Finite Group Databasebeta

← #89733⋊C3 #13051861

Subgroups — #89733⋊C3 #13051861

Order 1344.

476 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C22×(C22⋊D7)⋊C3C211yesmaximal, characteristic
2C2×C22⋊C4×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
3C2×(C22⋊Dic7)⋊C3C211yesmaximal, characteristic, complemented
4C2×(C2×AGL(1,𝔽7))⋊C4C211yesmaximal, characteristic, complemented
5C2×(C2×AGL(1,𝔽7))⋊C4C211yesmaximal, characteristic, complemented
6C2×C4⋊(ΓL(1,𝔽8)⋊C4)C211yesmaximal, characteristic, complemented
7ΓL(1,𝔽8)⋊#95C211yesmaximal, characteristic, complemented
8#89733C311yesmaximal, characteristic, complemented, Hall
9C24×ΓL(1,𝔽8)C2211yescharacteristic
10C23×AGL(1,𝔽7)C2211yescharacteristic
11C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
12C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
13C2×(C22⋊D7)⋊C3—22no—
14C2×(C22⋊D7)⋊C3—22no—
15C2×(C22⋊D7)⋊C3—22no—
16C22×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
17C22×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
18C22×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
19C22⋊C4×ΓL(1,𝔽8)—22no—
20(C22⋊Dic7)⋊C3—22no—
21(C2×AGL(1,𝔽7))⋊C4—22no—
22(C2×AGL(1,𝔽7))⋊C4—22no—
23C4⋊(ΓL(1,𝔽8)⋊C4)—22no—
24C22×C22⋊D7C2×C311yescharacteristic
25C2×C7×C22⋊C4C2×C311yescharacteristic, complemented
26C2×C22⋊Dic7C2×C311yescharacteristic, complemented
27C2×(C2×D7)⋊C4C2×C311yescharacteristic, complemented
28C2×(C2×D7)⋊C4C2×C311yescharacteristic, complemented
29C2×C4⋊Dic7C2×C311yescharacteristic, complemented
30C7⋊#95C2×C311yescharacteristic, complemented
31C3×#394—71nomaximal, Hall
32C23×ΓL(1,𝔽8)—21no—
33C23×ΓL(1,𝔽8)—21no—
34C23×ΓL(1,𝔽8)—21no—
35C23×ΓL(1,𝔽8)—21no—
36C23×ΓL(1,𝔽8)—21no—
37C23×ΓL(1,𝔽8)—21no—
38C23×ΓL(1,𝔽8)D412yes—
39C23×ΓL(1,𝔽8)C2311yescharacteristic
40C22×AGL(1,𝔽7)—21no—
41C22×AGL(1,𝔽7)—21no—
42C22×AGL(1,𝔽7)—21no—
43C22×AGL(1,𝔽7)—21no—
44C22×AGL(1,𝔽7)—21no—
45C22×AGL(1,𝔽7)—21no—
46C22×AGL(1,𝔽7)D412yes—
47C2×C4×ΓL(1,𝔽8)—21no—
48C2×C4×ΓL(1,𝔽8)—21no—
49C2×C4×ΓL(1,𝔽8)—21no—
50C2×C4×ΓL(1,𝔽8)—21no—
51C2×C4×ΓL(1,𝔽8)—21no—
52C2×C4×ΓL(1,𝔽8)D412yescomplemented
53(C22⋊D7)⋊C3—28no—
54C2×ΓL(1,𝔽8)⋊C4—21no—
55C2×ΓL(1,𝔽8)⋊C4—21no—
56C2×ΓL(1,𝔽8)⋊C4—21no—
57C2×ΓL(1,𝔽8)⋊C4—21no—
58C2×ΓL(1,𝔽8)⋊C4—21no—
59C2×ΓL(1,𝔽8)⋊C4—21no—
60C2×ΓL(1,𝔽8)⋊C4—21no—
61C2×ΓL(1,𝔽8)⋊C4—21no—
62C2×ΓL(1,𝔽8)⋊C4D412yes—
63C24×C7C22×C311yescharacteristic
64C23×D7C22×C311yescharacteristic
65C22×C4×C7C22×C311yescharacteristic
66C22×C4×C7C22×C311yescharacteristic
67C2×C22⋊D7—22no—
68C2×C22⋊D7—22no—
69C2×C22⋊D7—22no—
70C22×Dic7C22×C311yescharacteristic
71C22×Dic7C22×C311yescharacteristic
72C22×Dic7C22×C311yescharacteristic
73C7×C22⋊C4—22no—
74C22⋊Dic7—22no—
75(C2×D7)⋊C4—22no—
76(C2×D7)⋊C4—22no—
77C4⋊Dic7—22no—
78C22×C3×D4—71no—
79C2×C3×C4⋊C4—71no—
80C2×C3×C22⋊C4—71no—
81C2×C3×C22⋊C4—71no—
82C2×C3×C22⋊C4—71no—
83C2×C3×C22⋊C4—71no—
84C3×#95—71no—
85C22×ΓL(1,𝔽8)—22no—
86C22×ΓL(1,𝔽8)—22no—
87C22×ΓL(1,𝔽8)—22no—
88C22×ΓL(1,𝔽8)—41no—
89C22×ΓL(1,𝔽8)—41no—
90C22×ΓL(1,𝔽8)—41no—
91C22×ΓL(1,𝔽8)—41no—
92C22×ΓL(1,𝔽8)C2×D411yescharacteristic
93C22×ΓL(1,𝔽8)C2×D411yescharacteristic
94C22×ΓL(1,𝔽8)C2×D411yescharacteristic
95C22×ΓL(1,𝔽8)C2×D411yescharacteristic
96C22×ΓL(1,𝔽8)#4111yescharacteristic
97C22×ΓL(1,𝔽8)#4111yescharacteristic
98C22×ΓL(1,𝔽8)#4111yescharacteristic
99C2×AGL(1,𝔽7)—22no—
100C2×AGL(1,𝔽7)—22no—
101C2×AGL(1,𝔽7)—22no—
102C2×AGL(1,𝔽7)—41no—
103C2×AGL(1,𝔽7)—41no—
104C2×AGL(1,𝔽7)—41no—
105C2×AGL(1,𝔽7)—41no—
106C4×ΓL(1,𝔽8)—22no—
107C4×ΓL(1,𝔽8)—41no—
108ΓL(1,𝔽8)⋊C4—22no—
109ΓL(1,𝔽8)⋊C4—41no—
110ΓL(1,𝔽8)⋊C4—41no—
111#394—71noSylow
112C23×C7—21no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7C3×D412yes—
119C23×C7C23×C311yescharacteristic
120C22×D7—21no—
121C22×D7—21no—
122C22×D7—21no—
123C22×D7—21no—
124C22×D7—21no—
125C22×D7—21no—
126C22×D7C3×D412yes—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7C3×D412yescomplemented
133C22⋊D7—28no—
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×Dic7C3×D412yes—
143C24×C3—71no—
144C24×C3—71no—
145C2×C3×D4—142no—
146C2×C3×D4—142no—
147C2×C3×D4—142no—
148C22×C3×C4—71no—
149C22×C3×C4—71no—
150C22×C3×C4—71no—
151C22×C3×C4—71no—
152C22×C3×C4—71no—
153C3×C4⋊C4—142no—
154C3×C22⋊C4—142no—
155C3×C22⋊C4—142no—
156C3×C22⋊C4—142no—
157C3×C22⋊C4—142no—
158C2×ΓL(1,𝔽8)—42no—
159C2×ΓL(1,𝔽8)#12011yescharacteristic
160C2×ΓL(1,𝔽8)#12111yescharacteristic
161C2×ΓL(1,𝔽8)#12111yescharacteristic
162C2×ΓL(1,𝔽8)#12111yescharacteristic
163C2×ΓL(1,𝔽8)#12311yescharacteristic
164C2×ΓL(1,𝔽8)#12311yescharacteristic
165C2×ΓL(1,𝔽8)#12411yescharacteristic
166AGL(1,𝔽7)—42no—
167C22×D4—71no—
168C2×C4⋊C4—71no—
169C2×C22⋊C4—71no—
170C2×C22⋊C4—71no—
171C2×C22⋊C4—71no—
172C2×C22⋊C4AGL(1,𝔽7)11yescharacteristic, complemented
173#95—71no—
174C22×C7—22no—
175C22×C7—22no—
176C22×C7—22no—
177C22×C7—41no—
178C22×C7—41no—
179C22×C7—41no—
180C22×C7—41no—
181C22×C7C2×C3×D411yescharacteristic
182C22×C7C2×C3×D411yescharacteristic
183C22×C7C2×C3×D411yescharacteristic
184C22×C7C2×C3×D411yescharacteristic
185C22×C7C3×#4111yescharacteristic
186C22×C7C3×#4111yescharacteristic
187C22×C7C3×#4111yescharacteristic
188C2×D7—22no—
189C2×D7—22no—
190C2×D7—22no—
191C2×D7—41no—
192C2×D7—41no—
193C2×D7—41no—
194C2×D7—41no—
195C4×C7—22no—
196C4×C7—41no—
197Dic7—22no—
198Dic7—41no—
199Dic7—41no—
200C23×C3—72no—
201C23×C3—72no—
202C23×C3—71no—
203C23×C3—141no—
204C23×C3—141no—
205C23×C3—141no—
206C23×C3—141no—
207C23×C3—141no—
208C23×C3—141no—
209C23×C3—141no—
210C23×C3—141no—
211C23×C3—141no—
212C23×C3—141no—
213C23×C3—141no—
214C23×C3—141no—
215C3×D4—148no—
216C2×C3×C4—72no—
217C2×C3×C4—72no—
218C2×C3×C4—141no—
219C2×C3×C4—141no—
220C2×C3×C4—141no—
221C2×C3×C4—141no—
222C2×C3×C4—141no—
223C2×C3×C4—141no—
224C2×C3×C4—141no—
225C2×C3×C4—141no—
226C2×C3×C4—141no—
227C2×C3×C4—141no—
228C2×C3×C4—141no—
229C2×C3×C4—141no—
230C2×C3×C4—141no—
231ΓL(1,𝔽8)#39411yescharacteristic, complemented, Hall
232C24—71no—
233C24C2×AGL(1,𝔽7)11yescharacteristic
234C2×D4—142no—
235C2×D4—142no—
236C2×D4—142no—
237C22×C4—71no—
238C22×C4—71no—
239C22×C4—71no—
240C22×C4C2×AGL(1,𝔽7)11yescharacteristic
241C22×C4C2×AGL(1,𝔽7)11yescharacteristic
242C4⋊C4—142no—
243C22⋊C4—22no—
244C22⋊C4—142no—
245C22⋊C4—142no—
246C22⋊C4—142no—
247C2×C7—42no—
248C2×C7C3×#12011yescharacteristic
249C2×C7C3×#12111yescharacteristic
250C2×C7C3×#12111yescharacteristic
251C2×C7C3×#12111yescharacteristic
252C2×C7C3×#12311yescharacteristic
253C2×C7C3×#12311yescharacteristic
254C2×C7C3×#12411yescharacteristic
255D7—42no—
256C22×C3—71no—
257C22×C3—71no—
258C22×C3—71no—
259C22×C3—71no—
260C22×C3—71no—
261C22×C3—71no—
262C22×C3—71no—
263C22×C3—142no—
264C22×C3—142no—
265C22×C3—142no—
266C22×C3—142no—
267C22×C3—142no—
268C22×C3—142no—
269C22×C3—281no—
270C22×C3—281no—
271C22×C3—281no—
272C22×C3—281no—
273C22×C3—281no—
274C22×C3—281no—
275C22×C3—281no—
276C22×C3—281no—
277C3×C4—142no—
278C3×C4—142no—
279C3×C4—281no—
280C3×C4—281no—
281C3×C4—281no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—72no—
289C23—141no—
290C23—141no—
291C23—141no—
292C23—141no—
293C23—141no—
294C23—141no—
295C23C4⋊AGL(1,𝔽7)12yes—
296C23C22×AGL(1,𝔽7)11yescharacteristic
297D4—148no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—72no—
304C2×C4—141no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—141no—
308C2×C4—141no—
309C2×C4—141no—
310C2×C4—141no—
311C2×C4—141no—
312C2×C4(C22⋊D7)⋊C312yescomplemented
313C7C3×#39411yescharacteristic, complemented, Sylow
314C2×C3—71no—
315C2×C3—71no—
316C2×C3—71no—
317C2×C3—71no—
318C2×C3—71no—
319C2×C3—71no—
320C2×C3—71no—
321C2×C3—282no—
322C2×C3—282no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—142no—
331C22—142no—
332C22—142no—
333C22—281no—
334C22—281no—
335C22—281no—
336C22—281no—
337C22C2×C4⋊AGL(1,𝔽7)11yescharacteristic
338C22#1157⋊C311yescharacteristic
339C22D4×AGL(1,𝔽7)11yescharacteristic
340C22D4×AGL(1,𝔽7)11yescharacteristic
341C22#1159⋊C311yescharacteristic
342C22#1159⋊C311yescharacteristic
343C22C2×(C22⋊D7)⋊C311yescharacteristic
344C4—22no—
345C4—41no—
346C4—142no—
347C4—281no—
348C4—281no—
349C3—71noSylow
350C2—42no—
351C2—282no—
352C2#6445⋊C311yescharacteristic
353C2#6446⋊C311yescharacteristic
354C2#6448⋊C311yescharacteristic
355C2#6449⋊C311yescharacteristic
356C2#6493⋊C311yescharacteristic
357C2#6501⋊C311yescharacteristic
358C2#6502⋊C311yescharacteristic
subgroupG/H
center, FrattiniC23C22×AGL(1,𝔽7)
commutator, socle, cosocleC23×C7C23×C3
FittingC2×C7×C22⋊C4C2×C3
prime radicalC22×(C22⋊D7)⋊C3C2
hypercenterC2×C22⋊C4AGL(1,𝔽7)
hypocenterC7C3×#394
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C4×ΓL(1,𝔽8)C2semidirect
3. C2×(C22⋊Dic7)⋊C3C2semidirect
4. C2×(C2×AGL(1,𝔽7))⋊C4C2semidirect
5. C2×(C2×AGL(1,𝔽7))⋊C4C2semidirect
6. C2×C4⋊(ΓL(1,𝔽8)⋊C4)C2semidirect
7. ΓL(1,𝔽8)⋊#95C2semidirect
8. #89733C3semidirect
25. C2×C7×C22⋊C4C2×C3semidirect
26. C2×C22⋊Dic7C2×C3semidirect
27. C2×(C2×D7)⋊C4C2×C3semidirect
28. C2×(C2×D7)⋊C4C2×C3semidirect
29. C2×C4⋊Dic7C2×C3semidirect
30. C7⋊#95C2×C3semidirect
52. C2×C4×ΓL(1,𝔽8)D4semidirect
132. C2×C4×C7C3×D4semidirect
172. C2×C22⋊C4AGL(1,𝔽7)semidirect
231. ΓL(1,𝔽8)#394semidirect
312. C2×C4(C22⋊D7)⋊C3semidirect
313. C7C3×#394semidirect

Maximal subgroups

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

subgroupindex
C22×(C22⋊D7)⋊C32
C2×C22⋊C4×ΓL(1,𝔽8)2
C2×(C22⋊Dic7)⋊C32
C2×(C2×AGL(1,𝔽7))⋊C42
C2×C4⋊(ΓL(1,𝔽8)⋊C4)2
ΓL(1,𝔽8)⋊#952
#897333
C3×#3947