Finite Group Databasebeta

← #13245942

Subgroups — #13245942

Order 1792.

746 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 746 conjugacy classes of subgroups — too many to draw. Only its 143 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×#3501C211yesmaximal, characteristic
2#11729585C212yesmaximal, complemented
3#11729576C212yesmaximal, complemented
4#11729369C211yesmaximal, characteristic
5#11729357C211yesmaximal, characteristic
6C7×#554—22no—
7C7×#554—24no—
8C7×D4×Q8—22no—
9C7×#544C412yes—
10C7×#543—22no—
11C7×#533C2212yes—
12C2×C7×C22⋊Q8—22no—
13C7×#518C412yes—
14C2×C4×C7×D4C2211yescharacteristic
15C2×C22⋊(C7⋊C8)—22no—
16#89698C2212yes—
17C2×C4⋊(C7⋊C8)C2211yescharacteristic
18C2×#6377C2211yescharacteristic
19#89342—22no—
20#89342—22no—
21#89338—24no—
22#89320—22no—
23#8457—71nomaximal, Sylow
24C22×C7×Q8—22no—
25C22×C7×D4C2×C411yescharacteristic
26C23×C4×C7D412yes—
27C7×C4⋊Q8—24no—
28C7×C4⋊Q8D412yes—
29C7×C4⋊Q8D412yes—
30C7×#125—24no—
31C7×#125D414yes—
32C7×C22⋊Q8—24no—
33C7×C22⋊Q8—24no—
34C7×C22⋊Q8—28no—
35C7×C22⋊Q8—28no—
36C7×C22⋊Q8—24no—
37C7×C22⋊Q8—22no—
38C7×C22⋊Q8—22no—
39C4×C7×Q8—22no—
40C4×C7×Q8—22no—
41C4×C7×D4—24no—
42C4×C7×D4—24no—
43C4×C7×D4—22no—
44C4×C7×D4—22no—
45C7×#117C2×C414yes—
46C2×C7×C4⋊C4—22no—
47C2×C7×C4⋊C4—24no—
48C2×C7×C4⋊C4C2×C411yescharacteristic
49C2×C7×C22⋊C4—24no—
50C2×C7×C22⋊C4D412yes—
51C2×C42×C7C2311yescharacteristic
52C22×C7⋊C8—21no—
53C22×C7⋊C8—21no—
54C22⋊(C7⋊C8)—22no—
55C22⋊(C7⋊C8)—44no—
56C4⋊(C7⋊C8)—22no—
57C4⋊(C7⋊C8)—22no—
58#6377—22no—
59#6377—22no—
60#3501D711yescharacteristic
61#1955—71no—
62#1552—72no—
63#1531—72no—
64#1466—71no—
65C24×C7C22⋊C412yes—
66C2×C7×Q8—24no—
67C2×C7×Q8—22no—
68C2×C7×Q8—22no—
69C2×C7×Q8—82no—
70C2×C7×Q8—82no—
71C2×C7×D4—24no—
72C2×C7×D4—22no—
73C22×C4×C7—24no—
74C22×C4×C7—24no—
75C22×C4×C7—22no—
76C22×C4×C7—22no—
77C22×C4×C7—42no—
78C22×C4×C7—42no—
79C22×C4×C7C22⋊C412yes—
80C22×C4×C7C22×C411yescharacteristic
81C22×C4×C7C2×D411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C7×C4⋊C4—24no—
84C7×C4⋊C4—28no—
85C7×C4⋊C4—22no—
86C7×C4⋊C4—24no—
87C7×C4⋊C4—22no—
88C7×C4⋊C4—24no—
89C7×C4⋊C4—22no—
90C7×C4⋊C4C22⋊C414yes—
91C7×C4⋊C4C22⋊C414yes—
92C7×C22⋊C4—28no—
93C7×C22⋊C4—24no—
94C7×C22⋊C4—24no—
95C42×C7—24no—
96C42×C7C2×D412yes—
97C42×C7C2×D412yes—
98C2×C7⋊C8—22no—
99C2×C7⋊C8—41no—
100C2×C7⋊C8—41no—
101C2×C7⋊C8—42no—
102C2×C7⋊C8—42no—
103C2×C7⋊C8—41no—
104#554—22no—
105#554—24no—
106D4×Q8—22no—
107#544Dic712yes—
108#543—22no—
109#533C2×D712yes—
110C2×C22⋊Q8—22no—
111#518Dic712yes—
112C2×C4×D4C2×D711yescharacteristic
113#432—72no—
114C2×C4⋊C8—71no—
115C2×C22⋊C8—142no—
116C2×#97—71no—
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×Q8—42no—
128C7×Q8—42no—
129C7×Q8—82no—
130C7×Q8—84no—
131C7×D4—28no—
132C2×C4×C7—22no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—24no—
136C2×C4×C7—24no—
137C2×C4×C7—22no—
138C2×C4×C7—41no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—82no—
143C2×C4×C7—84no—
144C2×C4×C7—84no—
145C2×C4×C7C2×C22⋊C412yes—
146C2×C4×C7#12012yes—
147C2×C4×C7#12011yescharacteristic
148C2×C4×C7#12011yescharacteristic
149C7⋊C8—42no—
150C7⋊C8—82no—
151C7⋊C8—81no—
152C22×Q8—22no—
153C22×D4C2×Dic711yescharacteristic
154C23×C4C22⋊D712yes—
155C22×C8—141no—
156C22×C8—141no—
157C4⋊Q8—24no—
158C4⋊Q8C22⋊D712yes—
159C4⋊Q8C22⋊D712yes—
160#125—24no—
161#125C22⋊D714yes—
162C22⋊Q8—24no—
163C22⋊Q8—24no—
164C22⋊Q8—28no—
165C22⋊Q8—28no—
166C22⋊Q8—24no—
167C22⋊Q8—22no—
168C22⋊Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×D4—24no—
172C4×D4—24no—
173C4×D4—22no—
174C4×D4—22no—
175#117C2×Dic714yes—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4C2×Dic711yescharacteristic
179C2×C22⋊C4—24no—
180C2×C22⋊C4C22⋊D712yes—
181C2×C42C22×D711yescharacteristic
182C4⋊C8—142no—
183C4⋊C8—142no—
184C22⋊C8—142no—
185C22⋊C8—284no—
186#97—142no—
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—44no—
202C4×C7—44no—
203C4×C7#45311yescharacteristic
204C4×C7#45511yescharacteristic
205C24C22⋊Dic712yes—
206C2×Q8—24no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—82no—
210C2×Q8—82no—
211C2×D4—24no—
212C2×D4—22no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4C22⋊Dic712yes—
220C22×C4C22×Dic711yescharacteristic
221C22×C4C2×C22⋊D711yescharacteristic
222C22×C4C2×C22⋊D711yescharacteristic
223C2×C8—142no—
224C2×C8—281no—
225C2×C8—281no—
226C2×C8—282no—
227C2×C8—282no—
228C2×C8—281no—
229C4⋊C4—24no—
230C4⋊C4—28no—
231C4⋊C4—22no—
232C4⋊C4—24no—
233C4⋊C4—22no—
234C4⋊C4—24no—
235C4⋊C4—22no—
236C4⋊C4C22⋊Dic714yes—
237C4⋊C4C22⋊Dic714yes—
238C22⋊C4—28no—
239C22⋊C4—24no—
240C22⋊C4—24no—
241C42—24no—
242C42C2×C22⋊D712yes—
243C42C2×C22⋊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#641014yes—
254C23C2×C22⋊Dic711yescharacteristic
255Q8—42no—
256Q8—42no—
257Q8—82no—
258Q8—84no—
259D4—28no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—82no—
271C2×C4—84no—
272C2×C4—84no—
273C2×C4C2×C22⋊Dic712yes—
274C2×C4#651612yes—
275C2×C4#651611yescharacteristic
276C2×C4#651611yescharacteristic
277C8—282no—
278C8—562no—
279C8—561no—
280C7#845711yescharacteristic, complemented, Sylow
281C22—21no—
282C22—21no—
283C22—24no—
284C22—44no—
285C22—42no—
286C22—42no—
287C22—82no—
288C22C2×#641012yes—
289C22#9001911yescharacteristic
290C4—22no—
291C4—21no—
292C4—22no—
293C4—42no—
294C4—44no—
295C4—44no—
296C4#9001311yescharacteristic
297C4#9001411yescharacteristic
298C2—22no—
299C2—44no—
300C2#1172988711yescharacteristic
301C2#1172989411yescharacteristic
302C2#1172990011yescharacteristic
subgroupG/H
centerC22#90019
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#3501C2
socleC22×C7#379
prime radicalC22×C7×D4C2×C4
cosocleC2×C42×C7C23
hypercenter#3501D7
hypocenterC7#8457
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
2. #11729585C2semidirect
3. #11729576C2semidirect
280. C7#8457semidirect

Maximal subgroups

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

subgroupindex
C7×#35012
#117295852
#117295762
#117293692
#117293572
#84577