Finite Group Databasebeta

← #31917794

Subgroups — #31917794

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
1#22590025C212yesmaximal
2#22586509C214yesmaximal, complemented
3C71×#3152C211yesmaximal, characteristic
4#17985983—22no—
5#17985798C2214yes—
6#17985796—22no—
7C2×#137×C71—22no—
8#532×C71C2211yescharacteristic
9C2×C4⋊Q8×C71C411yescharacteristic
10#497×C71—22no—
11#494×C71C2212yes—
12#492×C71C412yes—
13C4×M16×C71C411yescharacteristic
14C2×#11765849—42no—
15#14556047—24no—
16#14556045—42no—
17C2×#41×C71D412yes—
18C22×Q8×C71—21no—
19#137×C71—44no—
20#137×C71—44no—
21C2×Q16×C71—24no—
22C2×SD16×C71—24no—
23C2×M16×C71—21no—
24C4⋊Q8×C71—21no—
25C4⋊Q8×C71—41no—
26C4⋊Q8×C71C2×C412yes—
27C4⋊Q8×C71C2311yescharacteristic
28#127×C71C2×C411yescharacteristic
29#124×C71D412yes—
30#121×C71—41no—
31C4×D4×C71—41no—
32C2×C4⋊C4×C71—41no—
33C2×C42×C71C2×C411yescharacteristic
34#97×C71—21no—
35C4×C8×C71C2×C412yes—
36#11765849—44no—
37#11765849—82no—
38C2×C71⋊C8—82no—
39#41×C71—42no—
40#41×C71—42no—
41C2×Q8×C71—22no—
42C2×Q8×C71—42no—
43C2×Q8×C71—42no—
44C2×Q8×C71—41no—
45C2×Q8×C71C2×D412yes—
46C2×D4×C71—22no—
47C2×D4×C71—41no—
48C2×D4×C71C22⋊C412yes—
49C22×C4×C71—21no—
50C22×C4×C71—22no—
51C22×C4×C71—41no—
52C22×C4×C71C22⋊C411yescharacteristic
53Q16×C71—48no—
54SD16×C71—48no—
55M16×C71—24no—
56C2×C8×C71—22no—
57C4⋊C4×C71—41no—
58C4⋊C4×C71—42no—
59C4⋊C4×C71—42no—
60C22⋊C4×C71—42no—
61C42×C71—21no—
62C42×C71C22⋊C411yescharacteristic
63C42×C71C22×C411yescharacteristic
64C23×C71—22no—
65C23×C71#9911yescharacteristic
66Q8×C71—42no—
67Q8×C71—44no—
68Q8×C71—82no—
69D4×C71—42no—
70D4×C71—82no—
71D4×C71—82no—
72C2×C4×C71—21no—
73C2×C4×C71—22no—
74C2×C4×C71—21no—
75C2×C4×C71—22no—
76C2×C4×C71—42no—
77C2×C4×C71—42no—
78C2×C4×C71—42no—
79C2×C4×C71—42no—
80C2×C4×C71—81no—
81C2×C4×C71—81no—
82C2×C4×C71#9911yescharacteristic
83C2×C4×C71C2×C22⋊C411yescharacteristic
84C8×C71—24no—
85C71⋊C8—84no—
86C22×C71—21no—
87C22×C71—41no—
88C22×C71—42no—
89C22×C71—82no—
90C22×C71C2×#9911yescharacteristic
91C4×C71—22no—
92C4×C71—21no—
93C4×C71—41no—
94C4×C71—42no—
95C4×C71—82no—
96C4×C71#35312yes—
97#13592—711nomaximal, Sylow
98C2×C71—21no—
99C2×C71—41no—
100C2×C71—82no—
101C2×C71C2×#35311yescharacteristic
102#3152D7111yescharacteristic
103#2132—712no—
104#1411—714no—
105C71#1359211yescharacteristic, complemented, Sylow
106C2×#137—22no—
107#532C2×D7111yescharacteristic
108C2×C4⋊Q8Dic7111yescharacteristic
109#497—22no—
110#494C2×D7112yes—
111#492Dic7112yes—
112#413—1422no—
113C4×M16Dic7111yescharacteristic
114#356—714no—
115#355—1422no—
116C2×#41C22⋊D7112yes—
117C22×Q8—21no—
118#137—44no—
119#137—44no—
120C2×Q16—24no—
121C2×SD16—24no—
122C2×M16—21no—
123C2×M16—2842no—
124C4⋊Q8—21no—
125C4⋊Q8—41no—
126C4⋊Q8C2×Dic7112yes—
127C4⋊Q8C22×D7111yescharacteristic
128#127C2×Dic7111yescharacteristic
129#124C22⋊D7112yes—
130#121—41no—
131C4×D4—41no—
132C2×C4⋊C4—41no—
133C2×C42C2×Dic7111yescharacteristic
134#101—1424no—
135#100—2842no—
136#97—21no—
137C4×C8C2×Dic7112yes—
138#41—42no—
139#41—42no—
140C2×Q8—22no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—41no—
144C2×Q8C2×C22⋊D7112yes—
145C2×D4—22no—
146C2×D4—41no—
147C2×D4C22⋊Dic7112yes—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—41no—
151C22×C4C22⋊Dic7111yescharacteristic
152Q16—48no—
153SD16—48no—
154M16—24no—
155M16—2844no—
156M16—5682no—
157C2×C8—22no—
158C2×C8—5682no—
159C4⋊C4—41no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C22⋊C4—42no—
163C42—21no—
164C42C22⋊Dic7111yescharacteristic
165C42C22×Dic7111yescharacteristic
166C23—22no—
167C23#1455604311yescharacteristic
168Q8—42no—
169Q8—44no—
170Q8—82no—
171D4—42no—
172D4—82no—
173D4—82no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4#1455604311yescharacteristic
185C2×C4C2×C22⋊Dic7111yescharacteristic
186C8—24no—
187C8—5684no—
188C22—21no—
189C22—41no—
190C22—42no—
191C22—82no—
192C22C2×#1455604311yescharacteristic
193C4—22no—
194C4—21no—
195C4—41no—
196C4—42no—
197C4—82no—
198C4#1798579212yes—
199C2—21no—
200C2—41no—
201C2—82no—
202C2C2×#1798579211yescharacteristic
subgroupG/H
centerC2C2×#17985792
commutatorC42×C71C22×C4
FrattiniC4⋊Q8C22×D71
FittingC71×#3152C2
socleC2×C71C2×#353
prime radical#532×C71C22
cosocleC4⋊Q8×C71C23
hypercenter#3152D71
hypocenterC71#13592
LCSG ▷ C42×C71 ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #3152

Complements

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

NH ≅ G/Ntype
2. #22586509C2semidirect
105. C71#13592semidirect

Maximal subgroups

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

subgroupindex
#225900252
#225865092
C71×#31522
#1359271