Finite Group Databasebeta

← #31898788

Subgroups — #31898788

Order 18176.

500 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#17986409C212yesmaximal
2#22589159C211yesmaximal, characteristic
3C2×#17985741C212yesmaximal
4#22586221C211yesmaximal, characteristic
5C71×#2822C211yesmaximal, characteristic
6C2×#14556128C2211yescharacteristic
7#17986409—24no—
8#17986409—24no—
9C2×#14556061—22no—
10C2×#14556059—22no—
11C2×#14556051—22no—
12C2×#14556039C2211yescharacteristic
13C2×#14556039C2212yes—
14#17985741—24no—
15C2×#125×C71C2212yes—
16#400×C71—22no—
17#398×C71—22no—
18#398×C71—22no—
19#382×C71C2211yescharacteristic
20C22×Dic142—21no—
21C22×C71⋊C8—22no—
22C22×C71⋊C8—21no—
23#14556128—22no—
24#14556128—21no—
25#14556128D412yes—
26C2×C4⋊Dic71—22no—
27#14556061—28no—
28#14556059—28no—
29#14556051—44no—
30#14556039—22no—
31#14556039—22no—
32#14556039D414yes—
33#125×C71—22no—
34#125×C71—22no—
35#125×C71—22no—
36#125×C71—22no—
37C2×C4⋊C4×C71—22no—
38C2×C4⋊C4×C71—22no—
39C2×C4⋊C4×C71—22no—
40C2×C4⋊C4×C71D412yes—
41C2×C4⋊C4×C71D412yes—
42C2×C4⋊C4×C71D412yes—
43C2×C42×C71C2311yescharacteristic
44#95×C71—24no—
45#95×C71—22no—
46#95×C71—22no—
47C2×Dic142—22no—
48C2×Dic142—42no—
49C2×Dic142—42no—
50C2×Dic142—41no—
51C22×Dic71—41no—
52C2×C71⋊C8—28no—
53C2×C71⋊C8—24no—
54C2×C71⋊C8—22no—
55C2×C71⋊C8—41no—
56C2×C71⋊C8—41no—
57C4⋊Dic71—24no—
58C4⋊Dic71—42no—
59C22×C4×C71—22no—
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—42no—
69C4⋊C4×C71—42no—
70C4⋊C4×C71—42no—
71C4⋊C4×C71—42no—
72C4⋊C4×C71—42no—
73C4⋊C4×C71—82no—
74C42×C71C2×D411yescharacteristic
75C42×C71C2×D412yes—
76C42×C71#4111yescharacteristic
77C23×C71#12011yescharacteristic
78C2×C4×C71—21no—
79C2×C4×C71—21no—
80C2×C4×C71—24no—
81C2×C4×C71—42no—
82C2×C4×C71—42no—
83C2×C4×C71—42no—
84C2×C4×C71—42no—
85C2×C4×C71—42no—
86C2×C4×C71—44no—
87C2×C4×C71—44no—
88C2×C4×C71#12012yes—
89C2×C4×C71#12111yescharacteristic
90C2×C4×C71#12112yes—
91C2×C4×C71#12711yescharacteristic
92C2×Dic71—42no—
93C2×Dic71—81no—
94C2×Dic71—81no—
95Dic142—44no—
96Dic142—82no—
97C71⋊C8—28no—
98C71⋊C8—42no—
99C22×C71#39211yescharacteristic
100C22×C71#45311yescharacteristic
101C22×C71#45411yescharacteristic
102C22×C71#45614yes—
103C4×C71—22no—
104C4×C71—22no—
105C4×C71—21no—
106C4×C71—21no—
107C4×C71—82no—
108C4×C71—82no—
109C4×C71—82no—
110Dic71—82no—
111#10087—711nomaximal, Sylow
112C2×C71#166012yes—
113C2×C71#166312yes—
114C2×C71#200512yes—
115C2×C71#200611yescharacteristic
116C2×#501—712no—
117#2822D7111yescharacteristic
118#1960—711no—
119C2×#330—712no—
120#1295—711no—
121C71#1008711yescharacteristic, complemented, Sylow
122C2×C4⋊Q8—711no—
123C2×#125C2×D7112yes—
124#501—1424no—
125#501—1424no—
126C2×#107—1422no—
127C2×#106—1422no—
128C2×#103—1422no—
129C2×#97—712no—
130C2×#97—711no—
131#400—22no—
132#398—22no—
133#398—22no—
134#382C2×D7111yescharacteristic
135#330—1424no—
136C22×Q8—1421no—
137C22×C8—1421no—
138C22×C8—1422no—
139C4⋊Q8—712no—
140C4⋊Q8—1422no—
141C4⋊Q8—1421no—
142#125—22no—
143#125—22no—
144#125—22no—
145#125—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—1422no—
150C2×C4⋊C4C22⋊D7112yes—
151C2×C4⋊C4C22⋊D7112yes—
152C2×C4⋊C4C22⋊D7112yes—
153C2×C42C22×D7111yescharacteristic
154#107—1428no—
155#106—1428no—
156#103—2844no—
157#97—714no—
158#97—1422no—
159#97—1422no—
160#95—24no—
161#95—22no—
162#95—22no—
163C2×Q8—2842no—
164C2×Q8—2842no—
165C2×Q8—2841no—
166C2×Q8—1422no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—2841no—
171C22×C4C2×C22⋊D7112yes—
172C22×C4C2×C22⋊D7111yescharacteristic
173C2×C8—2841no—
174C2×C8—2841no—
175C2×C8—1422no—
176C2×C8—1428no—
177C2×C8—1424no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—82no—
188C4⋊C4—2842no—
189C4⋊C4—1424no—
190C42D4×D7111yescharacteristic
191C42D4×D7112yes—
192C42#1176586411yescharacteristic
193C23#1455609711yescharacteristic
194Q8—2844no—
195Q8—5682no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—24no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—2842no—
207C2×C4—5681no—
208C2×C4—5681no—
209C2×C4#1455609612yes—
210C2×C4#1455609911yescharacteristic
211C2×C4#1455610012yes—
212C2×C4#1455612411yescharacteristic
213C8—2842no—
214C8—1428no—
215C22#1798590711yescharacteristic
216C22#1798616711yescharacteristic
217C22#1798617211yescharacteristic
218C22#1798618214yes—
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—82no—
224C4—82no—
225C4—82no—
226C4—5682no—
227C2#2258765112yes—
228C2#2258766812yes—
229C2#2258937412yes—
230C2#2258938211yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2822C2
socle, prime radicalC23×C71#120
hypercenter#2822D71
hypocenterC71#10087
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
121. C71#10087semidirect

Maximal subgroups

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

subgroupindex
C2×#179864092
#225891592
C2×#179857412
#225862212
C71×#28222
#1008771