Finite Group Databasebeta

← #356×D157 #34101815

Subgroups — #356×D157 #34101815

Order 20096.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C4⋊Q8×D157C211yesmaximal, characteristic
2#101×D157C212yesmaximal
3#23757072C211yesmaximal, characteristic, complemented
4#23757071C212yesmaximal, complemented
5#356×C157C211yesmaximal, characteristic, complemented
6C2×Q8×D157C411yescharacteristic
7C2×Q8×D157C2211yescharacteristic
8M16×D157—22no—
9#19097771C411yescharacteristic
10#19097770—22no—
11#19097769C411yescharacteristic
12#19097769C2211yescharacteristic
13C4⋊C4×D157—22no—
14C42×D157C411yescharacteristic
15#19097690C2212yes—
16#19097689—22no—
17C4⋊Q8×C157C2211yescharacteristic
18#101×C157C2212yes—
19C2×Dic314—21no—
20C2×Dic314C2×C411yescharacteristic
21C2×Dic314C2×C411yescharacteristic
22C2×Dic314D412yes—
23Q8×D157—41no—
24Q8×D157—41no—
25Q8×D157—41no—
26Q8×D157—41no—
27C2×C4×D157—21no—
28C2×C4×D157—21no—
29C2×C4×D157C2×C411yescharacteristic
30C2×C4×D157D412yes—
31#14578938—22no—
32#14578937—42no—
33C8×D157—42no—
34C4⋊Dic157—22no—
35#14578933—22no—
36#14578933—22no—
37C4×Dic157—21no—
38C4×Dic157C2×C411yescharacteristic
39C2×Q8×C157C2×C411yescharacteristic
40C2×Q8×C157C2311yescharacteristic
41M16×C157—22no—
42C4⋊C4×C157—22no—
43C42×C157C2×C411yescharacteristic
44C22×D157C22⋊C411yescharacteristic
45Dic314—41no—
46Dic314—42no—
47Dic314—42no—
48Dic314—41no—
49C2×Dic157—21no—
50C2×Dic157—21no—
51C2×Dic157C22⋊C411yescharacteristic
52C2×Dic157C22⋊C412yes—
53C4×D157—24no—
54C4×D157—21no—
55C4×D157—21no—
56C4×D157—22no—
57C4×D157—22no—
58C4×D157—41no—
59C4×D157—41no—
60#11929434—42no—
61Q8×C157—41no—
62Q8×C157—41no—
63C2×C4×C157—21no—
64C2×C4×C157—21no—
65C2×C4×C157C22×C411yescharacteristic
66C2×C4×C157C2×D412yes—
67C8×C157—42no—
68C22×C157C2×C22⋊C411yescharacteristic
69C2×D157—22no—
70C2×D157#9911yescharacteristic
71C2×D157#9911yescharacteristic
72C4×C157—22no—
73C4×C157—22no—
74C4×C157—21no—
75C4×C157—41no—
76Dic157—22no—
77Dic157—22no—
78Dic157—21no—
79Dic157—41no—
80C2×C157—21no—
81C2×C157C2×#9911yescharacteristic
82D157—21no—
83D157#35612yesdirect product factor
84C157C2×#35611yescharacteristic, complemented, Sylow
85C2×#356—1571nomaximal, Sylow
86C2×C4⋊Q8—1571no—
87C2×#101—1572no—
88#356—1571no—
89#356—1572no—
90#356D15711yescharacteristic, direct product factor
91C22×Q8—1571no—
92C22×Q8—1571no—
93C2×M16—3142no—
94C4⋊Q8—3142no—
95C4⋊Q8—1571no—
96C4⋊Q8—1571no—
97C4⋊Q8—1571no—
98C4⋊Q8C2×D15711yescharacteristic
99C2×C4⋊C4—3142no—
100C2×C42—1571no—
101#101—3142no—
102#101—1572no—
103#101C2×D15712yes—
104C2×Q8—3141no—
105C2×Q8—6281no—
106C2×Q8—6281no—
107C2×Q8—6281no—
108C2×Q8—6281no—
109C2×Q8—1571no—
110C2×Q8—1571no—
111C2×Q8—1572no—
112C2×Q8C4×D15711yescharacteristic
113C2×Q8C22×D15711yescharacteristic
114C22×C4—3141no—
115C22×C4—3141no—
116C22×C4—1571no—
117C22×C4—1572no—
118M16—22no—
119M16—3142no—
120M16—6282no—
121C2×C8—6282no—
122C4⋊C4—22no—
123C4⋊C4—3142no—
124C4⋊C4—3142no—
125C4⋊C4—3142no—
126C42—3141no—
127C42—1571no—
128C42C4×D15711yescharacteristic
129C23—1571no—
130Q8—41no—
131Q8—41no—
132Q8—6282no—
133Q8—6281no—
134Q8—6281no—
135Q8—6282no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—3144no—
139C2×C4—3142no—
140C2×C4—3142no—
141C2×C4—3141no—
142C2×C4—3141no—
143C2×C4—3141no—
144C2×C4—3141no—
145C2×C4—6281no—
146C2×C4—6281no—
147C2×C4—1571no—
148C2×C4—1572no—
149C2×C4C2×C4×D15711yescharacteristic
150C2×C4D4×D15712yes—
151C8—42no—
152C8—6282no—
153C22—3142no—
154C22—1571no—
155C22—1571no—
156C22C22⋊C4×D15711yescharacteristic
157C4—22no—
158C4—22no—
159C4—21no—
160C4—41no—
161C4—3142no—
162C4—3141no—
163C4—3142no—
164C4—6281no—
165C2—21no—
166C2—3141no—
167C2—1572no—
168C2#99×D15711yescharacteristic
subgroupG/H
centerC2#99×D157
commutatorC2×C4×C157C22×C4
FrattiniC2×Q8C22×D157
Fitting#356×C157C2
socleC2×C157C2×#99
prime radicalC22×D157C22⋊C4
cosocleC2×Q8×C157C23
hypercenter#356D157
hypocenterC157C2×#356
LCSG ▷ C2×C4×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
3. #23757072C2semidirect
4. #23757071C2semidirect
5. #356×C157C2semidirect
83. D157#356direct (N × H)
84. C157C2×#356semidirect
90. #356D157direct (N × H)

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1572
#101×D1572
#237570722
#237570712
#356×C1572
C2×#356157