Finite Group Databasebeta

← #31910715

Subgroups — #31910715

Order 18176.

354 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 71 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#22590518C212yesmaximal, complemented
2#22589828C211yesmaximal, characteristic, complemented
3#22586349C212yesmaximal, complemented
4#22586280C211yesmaximal, characteristic, complemented
5C71×#3245C211yesmaximal, characteristic
6C2×#14556062C2211yescharacteristic
7#17985897—21no—
8#17985824—22no—
9#17985800—22no—
10C22⋊(C71⋊C16)C2212yes—
11#17985762—21no—
12#17985762C2211yescharacteristic
13#532×C71—21no—
14#474×C71—22no—
15#466×C71—21no—
16#466×C71C2212yes—
17#459×C71—22no—
18C2×C4⋊C8×C71C2211yescharacteristic
19C2×C4⋊Dic71—21no—
20C2×C4⋊Dic71—21no—
21C2×Dic71⋊C4—41no—
22C2×C71⋊C16—22no—
23#14556062—22no—
24C71⋊#95—41no—
25C2×#41×C71—41no—
26C2×D8×C71—21no—
27C2×D8×C71D412yes—
28C22×C8×C71D411yescharacteristic
29C22×C8×C71C2311yescharacteristic
30C4⋊Q8×C71—21no—
31#127×C71—21no—
32#124×C71—41no—
33#121×C71—22no—
34#121×C71—41no—
35C4×D4×C71—22no—
36C4×D4×C71—41no—
37C2×C42×C71D411yescharacteristic
38#107×C71—21no—
39#107×C71D412yes—
40C4⋊C8×C71—22no—
41#102×C71—22no—
42#102×C71—22no—
43C22×Dic71—41no—
44C22×Dic71—41no—
45C4⋊Dic71—22no—
46C4⋊Dic71—41no—
47C4⋊Dic71—41no—
48C4⋊Dic71—41no—
49Dic71⋊C4—82no—
50C71⋊C16—42no—
51#41×C71—82no—
52#41×C71—82no—
53C2×Q8×C71—41no—
54C2×D4×C71—22no—
55C2×D4×C71—41no—
56C2×D4×C71—41no—
57C2×D4×C71—42no—
58C22×C4×C71—42no—
59C22×C4×C71D811yescharacteristic
60C22×C4×C71D811yescharacteristic
61C22×C4×C71C2×D411yescharacteristic
62D8×C71—42no—
63D8×C71—81no—
64C2×C8×C71—21no—
65C2×C8×C71—41no—
66C2×C8×C71—41no—
67C2×C8×C71C2×D412yes—
68C4⋊C4×C71—22no—
69C4⋊C4×C71—41no—
70C22⋊C4×C71—42no—
71C22⋊C4×C71—42no—
72C42×C71—21no—
73C42×C71—21no—
74C23×C71—42no—
75C23×C71C2×D811yescharacteristic
76Q8×C71—82no—
77D4×C71—42no—
78D4×C71—42no—
79D4×C71—82no—
80D4×C71—82no—
81D4×C71—82no—
82D4×C71—82no—
83C2×C4×C71—21no—
84C2×C4×C71—21no—
85C2×C4×C71—21no—
86C2×C4×C71—21no—
87C2×C4×C71—21no—
88C2×C4×C71—41no—
89C2×C4×C71—42no—
90C2×C4×C71—84no—
91C2×C4×C71D1612yes—
92C2×C4×C71#12011yescharacteristic
93C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
94C2×Dic71—42no—
95C2×Dic71—81no—
96C2×Dic71—81no—
97C2×Dic71—81no—
98C2×Dic71—81no—
99C2×Dic71—81no—
100C8×C71—22no—
101C8×C71—41no—
102C22×C71—21no—
103C22×C71—21no—
104C22×C71—82no—
105C22×C71—82no—
106C22×C71—82no—
107C22×C71#44711yescharacteristic
108C22×C71C2×D1611yescharacteristic
109C22×C71#50911yescharacteristic
110C4×C71—21no—
111C4×C71—22no—
112C4×C71—41no—
113C4×C71—82no—
114C4×C71#45611yescharacteristic
115C4×C71#45811yescharacteristic
116Dic71—82no—
117Dic71—161no—
118#12288—711nomaximal, Sylow
119C2×C71—21no—
120C2×C71—21no—
121C2×C71—162no—
122C2×C71#161011yescharacteristic
123C2×C71#218411yescharacteristic
124C2×C71#219111yescharacteristic
125#3245D7111yescharacteristic
126#2232—712no—
127#2086—711no—
128#1347—712no—
129#1322—711no—
130C71#1228811yescharacteristic, complemented, Sylow
131#532—21no—
132#474—22no—
133#466—21no—
134#466C2×D7112yes—
135#459—22no—
136C2×#107—711no—
137C2×C4⋊C8C2×D7111yescharacteristic
138#389—1421no—
139#366—1422no—
140#357—1422no—
141C22⋊C16—712no—
142#340—711no—
143#340—1421no—
144C2×#41—41no—
145C2×D8—21no—
146C2×D8C22⋊D7112yes—
147C22×C8C22⋊D7111yescharacteristic
148C22×C8C22×D7111yescharacteristic
149C4⋊Q8—21no—
150#127—21no—
151#124—41no—
152#121—22no—
153#121—41no—
154C4×D4—22no—
155C4×D4—41no—
156C2×C4⋊C4—2841no—
157C2×C4⋊C4—1421no—
158C2×C4⋊C4—1421no—
159C2×C42C22⋊D7111yescharacteristic
160C2×C16—1422no—
161#107—21no—
162#107—1422no—
163#107C22⋊D7112yes—
164C4⋊C8—22no—
165#102—22no—
166#102—22no—
167#95—2841no—
168#41—82no—
169#41—82no—
170C2×Q8—41no—
171C2×D4—22no—
172C2×D4—41no—
173C2×D4—41no—
174C2×D4—42no—
175C22×C4—42no—
176C22×C4—2841no—
177C22×C4—2841no—
178C22×C4#1176585011yescharacteristic
179C22×C4#1176585011yescharacteristic
180C22×C4C2×C22⋊D7111yescharacteristic
181D8—42no—
182D8—81no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8C2×C22⋊D7112yes—
187C4⋊C4—22no—
188C4⋊C4—41no—
189C4⋊C4—2841no—
190C4⋊C4—2841no—
191C4⋊C4—2841no—
192C4⋊C4—5682no—
193C4⋊C4—1422no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C42—21no—
197C42—21no—
198C16—2842no—
199C23—42no—
200C23C2×#1176585011yescharacteristic
201Q8—82no—
202D4—42no—
203D4—42no—
204D4—82no—
205D4—82no—
206D4—82no—
207D4—82no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—84no—
216C2×C4—2842no—
217C2×C4—5681no—
218C2×C4—5681no—
219C2×C4—5681no—
220C2×C4—5681no—
221C2×C4—5681no—
222C2×C4#1455606912yes—
223C2×C4#1455609711yescharacteristic
224C2×C4#1455615411yescharacteristic
225C8—22no—
226C8—41no—
227C22—21no—
228C22—21no—
229C22—82no—
230C22—82no—
231C22—82no—
232C22#1798613311yescharacteristic
233C22C2×#1455606911yescharacteristic
234C22#1798643511yescharacteristic
235C4—22no—
236C4—21no—
237C4—41no—
238C4—82no—
239C4—5682no—
240C4—11361no—
241C4#1798618211yescharacteristic
242C4#1798618911yescharacteristic
243C2—21no—
244C2—21no—
245C2—162no—
246C2#2258734311yescharacteristic
247C2#2259025211yescharacteristic
248C2#2259029111yescharacteristic
subgroupG/H
centerC22#17986133
commutator, cosocleC22×C8×C71C23
FrattiniC22×C8C22×D71
Fitting, prime radicalC71×#3245C2
socleC22×C71#447
hypercenter#3245D71
hypocenterC71#12288
LCSG ▷ C22×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #3245

Complements

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

NH ≅ G/Ntype
1. #22590518C2semidirect
2. #22589828C2semidirect
3. #22586349C2semidirect
4. #22586280C2semidirect
130. C71#12288semidirect

Maximal subgroups

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

subgroupindex
#225905182
#225898282
#225863492
#225862802
C71×#32452
#1228871