Finite Group Databasebeta

← #13249542

Subgroups — #13249542

Order 1792.

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

normal subgroups only

This group has 756 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×#2854C211yesmaximal, characteristic
2C2×#89704C211yesmaximal, characteristic, complemented
3C2×#89332C212yesmaximal
4#11729285C211yesmaximal, characteristic, complemented
5#11725430C212yesmaximal, complemented
6C2×C7×#123—21no—
7C2×C7×#121C2211yescharacteristic
8C22×C42×C7C2211yescharacteristic
9C22×C4×Dic7—21no—
10C7×#390C412yescomplemented
11C7×#389C412yescomplemented
12C7×#388—22no—
13C7×#386—21no—
14C7×#380C2211yescharacteristic
15C2×C7⋊#95—21no—
16C2×C7⋊#95C2212yes—
17C2×C22⋊(C7⋊C8)C2212yes—
18#89704—24no—
19#89332—24no—
20#89319—22no—
21#9149—71nomaximal, Sylow
22C22×C7×D4—21no—
23C22×C7×D4C2×C411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7C2311yescharacteristic
27C23×Dic7—22no—
28C7×#123—44no—
29C7×#121—22no—
30C7×#121—22no—
31C2×C7×C4⋊C4—22no—
32C2×C7×C4⋊C4C2×C411yescharacteristic
33C2×C7×C4⋊C4C2×C412yes—
34C2×C7×C22⋊C4—21no—
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—21no—
38C2×C42×C7—22no—
39C2×C42×C7—22no—
40C2×C42×C7C2×C412yes—
41C2×C42×C7D412yes—
42C2×C4×Dic7—24no—
43C2×C4×Dic7—28no—
44C22×C7⋊C8—22no—
45C7×#95—22no—
46C7×#95—21no—
47C7⋊#95—24no—
48C7⋊#95—22no—
49C7⋊#95D414yes—
50C22⋊(C7⋊C8)—22no—
51C22⋊(C7⋊C8)—22no—
52#2854D711yescharacteristic
53C2×#376—71no—
54#1817—71no—
55C2×#327—72no—
56#1294—72no—
57C24×C7—21no—
58C24×C7C2×D411yescharacteristic
59C2×C7×D4—22no—
60C2×C7×D4—42no—
61C2×C7×D4—41no—
62C2×C7×D4—41no—
63C2×C7×D4—42no—
64C2×C7×D4—42no—
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—24no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—22no—
71C22×C4×C7—22no—
72C22×C4×C7—22no—
73C22×C4×C7—21no—
74C22×C4×C7—42no—
75C22×C4×C7C22⋊C412yes—
76C22×C4×C7C22⋊C412yes—
77C22×C4×C7C22×C411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C22×C4×C7C2×D412yes—
80C22×C4×C7#4112yes—
81C22×Dic7—28no—
82C22×Dic7—24no—
83C22×Dic7—44no—
84C22×Dic7—42no—
85C22×Dic7—42no—
86C7×C4⋊C4—22no—
87C7×C4⋊C4—42no—
88C7×C4⋊C4—44no—
89C7×C22⋊C4—44no—
90C7×C22⋊C4—42no—
91C7×C22⋊C4—44no—
92C7×C22⋊C4—42no—
93C42×C7—24no—
94C42×C7—24no—
95C4×Dic7—216no—
96C2×C7⋊C8—44no—
97C2×C7⋊C8—42no—
98C2×#123—21no—
99C2×#121C2×D711yescharacteristic
100C22×C42—141no—
101C22×C42C2×D711yescharacteristic
102C2×C22⋊C8—72no—
103#390Dic712yescomplemented
104#389Dic712yescomplemented
105#388—22no—
106#386—21no—
107#380C2×D711yescharacteristic
108#376—144no—
109C2×#95—72no—
110C2×#95—141no—
111#336—142no—
112#327—144no—
113C23×C7—22no—
114C23×C7—22no—
115C23×C7—41no—
116C23×C7—42no—
117C23×C7—41no—
118C23×C7—41no—
119C23×C7—42no—
120C23×C7C2×C22⋊C411yescharacteristic
121C23×C7#12011yescharacteristic
122C23×C7#12311yescharacteristic
123C23×C7#13512yes—
124C23×C7AGL(1,ℤ/8)12yes—
125C7×D4—82no—
126C7×D4—82no—
127C7×D4—82no—
128C2×C4×C7—24no—
129C2×C4×C7—24no—
130C2×C4×C7—22no—
131C2×C4×C7—22no—
132C2×C4×C7—22no—
133C2×C4×C7—22no—
134C2×C4×C7—22no—
135C2×C4×C7—22no—
136C2×C4×C7—22no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7—41no—
142C2×C4×C7—42no—
143C2×C4×C7—41no—
144C2×C4×C7—42no—
145C2×C4×C7—41no—
146C2×C4×C7—42no—
147C2×C4×C7#10414yes—
148C2×C4×C7C4×D412yes—
149C2×C4×C7#12112yes—
150C2×Dic7—28no—
151C2×Dic7—44no—
152C2×Dic7—48no—
153C2×Dic7—48no—
154C2×Dic7—44no—
155C7⋊C8—82no—
156C22×D4—21no—
157C22×D4C2×Dic711yescharacteristic
158C23×C4—142no—
159C23×C4C22⋊D711yescharacteristic
160C23×C4C22⋊D711yescharacteristic
161C23×C4C22×D711yescharacteristic
162C22×C8—142no—
163#123—44no—
164#121—22no—
165#121—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4C2×Dic711yescharacteristic
168C2×C4⋊C4C2×Dic712yes—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—21no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—144no—
176C2×C42—148no—
177C2×C42C2×Dic712yes—
178C2×C42C22⋊D712yes—
179C22⋊C8—142no—
180C22⋊C8—142no—
181#95—22no—
182#95—21no—
183#95—74no—
184#95—144no—
185#95—142no—
186C22×C7—21no—
187C22×C7—21no—
188C22×C7—21no—
189C22×C7—21no—
190C22×C7—24no—
191C22×C7—24no—
192C22×C7—81no—
193C22×C7—81no—
194C22×C7—81no—
195C22×C7—82no—
196C22×C7—82no—
197C22×C7#38611yescharacteristic
198C22×C7#41012yes—
199C22×C7#41611yescharacteristic
200C22×C7#41811yescharacteristic
201C22×C7C2×#10412yes—
202C22×C7#44912yes—
203C22×C7#48212yes—
204C4×C7—24no—
205C4×C7—22no—
206C4×C7—22no—
207C4×C7—42no—
208C4×C7—81no—
209C4×C7—82no—
210Dic7—48no—
211C24—21no—
212C24C2×C22⋊D711yescharacteristic
213C2×D4—22no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—42no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—24no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—42no—
229C22×C4—148no—
230C22×C4—144no—
231C22×C4—284no—
232C22×C4—282no—
233C22×C4—282no—
234C22×C4C22⋊Dic712yes—
235C22×C4C22⋊Dic712yes—
236C22×C4D4×D712yes—
237C22×C4#115912yes—
238C22×C4C22×Dic711yescharacteristic
239C22×C4C2×C22⋊D711yescharacteristic
240C2×C8—284no—
241C2×C8—282no—
242C4⋊C4—22no—
243C4⋊C4—42no—
244C4⋊C4—44no—
245C22⋊C4—44no—
246C22⋊C4—42no—
247C22⋊C4—44no—
248C22⋊C4—42no—
249C42—24no—
250C42—24no—
251C42—1416no—
252C2×C7—22no—
253C2×C7—22no—
254C2×C7—161no—
255C2×C7#150012yes—
256C2×C7#150512yes—
257C2×C7#187511yescharacteristic
258C2×C7#189611yescharacteristic
259C2×C7#189911yescharacteristic
260C23—22no—
261C23—22no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—42no—
266C23—42no—
267C23#649512yes—
268C23#649811yescharacteristic
269C23#650011yescharacteristic
270C23#651312yes—
271C23C2×C22⋊Dic711yescharacteristic
272D4—82no—
273D4—82no—
274D4—82no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—148no—
295C2×C4—284no—
296C2×C4—288no—
297C2×C4—288no—
298C2×C4—284no—
299C2×C4#641114yes—
300C2×C4D4×Dic712yes—
301C2×C4#650212yes—
302C8—562no—
303C7#914911yescharacteristic, complemented, Sylow
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—24no—
309C22—24no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22—82no—
314C22—82no—
315C22#8980712yes—
316C22#8980912yes—
317C22#8998711yescharacteristic
318C22#8999211yescharacteristic
319C22#9000511yescharacteristic
320C22C2×#641112yes—
321C22#9001212yes—
322C4—22no—
323C4—22no—
324C4—24no—
325C4—42no—
326C4—81no—
327C4—82no—
328C4—288no—
329C2—22no—
330C2—22no—
331C2—161no—
332C2#1172943211yescharacteristic
333C2#1172945911yescharacteristic
334C2#1172946511yescharacteristic
335C2#1172950812yes—
336C2#1172956612yes—
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2854C2
socleC23×C7C2×C22⋊C4
prime radicalC2×C7×#121C22
cosocleC23×C4×C7C23
hypercenter#2854D7
hypocenterC7#9149
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
2. C2×#89704C2semidirect
4. #11729285C2semidirect
5. #11725430C2semidirect
10. C7×#390C4semidirect
11. C7×#389C4semidirect
103. #390Dic7semidirect
104. #389Dic7semidirect
303. C7#9149semidirect

Maximal subgroups

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

subgroupindex
C7×#28542
C2×#897042
C2×#893322
#117292852
#117254302
#91497