Finite Group Databasebeta

← #13242303

Subgroups — #13242303

Order 1792.

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

normal subgroups only

This group has 444 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
1#11730759C211yesmaximal, characteristic
2C7×#1745C211yesmaximal, characteristic, complemented
3#11727142C211yesmaximal, characteristic, complemented
4#11725285C212yesmaximal
5#11725283C212yesmaximal, complemented
6C2×#6439—21no—
7C2×C7×#97—21no—
8C43×C7C2211yescharacteristic
9#89712—21no—
10#89712—21no—
11#89712—21no—
12#89711C411yescharacteristic
13#89711C411yescharacteristic
14#89711C2211yescharacteristic
15#89700C411yescharacteristic
16#89700C411yescharacteristic
17#89700C2211yescharacteristic
18C2×C4×C7⋊C8—21no—
19C7×#336—21no—
20C7×#336C2212yes—
21#89319—21no—
22#89319C2212yes—
23#7692—71nomaximal, Sylow
24C22×D28C2×C411yescharacteristic
25C22×C7×C8—22no—
26C2×C42×C7C2×C411yescharacteristic
27C2×C42×C7C2×C411yescharacteristic
28C2×C42×C7D411yescharacteristic
29C2×C42×C7D411yescharacteristic
30C2×C42×C7D411yescharacteristic
31C2×C42×C7D411yescharacteristic
32C2×C42×C7C2311yescharacteristic
33C2×(C2×D7)⋊C4—21no—
34C2×(C2×D7)⋊C4—21no—
35C2×(C2×D7)⋊C4—21no—
36C2×(C2×D7)⋊C4—21no—
37C2×(C2×D7)⋊C4—21no—
38C2×(C2×D7)⋊C4—21no—
39C2×C4⋊Dic7C2×C411yescharacteristic
40C2×C4⋊Dic7C2×C411yescharacteristic
41C2×C4⋊Dic7C2×C411yescharacteristic
42C2×Dic7⋊C4—21no—
43C2×Dic7⋊C4—21no—
44C2×Dic7⋊C4—21no—
45C22×C7⋊C8—22no—
46#6439—44no—
47C7×#97—24no—
48C7⋊#95—21no—
49C7⋊#95—21no—
50C7⋊#95—21no—
51C4×C7⋊C8—24no—
52#2860—71no—
53#1745D711yescharacteristic, complemented
54#1744—71no—
55#1271—72no—
56#1270—72no—
57C23×D7—21no—
58C22×C4×C7C22⋊C411yescharacteristic
59C22×C4×C7C22⋊C411yescharacteristic
60C22×C4×C7C22⋊C411yescharacteristic
61C22×C4×C7C22⋊C411yescharacteristic
62C22×C4×C7C22×C411yescharacteristic
63C22×C4×C7C2×D411yescharacteristic
64C22×C4×C7C2×D411yescharacteristic
65C22×Dic7—21no—
66C22×Dic7—21no—
67C22×Dic7—21no—
68C2×D28—22no—
69C2×D28—41no—
70C2×D28—41no—
71C2×C7×C8—24no—
72C2×C7×C8—42no—
73C2×C7×C8—42no—
74C42×C7—22no—
75C42×C7—22no—
76C42×C7—22no—
77C42×C7—22no—
78C42×C7—22no—
79C42×C7—41no—
80C42×C7—41no—
81(C2×D7)⋊C4—42no—
82(C2×D7)⋊C4—42no—
83(C2×D7)⋊C4—42no—
84(C2×D7)⋊C4—42no—
85(C2×D7)⋊C4—42no—
86(C2×D7)⋊C4—42no—
87C4⋊Dic7—22no—
88C4⋊Dic7—41no—
89C4⋊Dic7—41no—
90Dic7⋊C4—42no—
91Dic7⋊C4—42no—
92Dic7⋊C4—42no—
93C2×C7⋊C8—24no—
94C2×C7⋊C8—42no—
95C2×C7⋊C8—42no—
96C2×#126—141no—
97C2×#97—21no—
98C2×C4×C8—141no—
99#390—71no—
100#390—71no—
101#390—71no—
102#389—71no—
103#389—71no—
104#389—71no—
105#388—141no—
106#388—141no—
107#388—141no—
108C43C2×D711yescharacteristic
109#336—21no—
110#336—72no—
111#336—141no—
112#336C2×D712yes—
113C23×C7C2×C22⋊C411yescharacteristic
114C22×D7—41no—
115C22×D7—41no—
116C22×D7—41no—
117C22×D7—41no—
118C22×D7—41no—
119C22×D7—41no—
120C22×D7—41no—
121C2×C4×C7—21no—
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—21no—
137C2×C4×C7#10412yes—
138C2×C4×C7#10412yes—
139C2×C4×C7#13112yes—
140C2×C4×C7#13512yes—
141C2×C4×C7AGL(1,ℤ/8)12yes—
142C2×Dic7—41no—
143C2×Dic7—41no—
144C2×Dic7—41no—
145C2×Dic7—41no—
146C2×Dic7—41no—
147C2×Dic7—41no—
148C2×Dic7—41no—
149C2×Dic7—41no—
150C2×Dic7—41no—
151D28—82no—
152C7×C8—44no—
153C7⋊C8—44no—
154C22×D4—71no—
155C22×C8—22no—
156C22×C8—142no—
157#126—284no—
158C2×C4⋊C4—71no—
159C2×C4⋊C4—71no—
160C2×C4⋊C4—71no—
161C2×C4⋊C4—141no—
162C2×C4⋊C4—141no—
163C2×C4⋊C4—141no—
164C2×C22⋊C4—141no—
165C2×C22⋊C4—141no—
166C2×C22⋊C4—141no—
167C2×C22⋊C4—141no—
168C2×C22⋊C4—141no—
169C2×C22⋊C4—141no—
170C2×C42C4×D711yescharacteristic
171C2×C42C4×D711yescharacteristic
172C2×C42D2811yescharacteristic
173C2×C42D2811yescharacteristic
174C2×C42C22⋊D711yescharacteristic
175C2×C42C22⋊D711yescharacteristic
176C2×C42C22×D711yescharacteristic
177#97—24no—
178C4×C8—144no—
179#95—141no—
180#95—141no—
181#95—141no—
182C22×C7#40811yescharacteristic
183C22×C7#41011yescharacteristic
184C22×C7#41311yescharacteristic
185C22×C7#41611yescharacteristic
186C22×C7#41811yescharacteristic
187C22×C7C2×#10411yescharacteristic
188C22×C7C2×#10411yescharacteristic
189C2×D7—81no—
190C2×D7—81no—
191C2×D7—81no—
192C2×D7—81no—
193C2×D7—81no—
194C2×D7—81no—
195C2×D7—81no—
196C4×C7—22no—
197C4×C7—22no—
198C4×C7—22no—
199C4×C7—22no—
200C4×C7—22no—
201C4×C7—41no—
202C4×C7—41no—
203Dic7—81no—
204Dic7—81no—
205Dic7—81no—
206C24—141no—
207C2×D4—142no—
208C2×D4—281no—
209C2×D4—281no—
210C22×C4—141no—
211C22×C4—141no—
212C22×C4—141no—
213C22×C4(C2×D7)⋊C411yescharacteristic
214C22×C4(C2×D7)⋊C411yescharacteristic
215C22×C4(C2×D7)⋊C411yescharacteristic
216C22×C4(C2×D7)⋊C411yescharacteristic
217C22×C4C2×C4×D711yescharacteristic
218C22×C4C2×D2811yescharacteristic
219C22×C4C2×C22⋊D711yescharacteristic
220C2×C8—24no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—144no—
224C2×C8—282no—
225C2×C8—282no—
226C4⋊C4—142no—
227C4⋊C4—281no—
228C4⋊C4—281no—
229C4⋊C4—282no—
230C4⋊C4—282no—
231C4⋊C4—282no—
232C22⋊C4—282no—
233C22⋊C4—282no—
234C22⋊C4—282no—
235C22⋊C4—282no—
236C22⋊C4—282no—
237C22⋊C4—282no—
238C42—22no—
239C42—22no—
240C42—22no—
241C42—22no—
242C42—22no—
243C42—41no—
244C42—41no—
245C2×C7#147311yescharacteristic
246C2×C7#148411yescharacteristic
247C2×C7#149411yescharacteristic
248C2×C7#150011yescharacteristic
249C2×C7#150511yescharacteristic
250C2×C7#152411yescharacteristic
251C2×C7#155311yescharacteristic
252D7—161no—
253C23—281no—
254C23—281no—
255C23—281no—
256C23—281no—
257C23—281no—
258C23—281no—
259C23—281no—
260C23C2×(C2×D7)⋊C411yescharacteristic
261D4—562no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
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—281no—
279C2×C4—281no—
280C2×C4—281no—
281C2×C4—281no—
282C2×C4—281no—
283C2×C4—281no—
284C2×C4—281no—
285C2×C4—281no—
286C2×C4—281no—
287C2×C4#637912yes—
288C2×C4#639912yes—
289C2×C4#647012yes—
290C2×C4#647112yes—
291C2×C4#651312yes—
292C8—44no—
293C8—284no—
294C7#769211yescharacteristic, complemented, Sylow
295C22—561no—
296C22—561no—
297C22—561no—
298C22—561no—
299C22—561no—
300C22—561no—
301C22—561no—
302C22C2×#637911yescharacteristic
303C22#8977411yescharacteristic
304C22#8977811yescharacteristic
305C22#8990111yescharacteristic
306C22#8990311yescharacteristic
307C22#8990411yescharacteristic
308C22C2×#639911yescharacteristic
309C4—22no—
310C4—22no—
311C4—22no—
312C4—22no—
313C4—22no—
314C4—41no—
315C4—41no—
316C4—561no—
317C4—561no—
318C4—561no—
319C2—1121no—
320C2#1172718511yescharacteristic
321C2#1172760711yescharacteristic
322C2#1172761211yescharacteristic
323C2#1172855411yescharacteristic
324C2#1172857111yescharacteristic
325C2#1172858311yescharacteristic
326C2#1172858611yescharacteristic
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1745C2
socleC23×C7C2×C22⋊C4
prime radicalC22×D28C2×C4
cosocleC2×C42×C7C23
hypercenter#1745D7
hypocenterC7#7692
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C43 ◁ #1745

Complements

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

NH ≅ G/Ntype
2. C7×#1745C2semidirect
3. #11727142C2semidirect
5. #11725283C2semidirect
53. #1745D7semidirect
294. C7#7692semidirect

Maximal subgroups

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

subgroupindex
#117307592
C7×#17452
#117271422
#117252852
#117252832
#76927