Finite Group Databasebeta

← #22594287

Subgroups — #22594287

Order 9088.

416 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14556127C211yesmaximal, characteristic
2C2×#14556118C211yesmaximal, characteristic
3#17985941C212yesmaximal
4#17985940C212yesmaximal
5#17985939C211yesmaximal, characteristic
6#17985938C211yesmaximal, characteristic
7#17985937C211yesmaximal, characteristic
8#17985903C212yesmaximal
9#17985903C212yesmaximal
10#17985844C211yesmaximal, characteristic
11#400×C71C211yesmaximal, characteristic
12C22×Dic142C2212yes—
13#14556127—24no—
14#14556118—24no—
15C2×C4⋊Dic71C2211yescharacteristic
16C2×C4⋊Dic71C2211yescharacteristic
17C2×Dic71⋊C4C2211yescharacteristic
18C2×Dic71⋊C4C2211yescharacteristic
19C2×Dic71⋊C4C2211yescharacteristic
20C2×Dic71⋊C4C2211yescharacteristic
21C2×Dic71⋊C4C2212yes—
22C2×C4×Dic71C2212yes—
23C2×C4×Dic71C2211yescharacteristic
24C71⋊#95C2211yescharacteristic
25C71⋊#95C2212yes—
26C71⋊#95C2212yes—
27C71⋊#95C2212yes—
28C71⋊#95C2212yes—
29C71⋊#95C2212yes—
30C71⋊#95C2212yes—
31C71⋊#95C2212yes—
32C2×C4⋊C4×C71C2211yescharacteristic
33C2×C4⋊C4×C71C2211yescharacteristic
34#95×C71C2211yescharacteristic
35#95×C71C2212yes—
36#95×C71C2212yes—
37C2×Dic142—24no—
38C2×Dic142—28no—
39C22×Dic71C2312yes—
40C22×Dic71C2312yes—
41C22×Dic71C2311yescharacteristic
42C22×Dic71C2311yescharacteristic
43C22×Dic71C2311yescharacteristic
44C22×Dic71C2311yescharacteristic
45C4⋊Dic71—22no—
46C4⋊Dic71—22no—
47Dic71⋊C4—22no—
48Dic71⋊C4—22no—
49Dic71⋊C4—22no—
50Dic71⋊C4—22no—
51Dic71⋊C4—24no—
52C4×Dic71—24no—
53C4×Dic71D414yes—
54C22×C4×C71C2312yes—
55C22×C4×C71C2312yes—
56C22×C4×C71C2311yescharacteristic
57C22×C4×C71C2311yescharacteristic
58C22×C4×C71C2311yescharacteristic
59C4⋊C4×C71—22no—
60C4⋊C4×C71—22no—
61C23×C71C2411yescharacteristic
62C2×C4×C71—22no—
63C2×C4×C71—21no—
64C2×C4×C71—21no—
65C2×C4×C71—22no—
66C2×C4×C71—22no—
67C2×C4×C71—24no—
68C2×C4×C71—21no—
69C2×C4×C71—22no—
70C2×C4×C71—22no—
71C2×C4×C71—22no—
72C2×C4×C71—21no—
73C2×C4×C71C2×D412yes—
74C2×Dic71—21no—
75C2×Dic71—21no—
76C2×Dic71—21no—
77C2×Dic71—21no—
78C2×Dic71—22no—
79C2×Dic71—22no—
80C2×Dic71—24no—
81C2×Dic71—22no—
82C2×Dic71—21no—
83C2×Dic71—22no—
84C2×Dic71—21no—
85C2×Dic71—22no—
86C2×Dic71C2×D412yes—
87C2×Dic71C2×D412yes—
88C2×Dic71#4114yes—
89Dic142—48no—
90C22×C71C22×D411yescharacteristic
91C22×C71C2×#4112yes—
92C22×C712−1+412yes—
93C22×C712−1+411yescharacteristic
94C22×C712−1+411yescharacteristic
95C4×C71—22no—
96C4×C71—42no—
97C4×C71—41no—
98C4×C71—41no—
99C4×C71—42no—
100Dic71—22no—
101Dic71—42no—
102Dic71—41no—
103Dic71—41no—
104Dic71—42no—
105Dic71#12414yes—
106C2×C71C2×#12411yescharacteristic
107C2×C71#53611yescharacteristic
108C2×C71#53611yescharacteristic
109C2×C71#54112yes—
110C2×C71#54112yes—
111#2642—711nomaximal, Sylow
112C71#264211yescharacteristic, complemented, Sylow
113C2×C4⋊Q8—711no—
114C2×#125—711no—
115#400—711no—
116#400—711no—
117#400—711no—
118#400—712no—
119#400—712no—
120#400D7111yescharacteristic
121#391—712no—
122#391—712no—
123#376—711no—
124C22×Q8—712no—
125C4⋊Q8—1424no—
126#125—1424no—
127C2×C4⋊C4—711no—
128C2×C4⋊C4—711no—
129C2×C4⋊C4—711no—
130C2×C4⋊C4—711no—
131C2×C4⋊C4—711no—
132C2×C4⋊C4—712no—
133C2×C4⋊C4—711no—
134C2×C4⋊C4C2×D7111yescharacteristic
135C2×C4⋊C4C2×D7111yescharacteristic
136C2×C42—711no—
137C2×C42—712no—
138#95—712no—
139#95—711no—
140#95—712no—
141#95—712no—
142#95—712no—
143#95—712no—
144#95—712no—
145#95—712no—
146#95C2×D7111yescharacteristic
147#95C2×D7112yes—
148#95C2×D7112yes—
149C2×Q8—1424no—
150C2×Q8—1428no—
151C22×C4—711no—
152C22×C4—711no—
153C22×C4—712no—
154C22×C4—711no—
155C22×C4—711no—
156C22×C4—712no—
157C22×C4C22×D7112yes—
158C22×C4C22×D7111yescharacteristic
159C22×C4C22×D7111yescharacteristic
160C22×C4C22×D7112yes—
161C22×C4C22×D7111yescharacteristic
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—1422no—
165C4⋊C4—1422no—
166C4⋊C4—1422no—
167C4⋊C4—1424no—
168C4⋊C4—1422no—
169C4⋊C4—1422no—
170C4⋊C4—1422no—
171C42—714no—
172C42—1424no—
173C23C23×D7111yescharacteristic
174Q8—2848no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—712no—
187C2×C4—714no—
188C2×C4—712no—
189C2×C4—1421no—
190C2×C4—1421no—
191C2×C4—1422no—
192C2×C4—1422no—
193C2×C4—1424no—
194C2×C4—1422no—
195C2×C4—1421no—
196C2×C4—1422no—
197C2×C4—1421no—
198C2×C4—1422no—
199C2×C4—1421no—
200C2×C4—1421no—
201C2×C4D4×D7112yes—
202C22C2×D4×D7111yescharacteristic
203C22#41×D7112yes—
204C22#1455617512yes—
205C22#1455617511yescharacteristic
206C22#1455617611yescharacteristic
207C4—22no—
208C4—41no—
209C4—41no—
210C4—42no—
211C4—42no—
212C4—2841no—
213C4—2841no—
214C4—2842no—
215C4—2842no—
216C4—714no—
217C4—1422no—
218C2#124×D7111yescharacteristic
219C2#1798657111yescharacteristic
220C2#1798657211yescharacteristic
221C2#1798660612yes—
222C2#1798660812yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#400×C71C2
hypercenter#400D71
hypocenterC71#2642
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
112. C71#2642semidirect