Finite Group Databasebeta

← #31917788

Subgroups — #31917788

Order 18176.

398 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 79 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×#17985798C212yesmaximal, complemented
2#22586509C212yesmaximal, complemented
3#22586507C212yesmaximal, complemented
4C71×#3152C211yesmaximal, characteristic
5C2×#14556047—22no—
6#17985798—22no—
7#17985798C2214yes—
8C2×#137×C71—22no—
9#532×C71C411yescharacteristic
10C2×C4⋊Q8×C71C2211yescharacteristic
11#497×C71—22no—
12#494×C71C411yescharacteristic
13#494×C71C2211yescharacteristic
14#492×C71C411yescharacteristic
15#492×C71C2211yescharacteristic
16C4×M16×C71C411yescharacteristic
17C2×#11765849—42no—
18#14556047—24no—
19#14556047—42no—
20C2×#41×C71—21no—
21C22×Q8×C71D412yes—
22#137×C71—44no—
23#137×C71—44no—
24C2×Q16×C71—22no—
25C2×Q16×C71—22no—
26C2×SD16×C71—22no—
27C2×SD16×C71—22no—
28C2×M16×C71—21no—
29C4⋊Q8×C71—41no—
30C4⋊Q8×C71C2×C411yescharacteristic
31C4⋊Q8×C71C2×C411yescharacteristic
32C4⋊Q8×C71D412yes—
33C4⋊Q8×C71C2311yescharacteristic
34#127×C71C2×C411yescharacteristic
35#124×C71—21no—
36#121×C71—41no—
37C4×D4×C71—41no—
38C2×C4⋊C4×C71—41no—
39C2×C42×C71C2×C411yescharacteristic
40#97×C71—21no—
41C4×C8×C71C2×C411yescharacteristic
42C4×C8×C71C2×C411yescharacteristic
43#11765849—44no—
44#11765849—82no—
45C2×C71⋊C8—82no—
46#41×C71—42no—
47#41×C71—42no—
48C2×Q8×C71—22no—
49C2×Q8×C71—21no—
50C2×Q8×C71—42no—
51C2×Q8×C71—42no—
52C2×Q8×C71C22⋊C412yes—
53C2×Q8×C71C2×D412yes—
54C2×D4×C71—21no—
55C2×D4×C71—41no—
56C2×D4×C71—41no—
57C22×C4×C71—21no—
58C22×C4×C71—22no—
59C22×C4×C71—41no—
60C22×C4×C71C22⋊C411yescharacteristic
61Q16×C71—44no—
62Q16×C71—44no—
63SD16×C71—44no—
64SD16×C71—44no—
65M16×C71—24no—
66C2×C8×C71—21no—
67C2×C8×C71—21no—
68C4⋊C4×C71—41no—
69C4⋊C4×C71—42no—
70C4⋊C4×C71—41no—
71C4⋊C4×C71—41no—
72C22⋊C4×C71—42no—
73C42×C71—21no—
74C42×C71C22⋊C411yescharacteristic
75C42×C71C22×C411yescharacteristic
76C23×C71—41no—
77C23×C71#9911yescharacteristic
78Q8×C71—42no—
79Q8×C71—42no—
80Q8×C71—42no—
81Q8×C71—82no—
82D4×C71—42no—
83D4×C71—82no—
84D4×C71—82no—
85C2×C4×C71—22no—
86C2×C4×C71—21no—
87C2×C4×C71—21no—
88C2×C4×C71—22no—
89C2×C4×C71—22no—
90C2×C4×C71—42no—
91C2×C4×C71—42no—
92C2×C4×C71—42no—
93C2×C4×C71—41no—
94C2×C4×C71—81no—
95C2×C4×C71—81no—
96C2×C4×C71#9911yescharacteristic
97C2×C4×C71C2×C22⋊C411yescharacteristic
98C8×C71—22no—
99C8×C71—22no—
100C71⋊C8—84no—
101C22×C71—41no—
102C22×C71—82no—
103C22×C71—81no—
104C22×C71#35312yes—
105C22×C71C2×#9911yescharacteristic
106C4×C71—22no—
107C4×C71—21no—
108C4×C71—21no—
109C4×C71—41no—
110C4×C71—42no—
111C4×C71—42no—
112C4×C71—81no—
113#13590—711nomaximal, Sylow
114C2×C71—21no—
115C2×C71—22no—
116C2×C71—161no—
117C2×C71C2×#35311yescharacteristic
118#3152D7111yescharacteristic
119C2×#356—712no—
120#1411—712no—
121#1410—712no—
122C71#1359011yescharacteristic, complemented, Sylow
123C2×#137—22no—
124#532Dic7111yescharacteristic
125C2×C4⋊Q8C2×D7111yescharacteristic
126#497—22no—
127#494Dic7111yescharacteristic
128#494C2×D7111yescharacteristic
129#492Dic7111yescharacteristic
130#492C2×D7111yescharacteristic
131C2×#101—1422no—
132C4×M16Dic7111yescharacteristic
133#356—714no—
134#356—1422no—
135C2×#41—21no—
136C22×Q8C22⋊D7112yes—
137#137—44no—
138#137—44no—
139C2×Q16—22no—
140C2×Q16—22no—
141C2×SD16—22no—
142C2×SD16—22no—
143C2×M16—21no—
144C2×M16—2842no—
145C4⋊Q8—41no—
146C4⋊Q8C2×Dic7111yescharacteristic
147C4⋊Q8C2×Dic7111yescharacteristic
148C4⋊Q8C22⋊D7112yes—
149C4⋊Q8C22×D7111yescharacteristic
150#127C2×Dic7111yescharacteristic
151#124—21no—
152#121—41no—
153C4×D4—41no—
154C2×C4⋊C4—41no—
155C2×C42C2×Dic7111yescharacteristic
156#101—2842no—
157#101—1424no—
158#97—21no—
159C4×C8C2×Dic7111yescharacteristic
160C4×C8C2×Dic7111yescharacteristic
161#41—42no—
162#41—42no—
163C2×Q8—22no—
164C2×Q8—21no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8C22⋊Dic7112yes—
168C2×Q8C2×C22⋊D7112yes—
169C2×D4—21no—
170C2×D4—41no—
171C2×D4—41no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—41no—
175C22×C4C22⋊Dic7111yescharacteristic
176Q16—44no—
177Q16—44no—
178SD16—44no—
179SD16—44no—
180M16—24no—
181M16—2844no—
182M16—5682no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—5682no—
186C4⋊C4—41no—
187C4⋊C4—42no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C22⋊C4—42no—
191C42—21no—
192C42C22⋊Dic7111yescharacteristic
193C42C22×Dic7111yescharacteristic
194C23—41no—
195C23#1455604311yescharacteristic
196Q8—42no—
197Q8—42no—
198Q8—42no—
199Q8—82no—
200D4—42no—
201D4—82no—
202D4—82no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4#1455604311yescharacteristic
215C2×C4C2×C22⋊Dic7111yescharacteristic
216C8—22no—
217C8—22no—
218C8—5684no—
219C22—41no—
220C22—82no—
221C22—81no—
222C22#1798579212yes—
223C22C2×#1455604311yescharacteristic
224C4—22no—
225C4—21no—
226C4—21no—
227C4—41no—
228C4—42no—
229C4—42no—
230C4—81no—
231C2—21no—
232C2—22no—
233C2—161no—
234C2C2×#1798579211yescharacteristic
subgroupG/H
centerC2C2×#17985792
commutatorC42×C71C22×C4
FrattiniC4⋊Q8C22×D71
FittingC71×#3152C2
socleC2×C71C2×#353
prime radical#532×C71C4
cosocleC4⋊Q8×C71C23
hypercenter#3152D71
hypocenterC71#13590
LCSG ▷ C42×C71 ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ #3152

Complements

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

NH ≅ G/Ntype
1. C2×#17985798C2semidirect
2. #22586509C2semidirect
3. #22586507C2semidirect
122. C71#13590semidirect

Maximal subgroups

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

subgroupindex
C2×#179857982
#225865092
#225865072
C71×#31522
#1359071