Finite Group Databasebeta

← #31894842

Subgroups — #31894842

Order 18176.

574 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 135 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#22593339C211yesmaximal, characteristic
2#22588336C211yesmaximal, characteristic, complemented
3#22587007C212yesmaximal
4#22586242C212yesmaximal, complemented
5C71×#2982C211yesmaximal, characteristic, complemented
6C2×#14556125C2211yescharacteristic
7C2×#14556117—21no—
8#17986023C2212yes—
9C2×#14556047—21no—
10#17985916C414yes—
11#17985901—22no—
12#17985900—21no—
13C4×C4⋊Dic71C2211yescharacteristic
14#17985768—21no—
15#17985742—24no—
16C2×C4×Q8×C71C2211yescharacteristic
17C8×Q8×C71—24no—
18#431×C71—22no—
19#431×C71C2212yes—
20C22×D284—21no—
21C22×D284—21no—
22C22×D284C2×C412yes—
23C2×#11765849—22no—
24#14556125—22no—
25#14556125—22no—
26#14556117—42no—
27#14556117—42no—
28C2×C4⋊Dic71—21no—
29C2×(C2×D71)⋊C4—24no—
30C2×(C2×D71)⋊C4—24no—
31C2×C4×Dic71—21no—
32C4⋊(C71⋊C8)—22no—
33C4⋊(C71⋊C8)—22no—
34#14556047—42no—
35C71⋊#95C2×C412yes—
36C22×Q8×C71D411yescharacteristic
37C4×Q8×C71—22no—
38C4×Q8×C71—22no—
39C2×C4⋊C4×C71C2×C412yes—
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C42×C71D412yes—
42C2×C42×C71C2311yescharacteristic
43C4⋊C8×C71—24no—
44C4⋊C8×C71D414yes—
45C22⋊C8×C71—22no—
46C4×C8×C71—24no—
47C4×C8×C71—22no—
48C23×D71—22no—
49C2×D284—24no—
50C2×D284—44no—
51C2×D284—41no—
52C2×D284—42no—
53C2×D284—41no—
54C2×D284—41no—
55C2×D284—41no—
56C2×D284—42no—
57C2×D284—82no—
58C22×Dic71—22no—
59#11765849—42no—
60#11765849—42no—
61C2×C71⋊C8—42no—
62C4⋊Dic71—42no—
63(C2×D71)⋊C4—48no—
64(C2×D71)⋊C4—84no—
65C4×Dic71—41no—
66C4×Dic71—41no—
67C2×Q8×C71—22no—
68C2×Q8×C71—22no—
69C2×Q8×C71D812yes—
70C2×Q8×C71SD1612yes—
71C22×C4×C71C22⋊C412yes—
72C22×C4×C71C22⋊C412yes—
73C22×C4×C71C22×C411yescharacteristic
74C22×C4×C71C2×D411yescharacteristic
75C22×C4×C71C2×D411yescharacteristic
76C2×C8×C71—24no—
77C4⋊C4×C71—24no—
78C4⋊C4×C71—22no—
79C42×C71—22no—
80C42×C71—22no—
81C42×C71C2×D412yes—
82C42×C71#4112yes—
83C23×C71C2×C22⋊C411yescharacteristic
84C22×D71—44no—
85C22×D71—44no—
86C22×D71—42no—
87C22×D71—42no—
88C22×D71—42no—
89Q8×C71—24no—
90Q8×C71—24no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71—22no—
94C2×C4×C71—22no—
95C2×C4×C71—41no—
96C2×C4×C71—42no—
97C2×C4×C71#10214yes—
98C2×C4×C71C4×D412yes—
99C2×C4×C71#12011yescharacteristic
100C2×C4×C71#12112yes—
101C2×C4×C71#12311yescharacteristic
102C2×C4×C71C2×D811yescharacteristic
103C2×C4×C71C2×SD1611yescharacteristic
104C2×C4×C71#13512yes—
105C2×Dic71—42no—
106C2×Dic71—42no—
107C2×Dic71—42no—
108D284—84no—
109D284—82no—
110D284—82no—
111D284—82no—
112D284—82no—
113C8×C71—44no—
114C71⋊C8—82no—
115C22×C71—21no—
116C22×C71—21no—
117C22×C71#38611yescharacteristic
118C22×C71C2×#10211yescharacteristic
119C22×C71#41611yescharacteristic
120C2×D71—84no—
121C2×D71—84no—
122C2×D71—82no—
123C2×D71—82no—
124C2×D71—82no—
125C4×C71—21no—
126C4×C71—24no—
127C4×C71—22no—
128C4×C71—42no—
129C4×C71#45412yes—
130C4×C71#46512yes—
131C4×C71#46612yes—
132Dic71—82no—
133#9398—711nomaximal, Sylow
134C2×C71—21no—
135C2×C71—21no—
136C2×C71#188811yescharacteristic
137C2×C71#189311yescharacteristic
138C2×C71#191411yescharacteristic
139D71—162no—
140#2982D7111yescharacteristic, complemented
141#2548—711no—
142#1801—711no—
143#1545—712no—
144#1303—712no—
145C71#939811yescharacteristic, complemented, Sylow
146C2×#127—711no—
147C2×#124—1421no—
148C2×C4×Q8C2×D7111yescharacteristic
149C8×Q8—24no—
150#431—22no—
151#431C2×D7112yes—
152#423—712no—
153C2×#101—1421no—
154#394—714no—
155#390—1421no—
156#390—1422no—
157C4×C4⋊C4—711no—
158#343—1421no—
159#331—1424no—
160C22×Q8D28411yescharacteristic
161C22×D4—712no—
162C22×D4—1421no—
163C22×D4—1421no—
164C2×M16—1422no—
165#127—1422no—
166#127—1422no—
167#124—2842no—
168#124—2842no—
169C4×Q8—22no—
170C4×Q8—22no—
171C2×C4⋊C4—1421no—
172C2×C4⋊C4C4×D7112yes—
173C2×C4⋊C4D28411yescharacteristic
174C2×C22⋊C4—1424no—
175C2×C22⋊C4—1424no—
176C2×C42—1421no—
177C2×C42C22⋊D7112yes—
178C2×C42C22×D7111yescharacteristic
179C4⋊C8—24no—
180C4⋊C8—1422no—
181C4⋊C8—1422no—
182C4⋊C8C22⋊D7114yes—
183#101—2842no—
184C22⋊C8—22no—
185C4×C8—24no—
186C4×C8—22no—
187#95—712no—
188C24—1422no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8D56812yes—
192C2×Q8#1176585412yes—
193C2×D4—2844no—
194C2×D4—2841no—
195C2×D4—2842no—
196C2×D4—2841no—
197C2×D4—2841no—
198C2×D4—2841no—
199C2×D4—2842no—
200C2×D4—5682no—
201C2×D4—1424no—
202C22×C4—1422no—
203C22×C4(C2×D71)⋊C412yes—
204C22×C4(C2×D71)⋊C412yes—
205C22×C4C2×C4×D7111yescharacteristic
206C22×C4C2×C22⋊D7111yescharacteristic
207C22×C4C2×D28411yescharacteristic
208M16—2842no—
209M16—2842no—
210C2×C8—24no—
211C2×C8—2842no—
212C4⋊C4—24no—
213C4⋊C4—22no—
214C4⋊C4—2842no—
215C22⋊C4—2848no—
216C22⋊C4—5684no—
217C42—22no—
218C42—22no—
219C42—2841no—
220C42—2841no—
221C42C2×C22⋊D7112yes—
222C42#1176586512yes—
223C23—2844no—
224C23—2844no—
225C23—2842no—
226C23—2842no—
227C23—2842no—
228C23C2×(C2×D71)⋊C411yescharacteristic
229Q8—24no—
230Q8—24no—
231D4—5684no—
232D4—5682no—
233D4—5682no—
234D4—5682no—
235D4—5682no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—2842no—
243C2×C4—2842no—
244C2×C4—2842no—
245C2×C4#1455605014yes—
246C2×C4C4×C22⋊D7112yes—
247C2×C4#1455609711yescharacteristic
248C2×C4#1455610212yes—
249C2×C4#1455611111yescharacteristic
250C2×C4C2×D56811yescharacteristic
251C2×C4C2×#1176585411yescharacteristic
252C2×C4#1455615012yes—
253C8—44no—
254C8—5682no—
255C22—21no—
256C22—21no—
257C22—5684no—
258C22—5684no—
259C22—5682no—
260C22—5682no—
261C22—5682no—
262C22#1798587811yescharacteristic
263C22C2×#1455605011yescharacteristic
264C22#1798599611yescharacteristic
265C4—21no—
266C4—24no—
267C4—22no—
268C4—42no—
269C4—5682no—
270C4#1798617212yes—
271C4#1798622812yes—
272C4#1798623512yes—
273C2—21no—
274C2—21no—
275C2—11362no—
276C2#2258874311yescharacteristic
277C2#2258876811yescharacteristic
278C2#2258888711yescharacteristic
subgroupG/H
centerC22#17985878
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2982C2
socleC22×C71#386
prime radicalC2×#14556125C22
cosocleC2×C42×C71C23
hypercenter#2982D71
hypocenterC71#9398
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. #22588336C2semidirect
4. #22586242C2semidirect
5. C71×#2982C2semidirect
140. #2982D71semidirect
145. C71#9398semidirect

Maximal subgroups

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

subgroupindex
#225933392
#225883362
#225870072
#225862422
C71×#29822
#939871