Finite Group Databasebeta

← #13249579

Subgroups — #13249579

Order 1792.

648 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 127 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#11739525C211yesmaximal, characteristic
2C2×#89709C211yesmaximal, characteristic, complemented
3C7×#1817C211yesmaximal, characteristic, complemented
4C2×#89265C212yesmaximal
5#11725481C212yesmaximal, complemented
6C22×C42×C7C2211yescharacteristic
7C2×#6493C2211yescharacteristic
8C2×#6492—21no—
9C2×C7×C22⋊C8C2212yes—
10#89985—21no—
11#89984C2211yescharacteristic
12C2×C7⋊#95—21no—
13C2×C7⋊#95—21no—
14C2×C7⋊#95C2212yes—
15#89712—22no—
16#89711C412yescomplemented
17#89709—24no—
18#89700C412yescomplemented
19C7×#336—22no—
20#89265—24no—
21#9157—71nomaximal, Sylow
22C23×C4×C7D411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7C2311yescharacteristic
25C22×C22⋊D7—21no—
26C23×Dic7—22no—
27C22×D28C2×C411yescharacteristic
28C22×C7×C8—22no—
29C2×C42×C7—22no—
30C2×C42×C7—22no—
31C2×C42×C7C2×C412yes—
32C2×C42×C7D412yes—
33C2×C22⋊Dic7—21no—
34#6493—22no—
35#6493—22no—
36#6492—44no—
37C2×(C2×D7)⋊C4—22no—
38C2×(C2×D7)⋊C4—22no—
39C2×(C2×D7)⋊C4—21no—
40C2×C4⋊Dic7C2×C411yescharacteristic
41C2×C4⋊Dic7C2×C412yes—
42C2×Dic7⋊C4—22no—
43C7×C22⋊C8—22no—
44C7×C22⋊C8—22no—
45C7⋊#95—24no—
46C7⋊#95—24no—
47C7⋊#95—22no—
48C7⋊#95—22no—
49C7⋊#95—21no—
50C7⋊#95D414yes—
51#2854—71no—
52C2×#383—71no—
53#1817D711yescharacteristic, complemented
54C2×#327—72no—
55#1294—72no—
56C24×C7C2×D411yescharacteristic
57C23×D7—21no—
58C22×C4×C7—21no—
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—22no—
63C22×C4×C7—22no—
64C22×C4×C7—42no—
65C22×C4×C7—42no—
66C22×C4×C7C22⋊C412yes—
67C22×C4×C7C22⋊C412yes—
68C22×C4×C7C22×C411yescharacteristic
69C22×C4×C7C2×D411yescharacteristic
70C22×C4×C7C2×D412yes—
71C22×C4×C7#4112yes—
72C2×C22⋊D7—41no—
73C2×C22⋊D7—41no—
74C2×C22⋊D7—42no—
75C2×C22⋊D7—42no—
76C22×Dic7—21no—
77C22×Dic7—22no—
78C22×Dic7—24no—
79C22×Dic7—44no—
80C22×Dic7—44no—
81C22×Dic7—42no—
82C22×Dic7—42no—
83C2×D28—22no—
84C2×D28—42no—
85C2×C7×C8—44no—
86C2×C7×C8—42no—
87C42×C7—24no—
88C42×C7—24no—
89C22⋊Dic7—42no—
90(C2×D7)⋊C4—44no—
91(C2×D7)⋊C4—44no—
92(C2×D7)⋊C4—42no—
93C4⋊Dic7—22no—
94C4⋊Dic7—42no—
95Dic7⋊C4—44no—
96C2×#123—141no—
97C2×#121—71no—
98C22×C42C2×D711yescharacteristic
99C2×C22⋊C8C2×D712yes—
100#390—72no—
101#389—72no—
102#388—142no—
103#386—141no—
104#383—144no—
105#380—71no—
106C2×#95—72no—
107C2×#95—141no—
108C2×#95—141no—
109#336—22no—
110#327—144no—
111C23×C7—22no—
112C23×C7—22no—
113C23×C7C2×C22⋊C411yescharacteristic
114C23×C7#12011yescharacteristic
115C23×C7#12311yescharacteristic
116C23×C7#13512yes—
117C23×C7AGL(1,ℤ/8)12yes—
118C22×D7—41no—
119C22×D7—42no—
120C22×D7—41no—
121C22×D7—41no—
122C22×D7—42no—
123C2×C4×C7—22no—
124C2×C4×C7—22no—
125C2×C4×C7—22no—
126C2×C4×C7—22no—
127C2×C4×C7—22no—
128C2×C4×C7—22no—
129C2×C4×C7—22no—
130C2×C4×C7—22no—
131C2×C4×C7—42no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7C4×D412yes—
138C2×C4×C7#12112yes—
139C22⋊D7—82no—
140C22⋊D7—82no—
141C2×Dic7—44no—
142C2×Dic7—44no—
143C2×Dic7—41no—
144C2×Dic7—42no—
145C2×Dic7—41no—
146C2×Dic7—41no—
147C2×Dic7—42no—
148C2×Dic7—42no—
149C2×Dic7—44no—
150C2×Dic7—84no—
151C2×Dic7—84no—
152D28—82no—
153C7×C8—82no—
154C22×D4—71no—
155C22×D4—141no—
156C23×C4—142no—
157C23×C4D2811yescharacteristic
158C23×C4D2811yescharacteristic
159C23×C4C22×D711yescharacteristic
160C22×C8—22no—
161#123—284no—
162#121—142no—
163#121—142no—
164C2×C4⋊C4—71no—
165C2×C4⋊C4—72no—
166C2×C4⋊C4—142no—
167C2×C22⋊C4—141no—
168C2×C22⋊C4—142no—
169C2×C22⋊C4—142no—
170C2×C22⋊C4—141no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42C4×D712yes—
174C2×C42C22⋊D712yes—
175C22⋊C8—22no—
176C22⋊C8—22no—
177#95—74no—
178#95—144no—
179#95—144no—
180#95—142no—
181#95—142no—
182#95—141no—
183C22×C7—21no—
184C22×C7—21no—
185C22×C7—21no—
186C22×C7—21no—
187C22×C7—24no—
188C22×C7—24no—
189C22×C7#38611yescharacteristic
190C22×C7#41012yes—
191C22×C7#41611yescharacteristic
192C22×C7#41811yescharacteristic
193C22×C7#42112yes—
194C22×C7#44912yes—
195C22×C7#48212yes—
196C2×D7—81no—
197C2×D7—81no—
198C2×D7—81no—
199C2×D7—82no—
200C2×D7—82no—
201C4×C7—22no—
202C4×C7—22no—
203C4×C7—42no—
204C4×C7—42no—
205Dic7—84no—
206Dic7—81no—
207Dic7—82no—
208C24—141no—
209C24C2×D2811yescharacteristic
210C2×D4—142no—
211C2×D4—282no—
212C2×D4—281no—
213C2×D4—281no—
214C2×D4—282no—
215C2×D4—282no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—144no—
225C22×C4—142no—
226C22×C4—141no—
227C22×C4—284no—
228C22×C4—284no—
229C22×C4—282no—
230C22×C4—282no—
231C22×C4(C2×D7)⋊C412yes—
232C22×C4(C2×D7)⋊C412yes—
233C22×C4C2×C4×D711yescharacteristic
234C22×C4D4×D712yes—
235C22×C4#115912yes—
236C22×C4C2×C22⋊D711yescharacteristic
237C2×C8—44no—
238C2×C8—42no—
239C4⋊C4—142no—
240C4⋊C4—282no—
241C4⋊C4—284no—
242C22⋊C4—284no—
243C22⋊C4—284no—
244C22⋊C4—282no—
245C22⋊C4—282no—
246C42—24no—
247C42—24no—
248C2×C7—22no—
249C2×C7—22no—
250C2×C7#151412yes—
251C2×C7#152112yes—
252C2×C7#187511yescharacteristic
253C2×C7#189711yescharacteristic
254C2×C7#189911yescharacteristic
255D7—161no—
256C23—22no—
257C23—22no—
258C23—281no—
259C23—281no—
260C23—282no—
261C23—281no—
262C23—282no—
263C23#644511yescharacteristic
264C23#644911yescharacteristic
265C23#646712yes—
266C23#647112yes—
267C23C2×(C2×D7)⋊C411yescharacteristic
268D4—562no—
269D4—562no—
270D4—562no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—284no—
286C2×C4—284no—
287C2×C4—281no—
288C2×C4—282no—
289C2×C4—281no—
290C2×C4—282no—
291C2×C4—281no—
292C2×C4—284no—
293C2×C4—282no—
294C2×C4—564no—
295C2×C4—564no—
296C2×C4(C22⋊D7)⋊C412yes—
297C2×C4#650212yes—
298C8—82no—
299C7#915711yescharacteristic, complemented, Sylow
300C22—21no—
301C22—21no—
302C22—24no—
303C22—21no—
304C22—21no—
305C22—24no—
306C22—561no—
307C22—561no—
308C22—562no—
309C22—561no—
310C22—562no—
311C22#8950612yes—
312C22#8950912yes—
313C22#8973011yescharacteristic
314C22#8977812yes—
315C22#8988511yescharacteristic
316C22#8990311yescharacteristic
317C22#8991112yes—
318C4—22no—
319C4—22no—
320C4—42no—
321C4—42no—
322C4—564no—
323C4—561no—
324C4—562no—
325C2—22no—
326C2—22no—
327C2—1121no—
328C2#1172728911yescharacteristic
329C2#1172768711yescharacteristic
330C2#1172772011yescharacteristic
331C2#1172854512yes—
332C2#1172858912yes—
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1817C2
socleC23×C7C2×C22⋊C4
prime radicalC2×#6493C22
cosocleC23×C4×C7C23
hypercenter#1817D7
hypocenterC7#9157
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. C2×#89709C2semidirect
3. C7×#1817C2semidirect
5. #11725481C2semidirect
16. #89711C4semidirect
18. #89700C4semidirect
53. #1817D7semidirect
299. C7#9157semidirect

Maximal subgroups

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

subgroupindex
#117395252
C2×#897092
C7×#18172
C2×#892652
#117254812
#91577