Finite Group Databasebeta

← #13247203

Subgroups — #13247203

Order 1792.

482 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 115 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#11739505C211yesmaximal, characteristic
2C7×#1749C211yesmaximal, characteristic
3#11729283C211yesmaximal, characteristic
4#11725465C212yesmaximal, complemented
5#11725465C212yesmaximal, complemented
6C22×C42×C7C2211yescharacteristic
7C2×C7×C22⋊C8—21no—
8C2×C7×#97C2212yes—
9#89984C411yescharacteristic
10#89984C411yescharacteristic
11#89983—21no—
12C4×C22⋊Dic7—21no—
13C2×C22⋊(C7⋊C8)—21no—
14#89709—21no—
15#89708C2211yescharacteristic
16#89708C2211yescharacteristic
17C4×C4⋊Dic7C411yescharacteristic
18C4×C4⋊Dic7C411yescharacteristic
19#89704—21no—
20C2×#6377C2212yes—
21C7×#336—21no—
22#89319—21no—
23#8720—71nomaximal, Sylow
24C23×C4×C7C2×C411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C22×C7×C8—22no—
28C2×C42×C7—24no—
29C2×C42×C7C2×C411yescharacteristic
30C2×C42×C7D411yescharacteristic
31C2×C42×C7D411yescharacteristic
32C2×C42×C7C2311yescharacteristic
33C2×C22⋊Dic7—21no—
34C2×C22⋊Dic7—21no—
35C2×C4⋊Dic7C2×C411yescharacteristic
36C2×C4⋊Dic7C2×C411yescharacteristic
37C2×C4⋊Dic7C2×C411yescharacteristic
38C2×C4⋊Dic7C2×C411yescharacteristic
39C2×C4×Dic7—21no—
40C2×C4×Dic7—21no—
41C22×C7⋊C8—22no—
42C7×C22⋊C8—24no—
43C7×#97—24no—
44C7⋊#95—21no—
45C7⋊#95—21no—
46C7⋊#95—21no—
47C7⋊#95—21no—
48C7⋊#95—21no—
49C7⋊#95—21no—
50C7⋊#95—21no—
51C7⋊#95—21no—
52C22⋊(C7⋊C8)—24no—
53#6377—24no—
54#2283—71no—
55#1749—71no—
56#1749D711yescharacteristic
57#1295—72no—
58#1295—72no—
59C24×C7C22⋊C411yescharacteristic
60C22×C4×C7—22no—
61C22×C4×C7—22no—
62C22×C4×C7—22no—
63C22×C4×C7—22no—
64C22×C4×C7—22no—
65C22×C4×C7—41no—
66C22×C4×C7—41no—
67C22×C4×C7—41no—
68C22×C4×C7—41no—
69C22×C4×C7C22⋊C411yescharacteristic
70C22×C4×C7C22⋊C411yescharacteristic
71C22×C4×C7C22⋊C411yescharacteristic
72C22×C4×C7C22×C411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C22×Dic7—21no—
76C22×Dic7—21no—
77C22×Dic7—21no—
78C22×Dic7—21no—
79C2×C7×C8—24no—
80C2×C7×C8—42no—
81C2×C7×C8—42no—
82C42×C7—22no—
83C42×C7—22no—
84C42×C7—22no—
85C42×C7#4114yes—
86C22⋊Dic7—42no—
87C22⋊Dic7—42no—
88C4⋊Dic7—41no—
89C4⋊Dic7—41no—
90C4⋊Dic7—41no—
91C4⋊Dic7—41no—
92C4×Dic7—42no—
93C4×Dic7—42no—
94C2×C7⋊C8—24no—
95C2×C7⋊C8—42no—
96C2×C7⋊C8—42no—
97C22×C42C2×D711yescharacteristic
98C2×C22⋊C8—21no—
99C2×C22⋊C8—141no—
100C2×#97—72no—
101C2×#97C2×D712yes—
102#383—141no—
103#382—71no—
104#382—71no—
105#381—141no—
106#380—71no—
107#380—71no—
108C4×C4⋊C4—71no—
109C4×C4⋊C4—71no—
110C4×C22⋊C4—141no—
111#376—141no—
112#336—21no—
113#336—141no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7#10412yes—
121C23×C7C2×C22⋊C411yescharacteristic
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—44no—
137C2×C4×C7—44no—
138C2×C4×C7—44no—
139C2×C4×C7#10412yes—
140C2×C4×C7#11712yes—
141C2×C4×C7#12312yes—
142C2×C4×C7#12312yes—
143C2×Dic7—41no—
144C2×Dic7—41no—
145C2×Dic7—41no—
146C2×Dic7—41no—
147C2×Dic7—41no—
148C2×Dic7—41no—
149C2×Dic7—41no—
150C2×Dic7—41no—
151C2×Dic7—41no—
152C2×Dic7—41no—
153C2×Dic7—41no—
154C2×Dic7—41no—
155C7×C8—44no—
156C7⋊C8—44no—
157C23×C4C4×D711yescharacteristic
158C23×C4D2811yescharacteristic
159C23×C4C22⋊D711yescharacteristic
160C22×C8—22no—
161C22×C8—142no—
162C2×C4⋊C4—71no—
163C2×C4⋊C4—71no—
164C2×C4⋊C4—71no—
165C2×C4⋊C4—71no—
166C2×C22⋊C4—141no—
167C2×C22⋊C4—141no—
168C2×C42—24no—
169C2×C42—141no—
170C2×C42—141no—
171C2×C42C4×D711yescharacteristic
172C2×C42D2811yescharacteristic
173C2×C42C22⋊D711yescharacteristic
174C2×C42C22×D711yescharacteristic
175C22⋊C8—24no—
176C22⋊C8—144no—
177#97—24no—
178#97—144no—
179#95—141no—
180#95—141no—
181#95—141no—
182#95—141no—
183#95—141no—
184#95—141no—
185#95—141no—
186#95—141no—
187C22×C7—22no—
188C22×C7—22no—
189C22×C7—22no—
190C22×C7—41no—
191C22×C7—41no—
192C22×C7—41no—
193C22×C7—41no—
194C22×C7#38111yescharacteristic
195C22×C7#41311yescharacteristic
196C22×C7#41311yescharacteristic
197C22×C7C2×#10411yescharacteristic
198C22×C7C2×#10411yescharacteristic
199C22×C7#42111yescharacteristic
200C22×C7#42111yescharacteristic
201C4×C7—22no—
202C4×C7—22no—
203C4×C7—22no—
204C4×C7—22no—
205C4×C7—41no—
206C4×C7—41no—
207Dic7—81no—
208Dic7—81no—
209Dic7—81no—
210Dic7—81no—
211C24(C2×D7)⋊C411yescharacteristic
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—141no—
222C22×C4—141no—
223C22×C4—141no—
224C22×C4—141no—
225C22×C4(C2×D7)⋊C411yescharacteristic
226C22×C4(C2×D7)⋊C411yescharacteristic
227C22×C4(C2×D7)⋊C411yescharacteristic
228C22×C4C2×C4×D711yescharacteristic
229C22×C4C2×D2811yescharacteristic
230C22×C4C2×C22⋊D711yescharacteristic
231C2×C8—24no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—144no—
235C2×C8—282no—
236C2×C8—282no—
237C4⋊C4—281no—
238C4⋊C4—281no—
239C4⋊C4—281no—
240C4⋊C4—281no—
241C22⋊C4—282no—
242C22⋊C4—282no—
243C42—22no—
244C42—22no—
245C42—22no—
246C42—282no—
247C42—282no—
248C42#115914yes—
249C2×C7—42no—
250C2×C7#152411yescharacteristic
251C2×C7#152911yescharacteristic
252C2×C7#153211yescharacteristic
253C2×C7#153511yescharacteristic
254C2×C7#182211yescharacteristic
255C2×C7#182611yescharacteristic
256C2×C7#182611yescharacteristic
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23#637912yes—
264C23C2×(C2×D7)⋊C411yescharacteristic
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—281no—
283C2×C4—281no—
284C2×C4—281no—
285C2×C4—281no—
286C2×C4—281no—
287C2×C4—281no—
288C2×C4—281no—
289C2×C4—281no—
290C2×C4—281no—
291C2×C4—281no—
292C2×C4—281no—
293C2×C4—281no—
294C2×C4#639912yes—
295C2×C4#644012yes—
296C2×C4#644912yes—
297C2×C4#649812yes—
298C8—44no—
299C8—284no—
300C7#872011yescharacteristic, complemented, Sylow
301C22—22no—
302C22—22no—
303C22—22no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22C2×#637911yescharacteristic
309C22#8971511yescharacteristic
310C22#8977711yescharacteristic
311C22#8990411yescharacteristic
312C22#8990411yescharacteristic
313C22C2×#639911yescharacteristic
314C22#8991111yescharacteristic
315C4—22no—
316C4—22no—
317C4—22no—
318C4—22no—
319C4—41no—
320C4—41no—
321C4—561no—
322C4—561no—
323C4—561no—
324C4—561no—
325C2—42no—
326C2#1172719611yescharacteristic
327C2#1172755311yescharacteristic
328C2#1172761211yescharacteristic
329C2#1172772811yescharacteristic
330C2#1172772911yescharacteristic
331C2#1172855911yescharacteristic
332C2#1172856411yescharacteristic
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1749C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C22⋊C4
cosocleC2×C42×C7C23
hypercenter#1749D7
hypocenterC7#8720
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
4. #11725465C2semidirect
5. #11725465C2semidirect
300. C7#8720semidirect

Maximal subgroups

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

subgroupindex
#117395052
C7×#17492
#117292832
#117254652
#87207