Finite Group Databasebeta

← #31888102

Subgroups — #31888102

Order 18176.

562 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 167 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#22590804C211yesmaximal, characteristic
2#22588086C211yesmaximal, characteristic, complemented
3C2×#17985726C212yesmaximal
4#22586179C212yesmaximal, complemented
5C71×#1767C211yesmaximal, characteristic, complemented
6C2×C4×D284C2211yescharacteristic
7C2×#14556085—21no—
8C2×C4×C71⋊C8C2212yes—
9#17985901C412yes—
10#17985897C412yes—
11#17985890—22no—
12C4×C4⋊Dic71C2211yescharacteristic
13C4×(C2×D71)⋊C4—21no—
14#17985754—22no—
15#17985726C418yescomplemented
16C2×C4⋊C8×C71C2212yes—
17C2×#97×C71—21no—
18C43×C71C2211yescharacteristic
19#336×C71—21no—
20C22×D284C2×C411yescharacteristic
21C22×C4×D71—21no—
22C22×C71⋊C8—22no—
23C4×D284—22no—
24C4×D284C2×C414yes—
25#14556085—44no—
26C2×C4⋊Dic71C2×C412yes—
27C2×C4⋊Dic71C2×C411yescharacteristic
28C2×Dic71⋊C4—22no—
29C2×(C2×D71)⋊C4—22no—
30C2×(C2×D71)⋊C4—22no—
31C2×(C2×D71)⋊C4—21no—
32C2×C4×Dic71—21no—
33C4×C71⋊C8—22no—
34C4×C71⋊C8C2×C414yes—
35C71⋊#95—22no—
36C71⋊#95—21no—
37C22×C8×C71—22no—
38C2×C42×C71C2×C412yes—
39C2×C42×C71D411yescharacteristic
40C2×C42×C71D411yescharacteristic
41C2×C42×C71D412yes—
42C2×C42×C71C2311yescharacteristic
43C4⋊C8×C71—22no—
44C4⋊C8×C71C2×C414yes—
45#97×C71—42no—
46C23×D71—21no—
47C2×D284—24no—
48C2×D284—21no—
49C2×D284C4212yes—
50C22×Dic71—21no—
51C22×Dic71—22no—
52C2×C4×D71—24no—
53C2×C4×D71—41no—
54C2×C4×D71—42no—
55C2×C4×D71—41no—
56C2×C71⋊C8—24no—
57C2×C71⋊C8—44no—
58C4⋊Dic71—21no—
59C4⋊Dic71—24no—
60C4⋊Dic71C4212yes—
61Dic71⋊C4—44no—
62(C2×D71)⋊C4—24no—
63(C2×D71)⋊C4—44no—
64(C2×D71)⋊C4—44no—
65C4×Dic71—42no—
66C22×C4×C71C22⋊C412yes—
67C22×C4×C71C22⋊C412yes—
68C22×C4×C71C22×C411yescharacteristic
69C22×C4×C71C2×D411yescharacteristic
70C22×C4×C71C2×D411yescharacteristic
71C2×C8×C71—24no—
72C2×C8×C71—42no—
73C2×C8×C71—42no—
74C42×C71—24no—
75C42×C71—21no—
76C42×C71—22no—
77C42×C71—21no—
78C42×C71—42no—
79C42×C71C22×C412yes—
80C42×C71#4112yes—
81C23×C71C2×C22⋊C411yescharacteristic
82C22×D71—22no—
83C22×D71—42no—
84C22×D71—41no—
85C22×D71—41no—
86C22×D71—42no—
87C2×C4×C71—22no—
88C2×C4×C71—22no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71—22no—
94C2×C4×C71C2×C4211yescharacteristic
95C2×C4×C71#11711yescharacteristic
96C2×C4×C71C4×D412yes—
97C2×C4×C71C4×D412yes—
98C2×C4×C71#13114yes—
99C2×C4×C71AGL(1,ℤ/8)12yes—
100C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
101C2×C4×C71#13711yescharacteristic
102C2×Dic71—22no—
103C2×Dic71—41no—
104C2×Dic71—41no—
105C2×Dic71—42no—
106C2×Dic71—42no—
107C2×Dic71—42no—
108D284—24no—
109D284—42no—
110C4×D71—44no—
111C4×D71—44no—
112C8×C71—44no—
113C71⋊C8—44no—
114C22×C71C4×C22⋊C411yescharacteristic
115C22×C71#40812yes—
116C22×C71#41711yescharacteristic
117C22×C71#41811yescharacteristic
118C22×C71#42112yes—
119C2×D71—42no—
120C2×D71—42no—
121C2×D71—42no—
122C2×D71—82no—
123C2×D71—82no—
124C4×C71—22no—
125C4×C71—42no—
126C4×C71—42no—
127C4×C71#40512yes—
128C4×C71#43312yes—
129C4×C71#44012yes—
130C4×C71#44212yes—
131Dic71—42no—
132Dic71—82no—
133#8179—711nomaximal, Sylow
134C2×C71#149512yes—
135C2×C71#149612yes—
136C2×C71#175011yescharacteristic
137C2×C71#175911yescharacteristic
138C2×C71#176211yescharacteristic
139D71—82no—
140#2293—711no—
141#1767D7111yescharacteristic, complemented
142#1744—711no—
143C2×D4⋊C8—712no—
144#1279—712no—
145C71#817911yescharacteristic, complemented, Sylow
146C2×C4×D4—711no—
147C2×#117—1421no—
148C2×C4⋊C8C2×D7112yes—
149C2×#97—21no—
150C2×C4×C8—712no—
151#390—712no—
152#389—712no—
153#388—1422no—
154C4×C4⋊C4—711no—
155C4×C22⋊C4—1421no—
156C43C2×D7111yescharacteristic
157#336—21no—
158#336—1422no—
159D4⋊C8—718no—
160C22×D4—711no—
161C23×C4—1421no—
162C22×C8—22no—
163C22×C8—1422no—
164C4×D4—714no—
165C4×D4—1422no—
166#117—2844no—
167C2×C4⋊C4—711no—
168C2×C4⋊C4—712no—
169C2×C4⋊C4—1422no—
170C2×C22⋊C4—1422no—
171C2×C22⋊C4—1422no—
172C2×C22⋊C4—1421no—
173C2×C42—1421no—
174C2×C42C4×D7112yes—
175C2×C42D28412yes—
176C2×C42C22⋊D7111yescharacteristic
177C2×C42C22⋊D7111yescharacteristic
178C2×C42C22×D7111yescharacteristic
179C4⋊C8—22no—
180C4⋊C8C4×D7114yes—
181#97—42no—
182C4×C8—714no—
183C4×C8—1422no—
184#95—1422no—
185#95—1421no—
186C24—1421no—
187C2×D4—712no—
188C2×D4—1421no—
189C2×D4—1424no—
190C22×C4—2842no—
191C22×C4—2841no—
192C22×C4—2841no—
193C22×C4—1422no—
194C22×C4—1421no—
195C22×C4—1424no—
196C22×C4(C2×D71)⋊C412yes—
197C22×C4(C2×D71)⋊C412yes—
198C22×C4C2×C4×D7111yescharacteristic
199C22×C4C2×C22⋊D7111yescharacteristic
200C22×C4C2×D28411yescharacteristic
201C2×C8—24no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—2844no—
205C2×C8—1424no—
206C4⋊C4—2844no—
207C4⋊C4—712no—
208C4⋊C4—1424no—
209C4⋊C4—1421no—
210C22⋊C4—2844no—
211C22⋊C4—2844no—
212C22⋊C4—1424no—
213C42—24no—
214C42—22no—
215C42—21no—
216C42—21no—
217C42—42no—
218C42—2842no—
219C42C2×C4×D7112yes—
220C42#1176586512yes—
221C23—2842no—
222C23—2841no—
223C23—2841no—
224C23—2842no—
225C23—1422no—
226C23C2×(C2×D71)⋊C411yescharacteristic
227D4—2842no—
228D4—1424no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—2841no—
237C2×C4—2841no—
238C2×C4—2844no—
239C2×C4—2844no—
240C2×C4—2842no—
241C2×C4—2842no—
242C2×C4—2842no—
243C2×C4—1422no—
244C2×C4C42×D7111yescharacteristic
245C2×C4#1455608511yescharacteristic
246C2×C4C4×C22⋊D7112yes—
247C2×C4C4×D28412yes—
248C2×C4#1455613614yes—
249C2×C4#1455615312yes—
250C2×C4#1455615411yescharacteristic
251C2×C4#1455615911yescharacteristic
252C8—44no—
253C8—2844no—
254C22—2842no—
255C22—2842no—
256C22—2842no—
257C22—5682no—
258C22—5682no—
259C22C4×(C2×D71)⋊C411yescharacteristic
260C22#1798596712yes—
261C22#1798599811yescharacteristic
262C22#1798600311yescharacteristic
263C22#1798601512yes—
264C4—22no—
265C4—42no—
266C4—42no—
267C4—2842no—
268C4—5682no—
269C4#1798595612yes—
270C4#1798606112yes—
271C4#1798609712yes—
272C4#1798610912yes—
273C2—5682no—
274C2#2258680312yes—
275C2#2258680812yes—
276C2#2258810911yescharacteristic
277C2#2258815311yescharacteristic
278C2#2258816811yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D71
commutatorC2×C4×C71C2×C42
FrattiniC2×C42C22×D71
FittingC71×#1767C2
socleC23×C71C2×C22⋊C4
prime radicalC22×D284C2×C4
cosocleC2×C42×C71C23
hypercenter#1767D71
hypocenterC71#8179
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22×C4 ◁ C43 ◁ #1767

Complements

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

NH ≅ G/Ntype
2. #22588086C2semidirect
4. #22586179C2semidirect
5. C71×#1767C2semidirect
15. #17985726C4semidirect
141. #1767D71semidirect
145. C71#8179semidirect

Maximal subgroups

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

subgroupindex
#225908042
#225880862
C2×#179857262
#225861792
C71×#17672
#817971