Finite Group Databasebeta

← #11738153

Subgroups — #11738153

Order 896.

420 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 125 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#90302C212yesmaximal, complemented
2C7×#466C211yesmaximal, characteristic, complemented
3#89929C211yesmaximal, characteristic, complemented
4D8×Dic7C211yesmaximal, characteristic, complemented
5C8×C22⋊D7C211yesmaximal, characteristic, complemented
6C2×#6392C211yesmaximal, characteristic, complemented
7#89805C212yesmaximal, complemented
8D7×#107C211yesmaximal, characteristic, complemented
9#89646C211yesmaximal, characteristic, complemented
10#89546C212yesmaximal, complemented
11#89531C212yesmaximal, complemented
12C2×#1159—22no—
13C2×C7×D8C2211yescharacteristic
14C22×C7×C8C2211yescharacteristic
15C7×#121C2212yes—
16#6501C2212yes—
17#6498—22no—
18D4×Dic7C2212yes—
19C4×C22⋊D7C2211yescharacteristic
20C2×C4⋊Dic7C2212yes—
21C2×C8×D7C2211yescharacteristic
22D7×C4⋊C4C2212yes—
23#6451C2212yes—
24#6446—22no—
25#6441—22no—
26C7×#107C2211yescharacteristic
27C7×#102C2212yes—
28#6406C2212yes—
29#6406C2212yes—
30C22⋊(C7⋊C8)C2211yescharacteristic
31(C2×D7)⋊C8C2211yescharacteristic
32#6392C2211yescharacteristic
33#6392C2211yescharacteristic
34#6392C2212yes—
35Dic7⋊C8C2211yescharacteristic
36C8×Dic7C2211yescharacteristic
37#6381C2212yes—
38#6381C2212yes—
39#3334—71nomaximal, Sylow
40C2×C7×D4—22no—
41C2×C7×D4C2312yes—
42C22×C4×C7C2311yescharacteristic
43C2×C22⋊D7—22no—
44C2×C22⋊D7D411yescharacteristic
45C22×Dic7—22no—
46C22×Dic7—22no—
47#1159—48no—
48C2×C4×D7—22no—
49C2×C4×D7C2311yescharacteristic
50C2×Dic14—22no—
51C7×D8—22no—
52C2×C7×C8—22no—
53C2×C7×C8C2311yescharacteristic
54C2×C7×C8C2311yescharacteristic
55C7×C4⋊C4C2312yes—
56C7×C22⋊C4—22no—
57C22⋊Dic7—22no—
58C22⋊Dic7—22no—
59C22⋊Dic7D411yescharacteristic
60(C2×D7)⋊C4D411yescharacteristic
61C4⋊Dic7—22no—
62C4⋊Dic7C2312yes—
63C4⋊Dic7C2312yes—
64Dic7⋊C4—22no—
65Dic7⋊C4—22no—
66Dic7⋊C4D411yescharacteristic
67C4×Dic7C2311yescharacteristic
68C2×C7⋊C8C2311yescharacteristic
69C2×C7⋊C8C2311yescharacteristic
70C8×D7—22no—
71#547—72no—
72#500—71no—
73#480—72no—
74#480—72no—
75#474—72no—
76#466—71no—
77#466D711yescharacteristic, complemented
78C4×D8—71no—
79C8×D4—71no—
80C2×#107—71no—
81C2×#107—71no—
82C23×C7—22no—
83C23×C7C2×D411yescharacteristic
84C22×D7C2×D411yescharacteristic
85C7×D4—22no—
86C7×D4—22no—
87C7×D4—44no—
88C2×C4×C7—21no—
89C2×C4×C7—21no—
90C2×C4×C7—22no—
91C2×C4×C7C2×D411yescharacteristic
92C2×C4×C7C2411yescharacteristic
93C22⋊D7—44no—
94C22⋊D7D814yescomplemented
95C2×Dic7—22no—
96C2×Dic7—22no—
97C2×Dic7—44no—
98C2×Dic7—44no—
99C2×Dic7C2×D411yescharacteristic
100C2×Dic7C2×D411yescharacteristic
101C2×Dic7C2×D411yescharacteristic
102C4×D7—21no—
103C4×D7—21no—
104C4×D7—44no—
105Dic14—44no—
106C7×C8—21no—
107C7×C8#4112yescomplemented
108C7⋊C8—21no—
109C7⋊C8—21no—
110C2×#41—142no—
111C2×D8C2×D711yescharacteristic
112C22×C8—71no—
113C22×C8C2×D711yescharacteristic
114C4⋊Q8—72no—
115#123—142no—
116#123—142no—
117C22⋊Q8—142no—
118#121—72no—
119#121C2×D712yes—
120C4×D4—71no—
121C4×D4—72no—
122C2×C4⋊C4—72no—
123C2×C4⋊C4—72no—
124#107—71no—
125#107—71no—
126#107—72no—
127#107—72no—
128#107—72no—
129#107C2×D711yescharacteristic
130C4⋊C8—71no—
131#102—72no—
132#102—72no—
133#102C2×D712yes—
134C22⋊C8—71no—
135C22⋊C8—71no—
136C4×C8—71no—
137C22×C7—21no—
138C22×C7—21no—
139C22×C7—42no—
140C22×C7—42no—
141C22×C7—42no—
142C22×C7C2×D812yes—
143C22×C7C22×D411yescharacteristic
144C2×D7—21no—
145C2×D7—21no—
146C2×D7C2×D812yes—
147C4×C7—21no—
148C4×C7—42no—
149C4×C7C2×#4111yescharacteristic
150C4×C72−1+411yescharacteristic
151Dic7—21no—
152Dic7—21no—
153Dic7—42no—
154Dic7—42no—
155Dic7C2×D812yes—
156#41—288no—
157C2×Q8—142no—
158C2×D4—22no—
159C2×D4—71no—
160C2×D4—142no—
161C2×D4C22×D712yes—
162C22×C4—71no—
163C22×C4—142no—
164C22×C4—142no—
165C22×C4—142no—
166C22×C4C22×D711yescharacteristic
167D8—22no—
168C2×C8—22no—
169C2×C8—71no—
170C2×C8—71no—
171C2×C8—142no—
172C2×C8C22×D711yescharacteristic
173C2×C8C22×D711yescharacteristic
174C4⋊C4—71no—
175C4⋊C4—72no—
176C4⋊C4—72no—
177C4⋊C4—142no—
178C4⋊C4—142no—
179C4⋊C4—142no—
180C4⋊C4C22×D712yes—
181C22⋊C4—22no—
182C22⋊C4—71no—
183C22⋊C4—71no—
184C22⋊C4—142no—
185C22⋊C4—142no—
186C42—71no—
187C2×C7—22no—
188C2×C7—82no—
189C2×C7#54711yescharacteristic
190C2×C7C22×D811yescharacteristic
191C2×C7#57811yescharacteristic
192D7—22no—
193C23—22no—
194C23—71no—
195C23D4×D711yescharacteristic
196Q8—284no—
197D4—22no—
198D4—22no—
199D4—44no—
200D4—74no—
201D4—284no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—71no—
206C2×C4—71no—
207C2×C4—71no—
208C2×C4—141no—
209C2×C4—141no—
210C2×C4—142no—
211C2×C4—142no—
212C2×C4—284no—
213C2×C4—284no—
214C2×C4—284no—
215C2×C4D4×D711yescharacteristic
216C2×C4C23×D711yescharacteristic
217C8—21no—
218C8—141no—
219C8—141no—
220C8#115912yescomplemented
221C7#333411yescharacteristic, complemented, Sylow
222C22—21no—
223C22—21no—
224C22—42no—
225C22—42no—
226C22—42no—
227C22—72no—
228C22—141no—
229C22—141no—
230C22D7×D812yes—
231C22C2×D4×D711yescharacteristic
232C4—21no—
233C4—42no—
234C4—72no—
235C4—141no—
236C4—141no—
237C4—282no—
238C4—282no—
239C4C2×#115911yescharacteristic
240C4#655411yescharacteristic
241C2—22no—
242C2—82no—
243C2—142no—
244C2#9030211yescharacteristic
245C2C2×D7×D811yescharacteristic
246C2#9046211yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#466C2
socleC22×C7C22×D4
hypercenter#466D7
hypocenterC7#3334
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #90302C2semidirect
2. C7×#466C2semidirect
3. #89929C2semidirect
4. D8×Dic7C2semidirect
5. C8×C22⋊D7C2semidirect
6. C2×#6392C2semidirect
7. #89805C2semidirect
8. D7×#107C2semidirect
9. #89646C2semidirect
10. #89546C2semidirect
11. #89531C2semidirect
77. #466D7semidirect
94. C22⋊D7D8semidirect
107. C7×C8#41semidirect
220. C8#1159semidirect
221. C7#3334semidirect

Maximal subgroups

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

subgroupindex
#903022
C7×#4662
#899292
D8×Dic72
C8×C22⋊D72
C2×#63922
#898052
D7×#1072
#896462
#895462
#895312
#33347