Finite Group Databasebeta

← #11739072

Subgroups — #11739072

Order 896.

326 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 115 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#90412C211yesmaximal, characteristic
2#90412C211yesmaximal, characteristic
3C7×#487C211yesmaximal, characteristic
4#89866C211yesmaximal, characteristic
5#89862C211yesmaximal, characteristic
6#89569C212yesmaximal
7#89568C212yesmaximal
8#89565C212yesmaximal
9#89561C212yesmaximal
10C4×Dic28C211yesmaximal, characteristic
11#89451C211yesmaximal, characteristic
12C7×C4⋊Q8C2211yescharacteristic
13C7×C4⋊Q8C2211yescharacteristic
14Q8×Dic7—21no—
15Q8×Dic7C2212yes—
16#6507—21no—
17#6507C2212yes—
18C2×#1144C2212yes—
19C2×Dic28C2211yescharacteristic
20#6453—21no—
21#6453C2212yes—
22#6452—22no—
23#6452—22no—
24#6450—21no—
25#6450C2212yes—
26C4×Dic14C2211yescharacteristic
27C4×Dic14C2211yescharacteristic
28C4×Dic14C2211yescharacteristic
29C7×#103C2212yes—
30C7×#103C2212yes—
31C4×C7×C8C2211yescharacteristic
32#6409C2212yes—
33#6392C2211yescharacteristic
34#6390C2211yescharacteristic
35#6390C2211yescharacteristic
36#6389C2212yes—
37Dic7⋊C8C2212yes—
38#6384C2212yes—
39#6382C2212yes—
40C4⋊(C7⋊C8)C2211yescharacteristic
41#3374—71nomaximal, Sylow
42C2×C7×Q8—21no—
43C2×C7×Q8C2312yes—
44C2×Dic14D411yescharacteristic
45C2×Dic14C2311yescharacteristic
46C2×Dic14C2311yescharacteristic
47C2×C7×C8C2311yescharacteristic
48C2×C7×C8C2311yescharacteristic
49C7×C4⋊C4—21no—
50C7×C4⋊C4—21no—
51C7×C4⋊C4—21no—
52C7×C4⋊C4C2312yes—
53C42×C7C2311yescharacteristic
54#1144—24no—
55C4⋊Dic7—22no—
56C4⋊Dic7—22no—
57C4⋊Dic7D411yescharacteristic
58C4⋊Dic7C2311yescharacteristic
59C4⋊Dic7C2311yescharacteristic
60Dic7⋊C4—21no—
61Dic7⋊C4—21no—
62Dic7⋊C4—22no—
63Dic7⋊C4—22no—
64Dic7⋊C4—22no—
65Dic7⋊C4—22no—
66Dic7⋊C4D412yes—
67C4×Dic7—21no—
68C4×Dic7—21no—
69C4×Dic7—22no—
70C4×Dic7—22no—
71C4×Dic7C2312yes—
72C2×C7⋊C8C2312yes—
73Dic28—22no—
74#557—71no—
75#557—71no—
76#491—72no—
77#487D711yescharacteristic
78#479—72no—
79#475—71no—
80#462—71no—
81#462—72no—
82#446—71no—
83#441—72no—
84C4×Q16—71no—
85C7×Q8—22no—
86C7×Q8—22no—
87C7×Q8—42no—
88C2×C4×C7—22no—
89C2×C4×C7—22no—
90C2×C4×C7C2×D411yescharacteristic
91C2×C4×C7C2×D411yescharacteristic
92C2×C4×C7C2411yescharacteristic
93C2×Dic7—21no—
94C2×Dic7—21no—
95C2×Dic7—21no—
96C2×Dic7—21no—
97C2×Dic7C2×D412yes—
98C2×Dic7C2×D412yes—
99Dic14—22no—
100Dic14—21no—
101Dic14—21no—
102Dic14—21no—
103Dic14#4112yes—
104C7×C8—21no—
105C7×C8—21no—
106C7⋊C8—22no—
107C2×Q16—72no—
108C2×Q16—71no—
109C4⋊Q8—72no—
110C4⋊Q8—141no—
111C4⋊Q8C2×D711yescharacteristic
112C4⋊Q8C2×D711yescharacteristic
113#125—72no—
114#125—142no—
115#125—142no—
116#125—141no—
117C4×Q8—71no—
118C4×Q8—72no—
119C4×Q8—72no—
120C4×Q8—71no—
121C4×Q8—71no—
122C4×Q8—141no—
123C4×Q8—141no—
124#107—71no—
125#106—72no—
126C4⋊C8—72no—
127C4⋊C8—71no—
128#103—72no—
129#103—71no—
130#103—71no—
131#103—72no—
132#103C2×D712yes—
133#103C2×D712yes—
134#97—72no—
135C4×C8C2×D711yescharacteristic
136C22×C7C22×D411yescharacteristic
137C4×C7—21no—
138C4×C7—42no—
139C4×C7—42no—
140C4×C7#13712yes—
141C4×C7C2×#4111yescharacteristic
142C4×C72−1+411yescharacteristic
143Dic7—22no—
144Dic7—22no—
145Dic7—22no—
146Dic7—41no—
147Dic7—41no—
148Dic7—41no—
149C2×Q8—21no—
150C2×Q8—71no—
151C2×Q8—71no—
152C2×Q8—71no—
153C2×Q8C22×D712yes—
154Q16—142no—
155Q16—144no—
156C2×C8—72no—
157C2×C8C22×D711yescharacteristic
158C2×C8C22×D711yescharacteristic
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—72no—
163C4⋊C4—71no—
164C4⋊C4—71no—
165C4⋊C4—71no—
166C4⋊C4—142no—
167C4⋊C4—142no—
168C4⋊C4—142no—
169C4⋊C4—142no—
170C4⋊C4—142no—
171C4⋊C4—142no—
172C4⋊C4—141no—
173C4⋊C4—141no—
174C4⋊C4C22×D712yes—
175C42—72no—
176C42—142no—
177C42—142no—
178C42—141no—
179C42—141no—
180C42C22×D711yescharacteristic
181C2×C7#54811yescharacteristic
182C2×C7C2×#13711yescharacteristic
183C2×C7#57811yescharacteristic
184Q8—22no—
185Q8—22no—
186Q8—42no—
187Q8—72no—
188Q8—142no—
189Q8—141no—
190Q8—141no—
191Q8—141no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—72no—
195C2×C4—72no—
196C2×C4—141no—
197C2×C4—141no—
198C2×C4—141no—
199C2×C4—141no—
200C2×C4D4×D711yescharacteristic
201C2×C4D4×D711yescharacteristic
202C2×C4C23×D711yescharacteristic
203C8—21no—
204C8—21no—
205C8—142no—
206C7#337411yescharacteristic, complemented, Sylow
207C22C2×D4×D711yescharacteristic
208C4—21no—
209C4—42no—
210C4—42no—
211C4—142no—
212C4—142no—
213C4—142no—
214C4—281no—
215C4—281no—
216C4—281no—
217C4#647812yes—
218C4D7×#4111yescharacteristic
219C4#655411yescharacteristic
220C2#9036811yescharacteristic
221C2C2×#647811yescharacteristic
222C2#9045711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#487C2
socle, prime radicalC22×C7C22×D4
hypercenter#487D7
hypocenterC7#3374
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
206. C7#3374semidirect

Maximal subgroups

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

subgroupindex
#904122
C7×#4872
#898662
#898622
#895692
#895682
#895652
#895612
C4×Dic282
#894512
#33747