Finite Group Databasebeta

← #13335066

Subgroups — #13335066

Order 1792.

850 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 171 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×#563C211yesmaximal, characteristic
2#11740466C212yesmaximal, complemented
3#11740384C211yesmaximal, characteristic
4#11729711C218yesmaximal, complemented
5#11729489C212yesmaximal, complemented
6#11729473C211yesmaximal, characteristic
7C7×#563—24no—
8C7×#563—22no—
9C7×#563C2214yes—
10C2×C7×C4⋊Q8—21no—
11C2×C7×#126—22no—
12C2×C7×#125C2212yes—
13C2×C7×#124C2211yescharacteristic
14C2×C7×#123—21no—
15C2×C7×C22⋊Q8—22no—
16#90161C2211yescharacteristic
17#89997—21no—
18#89987—21no—
19#89841C2214yescomplemented
20#89837C2214yescomplemented
21#89835—24no—
22#89828—22no—
23#89827C2214yes—
24#89743—22no—
25#89742—22no—
26#89698C2212yes—
27C4×#1136C2211yescharacteristic
28#89338C2218yescomplemented
29#89335C2214yes—
30#18999—71nomaximal, Sylow
31C22×C7×Q8—21no—
32C22×C7×Q8D411yescharacteristic
33C22×C7×D4D411yescharacteristic
34C23×C4×C7—21no—
35C7×C4⋊Q8—24no—
36C7×C4⋊Q8—24no—
37C7×#126—28no—
38C7×#126—28no—
39C7×#125—22no—
40C7×#125—22no—
41C7×#125D414yes—
42C7×#125C2314yes—
43C7×#124—22no—
44C7×#124D412yes—
45C7×#124C2312yes—
46C7×#123—24no—
47C7×#123—24no—
48C7×C22⋊Q8—28no—
49C7×C22⋊Q8—28no—
50C2×C7×C4⋊C4—22no—
51C2×C7×C4⋊C4—22no—
52C2×C7×C4⋊C4—22no—
53C2×C7×C4⋊C4D412yes—
54C2×C7×C4⋊C4D412yes—
55C2×C7×C22⋊C4—22no—
56C2×C7×C22⋊C4—21no—
57C2×C7×C22⋊C4—22no—
58C2×C42×C7—21no—
59C2×C42×C7C2311yescharacteristic
60#6489—21no—
61#6487—22no—
62C2×#1136—21no—
63#6433—21no—
64#6432C2312yes—
65C4×Dic14—22no—
66#6409—24no—
67#6406—24no—
68C22⋊(C7⋊C8)—22no—
69#6384—24no—
70#6384—24no—
71#6382—24no—
72#6381—24no—
73C4⋊(C7⋊C8)—22no—
74#6377C2312yes—
75#6377C2314yes—
76C4×C7⋊C8—21no—
77C2×#563D711yescharacteristic
78#3236—72no—
79#3140—71no—
80#1664—78no—
81#1533—71no—
82#1528—72no—
83C24×C7—21no—
84C2×C7×Q8—24no—
85C2×C7×Q8—21no—
86C2×C7×Q8—42no—
87C2×C7×Q8—82no—
88C2×C7×Q8C2×D412yes—
89C2×C7×D4—21no—
90C2×C7×D4—42no—
91C2×C7×D4C2×D412yes—
92C22×C4×C7—22no—
93C22×C4×C7—21no—
94C22×C4×C7—22no—
95C22×C4×C7—24no—
96C22×C4×C7—82no—
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2×D412yes—
99C2×Dic14—22no—
100C7×C4⋊C4—28no—
101C7×C4⋊C4—24no—
102C7×C4⋊C4—24no—
103C7×C4⋊C4—22no—
104C7×C4⋊C4—22no—
105C7×C4⋊C4—24no—
106C7×C4⋊C4—24no—
107C7×C4⋊C4C2×D414yes—
108C7×C4⋊C4C2×D414yes—
109C7×C22⋊C4—28no—
110C7×C22⋊C4—24no—
111C7×C22⋊C4—24no—
112C7×C22⋊C4—24no—
113C42×C7—24no—
114C42×C7C2×D412yes—
115C42×C7C2×D411yescharacteristic
116C42×C7C2411yescharacteristic
117C22⋊Dic7—41no—
118C4⋊Dic7—22no—
119C4⋊Dic7—22no—
120Dic7⋊C4—42no—
121C4×Dic7—41no—
122#1136—42no—
123C2×C7⋊C8—24no—
124C2×C7⋊C8—22no—
125#563—24no—
126#563—22no—
127#563C2×D714yes—
128#533—71no—
129C2×C4⋊Q8—21no—
130C2×#126—22no—
131C2×#125C2×D712yes—
132C2×#124C2×D711yescharacteristic
133C2×#123—21no—
134C2×C22⋊Q8—22no—
135#501—74no—
136#491—74no—
137#489—74no—
138#488—142no—
139#484—142no—
140#483—142no—
141#479—144no—
142#432—72no—
143#419—141no—
144#418—141no—
145C4×M16—71no—
146#330—78no—
147#329—74no—
148C23×C7—22no—
149C23×C7—44no—
150C23×C7—42no—
151C23×C7#12011yescharacteristic
152C7×Q8—42no—
153C7×Q8—81no—
154C7×Q8—84no—
155C7×D4—42no—
156C7×D4—81no—
157C2×C4×C7—22no—
158C2×C4×C7—24no—
159C2×C4×C7—22no—
160C2×C4×C7—24no—
161C2×C4×C7—42no—
162C2×C4×C7—44no—
163C2×C4×C7—44no—
164C2×C4×C7—42no—
165C2×C4×C7—84no—
166C2×C4×C7#12011yescharacteristic
167C2×C4×C7#12012yes—
168C2×C4×C7C22×D411yescharacteristic
169C2×C4×C7C22×D412yes—
170C2×Dic7—42no—
171Dic14—44no—
172C7⋊C8—44no—
173C7⋊C8—42no—
174C22×Q8—21no—
175C22×Q8C22⋊D711yescharacteristic
176C22×D4C22⋊D711yescharacteristic
177C23×C4—21no—
178C2×M16—141no—
179C4⋊Q8—24no—
180C4⋊Q8—24no—
181C4⋊Q8—72no—
182#126—28no—
183#126—28no—
184#125—22no—
185#125—22no—
186#125—141no—
187#125C22⋊D714yes—
188#125C22×D714yes—
189#124—22no—
190#124C22⋊D712yes—
191#124C22×D712yes—
192#123—24no—
193#123—24no—
194C22⋊Q8—28no—
195C22⋊Q8—28no—
196C22⋊Q8—142no—
197C4×Q8—142no—
198#117—141no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4C22⋊D712yes—
203C2×C4⋊C4C22⋊D712yes—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C42—21no—
208C2×C42C22×D711yescharacteristic
209#107—144no—
210#106—144no—
211C4⋊C8—142no—
212#103—144no—
213#103—144no—
214#103—144no—
215#102—144no—
216C22⋊C8—142no—
217#97—74no—
218#97—72no—
219C4×C8—141no—
220C22×C7—22no—
221C22×C7—42no—
222C22×C7—44no—
223C22×C7—84no—
224C22×C7#45512yes—
225C22×C7C2×#12011yescharacteristic
226C4×C7—21no—
227C4×C7—21no—
228C4×C7—22no—
229C4×C7—22no—
230C4×C7—44no—
231C4×C7—42no—
232C4×C7—44no—
233Dic7—82no—
234C24—21no—
235C2×Q8—24no—
236C2×Q8—21no—
237C2×Q8—42no—
238C2×Q8—82no—
239C2×Q8—142no—
240C2×Q8C2×C22⋊D712yes—
241C2×D4—21no—
242C2×D4—42no—
243C2×D4C2×C22⋊D712yes—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—82no—
249C22×C4C2×C22⋊D711yescharacteristic
250C22×C4C2×C22⋊D712yes—
251M16—282no—
252C2×C8—142no—
253C2×C8—144no—
254C4⋊C4—28no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—142no—
262C4⋊C4—142no—
263C4⋊C4—282no—
264C4⋊C4C2×C22⋊D714yes—
265C4⋊C4C2×C22⋊D714yes—
266C22⋊C4—28no—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—281no—
271C42—24no—
272C42—281no—
273C42C2×C22⋊D711yescharacteristic
274C42C2×C22⋊D712yes—
275C42C23×D711yescharacteristic
276C2×C7—22no—
277C2×C7—82no—
278C2×C7C2×#45511yescharacteristic
279C2×C7#302012yes—
280C23—22no—
281C23—44no—
282C23—42no—
283C23#651611yescharacteristic
284Q8—42no—
285Q8—81no—
286Q8—84no—
287Q8—284no—
288D4—42no—
289D4—81no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—84no—
299C2×C4—282no—
300C2×C4#651611yescharacteristic
301C2×C4#651612yes—
302C2×C4C22×C22⋊D712yes—
303C2×C4C22×C22⋊D711yescharacteristic
304C8—284no—
305C8—282no—
306C7#1899911yescharacteristic, complemented, Sylow
307C22—22no—
308C22—42no—
309C22—44no—
310C22—84no—
311C22#9001412yes—
312C22C2×#651611yescharacteristic
313C4—21no—
314C4—21no—
315C4—22no—
316C4—22no—
317C4—44no—
318C4—42no—
319C4—44no—
320C4—562no—
321C2—22no—
322C2—82no—
323C2C2×#9001411yescharacteristic
324C2#1174156112yes—
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×#563C2
socleC22×C7C2×#120
prime radicalC22×C7×D4D4
hypercenterC2×#563D7
hypocenterC7#18999
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#563

Complements

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

NH ≅ G/Ntype
2. #11740466C2semidirect
4. #11729711C2semidirect
5. #11729489C2semidirect
19. #89841C22semidirect
20. #89837C22semidirect
28. #89338C22semidirect
306. C7#18999semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5632
#117404662
#117403842
#117297112
#117294892
#117294732
#189997