Finite Group Databasebeta

← #11735581

Subgroups — #11735581

Order 896.

460 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6451C211yesmaximal, characteristic
2C2×#6450C211yesmaximal, characteristic
3C7×#400C211yesmaximal, characteristic
4#89752C211yesmaximal, characteristic
5#89750C211yesmaximal, characteristic
6#89432C211yesmaximal, characteristic
7#89429C211yesmaximal, characteristic
8#89427C211yesmaximal, characteristic
9#89425C211yesmaximal, characteristic
10#89424C211yesmaximal, characteristic
11#89421C211yesmaximal, characteristic
12#89419C211yesmaximal, characteristic
13#89418C211yesmaximal, characteristic
14#89418C211yesmaximal, characteristic
15#89418C211yesmaximal, characteristic
16C22×Dic14C2211yescharacteristic
17C22×Dic14C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×C4⋊Dic7C2211yescharacteristic
21C2×C4⋊Dic7C2211yescharacteristic
22C2×Dic7⋊C4C2211yescharacteristic
23C2×Dic7⋊C4C2211yescharacteristic
24C2×Dic7⋊C4C2211yescharacteristic
25C2×Dic7⋊C4C2211yescharacteristic
26C2×Dic7⋊C4C2211yescharacteristic
27C2×Dic7⋊C4C2211yescharacteristic
28C2×Dic7⋊C4C2211yescharacteristic
29C2×Dic7⋊C4C2211yescharacteristic
30C2×C4×Dic7C2211yescharacteristic
31C2×C4×Dic7C2211yescharacteristic
32C2×C4×Dic7C2211yescharacteristic
33C2×C4×Dic7C2211yescharacteristic
34C2×C4×Dic7C2211yescharacteristic
35#6451—24no—
36#6450—24no—
37C7×#95C2211yescharacteristic
38C7×#95C2211yescharacteristic
39C7×#95C2211yescharacteristic
40C7×#95C2211yescharacteristic
41C7×#95C2211yescharacteristic
42C7⋊#95C2211yescharacteristic
43C7⋊#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53#2446—71nomaximal, Sylow
54C22×C4×C7C2311yescharacteristic
55C22×C4×C7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×Dic7C2311yescharacteristic
62C22×Dic7C2311yescharacteristic
63C22×Dic7C2311yescharacteristic
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C2×Dic14—22no—
70C2×Dic14—22no—
71C2×Dic14—22no—
72C2×Dic14—22no—
73C2×Dic14—22no—
74C2×Dic14D414yes—
75C7×C4⋊C4—22no—
76C7×C4⋊C4—22no—
77C4⋊Dic7—22no—
78C4⋊Dic7—22no—
79Dic7⋊C4—22no—
80Dic7⋊C4—22no—
81Dic7⋊C4—22no—
82Dic7⋊C4—22no—
83Dic7⋊C4—22no—
84Dic7⋊C4—22no—
85Dic7⋊C4—22no—
86Dic7⋊C4Q814yescomplemented
87C4×Dic7—22no—
88C4×Dic7—22no—
89C4×Dic7—22no—
90C4×Dic7—22no—
91C4×Dic7—22no—
92C2×C4⋊Q8—71no—
93C2×C4×Q8—71no—
94#400—71no—
95#400D711yescharacteristic
96#398—71no—
97#398—71no—
98#391—71no—
99#391—71no—
100#391—71no—
101#389—71no—
102#387—71no—
103#387—71no—
104#387—71no—
105#387—71no—
106C4×C4⋊C4—71no—
107C23×C7C2411yescharacteristic
108C2×C4×C7—21no—
109C2×C4×C7—21no—
110C2×C4×C7—21no—
111C2×C4×C7—21no—
112C2×C4×C7—21no—
113C2×C4×C7—21no—
114C2×C4×C7—21no—
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—21no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
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×C7C2×D412yes—
128C2×C4×C7C2×Q812yes—
129C2×Dic7—21no—
130C2×Dic7—21no—
131C2×Dic7—21no—
132C2×Dic7—21no—
133C2×Dic7—21no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7C2×D412yes—
147C2×Dic7C2×D412yes—
148C2×Dic7C2×Q812yes—
149C2×Dic7C2×Q812yes—
150C2×Dic7#4112yes—
151C2×Dic7#4112yes—
152C2×Dic7#4112yes—
153Dic14—28no—
154Dic14—44no—
155C22×Q8—71no—
156C22×Q8—71no—
157C4⋊Q8—144no—
158C4×Q8—144no—
159C2×C4⋊C4—71no—
160C2×C4⋊C4—71no—
161C2×C4⋊C4—71no—
162C2×C4⋊C4—71no—
163C2×C4⋊C4—71no—
164C2×C4⋊C4—71no—
165C2×C4⋊C4—71no—
166C2×C4⋊C4—71no—
167C2×C4⋊C4—71no—
168C2×C4⋊C4—71no—
169C2×C4⋊C4C2×D711yescharacteristic
170C2×C4⋊C4C2×D711yescharacteristic
171C2×C42—71no—
172C2×C42—71no—
173C2×C42—71no—
174C2×C42—71no—
175C2×C42—71no—
176#95—71no—
177#95—71no—
178#95—71no—
179#95—71no—
180#95—71no—
181#95—71no—
182#95—71no—
183#95—71no—
184#95—71no—
185#95—71no—
186#95—71no—
187#95C2×D711yescharacteristic
188#95C2×D711yescharacteristic
189#95C2×D711yescharacteristic
190#95C2×D711yescharacteristic
191#95C2×D711yescharacteristic
192C22×C7C22×D411yescharacteristic
193C22×C7C22×Q811yescharacteristic
194C22×C7C2×#4111yescharacteristic
195C22×C7C2×#4111yescharacteristic
196C22×C7C2×#4111yescharacteristic
197C22×C72−1+411yescharacteristic
198C22×C72−1+411yescharacteristic
199C4×C7—22no—
200C4×C7—22no—
201C4×C7—41no—
202C4×C7—41no—
203C4×C7—41no—
204C4×C7—41no—
205C4×C7—41no—
206Dic7—22no—
207Dic7—22no—
208Dic7—22no—
209Dic7—22no—
210Dic7—41no—
211Dic7—41no—
212Dic7—41no—
213Dic7#12314yes—
214C2×Q8—74no—
215C2×Q8—142no—
216C2×Q8—142no—
217C2×Q8—142no—
218C2×Q8—142no—
219C2×Q8—142no—
220C22×C4—71no—
221C22×C4—71no—
222C22×C4—71no—
223C22×C4—71no—
224C22×C4—71no—
225C22×C4—71no—
226C22×C4—71no—
227C22×C4—71no—
228C22×C4C22×D711yescharacteristic
229C22×C4C22×D711yescharacteristic
230C22×C4C22×D711yescharacteristic
231C22×C4C22×D711yescharacteristic
232C22×C4C22×D711yescharacteristic
233C22×C4C22×D711yescharacteristic
234C22×C4C22×D711yescharacteristic
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—74no—
238C4⋊C4—142no—
239C4⋊C4—142no—
240C4⋊C4—142no—
241C4⋊C4—142no—
242C4⋊C4—142no—
243C4⋊C4—142no—
244C4⋊C4—142no—
245C4⋊C4—142no—
246C4⋊C4—142no—
247C42—142no—
248C42—142no—
249C42—142no—
250C42—142no—
251C42—142no—
252C2×C7C2×#12311yescharacteristic
253C2×C7#53311yescharacteristic
254C2×C7#54111yescharacteristic
255C2×C7#54811yescharacteristic
256C2×C7D4×Q811yescharacteristic
257C2×C7#55611yescharacteristic
258C2×C7#55711yescharacteristic
259C23C23×D711yescharacteristic
260Q8—148no—
261Q8—284no—
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—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—72no—
282C2×C4—72no—
283C2×C4—72no—
284C2×C4—72no—
285C2×C4—72no—
286C2×C4—72no—
287C2×C4—72no—
288C2×C4—141no—
289C2×C4—141no—
290C2×C4—141no—
291C2×C4—141no—
292C2×C4—141no—
293C2×C4—141no—
294C2×C4—141no—
295C2×C4—141no—
296C2×C4—141no—
297C2×C4—141no—
298C2×C4—141no—
299C2×C4—141no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—141no—
305C2×C4D4×D712yes—
306C2×C4Q8×D712yes—
307C7#244611yescharacteristic, complemented, Sylow
308C22C2×D4×D711yescharacteristic
309C22C2×Q8×D711yescharacteristic
310C22D7×#4111yescharacteristic
311C22D7×#4111yescharacteristic
312C22D7×#4111yescharacteristic
313C22#655411yescharacteristic
314C22#655411yescharacteristic
315C4—22no—
316C4—22no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—74no—
323C4—142no—
324C4—142no—
325C4—142no—
326C4—142no—
327C4—281no—
328C4—281no—
329C4—281no—
330C2#9032311yescharacteristic
331C2D7×#12311yescharacteristic
332C2#9036811yescharacteristic
333C2#9037611yescharacteristic
334C2#9037911yescharacteristic
335C2#9039211yescharacteristic
336C2#9039311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#400C2
hypercenter#400D7
hypocenterC7#2446
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
86. Dic7⋊C4Q8semidirect
307. C7#2446semidirect

Maximal subgroups

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

subgroupindex
C2×#64512
C2×#64502
C7×#4002
#897522
#897502
#894322
#894292
#894272
#894252
#894242
#894212
#894192
#894182
#24467