Finite Group Databasebeta

← #11736659

Subgroups — #11736659

Order 896.

472 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 141 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×#6472C211yesmaximal, characteristic, complemented
2C2×#6470C211yesmaximal, characteristic, complemented
3#90217C211yesmaximal, characteristic, complemented
4#90201C211yesmaximal, characteristic
5C7×#428C211yesmaximal, characteristic, complemented
6#89774C211yesmaximal, characteristic, complemented
7C2×#6384C211yesmaximal, characteristic, complemented
8#89659C212yesmaximal, complemented
9#89658C212yesmaximal, complemented
10#89640C212yesmaximal, complemented
11#89636C212yesmaximal, complemented
12C2×#1157C2211yescharacteristic
13C2×#1157C2211yescharacteristic
14C22×Dic14C2211yescharacteristic
15C2×C7×M16C2211yescharacteristic
16C7×#117C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18#6493—22no—
19C22×C7⋊C8C2211yescharacteristic
20#6472—24no—
21#6470—24no—
22C2×Dic28C2212yes—
23C2×#1132C2212yes—
24C2×#1131C2212yes—
25C2×C8×D7C2212yes—
26#6460—21no—
27#6460C2212yes—
28#6460C2212yes—
29#6460C2212yes—
30#6458—21no—
31#6458C2212yes—
32#6449—22no—
33#6438—21no—
34#6432—21no—
35C7×#107C2212yes—
36C7×#106C2212yes—
37#6384C2212yes—
38#6384C2211yescharacteristic
39#6384C2212yes—
40#6384C2211yescharacteristic
41#6383C2211yescharacteristic
42#6383C2211yescharacteristic
43#3063—71nomaximal, Sylow
44C22×C4×C7—21no—
45C22×C4×C7C2311yescharacteristic
46C2×C22⋊D7—21no—
47C2×C22⋊D7D412yes—
48C22×Dic7—21no—
49#1157—22no—
50#1157—22no—
51#1157—22no—
52#1157—22no—
53C2×D28—22no—
54C2×D28D411yescharacteristic
55C2×D28C2311yescharacteristic
56C2×C4×D7—21no—
57C2×C4×D7C2312yes—
58C2×Dic14—22no—
59C2×Dic14—22no—
60C2×Dic14—21no—
61C2×Dic14D411yescharacteristic
62C2×Dic14C2312yes—
63C2×Dic14C2311yescharacteristic
64C7×M16D414yescomplemented
65C2×C7×C8C2312yes—
66C7×C4⋊C4—21no—
67C7×C4⋊C4C2311yescharacteristic
68C7×C4⋊C4C2312yes—
69C7×C4⋊C4C2311yescharacteristic
70C7×C22⋊C4—21no—
71C42×C7—21no—
72(C2×D7)⋊C4—22no—
73(C2×D7)⋊C4—22no—
74(C2×D7)⋊C4—22no—
75(C2×D7)⋊C4—22no—
76C4⋊Dic7—22no—
77C4⋊Dic7—22no—
78C4⋊Dic7—22no—
79C2×C7⋊C8—22no—
80C2×C7⋊C8C2311yescharacteristic
81C2×C7⋊C8C2311yescharacteristic
82Dic28—24no—
83#1132—24no—
84#1131—24no—
85C8×D7—24no—
86C2×#137—71no—
87C2×#131—71no—
88#537—71no—
89#536—71no—
90#470—72no—
91#468—72no—
92#467—72no—
93#465—72no—
94#428D711yescharacteristic, complemented
95#416—71no—
96C2×#103—71no—
97C23×C7C2×D411yescharacteristic
98C22×D7—21no—
99C22×D7C2×D412yes—
100C2×C4×C7—21no—
101C2×C4×C7—21no—
102C2×C4×C7—22no—
103C2×C4×C7—81no—
104C2×C4×C7C2×D412yes—
105C2×C4×C7C2×D411yescharacteristic
106C2×C4×C7#4112yes—
107C2×C4×C7C2411yescharacteristic
108C22⋊D7—24no—
109C22⋊D7—42no—
110C2×Dic7—22no—
111C2×Dic7—21no—
112C2×Dic7—42no—
113C2×Dic7C2×D412yes—
114D28—22no—
115D28—21no—
116D28—21no—
117D28—82no—
118C4×D7—22no—
119C4×D7—22no—
120C4×D7—42no—
121Dic14—22no—
122Dic14—22no—
123Dic14—21no—
124Dic14—22no—
125Dic14—21no—
126Dic14—42no—
127C7×C8C2×D414yes—
128C7⋊C8—21no—
129C7⋊C8—21no—
130C2×#41—71no—
131C2×#41—71no—
132C22×Q8—71no—
133#137—144no—
134C2×Q16—72no—
135C2×SD16—72no—
136#131—144no—
137C2×M16—72no—
138C2×M16C2×D711yescharacteristic
139C22×C8—71no—
140C22×C8—72no—
141C4⋊Q8—141no—
142#124—141no—
143#123—142no—
144C22⋊Q8—72no—
145C22⋊Q8—72no—
146C22⋊Q8—72no—
147C22⋊Q8—141no—
148#121—72no—
149#121—141no—
150#121—142no—
151#117C2×D711yescharacteristic
152C2×C4⋊C4C2×D711yescharacteristic
153#107C2×D712yes—
154#106C2×D712yes—
155#103—72no—
156#103—72no—
157#103—71no—
158#103—71no—
159#102—71no—
160#102—71no—
161C22×C7—21no—
162C22×C7—21no—
163C22×C7#12112yes—
164C22×C7C22×D411yescharacteristic
165C2×D7—22no—
166C2×D7—22no—
167C2×D7—22no—
168C2×D7—41no—
169C2×D7—41no—
170C2×D7—41no—
171C4×C7—42no—
172C4×C7—41no—
173C4×C7—41no—
174C4×C7#12112yes—
175C4×C7C22×D411yescharacteristic
176C4×C7C2×#4111yescharacteristic
177Dic7—22no—
178Dic7—42no—
179Dic7—41no—
180#41—142no—
181#41—142no—
182#41—142no—
183#41—142no—
184C2×Q8—71no—
185C2×Q8—72no—
186C2×Q8—71no—
187C2×Q8—141no—
188C2×Q8—142no—
189C2×Q8—142no—
190C2×D4—72no—
191C2×D4—71no—
192C2×D4—71no—
193C2×D4—142no—
194C2×D4—141no—
195C22×C4—21no—
196C22×C4—72no—
197C22×C4—141no—
198C22×C4—141no—
199C22×C4C22×D711yescharacteristic
200Q16—144no—
201SD16—144no—
202M16—144no—
203M16D2814yescomplemented
204C2×C8—71no—
205C2×C8—71no—
206C2×C8—142no—
207C2×C8—144no—
208C2×C8C22×D712yes—
209C4⋊C4—21no—
210C4⋊C4—142no—
211C4⋊C4—142no—
212C4⋊C4—142no—
213C4⋊C4C22×D711yescharacteristic
214C4⋊C4C22×D711yescharacteristic
215C4⋊C4C22×D712yes—
216C22⋊C4—21no—
217C22⋊C4—142no—
218C22⋊C4—142no—
219C22⋊C4—142no—
220C22⋊C4—142no—
221C42—21no—
222C2×C7—22no—
223C2×C7C2×#12111yescharacteristic
224C2×C7#57711yescharacteristic
225C2×C7#57811yescharacteristic
226D7—42no—
227D7—81no—
228C23—72no—
229C23—141no—
230C23D4×D711yescharacteristic
231Q8—142no—
232Q8—142no—
233Q8—141no—
234Q8—142no—
235Q8—141no—
236Q8—282no—
237D4—144no—
238D4—142no—
239D4—141no—
240D4—141no—
241D4—282no—
242D4—562no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—72no—
247C2×C4—81no—
248C2×C4—142no—
249C2×C4—142no—
250C2×C4—141no—
251C2×C4—142no—
252C2×C4—282no—
253C2×C4—282no—
254C2×C4C2×D2812yes—
255C2×C4D4×D711yescharacteristic
256C2×C4#116112yes—
257C2×C4C23×D711yescharacteristic
258C8—141no—
259C8—141no—
260C8C2×D2814yes—
261C7#306311yescharacteristic, complemented, Sylow
262C22—21no—
263C22—21no—
264C22—142no—
265C22—142no—
266C22—142no—
267C22—281no—
268C22—281no—
269C22—281no—
270C22#645812yes—
271C22C2×D4×D711yescharacteristic
272C4—42no—
273C4—41no—
274C4—41no—
275C4—142no—
276C4—281no—
277C4—282no—
278C4#645812yes—
279C4C22×D2811yescharacteristic
280C4C2×#116111yescharacteristic
281C2—22no—
282C2—282no—
283C2—561no—
284C2C2×#645811yescharacteristic
285C2#9046611yescharacteristic
286C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#428C2
socleC22×C7C22×D4
prime radical#90201C2
hypercenter#428D7
hypocenterC7#3063
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#6472C2semidirect
2. C2×#6470C2semidirect
3. #90217C2semidirect
5. C7×#428C2semidirect
6. #89774C2semidirect
7. C2×#6384C2semidirect
8. #89659C2semidirect
9. #89658C2semidirect
10. #89640C2semidirect
11. #89636C2semidirect
64. C7×M16D4semidirect
94. #428D7semidirect
203. M16D28semidirect
261. C7#3063semidirect

Maximal subgroups

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

subgroupindex
C2×#64722
C2×#64702
#902172
#902012
C7×#4282
#897742
C2×#63842
#896592
#896582
#896402
#896362
#30637