Finite Group Databasebeta

← #31917815

Subgroups — #31917815

Order 18176.

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

normal subgroups only

This group has 312 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
2#22586513C214yesmaximal, complemented
3C71×#3139C211yesmaximal, characteristic
4C2×#14556047—22no—
5#17985798—22no—
6#17985798C2214yes—
7#533×C71C411yescharacteristic
8C2×C4⋊Q8×C71C2211yescharacteristic
9#491×C71C412yes—
10#491×C71C2212yes—
11#487×C71—22no—
12#419×C71—22no—
13C4×M16×C71C411yescharacteristic
14C2×#11765849—42no—
15#14556047—24no—
16#14556047—42no—
17C22×Q8×C71D412yes—
18C2×M16×C71—21no—
19C4⋊Q8×C71—21no—
20C4⋊Q8×C71—41no—
21C4⋊Q8×C71C2×C411yescharacteristic
22C4⋊Q8×C71D412yes—
23C4⋊Q8×C71C2311yescharacteristic
24#125×C71C2×C412yes—
25C22⋊Q8×C71—41no—
26C4×Q8×C71—41no—
27#117×C71—21no—
28C2×C4⋊C4×C71—41no—
29C2×C42×C71C2×C411yescharacteristic
30#103×C71—24no—
31#103×C71—24no—
32#97×C71C2×C412yes—
33C4×C8×C71—21no—
34#11765849—44no—
35#11765849—82no—
36C2×C71⋊C8—82no—
37C2×Q8×C71—22no—
38C2×Q8×C71—41no—
39C2×Q8×C71—42no—
40C2×Q8×C71—42no—
41C2×Q8×C71C22⋊C412yes—
42C2×Q8×C71C2×D412yes—
43C22×C4×C71—21no—
44C22×C4×C71—22no—
45C22×C4×C71C22⋊C411yescharacteristic
46M16×C71—42no—
47C2×C8×C71—22no—
48C4⋊C4×C71—22no—
49C4⋊C4×C71—41no—
50C4⋊C4×C71—42no—
51C4⋊C4×C71—41no—
52C4⋊C4×C71—42no—
53C4⋊C4×C71—41no—
54C22⋊C4×C71—41no—
55C42×C71—21no—
56C42×C71—41no—
57C42×C71C22⋊C411yescharacteristic
58C42×C71C22×C411yescharacteristic
59C23×C71#9911yescharacteristic
60Q8×C71—42no—
61Q8×C71—42no—
62Q8×C71—82no—
63Q8×C71—82no—
64C2×C4×C71—22no—
65C2×C4×C71—21no—
66C2×C4×C71—21no—
67C2×C4×C71—22no—
68C2×C4×C71—22no—
69C2×C4×C71—42no—
70C2×C4×C71—42no—
71C2×C4×C71—42no—
72C2×C4×C71—42no—
73C2×C4×C71#9911yescharacteristic
74C2×C4×C71C2×C22⋊C411yescharacteristic
75C8×C71—42no—
76C71⋊C8—84no—
77C22×C71—41no—
78C22×C71#35412yes—
79C22×C71C2×#9911yescharacteristic
80C4×C71—22no—
81C4×C71—21no—
82C4×C71—21no—
83C4×C71—41no—
84C4×C71—42no—
85C4×C71—42no—
86C4×C71—82no—
87#13599—711nomaximal, Sylow
88C2×C71—21no—
89C2×C71—22no—
90C2×C71C2×#35411yescharacteristic
91#3139D7111yescharacteristic
92C2×#356—712no—
93#1413—714no—
94C71#1359911yescharacteristic, complemented, Sylow
95#533Dic7111yescharacteristic
96C2×C4⋊Q8C2×D7111yescharacteristic
97#491Dic7112yes—
98#491C2×D7112yes—
99#487—22no—
100#419—22no—
101C2×#101—1422no—
102C4×M16Dic7111yescharacteristic
103#356—714no—
104#356—1422no—
105C22×Q8C22⋊D7112yes—
106C2×M16—21no—
107C2×M16—2842no—
108C4⋊Q8—21no—
109C4⋊Q8—41no—
110C4⋊Q8C2×Dic7111yescharacteristic
111C4⋊Q8C22⋊D7112yes—
112C4⋊Q8C22×D7111yescharacteristic
113#125C2×Dic7112yes—
114C22⋊Q8—41no—
115C4×Q8—41no—
116#117—21no—
117C2×C4⋊C4—41no—
118C2×C42C2×Dic7111yescharacteristic
119#103—24no—
120#103—24no—
121#101—2842no—
122#101—1424no—
123#97C2×Dic7112yes—
124C4×C8—21no—
125C2×Q8—22no—
126C2×Q8—41no—
127C2×Q8—42no—
128C2×Q8—42no—
129C2×Q8C22⋊Dic7112yes—
130C2×Q8C2×C22⋊D7112yes—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4C22⋊Dic7111yescharacteristic
134M16—42no—
135M16—2844no—
136M16—5682no—
137C2×C8—22no—
138C2×C8—5682no—
139C4⋊C4—22no—
140C4⋊C4—41no—
141C4⋊C4—42no—
142C4⋊C4—41no—
143C4⋊C4—42no—
144C4⋊C4—41no—
145C22⋊C4—41no—
146C42—21no—
147C42—41no—
148C42C22⋊Dic7111yescharacteristic
149C42C22×Dic7111yescharacteristic
150C23#1455604311yescharacteristic
151Q8—42no—
152Q8—42no—
153Q8—82no—
154Q8—82no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4#1455604311yescharacteristic
165C2×C4C2×C22⋊Dic7111yescharacteristic
166C8—42no—
167C8—5684no—
168C22—41no—
169C22#1798579412yes—
170C22C2×#1455604311yescharacteristic
171C4—22no—
172C4—21no—
173C4—21no—
174C4—41no—
175C4—42no—
176C4—42no—
177C4—82no—
178C2—21no—
179C2—22no—
180C2C2×#1798579411yescharacteristic
subgroupG/H
centerC2C2×#17985794
commutatorC42×C71C22×C4
FrattiniC4⋊Q8C22×D71
FittingC71×#3139C2
socleC2×C71C2×#354
prime radicalC23×C71#99
cosocleC4⋊Q8×C71C23
hypercenter#3139D71
hypocenterC71#13599
LCSG ▷ C42×C71 ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ #3139

Complements

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

NH ≅ G/Ntype
2. #22586513C2semidirect
94. C71#13599semidirect

Maximal subgroups

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

subgroupindex
C2×#179857982
#225865132
C71×#31392
#1359971