Finite Group Databasebeta

← #31900948

Subgroups — #31900948

Order 18176.

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

normal subgroups only

This group has 402 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#22589310C212yesmaximal
2#22588509C211yesmaximal, characteristic
3C2×#17985741C212yesmaximal
4#22586221C211yesmaximal, characteristic
5C71×#2822C211yesmaximal, characteristic
6C2×#14556061—21no—
7C2×#14556059—21no—
8C2×#14556039C2212yes—
9C2×#14556039C2211yescharacteristic
10#17985869C2211yescharacteristic
11#17985761—22no—
12#17985761—22no—
13#17985741—24no—
14C2×#125×C71C2212yes—
15#400×C71—22no—
16#398×C71—21no—
17#398×C71—21no—
18#398×C71—22no—
19#382×C71C2211yescharacteristic
20C22×C71⋊C8—21no—
21C22×C71⋊C8—22no—
22C2×C4⋊Dic71—22no—
23C2×C4⋊Dic71—21no—
24#14556061—42no—
25#14556059—42no—
26#14556039—22no—
27#14556039—21no—
28#14556039—22no—
29#14556039D412yes—
30#125×C71—22no—
31#125×C71—22no—
32#125×C71—22no—
33#125×C71—22no—
34C2×C4⋊C4×C71—22no—
35C2×C4⋊C4×C71—22no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71D412yes—
39C2×C4⋊C4×C71D412yes—
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C4⋊C4×C71D411yescharacteristic
42C2×C42×C71C2311yescharacteristic
43#95×C71—22no—
44#95×C71—22no—
45#95×C71—21no—
46#95×C71—21no—
47#95×C71—22no—
48C22×Dic71—41no—
49C2×C71⋊C8—22no—
50C2×C71⋊C8—22no—
51C2×C71⋊C8—22no—
52C2×C71⋊C8—24no—
53C2×C71⋊C8—42no—
54C2×C71⋊C8—42no—
55C4⋊Dic71—41no—
56C4⋊Dic71—41no—
57C4⋊Dic71—44no—
58C22×C4×C71—21no—
59C22×C4×C71—21no—
60C22×C4×C71—22no—
61C22×C4×C71—22no—
62C22×C4×C71C2×D412yes—
63C22×C4×C71C2×D411yescharacteristic
64C4⋊C4×C71—22no—
65C4⋊C4×C71—22no—
66C4⋊C4×C71—22no—
67C4⋊C4×C71—22no—
68C4⋊C4×C71—41no—
69C4⋊C4×C71—41no—
70C4⋊C4×C71—42no—
71C4⋊C4×C71—42no—
72C4⋊C4×C71—42no—
73C4⋊C4×C71—42no—
74C4⋊C4×C71—81no—
75C4⋊C4×C71—81no—
76C42×C71C2×D411yescharacteristic
77C42×C71#4111yescharacteristic
78C42×C71#4112yes—
79C23×C71#12011yescharacteristic
80C2×C4×C71—21no—
81C2×C4×C71—22no—
82C2×C4×C71—22no—
83C2×C4×C71—21no—
84C2×C4×C71—42no—
85C2×C4×C71—42no—
86C2×C4×C71—41no—
87C2×C4×C71—41no—
88C2×C4×C71—42no—
89C2×C4×C71—42no—
90C2×C4×C71—42no—
91C2×C4×C71—42no—
92C2×C4×C71—42no—
93C2×C4×C71—42no—
94C2×C4×C71#12111yescharacteristic
95C2×C4×C71#12112yes—
96C2×C4×C71#12312yes—
97C2×C4×C71#12411yescharacteristic
98C2×Dic71—81no—
99C2×Dic71—82no—
100C71⋊C8—24no—
101C71⋊C8—44no—
102C22×C71#39411yescharacteristic
103C22×C71#45311yescharacteristic
104C22×C71#45412yes—
105C22×C71#45612yes—
106C22×C71#45611yescharacteristic
107C4×C71—22no—
108C4×C71—22no—
109C4×C71—21no—
110C4×C71—21no—
111C4×C71—81no—
112C4×C71—81no—
113C4×C71—82no—
114C4×C71—82no—
115Dic71—161no—
116#10429—711nomaximal, Sylow
117C2×C71#165711yescharacteristic
118C2×C71#166011yescharacteristic
119C2×C71#166312yes—
120C2×C71#203812yes—
121C2×C71#203911yescharacteristic
122#2822D7111yescharacteristic
123#1992—712no—
124#1847—711no—
125C2×#330—712no—
126#1295—711no—
127C71#1042911yescharacteristic, complemented, Sylow
128C2×#125C2×D7112yes—
129C2×#107—1421no—
130C2×#106—1421no—
131C2×#97—711no—
132C2×#97—712no—
133#400—22no—
134#398—21no—
135#398—21no—
136#398—22no—
137#384—711no—
138#382C2×D7111yescharacteristic
139#340—1422no—
140#340—1422no—
141#330—1424no—
142C22×C8—1422no—
143C22×C8—1421no—
144#125—22no—
145#125—22no—
146#125—22no—
147#125—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—1422no—
153C2×C4⋊C4—1421no—
154C2×C4⋊C4C22⋊D7112yes—
155C2×C4⋊C4C22⋊D7112yes—
156C2×C4⋊C4C22⋊D7111yescharacteristic
157C2×C4⋊C4C22⋊D7111yescharacteristic
158C2×C42C22×D7111yescharacteristic
159#107—2842no—
160#106—2842no—
161#97—712no—
162#97—1422no—
163#97—1422no—
164#97—1421no—
165#95—22no—
166#95—22no—
167#95—21no—
168#95—21no—
169#95—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—2841no—
175C22×C4C2×C22⋊D7111yescharacteristic
176C22×C4C2×C22⋊D7112yes—
177C2×C8—2842no—
178C2×C8—2842no—
179C2×C8—1424no—
180C2×C8—1422no—
181C2×C8—1422no—
182C2×C8—1422no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C4⋊C4—2841no—
196C4⋊C4—2841no—
197C4⋊C4—2844no—
198C42D4×D7111yescharacteristic
199C42#1176586411yescharacteristic
200C42#1176586412yes—
201C23#1455609711yescharacteristic
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—5681no—
217C2×C4—5682no—
218C2×C4#1455609911yescharacteristic
219C2×C4#1455610012yes—
220C2×C4#1455610912yes—
221C2×C4#1455611511yescharacteristic
222C8—2844no—
223C8—1424no—
224C22#1798591311yescharacteristic
225C22#1798616711yescharacteristic
226C22#1798617212yes—
227C22#1798618212yes—
228C22#1798618211yescharacteristic
229C4—22no—
230C4—22no—
231C4—21no—
232C4—21no—
233C4—81no—
234C4—82no—
235C4—82no—
236C4—81no—
237C4—11361no—
238C2#2258763811yescharacteristic
239C2#2258765111yescharacteristic
240C2#2258766812yes—
241C2#2258955412yes—
242C2#2258956111yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2822C2
socle, prime radicalC23×C71#120
hypercenter#2822D71
hypocenterC71#10429
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2822

Complements

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

NH ≅ G/Ntype
127. C71#10429semidirect

Maximal subgroups

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

subgroupindex
#225893102
#225885092
C2×#179857412
#225862212
C71×#28222
#1042971