Finite Group Databasebeta

← #11737379

Subgroups — #11737379

Order 896.

336 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 123 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#90275C211yesmaximal, characteristic
2#90258C211yesmaximal, characteristic, complemented
3C7×C8×Q8C211yesmaximal, characteristic, complemented
4#89621C211yesmaximal, characteristic, complemented
5#89618C211yesmaximal, characteristic
6#89617C212yesmaximal
7#89601C212yesmaximal, complemented
8C4×Dic28C212yesmaximal, complemented
9#89469C212yesmaximal
10C4×#1132C211yesmaximal, characteristic
11#89453C211yesmaximal, characteristic, complemented
12C4×C7×Q8C2211yescharacteristic
13#6509—21no—
14#6507—21no—
15C2×Dic28C2212yes—
16C2×#1132C2211yescharacteristic
17#6461—22no—
18#6455—21no—
19#6452—22no—
20#6450—21no—
21#6438C2212yes—
22C4×D28C2211yescharacteristic
23#6433C2212yes—
24#6432C2211yescharacteristic
25#6432C2211yescharacteristic
26C4×Dic14C2212yes—
27C4×Dic14C2212yes—
28C4×Dic14C2211yescharacteristic
29C7×C4⋊C8C2212yes—
30C7×C4⋊C8C2211yescharacteristic
31C4×C7×C8C2211yescharacteristic
32C4×C7×C8C2212yes—
33#6395C2212yes—
34#6395C2211yescharacteristic
35#6392C2212yes—
36#6391C2211yescharacteristic
37#6390C2212yes—
38#6390C2212yes—
39#6390C2212yes—
40#6390C2211yescharacteristic
41#3364—71nomaximal, Sylow
42C2×C7×Q8D411yescharacteristic
43C2×D28C2311yescharacteristic
44C2×C4×D7—21no—
45C2×Dic14—21no—
46C2×Dic14C2312yes—
47C2×Dic14C2311yescharacteristic
48C2×C7×C8C2311yescharacteristic
49C2×C7×C8C2311yescharacteristic
50C2×C7×C8C2312yes—
51C7×C4⋊C4D411yescharacteristic
52C7×C4⋊C4D412yes—
53C42×C7C2312yes—
54C42×C7C2311yescharacteristic
55(C2×D7)⋊C4—21no—
56(C2×D7)⋊C4—22no—
57(C2×D7)⋊C4—22no—
58C4⋊Dic7—21no—
59C4⋊Dic7—21no—
60C4⋊Dic7—21no—
61C4⋊Dic7C2311yescharacteristic
62C4⋊Dic7C2312yes—
63C4⋊Dic7C2311yescharacteristic
64Dic7⋊C4—21no—
65Dic7⋊C4—22no—
66Dic7⋊C4—22no—
67Dic7⋊C4—22no—
68Dic7⋊C4—22no—
69C4×Dic7—22no—
70C4×Dic7—21no—
71C4×Dic7—22no—
72Dic28—24no—
73#1132—22no—
74#557—71no—
75#556—71no—
76#488—72no—
77#487—71no—
78#479—72no—
79#474—71no—
80#464—72no—
81#462—71no—
82C8×Q8D711yescharacteristic, complemented
83C4×Q16—72no—
84C4×SD16—71no—
85C22×D7—21no—
86C7×Q8—22no—
87C2×C4×C7C2×D412yes—
88C2×C4×C7C2×D412yes—
89C2×C4×C7C2×D411yescharacteristic
90C2×C4×C7C2×D411yescharacteristic
91C2×C4×C7C2411yescharacteristic
92C2×Dic7—21no—
93C2×Dic7—21no—
94C2×Dic7—21no—
95C2×Dic7—22no—
96C2×Dic7—22no—
97D28—21no—
98D28—21no—
99C4×D7—42no—
100Dic14—21no—
101Dic14—22no—
102Dic14—22no—
103Dic14—42no—
104Dic14#4112yes—
105C7×C8—21no—
106C7×C8—22no—
107C7×C8—21no—
108C2×Q16—72no—
109C2×SD16—71no—
110C4⋊Q8—71no—
111C4⋊Q8—71no—
112C4⋊Q8—141no—
113#126—142no—
114#125—72no—
115#125—142no—
116#124—72no—
117C22⋊Q8—141no—
118C4×Q8—72no—
119C4×Q8—71no—
120C4×Q8—72no—
121C4×Q8—141no—
122C4×Q8C2×D711yescharacteristic
123C4×D4—71no—
124#117—141no—
125#107—72no—
126#106—71no—
127C4⋊C8C2×D711yescharacteristic
128C4⋊C8C2×D712yes—
129#103—71no—
130#103—72no—
131#103—72no—
132#103—72no—
133#102—71no—
134#102—72no—
135C4×C8C2×D711yescharacteristic
136C4×C8C2×D712yes—
137C22×C7C22×D411yescharacteristic
138C2×D7—41no—
139C2×D7—41no—
140C2×D7—41no—
141C4×C7—22no—
142C4×C7—22no—
143C4×C7—21no—
144C4×C7#13512yes—
145C4×C7C2×#4111yescharacteristic
146C4×C72−1+411yescharacteristic
147Dic7—22no—
148Dic7—42no—
149Dic7—42no—
150Dic7—41no—
151Dic7—41no—
152C2×Q8—71no—
153C2×Q8—72no—
154C2×Q8—141no—
155C2×Q8D2811yescharacteristic
156C2×D4—71no—
157C22×C4—141no—
158Q16—144no—
159SD16—142no—
160C2×C8C22×D711yescharacteristic
161C2×C8C22×D712yes—
162C2×C8C22×D711yescharacteristic
163C4⋊C4—71no—
164C4⋊C4—72no—
165C4⋊C4—71no—
166C4⋊C4—142no—
167C4⋊C4—141no—
168C4⋊C4—142no—
169C4⋊C4—142no—
170C4⋊C4—142no—
171C4⋊C4—141no—
172C4⋊C4—141no—
173C4⋊C4—141no—
174C4⋊C4D2811yescharacteristic
175C4⋊C4D2812yes—
176C22⋊C4—142no—
177C22⋊C4—141no—
178C22⋊C4—142no—
179C42—142no—
180C42—142no—
181C42—141no—
182C42C22×D711yescharacteristic
183C42C22×D712yes—
184C2×C7#54811yescharacteristic
185C2×C7C2×#13511yescharacteristic
186C2×C7#57811yescharacteristic
187D7—81no—
188C23—141no—
189Q8—22no—
190Q8—72no—
191Q8—141no—
192Q8—142no—
193Q8—142no—
194Q8—282no—
195D4—141no—
196D4—141no—
197C2×C4—141no—
198C2×C4—141no—
199C2×C4—142no—
200C2×C4—141no—
201C2×C4—142no—
202C2×C4—282no—
203C2×C4C2×D2812yes—
204C2×C4C2×D2811yescharacteristic
205C2×C4C2×D2812yes—
206C2×C4C2×D2811yescharacteristic
207C2×C4C23×D711yescharacteristic
208C8—21no—
209C8—21no—
210C8—22no—
211C7#336411yescharacteristic, complemented, Sylow
212C22—281no—
213C22—281no—
214C22—281no—
215C22C22×D2811yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—142no—
220C4—281no—
221C4—282no—
222C4—281no—
223C4—282no—
224C4#646712yes—
225C4#655111yescharacteristic
226C4D7×#4111yescharacteristic
227C2—561no—
228C2#9026711yescharacteristic
229C2C2×#646711yescharacteristic
230C2#9044411yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×C8×Q8C2
socleC22×C7C22×D4
prime radicalC2×D28C23
hypercenterC8×Q8D7
hypocenterC7#3364
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8

Complements

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

NH ≅ G/Ntype
2. #90258C2semidirect
3. C7×C8×Q8C2semidirect
4. #89621C2semidirect
7. #89601C2semidirect
8. C4×Dic28C2semidirect
11. #89453C2semidirect
82. C8×Q8D7semidirect
211. C7#3364semidirect

Maximal subgroups

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

subgroupindex
#902752
#902582
C7×C8×Q82
#896212
#896182
#896172
#896012
C4×Dic282
#894692
C4×#11322
#894532
#33647