Finite Group Databasebeta

← #31886750

Subgroups — #31886750

Order 18176.

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

normal subgroups only

This group has 394 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#22595372C211yesmaximal, characteristic
2#22589902C212yesmaximal
3#22586166C211yesmaximal, characteristic
4#22586164C212yesmaximal
5C71×#2095C211yesmaximal, characteristic
6C2×#14556126—21no—
7#17985929—21no—
8#17985920—22no—
9#17985897C2212yes—
10#17985884—22no—
11#17985761—22no—
12#17985761—22no—
13#17985754C2212yes—
14C2×#107×C71—21no—
15C2×#103×C71—21no—
16#391×C71C2211yescharacteristic
17#384×C71C2211yescharacteristic
18#336×C71C2211yescharacteristic
19C22×Dic142—21no—
20C22×C71⋊C8—22no—
21#14556126—42no—
22#14556126—42no—
23C2×C4⋊Dic71—21no—
24C2×C4⋊Dic71D412yes—
25C2×C4⋊Dic71D412yes—
26C2×Dic71⋊C4—22no—
27C2×Dic71⋊C4—22no—
28C2×C4×Dic71—21no—
29C71⋊#95—21no—
30C71⋊#95—22no—
31C71⋊#95—22no—
32C22×Q8×C71D411yescharacteristic
33C22×C8×C71—21no—
34C2×C4⋊C4×C71—21no—
35C2×C4⋊C4×C71—22no—
36C2×C4⋊C4×C71D411yescharacteristic
37C2×C42×C71C2311yescharacteristic
38#107×C71—42no—
39#103×C71—42no—
40#95×C71—22no—
41C2×Dic142—42no—
42C2×Dic142—81no—
43C2×Dic142—81no—
44C22×Dic71—22no—
45C22×Dic71—22no—
46C2×C71⋊C8—42no—
47C2×C71⋊C8—42no—
48C2×C71⋊C8—42no—
49C4⋊Dic71—22no—
50C4⋊Dic71—42no—
51C4⋊Dic71—81no—
52C4⋊Dic71Q1614yes—
53Dic71⋊C4—44no—
54Dic71⋊C4—42no—
55Dic71⋊C4—42no—
56C4×Dic71—42no—
57C2×Q8×C71—21no—
58C2×Q8×C71—41no—
59C2×Q8×C71—41no—
60C2×Q8×C71SD1612yes—
61C22×C4×C71—21no—
62C22×C4×C71—21no—
63C22×C4×C71—41no—
64C22×C4×C71C2×D412yes—
65C22×C4×C71C2×D411yescharacteristic
66C2×C8×C71—22no—
67C2×C8×C71—41no—
68C2×C8×C71—41no—
69C4⋊C4×C71—41no—
70C4⋊C4×C71—41no—
71C4⋊C4×C71—41no—
72C4⋊C4×C71—44no—
73C42×C71—41no—
74C23×C71#12011yescharacteristic
75Q8×C71—42no—
76Q8×C71—81no—
77C2×C4×C71—21no—
78C2×C4×C71—22no—
79C2×C4×C71—21no—
80C2×C4×C71—22no—
81C2×C4×C71—22no—
82C2×C4×C71—41no—
83C2×C4×C71—41no—
84C2×C4×C71—41no—
85C2×C4×C71—42no—
86C2×C4×C71—81no—
87C2×C4×C71—82no—
88C2×C4×C71C2×SD1611yescharacteristic
89C2×C4×C71C2×Q1612yes—
90C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
91C2×C4×C71#13712yes—
92C2×Dic71—24no—
93C2×Dic71—42no—
94C2×Dic71—42no—
95C2×Dic71—42no—
96C2×Dic71—42no—
97C2×Dic71—42no—
98Dic142—84no—
99C8×C71—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—81no—
112C4×C71—161no—
113Dic71—44no—
114Dic71—82no—
115#7829—711nomaximal, Sylow
116C2×C71#161311yescharacteristic
117C2×C71#161712yes—
118C2×C71#163311yescharacteristic
119C2×C71#163712yes—
120C2×C71#165711yescharacteristic
121#2734—711no—
122#2100—712no—
123#2095D7111yescharacteristic
124#1272—711no—
125#1271—712no—
126C71#782911yescharacteristic, complemented, Sylow
127C2×C4⋊Q8—1421no—
128C2×#107—21no—
129C2×#103—21no—
130#398—1421no—
131#395—1422no—
132#391C2×D7111yescharacteristic
133#389—712no—
134#387—1422no—
135#384C2×D7111yescharacteristic
136#340—1422no—
137#340—1422no—
138#336—712no—
139#336C2×D7111yescharacteristic
140C22×Q8—1421no—
141C22×Q8C22⋊D7111yescharacteristic
142C22×C8—21no—
143C22×C8—1422no—
144C4⋊Q8—2842no—
145C4⋊Q8—2842no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—712no—
149C2×C4⋊C4—712no—
150C2×C4⋊C4—1422no—
151C2×C4⋊C4—1422no—
152C2×C4⋊C4—1421no—
153C2×C4⋊C4C22⋊D7111yescharacteristic
154C2×C42—1421no—
155C2×C42C22×D7111yescharacteristic
156#107—42no—
157#103—42no—
158#95—22no—
159#95—1421no—
160#95—1422no—
161#95—1422no—
162C2×Q8—21no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—2842no—
166C2×Q8—5681no—
167C2×Q8—5681no—
168C2×Q8#1176585212yes—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—41no—
172C22×C4—1422no—
173C22×C4—1422no—
174C22×C4D4×D7112yes—
175C22×C4C2×C22⋊D7111yescharacteristic
176C2×C8—22no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—2842no—
180C2×C8—2842no—
181C2×C8—2842no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—44no—
186C4⋊C4—2842no—
187C4⋊C4—2844no—
188C4⋊C4—2842no—
189C4⋊C4—2842no—
190C4⋊C4—5681no—
191C4⋊C4—714no—
192C4⋊C4—1422no—
193C42—41no—
194C42—2842no—
195C23#1455609611yescharacteristic
196Q8—42no—
197Q8—81no—
198Q8—5684no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—81no—
209C2×C4—82no—
210C2×C4—2842no—
211C2×C4—2842no—
212C2×C4—2842no—
213C2×C4—2842no—
214C2×C4—2842no—
215C2×C4—1424no—
216C2×C4C2×#1176585211yescharacteristic
217C2×C4Q16×D7112yes—
218C2×C4#1455615311yescharacteristic
219C2×C4#1455616012yes—
220C8—42no—
221C8—5682no—
222C22#1798614911yescharacteristic
223C22#1798615312yes—
224C22#1798616611yescharacteristic
225C22#1798617012yes—
226C22#1798618811yescharacteristic
227C4—21no—
228C4—21no—
229C4—22no—
230C4—22no—
231C4—42no—
232C4—81no—
233C4—161no—
234C4—2844no—
235C4—5682no—
236C2#2258735911yescharacteristic
237C2#2258738012yes—
238C2#2258747811yescharacteristic
239C2#2258750312yes—
240C2#2258763611yescharacteristic
subgroupG/H
centerC23#14556096
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2095C2
socle, prime radicalC23×C71#120
hypercenter#2095D71
hypocenterC71#7829
LCSG ▷ C2×C42×C71 ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2095

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
#225953722
#225899022
#225861662
#225861642
C71×#20952
#782971