Finite Group Databasebeta

← #11726718

Subgroups — #11726718

Order 896.

380 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 79 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#90184C211yesmaximal, characteristic
2#90177C211yesmaximal, characteristic, complemented
3C2×#6380C211yesmaximal, characteristic, complemented
4C2×#6380C211yesmaximal, characteristic, complemented
5C2×#6380C211yesmaximal, characteristic, complemented
6C7×#342C211yesmaximal, characteristic, complemented
7#89322C211yesmaximal, characteristic, complemented
8C22×C22⋊D7C2211yescharacteristic
9C22×C22⋊D7C2211yescharacteristic
10C2×C7×C22⋊C4C2211yescharacteristic
11C2×C7×C22⋊C4C2211yescharacteristic
12C2×C7×C22⋊C4C2211yescharacteristic
13C2×C22⋊Dic7C2211yescharacteristic
14C2×C22⋊Dic7C2211yescharacteristic
15#6493—22no—
16C4×C22⋊D7—22no—
17#6449—21no—
18#6449C412yescomplemented
19#6445—21no—
20#6445C412yescomplemented
21(C22⋊D7)⋊C4—22no—
22D7×C22⋊C4—21no—
23#6440—21no—
24#6380—22no—
25#6380—22no—
26#6380—22no—
27#1913—71nomaximal, Sylow
28C24×C7C2311yescharacteristic
29C23×D7C2×C411yescharacteristic, complemented
30C23×D7D411yescharacteristic
31C22×C4×C7—21no—
32C22×C4×C7—21no—
33C22×C4×C7—21no—
34C2×C22⋊D7—21no—
35C2×C22⋊D7—21no—
36C2×C22⋊D7—21no—
37C2×C22⋊D7—21no—
38C2×C22⋊D7—21no—
39C2×C22⋊D7—21no—
40C2×C22⋊D7—21no—
41C2×C22⋊D7—21no—
42C2×C22⋊D7—21no—
43C2×C22⋊D7—21no—
44C2×C22⋊D7C2×C412yes—
45C2×C22⋊D7D412yes—
46C22×Dic7—21no—
47C22×Dic7—21no—
48C22×Dic7C2×C411yescharacteristic, complemented
49C22×Dic7D411yescharacteristic
50C2×D28—22no—
51C2×C4×D7—22no—
52C7×C22⋊C4—21no—
53C7×C22⋊C4—21no—
54C7×C22⋊C4—21no—
55C7×C22⋊C4—21no—
56C7×C22⋊C4C2×C412yescomplemented
57C7×C22⋊C4D412yescomplemented
58C22⋊Dic7—21no—
59C22⋊Dic7—21no—
60C22⋊Dic7—21no—
61C22⋊Dic7D412yescomplemented
62(C2×D7)⋊C4—22no—
63(C2×D7)⋊C4—22no—
64(C2×D7)⋊C4—22no—
65C4⋊Dic7—22no—
66Dic7⋊C4—22no—
67C4×Dic7—22no—
68#534—71no—
69#518—71no—
70C2×#99—71no—
71C2×#99—71no—
72C2×#99—71no—
73#342—71no—
74#342D711yescharacteristic, complemented
75C23×C7—81no—
76C23×C7C22×C411yescharacteristic
77C23×C7C2×D411yescharacteristic
78C23×C7C2×D411yescharacteristic
79C23×C7C2×D411yescharacteristic
80C23×C7C2×D411yescharacteristic
81C23×C7#4111yescharacteristic
82C23×C7#4111yescharacteristic
83C22×D7—21no—
84C22×D7—21no—
85C22×D7—21no—
86C22×D7—21no—
87C22×D7—21no—
88C22×D7—81no—
89C22×D7—81no—
90C22×D7C22⋊C412yes—
91C2×C4×C7—22no—
92C2×C4×C7—22no—
93C2×C4×C7—41no—
94C2×C4×C7—81no—
95C2×C4×C7—81no—
96C2×C4×C7—81no—
97C22⋊D7—22no—
98C22⋊D7—22no—
99C22⋊D7—22no—
100C22⋊D7—22no—
101C22⋊D7—41no—
102C22⋊D7—41no—
103C22⋊D7—41no—
104C22⋊D7—41no—
105C2×Dic7—21no—
106C2×Dic7—21no—
107C2×Dic7—21no—
108C2×Dic7—21no—
109C2×Dic7—21no—
110C2×Dic7—22no—
111C2×Dic7—41no—
112C2×Dic7—81no—
113C2×Dic7—81no—
114C2×Dic7C22⋊C412yes—
115D28—82no—
116C4×D7—82no—
117C22×D4—71no—
118C22×D4—71no—
119#123—72no—
120#123—141no—
121#121—142no—
122#120—72no—
123#120—141no—
124C4×D4—142no—
125C4×D4—142no—
126#117—141no—
127C2×C22⋊C4—71no—
128C2×C22⋊C4—71no—
129C2×C22⋊C4—141no—
130C2×C22⋊C4C2×D711yescharacteristic
131C2×C22⋊C4C2×D711yescharacteristic
132C2×C22⋊C4C2×D711yescharacteristic
133#99—142no—
134#99—142no—
135#99—142no—
136C22×C7—41no—
137C22×C7—41no—
138C22×C7—41no—
139C22×C7—41no—
140C22×C7—41no—
141C22×C7—41no—
142C22×C7—41no—
143C22×C7C2×C22⋊C411yescharacteristic
144C22×C7C4×D411yescharacteristic
145C22×C7C4×D411yescharacteristic
146C22×C7#12011yescharacteristic
147C22×C7#12111yescharacteristic
148C22×C7#12111yescharacteristic
149C22×C7#12311yescharacteristic
150C2×D7—22no—
151C2×D7—42no—
152C2×D7—42no—
153C2×D7—42no—
154C2×D7—41no—
155C2×D7—81no—
156C2×D7—81no—
157C2×D7—81no—
158C4×C7—42no—
159C4×C7—42no—
160C4×C7—81no—
161Dic7—22no—
162Dic7—42no—
163Dic7—41no—
164Dic7—81no—
165C24—71no—
166C24—71no—
167C24C22×D711yescharacteristic
168C2×D4—72no—
169C2×D4—72no—
170C2×D4—142no—
171C2×D4—141no—
172C2×D4—141no—
173C2×D4—141no—
174C2×D4—141no—
175C2×D4—141no—
176C2×D4—141no—
177C2×D4—141no—
178C2×D4—141no—
179C2×D4—141no—
180C2×D4—141no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—71no—
185C22×C4—71no—
186C22×C4—142no—
187C22×C4—141no—
188C22×C4—141no—
189C4⋊C4—142no—
190C4⋊C4—142no—
191C22⋊C4—21no—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—72no—
196C22⋊C4—142no—
197C22⋊C4—142no—
198C22⋊C4—142no—
199C22⋊C4—141no—
200C22⋊C4—141no—
201C22⋊C4—141no—
202C22⋊C4C4×D712yescomplemented
203C22⋊C4D2812yescomplemented
204C42—142no—
205C2×C7—21no—
206C2×C7—21no—
207C2×C7—21no—
208C2×C7—21no—
209C2×C7—21no—
210C2×C7—21no—
211C2×C7—21no—
212C2×C7#38611yescharacteristic
213D7—42no—
214D7—81no—
215C23—72no—
216C23—81no—
217C23—141no—
218C23—141no—
219C23—141no—
220C23—141no—
221C23—141no—
222C23—561no—
223C23—561no—
224C23C2×C4×D711yescharacteristic
225C23C2×D2811yescharacteristic
226C23#115711yescharacteristic
227C23D4×D711yescharacteristic
228C23D4×D711yescharacteristic
229C23D4×D711yescharacteristic
230C23#116111yescharacteristic
231D4—142no—
232D4—142no—
233D4—142no—
234D4—142no—
235D4—281no—
236D4—281no—
237D4—281no—
238D4—281no—
239D4—562no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—72no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—141no—
248C2×C4—141no—
249C2×C4—141no—
250C2×C4—141no—
251C2×C4—142no—
252C2×C4—141no—
253C2×C4—281no—
254C2×C4—562no—
255C2×C4—561no—
256C2×C4—561no—
257C7#191311yescharacteristic, complemented, Sylow
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—142no—
266C22—282no—
267C22—282no—
268C22—282no—
269C22—281no—
270C22—561no—
271C22—561no—
272C22—561no—
273C22C4×D2811yescharacteristic
274C22D7×C22⋊C411yescharacteristic
275C22#644511yescharacteristic
276C22#644711yescharacteristic
277C22#645611yescharacteristic
278C22#645711yescharacteristic
279C22#645811yescharacteristic
280C4—42no—
281C4—42no—
282C4—81no—
283C4—142no—
284C4—282no—
285C4—281no—
286C4—561no—
287C2—21no—
288C2—21no—
289C2—21no—
290C2—21no—
291C2—21no—
292C2—21no—
293C2—21no—
294C2—282no—
295C2—561no—
296C2#8943711yescharacteristic
subgroupG/H
centerC2#89437
commutatorC23×C7C22×C4
FrattiniC24C22×D7
FittingC7×#342C2
socleC2×C7#386
prime radical#90184C2
cosocleC24×C7C23
hypercenter#342D7
hypocenterC7#1913
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. #90177C2semidirect
3. C2×#6380C2semidirect
4. C2×#6380C2semidirect
5. C2×#6380C2semidirect
6. C7×#342C2semidirect
7. #89322C2semidirect
18. #6449C4semidirect
20. #6445C4semidirect
29. C23×D7C2×C4semidirect
48. C22×Dic7C2×C4semidirect
56. C7×C22⋊C4C2×C4semidirect
57. C7×C22⋊C4D4semidirect
61. C22⋊Dic7D4semidirect
74. #342D7semidirect
202. C22⋊C4C4×D7semidirect
203. C22⋊C4D28semidirect
257. C7#1913semidirect

Maximal subgroups

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

subgroupindex
#901842
#901772
C2×#63802
C7×#3422
#893222
#19137