Finite Group Databasebeta

← #31886770

Subgroups — #31886770

Order 18176.

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

normal subgroups only

This group has 438 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#22589878C212yesmaximal, complemented
2#22589873C211yesmaximal, characteristic
3#22586166C212yesmaximal, complemented
4#22586161C211yesmaximal, characteristic
5C71×#2804C211yesmaximal, characteristic
6C2×#14556061—22no—
7C2×#14556059—21no—
8C2×#14556052—22no—
9C2×#14556049—21no—
10#17985869C2211yescharacteristic
11#17985754C2212yes—
12#17985754C2211yescharacteristic
13#400×C71—21no—
14#393×C71—22no—
15#391×C71C2212yes—
16#390×C71C2211yescharacteristic
17#386×C71—21no—
18#385×C71—22no—
19C22×C71⋊C8—21no—
20C22×C71⋊C8—22no—
21C2×C4⋊Dic71—21no—
22C2×C4⋊Dic71—22no—
23#14556061—44no—
24#14556059—42no—
25#14556052—44no—
26#14556049—42no—
27C22×Q8×C71D412yes—
28C22×D4×C71D411yescharacteristic
29C23×C4×C71—21no—
30C2×C4⋊C4×C71—21no—
31C2×C4⋊C4×C71D412yes—
32C2×C4⋊C4×C71D411yescharacteristic
33C2×C22⋊C4×C71—22no—
34C2×C22⋊C4×C71—22no—
35C2×C22⋊C4×C71—21no—
36C2×C42×C71C2311yescharacteristic
37#95×C71—22no—
38#95×C71—21no—
39#95×C71—22no—
40#95×C71—22no—
41C22×Dic71—41no—
42C2×C71⋊C8—22no—
43C2×C71⋊C8—24no—
44C2×C71⋊C8—41no—
45C2×C71⋊C8—42no—
46C2×C71⋊C8—42no—
47C2×C71⋊C8—41no—
48C4⋊Dic71—41no—
49C4⋊Dic71—42no—
50C4⋊Dic71—41no—
51C4⋊Dic71—42no—
52C24×C71—21no—
53C2×Q8×C71—22no—
54C2×Q8×C71—22no—
55C2×Q8×C71—42no—
56C2×Q8×C71—42no—
57C2×D4×C71—21no—
58C2×D4×C71—41no—
59C2×D4×C71—41no—
60C2×D4×C71Q1612yes—
61C22×C4×C71—22no—
62C22×C4×C71—21no—
63C22×C4×C71—22no—
64C22×C4×C71—82no—
65C22×C4×C71—81no—
66C22×C4×C71C2×D411yescharacteristic
67C22×C4×C71C2×D412yes—
68C4⋊C4×C71—41no—
69C4⋊C4×C71—42no—
70C4⋊C4×C71—81no—
71C22⋊C4×C71—42no—
72C22⋊C4×C71—42no—
73C22⋊C4×C71—82no—
74C22⋊C4×C71—81no—
75C42×C71—41no—
76C23×C71—22no—
77C23×C71—42no—
78C23×C71—42no—
79C23×C71—41no—
80C23×C71—41no—
81C23×C71#12011yescharacteristic
82Q8×C71—44no—
83Q8×C71—82no—
84D4×C71—42no—
85D4×C71—81no—
86C2×C4×C71—21no—
87C2×C4×C71—22no—
88C2×C4×C71—21no—
89C2×C4×C71—22no—
90C2×C4×C71—42no—
91C2×C4×C71—42no—
92C2×C4×C71—42no—
93C2×C4×C71—42no—
94C2×C4×C71—41no—
95C2×C4×C71—42no—
96C2×C4×C71—42no—
97C2×C4×C71—42no—
98C2×C4×C71—84no—
99C2×C4×C71C2×Q1611yescharacteristic
100C2×C4×C71#13512yes—
101C2×C4×C71#13712yes—
102C2×C4×C71#13711yescharacteristic
103C2×Dic71—81no—
104C2×Dic71—82no—
105C71⋊C8—42no—
106C71⋊C8—44no—
107C22×C71—42no—
108C22×C71—44no—
109C22×C71—82no—
110C22×C71—81no—
111C22×C71—81no—
112C22×C71#45111yescharacteristic
113C22×C71#45212yes—
114C22×C71#45311yescharacteristic
115C22×C71#45312yes—
116C22×C71#45811yescharacteristic
117C4×C71—21no—
118C4×C71—21no—
119C4×C71—22no—
120C4×C71—22no—
121C4×C71—82no—
122C4×C71—81no—
123C4×C71—82no—
124Dic71—161no—
125#7833—711nomaximal, Sylow
126C2×C71—82no—
127C2×C71#161411yescharacteristic
128C2×C71#161812yes—
129C2×C71#163011yescharacteristic
130C2×C71#163212yes—
131C2×C71#165511yescharacteristic
132#2804D7111yescharacteristic
133#2095—712no—
134#2094—711no—
135#1272—712no—
136#1270—711no—
137C71#783311yescharacteristic, complemented, Sylow
138C2×#107—1422no—
139C2×#106—1421no—
140C2×#103—1422no—
141C2×#102—1421no—
142#400—21no—
143#393—22no—
144#391C2×D7112yes—
145#390C2×D7111yescharacteristic
146#386—21no—
147#385—22no—
148#384—711no—
149#336—711no—
150#336—712no—
151C22×Q8C22⋊D7112yes—
152C22×D4C22⋊D7111yescharacteristic
153C23×C4—21no—
154C22×C8—1421no—
155C22×C8—1422no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—1421no—
158C2×C4⋊C4—1422no—
159C2×C4⋊C4C22⋊D7112yes—
160C2×C4⋊C4C22⋊D7111yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—21no—
164C2×C42C22×D7111yescharacteristic
165#107—2844no—
166#106—2842no—
167#103—2844no—
168#102—2842no—
169#95—22no—
170#95—21no—
171#95—22no—
172#95—22no—
173C24—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×D4—21no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4#1176585512yes—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—82no—
186C22×C4—81no—
187C22×C4—2841no—
188C22×C4C2×C22⋊D7111yescharacteristic
189C22×C4C2×C22⋊D7112yes—
190C2×C8—2841no—
191C2×C8—2842no—
192C2×C8—2842no—
193C2×C8—2841no—
194C2×C8—1422no—
195C2×C8—1424no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—81no—
199C4⋊C4—2841no—
200C4⋊C4—2842no—
201C4⋊C4—2841no—
202C4⋊C4—2842no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—82no—
206C22⋊C4—81no—
207C42—41no—
208C23—22no—
209C23—42no—
210C23—42no—
211C23—41no—
212C23—41no—
213C23#1455609711yescharacteristic
214Q8—44no—
215Q8—82no—
216D4—42no—
217D4—81no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—84no—
231C2×C4—5681no—
232C2×C4—5682no—
233C2×C4C2×#1176585511yescharacteristic
234C2×C4#1455614812yes—
235C2×C4#1455615812yes—
236C2×C4#1455615911yescharacteristic
237C8—2842no—
238C8—2844no—
239C22—44no—
240C22—42no—
241C22—81no—
242C22—81no—
243C22—82no—
244C22#1798615511yescharacteristic
245C22#1798616112yes—
246C22#1798616711yescharacteristic
247C22#1798616712yes—
248C22#1798618911yescharacteristic
249C4—21no—
250C4—21no—
251C4—22no—
252C4—22no—
253C4—81no—
254C4—82no—
255C4—82no—
256C4—11361no—
257C2—82no—
258C2#2258736511yescharacteristic
259C2#2258738812yes—
260C2#2258746111yescharacteristic
261C2#2258747412yes—
262C2#2258762911yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2804C2
socleC23×C71#120
prime radicalC22×D4×C71D4
hypercenter#2804D71
hypocenterC71#7833
LCSG ▷ C2×C42×C71 ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2804

Complements

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

NH ≅ G/Ntype
1. #22589878C2semidirect
3. #22586166C2semidirect
137. C71#7833semidirect

Maximal subgroups

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

subgroupindex
#225898782
#225898732
#225861662
#225861612
C71×#28042
#783371