Finite Group Databasebeta

← #13264945

Subgroups — #13264945

Order 1792.

568 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2409C211yesmaximal, characteristic
2#11729371C211yesmaximal, characteristic
3#11729354C211yesmaximal, characteristic
4#11725454C214yesmaximal, complemented
5C23×C7×Q8C411yescharacteristic
6C2×C7×C22⋊Q8—21no—
7C2×C4×C7×D4C2211yescharacteristic
8C7×#395C2212yes—
9C7×#393C412yes—
10C7×#391—21no—
11C7×#386—21no—
12C7×#385—21no—
13C7×#380C411yescharacteristic
14C2×#6407—21no—
15C2×C22⋊(C7⋊C8)—21no—
16#89696—21no—
17#89325C2212yes—
18#89323C2212yes—
19#89320—21no—
20#11904—71nomaximal, Sylow
21C22×C7×Q8—21no—
22C22×C7×Q8—41no—
23C22×C7×Q8—41no—
24C22×C7×Q8—44no—
25C22×C7×Q8C2×C412yes—
26C22×C7×D4D411yescharacteristic
27C23×C4×C7—21no—
28C23×C4×C7C2×C411yescharacteristic
29C23×C4×C7D411yescharacteristic
30C7×C22⋊Q8—44no—
31C4×C7×D4—41no—
32C4×C7×D4—41no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—22no—
35C2×C7×C4⋊C4C2311yescharacteristic
36C2×C7×C22⋊C4—21no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4C2×C411yescharacteristic
39C2×C7×C22⋊C4D411yescharacteristic
40C2×C42×C7D411yescharacteristic
41C2×#1136—22no—
42C22×C7⋊C8—41no—
43C7×#95—22no—
44C7×#95—22no—
45C7×#95C2×C412yes—
46#6407—42no—
47C22⋊(C7⋊C8)—22no—
48C22⋊(C7⋊C8)—41no—
49C4⋊(C7⋊C8)—41no—
50C4⋊(C7⋊C8)—41no—
51#2409D711yescharacteristic
52#1800—71no—
53#1459—71no—
54#1354—74no—
55C24×C7C22⋊C411yescharacteristic
56C24×C7C22⋊C411yescharacteristic
57C2×C7×Q8—24no—
58C2×C7×Q8—42no—
59C2×C7×Q8—44no—
60C2×C7×Q8—42no—
61C2×C7×Q8—48no—
62C2×C7×Q8—44no—
63C2×C7×Q8—44no—
64C2×C7×Q8—41no—
65C2×C7×Q8—41no—
66C2×C7×D4—41no—
67C2×C7×D4—41no—
68C2×C7×D4SD1612yes—
69C2×C7×D4Q1612yes—
70C22×C4×C7—22no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—22no—
76C22×C4×C7—44no—
77C22×C4×C7—42no—
78C22×C4×C7—42no—
79C22×C4×C7—42no—
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22⋊C411yescharacteristic
82C22×C4×C7C22×C411yescharacteristic
83C22×C4×C7C2×D411yescharacteristic
84C22×C4×C7C2×D411yescharacteristic
85C7×C4⋊C4—41no—
86C7×C4⋊C4—42no—
87C7×C4⋊C4—44no—
88C7×C22⋊C4—41no—
89C7×C22⋊C4—41no—
90C7×C22⋊C4—44no—
91C7×C22⋊C4—42no—
92C42×C7—21no—
93C42×C7—21no—
94#1136—42no—
95#1136—42no—
96C2×C7⋊C8—42no—
97C2×C7⋊C8—42no—
98C2×C7⋊C8—82no—
99C23×Q8Dic711yescharacteristic
100C2×C22⋊Q8—21no—
101C2×C4×D4C2×D711yescharacteristic
102#423—141no—
103C2×#100—141no—
104C2×C22⋊C8—141no—
105#395C2×D712yes—
106#393Dic712yes—
107#391—21no—
108#386—21no—
109#385—21no—
110#380Dic711yescharacteristic
111#343—72no—
112#324—72no—
113#323—141no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—41no—
119C23×C7—41no—
120C23×C7—41no—
121C23×C7—41no—
122C23×C7#10312yes—
123C23×C7#10312yes—
124C23×C7C2×C22⋊C411yescharacteristic
125C7×Q8—44no—
126C7×Q8—44no—
127C7×Q8—44no—
128C7×Q8—44no—
129C7×D4—42no—
130C7×D4—42no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—22no—
134C2×C4×C7—24no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—42no—
139C2×C4×C7—42no—
140C2×C4×C7—48no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7—42no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—41no—
147C2×C4×C7—42no—
148C2×C4×C7—42no—
149C2×C4×C7—42no—
150C2×C4×C7—81no—
151C2×C4×C7#9911yescharacteristic
152C2×C4×C7#9911yescharacteristic
153C2×C4×C7C2×SD1611yescharacteristic
154C2×C4×C7C2×Q1611yescharacteristic
155C7⋊C8—82no—
156C7⋊C8—82no—
157C22×Q8—21no—
158C22×Q8—41no—
159C22×Q8—41no—
160C22×Q8—44no—
161C22×Q8C2×Dic712yes—
162C22×D4C22⋊D711yescharacteristic
163C23×C4—21no—
164C23×C4C2×Dic711yescharacteristic
165C23×C4C22⋊D711yescharacteristic
166C2×M16—142no—
167C22×C8—281no—
168C22⋊Q8—44no—
169C4×D4—41no—
170C4×D4—41no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4C22×D711yescharacteristic
174C2×C22⋊C4—22no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4C2×Dic711yescharacteristic
177C2×C22⋊C4C22⋊D711yescharacteristic
178C2×C42C22⋊D711yescharacteristic
179C4⋊C8—281no—
180C4⋊C8—281no—
181#100—282no—
182C22⋊C8—142no—
183C22⋊C8—281no—
184#95—22no—
185#95—22no—
186#95C2×Dic712yes—
187C22×C7—21no—
188C22×C7—21no—
189C22×C7—22no—
190C22×C7—22no—
191C22×C7—42no—
192C22×C7—42no—
193C22×C7—42no—
194C22×C7—41no—
195C22×C7—41no—
196C22×C7—41no—
197C22×C7—41no—
198C22×C7—81no—
199C22×C7C2×#9911yescharacteristic
200C22×C7C2×#10311yescharacteristic
201C22×C7#42111yescharacteristic
202C4×C7—21no—
203C4×C7—21no—
204C4×C7—21no—
205C4×C7—41no—
206C4×C7—44no—
207C4×C7—41no—
208C4×C7—41no—
209C4×C7—82no—
210C4×C7AΣL(1,𝔽16)12yes—
211C24C22⋊Dic711yescharacteristic
212C24C22⋊Dic711yescharacteristic
213C2×Q8—24no—
214C2×Q8—44no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—44no—
218C2×Q8—48no—
219C2×Q8—44no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4#114312yes—
225C2×D4#114412yes—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4C22⋊Dic711yescharacteristic
237C22×C4C22⋊Dic711yescharacteristic
238C22×C4C22×Dic711yescharacteristic
239C22×C4C2×C22⋊D711yescharacteristic
240C22×C4C2×C22⋊D711yescharacteristic
241M16—282no—
242M16—282no—
243C2×C8—282no—
244C2×C8—282no—
245C2×C8—562no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—44no—
252C22⋊C4—42no—
253C42—21no—
254C42—21no—
255C2×C7—21no—
256C2×C7—21no—
257C2×C7—42no—
258C2×C7—42no—
259C2×C7#152011yescharacteristic
260C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
261C2×C7#213511yescharacteristic
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23#640912yes—
271C23#640912yes—
272C23C2×C22⋊Dic711yescharacteristic
273Q8—44no—
274Q8—44no—
275Q8—44no—
276Q8—44no—
277D4—42no—
278D4—42no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—48no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—81no—
299C2×C4#640811yescharacteristic
300C2×C4#640811yescharacteristic
301C2×C4C2×#114311yescharacteristic
302C2×C4C2×#114411yescharacteristic
303C8—562no—
304C8—562no—
305C7#1190411yescharacteristic, complemented, Sylow
306C22—21no—
307C22—21no—
308C22—22no—
309C22—22no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22C2×#640811yescharacteristic
319C22C2×#640911yescharacteristic
320C22#9000811yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—44no—
326C4—41no—
327C4—41no—
328C4—82no—
329C4#8933112yes—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—42no—
334C2C2×#8933111yescharacteristic
335C2#1172944411yescharacteristic
336C2#1172962711yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2409C2
socleC22×C7C2×#99
prime radicalC22×C7×D4D4
cosocleC2×C7×C4⋊C4C23
hypercenter#2409D7
hypocenterC7#11904
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
4. #11725454C2semidirect
305. C7#11904semidirect

Maximal subgroups

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

subgroupindex
C7×#24092
#117293712
#117293542
#117254542
#119047