Finite Group Databasebeta

← #13245941

Subgroups — #13245941

Order 1792.

746 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 746 conjugacy classes of subgroups — too many to draw. Only its 119 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#11742834C211yesmaximal, characteristic
2C7×#1955C211yesmaximal, characteristic
3#11729357C211yesmaximal, characteristic
4#11728193C212yesmaximal, complemented
5#11727555C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7#90489C412yes—
8C2×#6487—22no—
9#90234—22no—
10#90230—24no—
11D4×Dic14—22no—
12#90191C412yes—
13#90174—22no—
14#90161C2212yes—
15C2×C7×C4⋊C8C2211yescharacteristic
16C2×C7×C22⋊C8—22no—
17C2×C7×#97C2211yescharacteristic
18#89698C2212yes—
19#89320—22no—
20#89281—22no—
21#89277—22no—
22#89260—24no—
23#8457—71nomaximal, Sylow
24C22×C7×D4C2×C411yescharacteristic
25C23×C4×C7D412yes—
26C22×Dic14—22no—
27C22×C7×C8—21no—
28C22×C7×C8—21no—
29C4×C7×D4—22no—
30C4×C7×D4—22no—
31C2×C7×C4⋊C4C2×C411yescharacteristic
32C2×C7×C22⋊C4D412yes—
33C2×C42×C7C2311yescharacteristic
34C2×C22⋊Dic7—24no—
35D4×Dic7—24no—
36D4×Dic7—24no—
37#6489C2×C414yes—
38C2×C4⋊Dic7—22no—
39#6487—24no—
40#6487—24no—
41#6487—24no—
42#6487—22no—
43#6487—22no—
44C2×Dic7⋊C4—24no—
45#6453—24no—
46#6451—24no—
47#6441—28no—
48#6441—28no—
49#6433D414yes—
50#6432D412yes—
51#6432D412yes—
52C4×Dic14—22no—
53C4×Dic14—22no—
54C7×C4⋊C8—22no—
55C7×C22⋊C8—44no—
56C7×#97—22no—
57C22⋊(C7⋊C8)—22no—
58C4⋊(C7⋊C8)—22no—
59#6377—22no—
60#3501—71no—
61#1955D711yescharacteristic
62#1552—72no—
63#1531—72no—
64#1466—71no—
65C24×C7C22⋊C412yes—
66C2×C7×D4—24no—
67C2×C7×D4—22no—
68C22×C4×C7—22no—
69C22×C4×C7—22no—
70C22×C4×C7—42no—
71C22×C4×C7—42no—
72C22×C4×C7C22⋊C412yes—
73C22×C4×C7C22×C411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C22×Dic7—24no—
77C22×Dic7—24no—
78C2×Dic14—22no—
79C2×Dic14—24no—
80C2×Dic14—22no—
81C2×Dic14—82no—
82C2×Dic14—82no—
83C2×C7×C8—22no—
84C2×C7×C8—41no—
85C2×C7×C8—41no—
86C2×C7×C8—42no—
87C2×C7×C8—41no—
88C7×C4⋊C4—22no—
89C7×C22⋊C4—24no—
90C42×C7C2×D412yes—
91C42×C7C2×D412yes—
92C22⋊Dic7—28no—
93C22⋊Dic7—24no—
94C4⋊Dic7—24no—
95C4⋊Dic7—22no—
96C4⋊Dic7—22no—
97C4⋊Dic7C22⋊C414yes—
98C4⋊Dic7C22⋊C414yes—
99Dic7⋊C4—28no—
100Dic7⋊C4—24no—
101Dic7⋊C4—24no—
102C4×Dic7—24no—
103C2×C7⋊C8—42no—
104#554—144no—
105#554—142no—
106D4×Q8—142no—
107#544—72no—
108#543—142no—
109#533—72no—
110C2×C22⋊Q8—142no—
111#518—72no—
112C2×C4×D4C2×D711yescharacteristic
113#432—72no—
114C2×C4⋊C8C2×D711yescharacteristic
115C2×C22⋊C8—22no—
116C2×#97C2×D711yescharacteristic
117#332—142no—
118#332—142no—
119#330—144no—
120#323—142no—
121C23×C7—22no—
122C23×C7—22no—
123C23×C7—42no—
124C23×C7—42no—
125C23×C7#10114yes—
126C23×C7C2×C22⋊C411yescharacteristic
127C7×D4—28no—
128C2×C4×C7—22no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—22no—
132C2×C4×C7—41no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—82no—
137C2×C4×C7C2×C22⋊C412yes—
138C2×C4×C7#12012yes—
139C2×C4×C7#12011yescharacteristic
140C2×C4×C7#12011yescharacteristic
141C2×Dic7—24no—
142C2×Dic7—24no—
143C2×Dic7—84no—
144C2×Dic7—84no—
145Dic14—42no—
146Dic14—42no—
147Dic14—82no—
148Dic14—84no—
149C7×C8—42no—
150C7×C8—81no—
151C7⋊C8—82no—
152C22×Q8—142no—
153C22×D4C4×D711yescharacteristic
154C23×C4C22⋊D712yes—
155C22×C8—21no—
156C22×C8—21no—
157C4⋊Q8—72no—
158C4⋊Q8—72no—
159C4⋊Q8—144no—
160#125—74no—
161#125—144no—
162C22⋊Q8—148no—
163C22⋊Q8—148no—
164C22⋊Q8—144no—
165C22⋊Q8—144no—
166C22⋊Q8—144no—
167C22⋊Q8—142no—
168C22⋊Q8—142no—
169C4×Q8—142no—
170C4×Q8—142no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—144no—
174C4×D4—144no—
175#117—74no—
176C2×C4⋊C4—144no—
177C2×C4⋊C4—142no—
178C2×C4⋊C4C4×D711yescharacteristic
179C2×C22⋊C4—144no—
180C2×C22⋊C4D2812yes—
181C2×C42C22×D711yescharacteristic
182C4⋊C8—22no—
183C4⋊C8—142no—
184C22⋊C8—44no—
185C22⋊C8—142no—
186#97—22no—
187#97—142no—
188C22×C7—21no—
189C22×C7—21no—
190C22×C7—24no—
191C22×C7—44no—
192C22×C7—42no—
193C22×C7—42no—
194C22×C7—82no—
195C22×C7#37911yescharacteristic
196C22×C7C2×#10112yes—
197C4×C7—21no—
198C4×C7—22no—
199C4×C7—22no—
200C4×C7—42no—
201C4×C7#45311yescharacteristic
202C4×C7#45511yescharacteristic
203Dic7—44no—
204Dic7—44no—
205C24(C2×D7)⋊C412yes—
206C2×Q8—144no—
207C2×Q8—142no—
208C2×Q8—142no—
209C2×Q8—562no—
210C2×Q8—562no—
211C2×D4—24no—
212C2×D4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—144no—
218C22×C4—144no—
219C22×C4(C2×D7)⋊C412yes—
220C22×C4C2×C4×D711yescharacteristic
221C22×C4C2×D2811yescharacteristic
222C22×C4C2×C22⋊D711yescharacteristic
223C2×C8—22no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—42no—
227C2×C8—41no—
228C2×C8—282no—
229C4⋊C4—22no—
230C4⋊C4—74no—
231C4⋊C4—74no—
232C4⋊C4—148no—
233C4⋊C4—144no—
234C4⋊C4—144no—
235C4⋊C4—142no—
236C4⋊C4—144no—
237C4⋊C4—142no—
238C22⋊C4—24no—
239C22⋊C4—148no—
240C22⋊C4—144no—
241C42—144no—
242C42D4×D712yes—
243C42D4×D712yes—
244C2×C7—22no—
245C2×C7—44no—
246C2×C7#178411yescharacteristic
247C2×C7#179211yescharacteristic
248C2×C7#179411yescharacteristic
249C23—22no—
250C23—22no—
251C23—42no—
252C23—42no—
253C23#639814yes—
254C23C2×(C2×D7)⋊C411yescharacteristic
255Q8—282no—
256Q8—282no—
257Q8—562no—
258Q8—564no—
259D4—28no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—82no—
269C2×C4—144no—
270C2×C4—144no—
271C2×C4—564no—
272C2×C4—564no—
273C2×C4D7×C22⋊C412yes—
274C2×C4#644512yes—
275C2×C4#650011yescharacteristic
276C2×C4#650011yescharacteristic
277C8—42no—
278C8—81no—
279C8—562no—
280C7#845711yescharacteristic, complemented, Sylow
281C22—21no—
282C22—21no—
283C22—24no—
284C22—44no—
285C22—42no—
286C22—42no—
287C22—82no—
288C22#8972711yescharacteristic
289C22C2×#639812yes—
290C4—21no—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—284no—
295C4—284no—
296C4#8983311yescharacteristic
297C4#8984911yescharacteristic
298C2—22no—
299C2—44no—
300C2#1172741611yescharacteristic
301C2#1172774411yescharacteristic
302C2#1172776411yescharacteristic
subgroupG/H
centerC22#89727
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1955C2
socleC22×C7#379
prime radicalC22×C7×D4C2×C4
cosocleC2×C42×C7C23
hypercenter#1955D7
hypocenterC7#8457
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1955

Complements

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

NH ≅ G/Ntype
4. #11728193C2semidirect
5. #11727555C2semidirect
280. C7#8457semidirect

Maximal subgroups

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

subgroupindex
#117428342
C7×#19552
#117293572
#117281932
#117275552
#84577