Finite Group Databasebeta

← #31886752

Subgroups — #31886752

Order 18176.

394 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 95 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
2#22589878C211yesmaximal, characteristic
3#22586166C211yesmaximal, characteristic
4#22586163C212yesmaximal
5C71×#2734C211yesmaximal, characteristic
6C2×#14556061—21no—
7C2×#14556052—21no—
8#17985869C2211yescharacteristic
9#17985761—22no—
10#17985761—22no—
11#17985754C2211yescharacteristic
12#17985754C2212yes—
13C2×C4⋊Q8×C71—21no—
14#398×C71—21no—
15#395×C71—22no—
16#391×C71C2211yescharacteristic
17#389×C71C2212yes—
18#387×C71—22no—
19C22×C71⋊C8—21no—
20C22×C71⋊C8—22no—
21C2×C4⋊Dic71—21no—
22C2×C4⋊Dic71—22no—
23#14556061—42no—
24#14556052—42no—
25C22×Q8×C71—21no—
26C22×Q8×C71D411yescharacteristic
27C4⋊Q8×C71—42no—
28C4⋊Q8×C71—42no—
29C2×C4⋊C4×C71—22no—
30C2×C4⋊C4×C71—22no—
31C2×C4⋊C4×C71—21no—
32C2×C4⋊C4×C71D412yes—
33C2×C4⋊C4×C71D412yes—
34C2×C4⋊C4×C71D411yescharacteristic
35C2×C42×C71—21no—
36C2×C42×C71C2311yescharacteristic
37#95×C71—21no—
38#95×C71—22no—
39#95×C71—22no—
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—
51C2×Q8×C71—21no—
52C2×Q8×C71—42no—
53C2×Q8×C71—41no—
54C2×Q8×C71—41no—
55C2×Q8×C71—81no—
56C2×Q8×C71—81no—
57C2×Q8×C71SD1612yes—
58C22×C4×C71—22no—
59C22×C4×C71—21no—
60C22×C4×C71—21no—
61C22×C4×C71—22no—
62C22×C4×C71C2×D411yescharacteristic
63C22×C4×C71C2×D412yes—
64C4⋊C4×C71—22no—
65C4⋊C4×C71—42no—
66C4⋊C4×C71—41no—
67C4⋊C4×C71—44no—
68C4⋊C4×C71—42no—
69C4⋊C4×C71—42no—
70C4⋊C4×C71—81no—
71C4⋊C4×C71Q1614yes—
72C42×C71—41no—
73C42×C71—42no—
74C23×C71#12011yescharacteristic
75Q8×C71—42no—
76Q8×C71—81no—
77Q8×C71—84no—
78C2×C4×C71—21no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—21no—
82C2×C4×C71—22no—
83C2×C4×C71—24no—
84C2×C4×C71—42no—
85C2×C4×C71—41no—
86C2×C4×C71—42no—
87C2×C4×C71—41no—
88C2×C4×C71—42no—
89C2×C4×C71—41no—
90C2×C4×C71—42no—
91C2×C4×C71—42no—
92C2×C4×C71—42no—
93C2×C4×C71C2×SD1611yescharacteristic
94C2×C4×C71C2×Q1612yes—
95C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
96C2×C4×C71#13712yes—
97C2×Dic71—81no—
98C2×Dic71—82no—
99C71⋊C8—42no—
100C71⋊C8—82no—
101C22×C71#45011yescharacteristic
102C22×C71#45112yes—
103C22×C71#45311yescharacteristic
104C22×C71#45412yes—
105C22×C71#45811yescharacteristic
106C4×C71—21no—
107C4×C71—21no—
108C4×C71—22no—
109C4×C71—22no—
110C4×C71—42no—
111C4×C71—44no—
112C4×C71—82no—
113C4×C71—81no—
114Dic71—161no—
115#7829—711nomaximal, Sylow
116C2×C71#161311yescharacteristic
117C2×C71#161712yes—
118C2×C71#163311yescharacteristic
119C2×C71#163712yes—
120C2×C71#165711yescharacteristic
121#2734D7111yescharacteristic
122#2100—712no—
123#2095—711no—
124#1272—711no—
125#1271—712no—
126C71#782911yescharacteristic, complemented, Sylow
127C2×C4⋊Q8—21no—
128C2×#107—1421no—
129C2×#103—1421no—
130#398—21no—
131#395—22no—
132#391C2×D7111yescharacteristic
133#389C2×D7112yes—
134#387—22no—
135#384—711no—
136#340—1422no—
137#340—1422no—
138#336—711no—
139#336—712no—
140C22×Q8—21no—
141C22×Q8C22⋊D7111yescharacteristic
142C22×C8—1422no—
143C22×C8—1421no—
144C4⋊Q8—42no—
145C4⋊Q8—42no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—1422no—
150C2×C4⋊C4—1421no—
151C2×C4⋊C4C22⋊D7112yes—
152C2×C4⋊C4C22⋊D7112yes—
153C2×C4⋊C4C22⋊D7111yescharacteristic
154C2×C42—21no—
155C2×C42C22×D7111yescharacteristic
156#107—2842no—
157#103—2842no—
158#95—21no—
159#95—22no—
160#95—22no—
161#95—22no—
162C2×Q8—21no—
163C2×Q8—42no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—81no—
167C2×Q8—81no—
168C2×Q8#1176585312yes—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—2841no—
174C22×C4C2×C22⋊D7111yescharacteristic
175C22×C4C2×C22⋊D7112yes—
176C2×C8—2841no—
177C2×C8—2841no—
178C2×C8—2842no—
179C2×C8—2842no—
180C2×C8—2842no—
181C2×C8—1422no—
182C4⋊C4—22no—
183C4⋊C4—41no—
184C4⋊C4—42no—
185C4⋊C4—44no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—81no—
189C4⋊C4—2841no—
190C4⋊C4—2841no—
191C4⋊C4—2844no—
192C4⋊C4#1176585514yes—
193C42—41no—
194C42—42no—
195C23#1455609711yescharacteristic
196Q8—42no—
197Q8—81no—
198Q8—84no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—5681no—
215C2×C4—5682no—
216C2×C4C2×#1176585311yescharacteristic
217C2×C4C2×#1176585512yes—
218C2×C4#1455615411yescharacteristic
219C2×C4#1455615912yes—
220C8—2842no—
221C8—5682no—
222C22#1798615011yescharacteristic
223C22#1798615512yes—
224C22#1798616711yescharacteristic
225C22#1798617212yes—
226C22#1798618911yescharacteristic
227C4—21no—
228C4—21no—
229C4—22no—
230C4—22no—
231C4—44no—
232C4—42no—
233C4—82no—
234C4—81no—
235C4—11361no—
236C2#2258736011yescharacteristic
237C2#2258738212yes—
238C2#2258748011yescharacteristic
239C2#2258750512yes—
240C2#2258763811yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2734C2
socle, prime radicalC23×C71#120
hypercenter#2734D71
hypocenterC71#7829
LCSG ▷ C2×C42×C71 ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2734

Complements

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

NH ≅ G/Ntype
126. C71#7829semidirect

Maximal subgroups

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

subgroupindex
#225899032
#225898782
#225861662
#225861632
C71×#27342
#782971