#8977711yescharacteristic
320C4—22no—
321C4—22no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—282no—
327C4—561no—
328C4—561no—
329C4—561no—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—81no—
334C2#1172719611yescharacteristic
335C2#1172782111yescharacteristic
336C2#1172788511yescharacteristic
subgroupG/H
centerC22#89715
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC4×C7×#100C2
socleC22×C7#381
prime radicalC22×C7×D4D4
cosocleC2×C42×C7C23
hypercenterC4×#100D7
hypocenterC7#8772
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
4. #11725472C2semidirect
5. #11725471C2semidirect
304. C7#8772semidirect

Maximal subgroups

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

subgroupindex
#117397812
C4×C7×#1002
#117293572
#117254722
#117254712
#87727