Finite Group Databasebeta

← #13245276

Subgroups — #13245276

Order 1792.

814 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C2×#90161C211yesmaximal, characteristic
2C7×#1749C211yesmaximal, characteristic
3#11729283C211yesmaximal, characteristic
4C2×#89260C214yesmaximal, complemented
5C22×C42×C7C2211yescharacteristic
6C2×#6489C412yes—
7C2×#6487—21no—
8C2×#6487—21no—
9#90161—28no—
10C2×#6433C2212yes—
11C2×#6432C411yescharacteristic
12C2×#6432C411yescharacteristic
13C2×C4×Dic14—21no—
14C2×C4×Dic14—21no—
15C2×C7×C22⋊C8—21no—
16C2×C7×#97C2212yes—
17C2×C22⋊(C7⋊C8)—21no—
18C2×#6377C2212yes—
19C7×#336—21no—
20#89319—21no—
21#89260—28no—
22#8338—71nomaximal, Sylow
23C23×C4×C7C2×C411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C22×Dic14—21no—
27C22×Dic14—21no—
28C22×C7×C8—22no—
29C2×C42×C7—24no—
30C2×C42×C7C2×C411yescharacteristic
31C2×C42×C7D411yescharacteristic
32C2×C42×C7D411yescharacteristic
33C2×C42×C7C2311yescharacteristic
34C2×C22⋊Dic7—22no—
35#6489—24no—
36#6489—24no—
37C2×C4⋊Dic7—21no—
38C2×C4⋊Dic7—21no—
39C2×C4⋊Dic7C2×C412yes—
40C2×C4⋊Dic7C2×C412yes—
41#6487—28no—
42#6487—28no—
43C2×Dic7⋊C4—22no—
44C2×Dic7⋊C4—22no—
45C2×C4×Dic7—22no—
46C22×C7⋊C8—22no—
47#6433—24no—
48#6433D418yes—
49#6432—24no—
50#6432—24no—
51C4×Dic14—28no—
52C4×Dic14—28no—
53C7×C22⋊C8—24no—
54C7×#97—24no—
55C22⋊(C7⋊C8)—24no—
56#6377—24no—
57C2×#533—71no—
58#1749—71no—
59#1749D711yescharacteristic
60C2×#330—74no—
61C24×C7C22⋊C411yescharacteristic
62C22×C4×C7—22no—
63C22×C4×C7—22no—
64C22×C4×C7—22no—
65C22×C4×C7—22no—
66C22×C4×C7—22no—
67C22×C4×C7—22no—
68C22×C4×C7—22no—
69C22×C4×C7—41no—
70C22×C4×C7—41no—
71C22×C4×C7C22⋊C411yescharacteristic
72C22×C4×C7C22⋊C411yescharacteristic
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22×C411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×Dic7—22no—
78C22×Dic7—22no—
79C2×Dic14—24no—
80C2×Dic14—24no—
81C2×Dic14—42no—
82C2×Dic14—42no—
83C2×Dic14—42no—
84C2×Dic14—42no—
85C2×C7×C8—24no—
86C2×C7×C8—42no—
87C2×C7×C8—42no—
88C42×C7—22no—
89C42×C7—22no—
90C42×C7—22no—
91C42×C7C2×D414yes—
92C22⋊Dic7—28no—
93C4⋊Dic7—24no—
94C4⋊Dic7—24no—
95C4⋊Dic7—22no—
96C4⋊Dic7—22no—
97C4⋊Dic7C22⋊C414yes—
98C4⋊Dic7C22⋊C414yes—
99Dic7⋊C4—28no—
100Dic7⋊C4—28no—
101C4×Dic7—28no—
102C2×C7⋊C8—24no—
103C2×C7⋊C8—42no—
104C2×C7⋊C8—42no—
105#533—148no—
106C2×C4⋊Q8—71no—
107C2×C4⋊Q8—71no—
108C2×#125—72no—
109C2×C22⋊Q8—141no—
110C2×C22⋊Q8—141no—
111C2×C4×Q8—141no—
112C2×C4×Q8—141no—
113C2×#117—72no—
114C22×C42C2×D711yescharacteristic
115C2×C22⋊C8—21no—
116C2×C22⋊C8—141no—
117C2×#97—72no—
118C2×#97C2×D712yes—
119#336—21no—
120#336—141no—
121#330—148no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7#10112yes—
129C23×C7C2×C22⋊C411yescharacteristic
130C2×C4×C7—28no—
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—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—44no—
144C2×C4×C7—44no—
145C2×C4×C7#10112yes—
146C2×C4×C7#10412yes—
147C2×C4×C7#10412yes—
148C2×C4×C7C2×C22⋊C412yes—
149C2×C4×C7#12012yes—
150C2×C4×C7#12012yes—
151C2×Dic7—24no—
152C2×Dic7—24no—
153C2×Dic7—42no—
154C2×Dic7—42no—
155C2×Dic7—42no—
156C2×Dic7—42no—
157Dic14—48no—
158Dic14—48no—
159C7×C8—44no—
160C7⋊C8—44no—
161C22×Q8—141no—
162C22×Q8—141no—
163C23×C4C4×D711yescharacteristic
164C23×C4D2811yescharacteristic
165C23×C4C22⋊D711yescharacteristic
166C22×C8—22no—
167C22×C8—142no—
168C4⋊Q8—144no—
169C4⋊Q8—144no—
170#125—78no—
171#125—144no—
172C22⋊Q8—148no—
173C22⋊Q8—148no—
174C4×Q8—148no—
175C4×Q8—148no—
176#117—144no—
177#117—144no—
178C2×C4⋊C4—72no—
179C2×C4⋊C4—72no—
180C2×C4⋊C4—141no—
181C2×C4⋊C4—141no—
182C2×C4⋊C4—142no—
183C2×C4⋊C4—142no—
184C2×C22⋊C4—142no—
185C2×C42—24no—
186C2×C42—142no—
187C2×C42C4×D711yescharacteristic
188C2×C42D2811yescharacteristic
189C2×C42C22⋊D711yescharacteristic
190C2×C42C22×D711yescharacteristic
191C22⋊C8—24no—
192C22⋊C8—144no—
193#97—24no—
194#97—144no—
195C22×C7—22no—
196C22×C7—22no—
197C22×C7—22no—
198C22×C7—41no—
199C22×C7—41no—
200C22×C7—41no—
201C22×C7—41no—
202C22×C7#37911yescharacteristic
203C22×C7C2×#10111yescharacteristic
204C22×C7C2×#10111yescharacteristic
205C22×C7C2×#10411yescharacteristic
206C22×C7C2×#10411yescharacteristic
207C22×C7#42111yescharacteristic
208C22×C7#42111yescharacteristic
209C4×C7—22no—
210C4×C7—22no—
211C4×C7—22no—
212C4×C7—22no—
213C4×C7—22no—
214C4×C7—22no—
215Dic7—44no—
216Dic7—44no—
217C24(C2×D7)⋊C411yescharacteristic
218C2×Q8—144no—
219C2×Q8—144no—
220C2×Q8—282no—
221C2×Q8—282no—
222C2×Q8—282no—
223C2×Q8—282no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—142no—
234C22×C4—142no—
235C22×C4(C2×D7)⋊C411yescharacteristic
236C22×C4(C2×D7)⋊C411yescharacteristic
237C22×C4(C2×D7)⋊C411yescharacteristic
238C22×C4C2×C4×D711yescharacteristic
239C22×C4C2×D2811yescharacteristic
240C22×C4C2×C22⋊D711yescharacteristic
241C2×C8—24no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—144no—
245C2×C8—282no—
246C2×C8—282no—
247C4⋊C4—74no—
248C4⋊C4—74no—
249C4⋊C4—144no—
250C4⋊C4—144no—
251C4⋊C4—142no—
252C4⋊C4—142no—
253C4⋊C4—148no—
254C4⋊C4—148no—
255C22⋊C4—148no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—148no—
260C42D4×D714yes—
261C2×C7—42no—
262C2×C7#152611yescharacteristic
263C2×C7#153111yescharacteristic
264C2×C7#153111yescharacteristic
265C2×C7#153711yescharacteristic
266C2×C7#178411yescharacteristic
267C2×C7#178911yescharacteristic
268C2×C7#178911yescharacteristic
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23#639812yes—
276C23C2×(C2×D7)⋊C411yescharacteristic
277Q8—288no—
278Q8—288no—
279C2×C4—28no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—144no—
295C2×C4—144no—
296C2×C4—282no—
297C2×C4—282no—
298C2×C4—282no—
299C2×C4—282no—
300C2×C4#637912yes—
301C2×C4#639812yes—
302C2×C4#639912yes—
303C2×C4D7×C22⋊C412yes—
304C2×C4#644512yes—
305C2×C4#650012yes—
306C8—44no—
307C8—284no—
308C7#833811yescharacteristic, complemented, Sylow
309C22—22no—
310C22—22no—
311C22—22no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22C2×#637911yescharacteristic
317C22#8972711yescharacteristic
318C22#8977711yescharacteristic
319C22C2×#639811yescharacteristic
320C22C2×#639811yescharacteristic
321C22C2×#639911yescharacteristic
322C22#8991111yescharacteristic
323C4—22no—
324C4—22no—
325C4—22no—
326C4—22no—
327C4—22no—
328C4—22no—
329C4—284no—
330C4—284no—
331C2—42no—
332C2#1172720811yescharacteristic
333C2#1172755511yescharacteristic
334C2#1172761611yescharacteristic
335C2#1172774411yescharacteristic
336C2#1172774511yescharacteristic
337C2#1172856711yescharacteristic
338C2#1172856911yescharacteristic
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#8338
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. C2×#89260C2semidirect
308. C7#8338semidirect

Maximal subgroups

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

subgroupindex
C2×#901612
C7×#17492
#117292832
C2×#892602
#83387