Finite Group Databasebeta

← #31906613

Subgroups — #31906613

Order 18176.

454 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 93 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#22590961C211yesmaximal, characteristic
2#324×D71C212yesmaximal
3#22586286C211yesmaximal, characteristic, complemented
4#22586283C212yesmaximal, complemented
5C71×#1325C211yesmaximal, characteristic, complemented
6C2×#14556086C411yescharacteristic
7C2×#14556085C411yescharacteristic
8C2×C4⋊C4×D71C2211yescharacteristic
9C22⋊C8×D71—22no—
10#17985894—22no—
11#17985870—22no—
12#17985868C411yescharacteristic
13#17985865C411yescharacteristic
14#17985864C2211yescharacteristic
15#17985725C2212yes—
16#17985724—22no—
17#384×C71C2211yescharacteristic
18#324×C71C2212yes—
19C22×C4×D71C2×C411yescharacteristic
20C22×C4×D71D412yes—
21C2×C8×D71—42no—
22#14556086—42no—
23#14556085—22no—
24#14556085C814yescomplemented
25C2×C4⋊Dic71C2×C411yescharacteristic
26C2×C4⋊Dic71C2×C411yescharacteristic
27C2×Dic71⋊C4—21no—
28C2×Dic71⋊C4—22no—
29C2×Dic71⋊C4D412yes—
30C4⋊C4×D71—42no—
31C2×(C2×D71)⋊C4—21no—
32C2×(C2×D71)⋊C4—21no—
33C2×C4×Dic71—21no—
34C2×C4×Dic71C2×C411yescharacteristic
35C22⋊(C71⋊C8)—22no—
36(C2×D71)⋊C8—42no—
37C71⋊#95—22no—
38C71⋊#95—22no—
39C2×C4⋊C4×C71—22no—
40C2×C4⋊C4×C71C2×C411yescharacteristic
41C2×C4⋊C4×C71C2311yescharacteristic
42C2×C42×C71C2×C411yescharacteristic
43C22⋊C8×C71—22no—
44C23×D71C22⋊C411yescharacteristic
45C22×Dic71—21no—
46C22×Dic71—21no—
47C22×Dic71C22⋊C411yescharacteristic
48C22×Dic71C22⋊C412yes—
49C2×C4×D71—22no—
50C2×C4×D71—22no—
51C2×C4×D71—22no—
52C2×C4×D71—22no—
53C2×C4×D71—82no—
54C2×C4×D71C2×C812yes—
55C2×C4×D71M1612yes—
56C2×C71⋊C8—42no—
57C8×D71—84no—
58C4⋊Dic71—41no—
59C4⋊Dic71—41no—
60Dic71⋊C4—24no—
61Dic71⋊C4—42no—
62Dic71⋊C4—82no—
63(C2×D71)⋊C4—24no—
64(C2×D71)⋊C4—42no—
65C4×Dic71—21no—
66C4×Dic71—42no—
67C4×Dic71C2×C812yescomplemented
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71C22×C411yescharacteristic
71C22×C4×C71C2×D412yes—
72C2×C8×C71—42no—
73C4⋊C4×C71—41no—
74C4⋊C4×C71—41no—
75C4⋊C4×C71—82no—
76C42×C71—21no—
77C42×C71C2×C812yescomplemented
78C23×C71C2×C22⋊C411yescharacteristic
79C22×D71—41no—
80C22×D71—41no—
81C22×D71C22⋊C812yes—
82C22×D71#9912yes—
83C22×D71#10112yes—
84C2×C4×C71—21no—
85C2×C4×C71—21no—
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×C4×C71—41no—
89C2×C4×C71—41no—
90C2×C4×C71—42no—
91C2×C4×C71—42no—
92C2×C4×C71—41no—
93C2×C4×C71C22×C811yescharacteristic
94C2×C4×C71C2×M1611yescharacteristic
95C2×Dic71—21no—
96C2×Dic71—21no—
97C2×Dic71—22no—
98C2×Dic71—22no—
99C2×Dic71—41no—
100C2×Dic71—41no—
101C2×Dic71—42no—
102C2×Dic71—42no—
103C2×Dic71—41no—
104C2×Dic71C22⋊C812yes—
105C4×D71—24no—
106C4×D71—24no—
107C4×D71—44no—
108C4×D71—82no—
109C8×C71—82no—
110C71⋊C8—82no—
111C22×C71—21no—
112C22×C71—21no—
113C22×C71C2×C22⋊C811yescharacteristic
114C22×C71C2×#9911yescharacteristic
115C22×C71C2×#10111yescharacteristic
116C2×D71—22no—
117C2×D71—22no—
118C2×D71—42no—
119C2×D71—42no—
120C2×D71#32414yes—
121C4×C71—21no—
122C4×C71—21no—
123C4×C71—42no—
124C4×C71—42no—
125C4×C71—81no—
126Dic71—22no—
127Dic71—42no—
128Dic71—42no—
129Dic71—81no—
130#11448—711nomaximal, Sylow
131C2×C71—21no—
132C2×C71—21no—
133C2×C71C2×#32411yescharacteristic
134C2×C71#212711yescharacteristic
135C2×C71#213311yescharacteristic
136D71—24no—
137#2317—711no—
138C2×#324—712no—
139#1325—711no—
140#1325D7111yescharacteristic, complemented
141#1324—712no—
142C71#1144811yescharacteristic, complemented, Sylow
143C2×#117—711no—
144C2×#117—711no—
145C22×C4⋊C4—711no—
146C2×C22⋊C8—1422no—
147#389—1422no—
148#385—1422no—
149#384—711no—
150#384C2×D7111yescharacteristic
151#382—711no—
152#382—711no—
153#324—712no—
154#324—1422no—
155#324C2×D7112yes—
156C23×C4—712no—
157C23×C4—711no—
158C22×C8—2842no—
159#117—2842no—
160#117—714no—
161#117—1422no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—2842no—
164C2×C4⋊C4—711no—
165C2×C4⋊C4—712no—
166C2×C4⋊C4—711no—
167C2×C4⋊C4—1422no—
168C2×C4⋊C4—1421no—
169C2×C4⋊C4C4×D7111yescharacteristic
170C2×C4⋊C4C22×D7111yescharacteristic
171C2×C22⋊C4—1421no—
172C2×C22⋊C4—1421no—
173C2×C42—711no—
174C2×C42—1421no—
175C2×C42C4×D7111yescharacteristic
176C22⋊C8—22no—
177C22⋊C8—2842no—
178C22⋊C8—1422no—
179#95—1422no—
180#95—1422no—
181C24—711no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—5682no—
185C22×C4—712no—
186C22×C4—711no—
187C22×C4—712no—
188C22×C4—712no—
189C22×C4—1422no—
190C22×C4—1422no—
191C22×C4—1422no—
192C22×C4—1422no—
193C22×C4—1421no—
194C22×C4—1421no—
195C22×C4C2×C4×D7111yescharacteristic
196C22×C4D4×D7112yes—
197C2×C8—42no—
198C2×C8—2842no—
199C2×C8—5684no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—82no—
203C4⋊C4—2841no—
204C4⋊C4—2842no—
205C4⋊C4—2841no—
206C4⋊C4—5682no—
207C4⋊C4—1424no—
208C22⋊C4—2842no—
209C22⋊C4—1424no—
210C42—21no—
211C42—2842no—
212C42—712no—
213C42—1421no—
214C42C8×D7112yescomplemented
215C23—2841no—
216C23—2841no—
217C23—712no—
218C23—712no—
219C23—712no—
220C23C22⋊C4×D7111yescharacteristic
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—2844no—
231C2×C4—2841no—
232C2×C4—2841no—
233C2×C4—2842no—
234C2×C4—2842no—
235C2×C4—2841no—
236C2×C4—5682no—
237C2×C4—712no—
238C2×C4—1424no—
239C2×C4—1424no—
240C2×C4—1422no—
241C2×C4—1422no—
242C2×C4—1421no—
243C2×C4—1421no—
244C2×C4C2×C8×D7111yescharacteristic
245C2×C4M16×D7111yescharacteristic
246C8—82no—
247C8—5682no—
248C22—21no—
249C22—21no—
250C22—2842no—
251C22—2842no—
252C22—714no—
253C22—1422no—
254C22—1422no—
255C22C22⋊C8×D7111yescharacteristic
256C22#99×D7111yescharacteristic
257C22#101×D7111yescharacteristic
258C4—21no—
259C4—21no—
260C4—42no—
261C4—42no—
262C4—81no—
263C4—2842no—
264C4—2842no—
265C4—5681no—
266C4—1422no—
267C2—21no—
268C2—21no—
269C2—1424no—
270C2#324×D7111yescharacteristic
271C2#2259000611yescharacteristic
272C2#2259002811yescharacteristic
subgroupG/H
centerC22#99×D71
commutatorC2×C4×C71C22×C8
FrattiniC2×C4⋊C4C22×D71
FittingC71×#1325C2
socleC22×C71C2×#99
prime radicalC23×D71C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#1325D71
hypocenterC71#11448
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
3. #22586286C2semidirect
4. #22586283C2semidirect
5. C71×#1325C2semidirect
24. #14556085C8semidirect
67. C4×Dic71C2×C8semidirect
77. C42×C71C2×C8semidirect
140. #1325D71semidirect
142. C71#11448semidirect
214. C42C8×D71semidirect

Maximal subgroups

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

subgroupindex
#225909612
#324×D712
#225862862
#225862832
C71×#13252
#1144871