Finite Group Databasebeta

← #31901950

Subgroups — #31901950

Order 18176.

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

normal subgroups only

This group has 406 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
1#22596612C211yesmaximal, characteristic
2#22588509C211yesmaximal, characteristic
3#22588508C212yesmaximal
4#22586222C212yesmaximal
5C71×#1295C211yesmaximal, characteristic
6C2×#14556062—22no—
7C2×#14556061—21no—
8C2×#14556060—22no—
9C2×#14556059—21no—
10C2×#14556039C2211yescharacteristic
11#17985941—22no—
12#17985938—21no—
13#17985920—22no—
14#17985919—21no—
15#17985869C2211yescharacteristic
16#17985865C2212yes—
17C2×#97×C71C2212yes—
18#382×C71C2211yescharacteristic
19C22×Dic142—21no—
20C22×C71⋊C8—21no—
21C2×C4⋊Dic71—21no—
22C2×C4⋊Dic71—21no—
23C2×C4⋊Dic71—22no—
24C2×C4⋊Dic71D412yes—
25C2×C4⋊Dic71D412yes—
26C2×Dic71⋊C4—22no—
27#14556062—44no—
28#14556061—42no—
29#14556060—44no—
30#14556059—42no—
31#14556039—21no—
32#14556039Q812yescomplemented
33C71⋊#95—21no—
34C71⋊#95—22no—
35C71⋊#95—21no—
36C71⋊#95—22no—
37C71⋊#95—22no—
38C22×C8×C71—22no—
39C2×C4⋊C4×C71D411yescharacteristic
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C42×C71C2311yescharacteristic
42#97×C71—22no—
43#97×C71Q814yescomplemented
44#95×C71—22no—
45C2×Dic142—82no—
46C2×Dic142—81no—
47C22×Dic71—22no—
48C22×Dic71—22no—
49C22×Dic71—41no—
50C2×C71⋊C8—22no—
51C2×C71⋊C8—22no—
52C2×C71⋊C8—22no—
53C4⋊Dic71—42no—
54C4⋊Dic71—42no—
55C4⋊Dic71—41no—
56C4⋊Dic71—42no—
57C4⋊Dic71—41no—
58C4⋊Dic71—42no—
59C4⋊Dic71—81no—
60Dic71⋊C4—82no—
61C22×C4×C71—21no—
62C22×C4×C71—21no—
63C22×C4×C71C2×D412yes—
64C22×C4×C71C2×D411yescharacteristic
65C2×C8×C71—24no—
66C2×C8×C71—24no—
67C2×C8×C71—24no—
68C4⋊C4×C71—41no—
69C4⋊C4×C71—41no—
70C42×C71C2×Q811yescharacteristic
71C42×C71C2×Q812yes—
72C42×C71#4111yescharacteristic
73C23×C71#12011yescharacteristic
74C2×C4×C71—21no—
75C2×C4×C71—22no—
76C2×C4×C71—22no—
77C2×C4×C71—21no—
78C2×C4×C71—41no—
79C2×C4×C71—41no—
80C2×C4×C71—42no—
81C2×C4×C71—42no—
82C2×C4×C71C22⋊Q812yes—
83C2×C4×C71C22⋊Q811yescharacteristic
84C2×C4×C71C4⋊Q812yes—
85C2×C4×C71C4⋊Q811yescharacteristic
86C2×Dic71—42no—
87C2×Dic71—42no—
88C2×Dic71—42no—
89C2×Dic71—42no—
90C2×Dic71—42no—
91C2×Dic71—42no—
92C2×Dic71—82no—
93C2×Dic71—81no—
94Dic142—84no—
95C8×C71—28no—
96C71⋊C8—24no—
97C22×C71#39511yescharacteristic
98C22×C71#45312yes—
99C22×C71#45311yescharacteristic
100C22×C71#45612yes—
101C22×C71#45611yescharacteristic
102C4×C71—21no—
103C4×C71—22no—
104C4×C71—22no—
105C4×C71—21no—
106C4×C71—81no—
107C4×C71—81no—
108Dic71—82no—
109Dic71—82no—
110Dic71—161no—
111#10583—711nomaximal, Sylow
112C2×C71#165511yescharacteristic
113C2×C71#166011yescharacteristic
114C2×C71#166012yes—
115C2×C71#206012yes—
116C2×C71#206011yescharacteristic
117#2844—711no—
118#1847—712no—
119#1847—711no—
120#1295—712no—
121#1295D7111yescharacteristic
122C71#1058311yescharacteristic, complemented, Sylow
123C2×#107—1422no—
124C2×#107—1421no—
125C2×#106—1422no—
126C2×#106—1421no—
127C2×#97—711no—
128C2×#97C2×D7112yes—
129#400—1422no—
130#400—1421no—
131#395—1422no—
132#395—1421no—
133#384—711no—
134#382—712no—
135#382C2×D7111yescharacteristic
136C22×Q8—1421no—
137C22×C8—22no—
138C22×C8—1421no—
139C2×C4⋊C4—712no—
140C2×C4⋊C4—712no—
141C2×C4⋊C4—1422no—
142C2×C4⋊C4—1421no—
143C2×C4⋊C4—1422no—
144C2×C4⋊C4—1421no—
145C2×C4⋊C4D28411yescharacteristic
146C2×C4⋊C4D28411yescharacteristic
147C2×C42C22×D7111yescharacteristic
148#107—2842no—
149#107—2844no—
150#106—2842no—
151#106—2844no—
152#97—22no—
153#97—712no—
154#97—1421no—
155#97Dic14214yescomplemented
156#95—22no—
157#95—1422no—
158#95—1421no—
159#95—1421no—
160#95—1422no—
161#95—1422no—
162C2×Q8—5682no—
163C2×Q8—5681no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—2841no—
167C22×C4—1422no—
168C22×C4—1422no—
169C22×C4D4×D7112yes—
170C22×C4C2×D28411yescharacteristic
171C2×C8—24no—
172C2×C8—24no—
173C2×C8—24no—
174C2×C8—1422no—
175C2×C8—1422no—
176C2×C8—1422no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—2842no—
180C4⋊C4—2842no—
181C4⋊C4—2842no—
182C4⋊C4—2841no—
183C4⋊C4—2842no—
184C4⋊C4—2841no—
185C4⋊C4—5682no—
186C4⋊C4—5681no—
187C42Q8×D7111yescharacteristic
188C42C2×Dic14212yes—
189C42#1176586411yescharacteristic
190C23#1455609811yescharacteristic
191Q8—5684no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—2842no—
201C2×C4—2842no—
202C2×C4—2842no—
203C2×C4—2842no—
204C2×C4—2842no—
205C2×C4—2842no—
206C2×C4—5682no—
207C2×C4—5681no—
208C2×C4#1455610512yes—
209C2×C4#1455610811yescharacteristic
210C2×C4#1455612712yes—
211C2×C4#1455612811yescharacteristic
212C8—28no—
213C8—1424no—
214C22#1798592011yescharacteristic
215C22#1798616812yes—
216C22#1798616911yescharacteristic
217C22#1798618312yes—
218C22#1798618411yescharacteristic
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—81no—
224C4—81no—
225C4—5682no—
226C4—5682no—
227C4—11361no—
228C2#2258763011yescharacteristic
229C2#2258765211yescharacteristic
230C2#2258765312yes—
231C2#2258968412yes—
232C2#2258968611yescharacteristic
subgroupG/H
centerC23#14556098
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1295C2
socle, prime radicalC23×C71#120
hypercenter#1295D71
hypocenterC71#10583
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
32. #14556039Q8semidirect
43. #97×C71Q8semidirect
122. C71#10583semidirect
155. #97Dic142semidirect

Maximal subgroups

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

subgroupindex
#225966122
#225885092
#225885082
#225862222
C71×#12952
#1058371