Finite Group Databasebeta

← #13272156

Subgroups — #13272156

Order 1792.

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

normal subgroups only

This group has 432 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#11741252C211yesmaximal, characteristic
2C7×#1757C211yesmaximal, characteristic
3#11729768C211yesmaximal, characteristic
4#11727904C212yesmaximal
5#11727812C212yesmaximal
6C2×C7×#131C2211yescharacteristic
7#90516C2212yes—
8C22×Dic28C411yescharacteristic, complemented
9C7×#413C2211yescharacteristic
10C7×#410C2211yescharacteristic
11C7×#405—21no—
12#90009C2212yes—
13#89909—21no—
14#89909—21no—
15#89892C412yescomplemented
16#89888—21no—
17#89888—21no—
18C2×#6393—21no—
19#89878C411yescharacteristic, complemented
20C7×#344—21no—
21C7⋊#338—21no—
22#89273—24no—
23#89267—24no—
24#13233—71nomaximal, Sylow
25C2×C7×#41C2311yescharacteristic
26C22×Dic14C2×C412yes—
27C7×#131—41no—
28C7×#131—41no—
29C2×C7×M16—21no—
30C2×C7×M16C2×C411yescharacteristic
31C2×C7×M16D411yescharacteristic
32C2×C7×M16D411yescharacteristic
33C22×C7×C8C2×C411yescharacteristic
34C22×C7×C8D411yescharacteristic
35C22×C7×C8D411yescharacteristic
36C7×#117—21no—
37#6507—22no—
38#6499—22no—
39#6498—22no—
40#6498—22no—
41#6489C2×C412yes—
42#6487—22no—
43#6487—22no—
44C2×#1136—22no—
45C2×Dic28—21no—
46C2×Dic28—21no—
47C2×Dic28—41no—
48C2×Dic28—41no—
49C7×#101D412yes—
50C7×#100—21no—
51C7×#99—21no—
52C7×#99D412yes—
53C7×#97—41no—
54C4×C7×C8—41no—
55#6410—22no—
56#6407—22no—
57#6393—41no—
58#6393—41no—
59#6392—21no—
60#6391—21no—
61#6390—22no—
62#6390—22no—
63#3059—71no—
64#2135—72no—
65#2123—72no—
66#1813—71no—
67#1757D711yescharacteristic
68C7×#41—41no—
69C7×#41—41no—
70C2×C7×Q8C2×D411yescharacteristic
71C2×C7×D4C2×D411yescharacteristic
72C2×C7×D4#4111yescharacteristic
73C2×C7×D4#4111yescharacteristic
74C22×C4×C7C22×C411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×Dic7—22no—
78C2×Dic14—22no—
79C2×Dic14—22no—
80C2×Dic14—42no—
81C2×Dic14—42no—
82C7×M16—22no—
83C7×M16—41no—
84C7×M16—41no—
85C7×M16—41no—
86C7×M16—41no—
87C2×C7×C8—21no—
88C2×C7×C8—21no—
89C2×C7×C8—21no—
90C2×C7×C8—21no—
91C2×C7×C8—21no—
92C2×C7×C8—21no—
93C2×C7×C8—41no—
94C2×C7×C8C22⋊C411yescharacteristic
95C2×C7×C8C22⋊C411yescharacteristic
96C2×C7×C8C22⋊C411yescharacteristic
97C2×C7×C8C22⋊C411yescharacteristic
98C7×C4⋊C4—41no—
99C7×C22⋊C4—22no—
100C42×C7—41no—
101C22⋊Dic7—22no—
102C22⋊Dic7—42no—
103C22⋊Dic7—42no—
104C4⋊Dic7—22no—
105Dic7⋊C4—42no—
106Dic7⋊C4—42no—
107C4×Dic7—22no—
108#1136—42no—
109#1136—42no—
110C2×C7⋊C8—42no—
111Dic28—42no—
112Dic28—41no—
113Dic28—41no—
114C22×Q16—71no—
115C2×#131C2×D711yescharacteristic
116#536—72no—
117#470—141no—
118#470—141no—
119#467—141no—
120#467—141no—
121C2×#108—141no—
122#427—71no—
123#419—72no—
124#413—72no—
125#413C2×D711yescharacteristic
126#410C2×D711yescharacteristic
127#405—21no—
128#356—144no—
129#352—144no—
130#344—21no—
131#338—141no—
132C23×C7C4×D411yescharacteristic
133C23×C7#12111yescharacteristic
134C23×C7#12411yescharacteristic
135C7×Q8—41no—
136C7×D4—41no—
137C7×D4—41no—
138C7×D4—41no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—42no—
146C2×C4×C7C2×C22⋊C411yescharacteristic
147C2×C4×C7C4×D411yescharacteristic
148C2×C4×C7#12111yescharacteristic
149C2×C4×C7#12711yescharacteristic
150C2×Dic7—42no—
151C2×Dic7—42no—
152C2×Dic7—82no—
153Dic14—42no—
154Dic14—42no—
155Dic14—82no—
156C7×C8—21no—
157C7×C8—21no—
158C7×C8—21no—
159C7×C8—42no—
160C7×C8#9911yescharacteristic
161C7×C8#9911yescharacteristic
162C7⋊C8—82no—
163C2×#41C22×D711yescharacteristic
164C22×Q8—72no—
165C2×Q16—141no—
166C2×Q16—141no—
167C2×Q16—281no—
168C2×Q16—281no—
169#131—41no—
170#131—41no—
171C2×M16—21no—
172C2×M16—142no—
173C2×M16C4×D711yescharacteristic
174C2×M16D2811yescharacteristic
175C2×M16C22⋊D711yescharacteristic
176C22×C8C4×D711yescharacteristic
177C22×C8D2811yescharacteristic
178C22×C8C22⋊D711yescharacteristic
179C4⋊Q8—142no—
180#124—142no—
181#123—142no—
182#123—142no—
183C22⋊Q8—142no—
184C22⋊Q8—142no—
185#117—21no—
186#117—72no—
187#108—281no—
188#108—281no—
189#107—141no—
190#106—141no—
191#103—142no—
192#103—142no—
193#101—142no—
194#101D2812yes—
195#100—21no—
196#100—142no—
197#99—21no—
198#99D2812yes—
199#97—41no—
200C4×C8—41no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7#39011yescharacteristic
208C4×C7—21no—
209C4×C7—21no—
210C4×C7—21no—
211C4×C7—82no—
212C4×C7C2×#9911yescharacteristic
213C4×C7#41011yescharacteristic
214Dic7—82no—
215Dic7—82no—
216#41—41no—
217#41—41no—
218C2×Q8—142no—
219C2×Q8—142no—
220C2×Q8—282no—
221C2×Q8—282no—
222C2×Q8C2×D2811yescharacteristic
223C2×D4C2×D2811yescharacteristic
224C2×D4#115711yescharacteristic
225C2×D4#115711yescharacteristic
226C22×C4—142no—
227C22×C4C2×C4×D711yescharacteristic
228C22×C4C2×D2811yescharacteristic
229C22×C4C2×C22⋊D711yescharacteristic
230Q16—282no—
231Q16—281no—
232Q16—281no—
233M16—22no—
234M16—41no—
235M16—41no—
236M16—41no—
237M16—41no—
238M16—282no—
239M16—282no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—41no—
247C2×C8—282no—
248C2×C8(C2×D7)⋊C411yescharacteristic
249C2×C8(C2×D7)⋊C411yescharacteristic
250C2×C8(C2×D7)⋊C411yescharacteristic
251C2×C8(C2×D7)⋊C411yescharacteristic
252C4⋊C4—41no—
253C4⋊C4—142no—
254C4⋊C4—282no—
255C4⋊C4—282no—
256C22⋊C4—22no—
257C22⋊C4—142no—
258C22⋊C4—282no—
259C22⋊C4—282no—
260C42—41no—
261C42—142no—
262C2×C7—21no—
263C2×C7—41no—
264C2×C7—41no—
265C2×C7—41no—
266C2×C7#196411yescharacteristic
267C23C4×D2811yescharacteristic
268C23#643811yescharacteristic
269C23#649311yescharacteristic
270Q8—41no—
271Q8—282no—
272Q8—282no—
273Q8—562no—
274D4—41no—
275D4—41no—
276D4—41no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—282no—
285C2×C4—282no—
286C2×C4—562no—
287C2×C4C4×D2811yescharacteristic
288C2×C4#643711yescharacteristic
289C2×C4C2×(C2×D7)⋊C411yescharacteristic
290C2×C4#649311yescharacteristic
291C8—21no—
292C8—21no—
293C8—21no—
294C8—42no—
295C8—562no—
296C8#638011yescharacteristic
297C8#638011yescharacteristic
298C7#1323311yescharacteristic, complemented, Sylow
299C22—21no—
300C22—21no—
301C22—21no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22#8971111yescharacteristic
306C4—21no—
307C4—21no—
308C4—21no—
309C4—82no—
310C4—562no—
311C4—562no—
312C4C2×#638011yescharacteristic
313C4#8977811yescharacteristic
314C2—21no—
315C2—41no—
316C2—41no—
317C2—41no—
318C2#1172721011yescharacteristic
subgroupG/H
centerC2#11727210
commutatorC22×C4×C7C22×C4
FrattiniC2×#41C22×D7
FittingC7×#1757C2
socleC2×C7#1964
prime radical, cosocleC2×C7×#41C23
hypercenter#1757D7
hypocenterC7#13233
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ #1757

Complements

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

NH ≅ G/Ntype
8. C22×Dic28C4semidirect
15. #89892C4semidirect
19. #89878C4semidirect
298. C7#13233semidirect

Maximal subgroups

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

subgroupindex
#117412522
C7×#17572
#117297682
#117279042
#117278122
#132337