Finite Group Databasebeta

← #31886721

Subgroups — #31886721

Order 18176.

418 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 87 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#22589903C212yesmaximal, complemented
2#22589873C211yesmaximal, characteristic
3#22586163C212yesmaximal, complemented
4#22586161C211yesmaximal, characteristic
5C71×#2724C211yesmaximal, characteristic
6C2×#14556059—21no—
7C2×#14556049—21no—
8#17985869C2211yescharacteristic
9#17985761—22no—
10#17985761—22no—
11#17985754C2212yes—
12#17985754C2211yescharacteristic
13C2×#123×C71—21no—
14#398×C71—21no—
15#397×C71—22no—
16#390×C71C2211yescharacteristic
17#389×C71C2212yes—
18#385×C71—22no—
19C22×C71⋊C8—21no—
20C22×C71⋊C8—22no—
21C2×C4⋊Dic71—21no—
22C2×C4⋊Dic71—22no—
23#14556059—42no—
24#14556049—42no—
25C22×D4×C71D411yescharacteristic
26C23×C4×C71—21no—
27#123×C71—42no—
28#123×C71—42no—
29C2×C4⋊C4×C71—22no—
30C2×C4⋊C4×C71—21no—
31C2×C4⋊C4×C71D412yes—
32C2×C4⋊C4×C71D412yes—
33C2×C4⋊C4×C71D411yescharacteristic
34C2×C22⋊C4×C71—22no—
35C2×C22⋊C4×C71—21no—
36C2×C22⋊C4×C71—22no—
37C2×C42×C71C2311yescharacteristic
38#95×C71—22no—
39#95×C71—21no—
40#95×C71—22no—
41C22×Dic71—41no—
42C2×C71⋊C8—22no—
43C2×C71⋊C8—41no—
44C2×C71⋊C8—41no—
45C2×C71⋊C8—42no—
46C2×C71⋊C8—42no—
47C2×C71⋊C8—42no—
48C4⋊Dic71—41no—
49C4⋊Dic71—41no—
50C4⋊Dic71—44no—
51C24×C71—21no—
52C2×D4×C71—21no—
53C2×D4×C71—41no—
54C2×D4×C71—41no—
55C2×D4×C71SD1612yes—
56C22×C4×C71—22no—
57C22×C4×C71—21no—
58C22×C4×C71—22no—
59C22×C4×C71—42no—
60C22×C4×C71—81no—
61C22×C4×C71—81no—
62C22×C4×C71C2×D411yescharacteristic
63C22×C4×C71C2×D412yes—
64C4⋊C4×C71—22no—
65C4⋊C4×C71—22no—
66C4⋊C4×C71—42no—
67C4⋊C4×C71—41no—
68C4⋊C4×C71—42no—
69C4⋊C4×C71—42no—
70C4⋊C4×C71—81no—
71C22⋊C4×C71—42no—
72C22⋊C4×C71—44no—
73C22⋊C4×C71—42no—
74C22⋊C4×C71—42no—
75C42×C71—41no—
76C23×C71—22no—
77C23×C71—42no—
78C23×C71—41no—
79C23×C71—42no—
80C23×C71—41no—
81C23×C71#12011yescharacteristic
82D4×C71—42no—
83D4×C71—81no—
84C2×C4×C71—21no—
85C2×C4×C71—22no—
86C2×C4×C71—22no—
87C2×C4×C71—21no—
88C2×C4×C71—42no—
89C2×C4×C71—42no—
90C2×C4×C71—42no—
91C2×C4×C71—42no—
92C2×C4×C71—41no—
93C2×C4×C71—42no—
94C2×C4×C71—42no—
95C2×C4×C71—42no—
96C2×C4×C71—84no—
97C2×C4×C71C2×SD1611yescharacteristic
98C2×C4×C71#13512yes—
99C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
100C2×C4×C71#13712yes—
101C2×Dic71—81no—
102C2×Dic71—82no—
103C71⋊C8—42no—
104C71⋊C8—82no—
105C22×C71—42no—
106C22×C71—44no—
107C22×C71—82no—
108C22×C71—81no—
109C22×C71—81no—
110C22×C71#45011yescharacteristic
111C22×C71#45212yes—
112C22×C71#45311yescharacteristic
113C22×C71#45412yes—
114C22×C71#45811yescharacteristic
115C4×C71—21no—
116C4×C71—21no—
117C4×C71—22no—
118C4×C71—22no—
119C4×C71—82no—
120C4×C71—81no—
121C4×C71—82no—
122Dic71—161no—
123#7824—711nomaximal, Sylow
124C2×C71—82no—
125C2×C71#161311yescharacteristic
126C2×C71#161512yes—
127C2×C71#163011yescharacteristic
128C2×C71#163512yes—
129C2×C71#165711yescharacteristic
130#2724D7111yescharacteristic
131#2100—712no—
132#2094—711no—
133#1271—712no—
134#1270—711no—
135C71#782411yescharacteristic, complemented, Sylow
136C2×#123—21no—
137C2×#106—1421no—
138C2×#102—1421no—
139#398—21no—
140#397—22no—
141#390C2×D7111yescharacteristic
142#389C2×D7112yes—
143#385—22no—
144#384—711no—
145#340—1422no—
146#340—1422no—
147#336—711no—
148#336—712no—
149C22×D4C22⋊D7111yescharacteristic
150C23×C4—21no—
151C22×C8—1422no—
152C22×C8—1421no—
153#123—42no—
154#123—42no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—1422no—
158C2×C4⋊C4—1421no—
159C2×C4⋊C4C22⋊D7112yes—
160C2×C4⋊C4C22⋊D7112yes—
161C2×C4⋊C4C22⋊D7111yescharacteristic
162C2×C22⋊C4—22no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—22no—
165C2×C42C22×D7111yescharacteristic
166#106—2842no—
167#102—2842no—
168#95—22no—
169#95—21no—
170#95—22no—
171C24—21no—
172C2×D4—21no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4#1176585312yes—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—42no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4—2841no—
183C22×C4C2×C22⋊D7111yescharacteristic
184C22×C4C2×C22⋊D7112yes—
185C2×C8—2841no—
186C2×C8—2841no—
187C2×C8—2842no—
188C2×C8—2842no—
189C2×C8—2842no—
190C2×C8—1422no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—81no—
198C4⋊C4—2841no—
199C4⋊C4—2841no—
200C4⋊C4—2844no—
201C22⋊C4—42no—
202C22⋊C4—44no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C42—41no—
206C23—22no—
207C23—42no—
208C23—41no—
209C23—42no—
210C23—41no—
211C23#1455609711yescharacteristic
212D4—42no—
213D4—81no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—84no—
227C2×C4—5681no—
228C2×C4—5682no—
229C2×C4C2×#1176585311yescharacteristic
230C2×C4#1455614812yes—
231C2×C4#1455615411yescharacteristic
232C2×C4#1455615812yes—
233C8—2842no—
234C8—5682no—
235C22—44no—
236C22—42no—
237C22—81no—
238C22—82no—
239C22—81no—
240C22#1798615011yescharacteristic
241C22#1798616112yes—
242C22#1798616711yescharacteristic
243C22#1798617212yes—
244C22#1798618911yescharacteristic
245C4—21no—
246C4—21no—
247C4—22no—
248C4—22no—
249C4—81no—
250C4—82no—
251C4—82no—
252C4—11361no—
253C2—82no—
254C2#2258736011yescharacteristic
255C2#2258737112yes—
256C2#2258746111yescharacteristic
257C2#2258749112yes—
258C2#2258763811yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2724C2
socleC23×C71#120
prime radicalC22×D4×C71D4
hypercenter#2724D71
hypocenterC71#7824
LCSG ▷ C2×C42×C71 ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2724

Complements

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

NH ≅ G/Ntype
1. #22589903C2semidirect
3. #22586163C2semidirect
135. C71#7824semidirect

Maximal subgroups

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

subgroupindex
#225899032
#225898732
#225861632
#225861612
C71×#27242
#782471