Finite Group Databasebeta

← #356×D71 #22590019

Subgroups — #356×D71 #22590019

Order 9088.

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×D71C211yesmaximal, characteristic
2#101×D71C212yesmaximal
3#17985798C211yesmaximal, characteristic, complemented
4#17985797C212yesmaximal, complemented
5#356×C71C211yesmaximal, characteristic, complemented
6C2×Q8×D71C411yescharacteristic
7C2×Q8×D71C2211yescharacteristic
8M16×D71—22no—
9#14556128C411yescharacteristic
10#14556127—22no—
11#14556126C411yescharacteristic
12#14556126C2211yescharacteristic
13C4⋊C4×D71—22no—
14C42×D71C411yescharacteristic
15#14556047C2212yes—
16#14556046—22no—
17C4⋊Q8×C71C2211yescharacteristic
18#101×C71C2212yes—
19C2×Dic142—21no—
20C2×Dic142C2×C411yescharacteristic
21C2×Dic142C2×C411yescharacteristic
22C2×Dic142D412yes—
23Q8×D71—41no—
24Q8×D71—41no—
25Q8×D71—41no—
26Q8×D71—41no—
27C2×C4×D71—21no—
28C2×C4×D71—21no—
29C2×C4×D71C2×C411yescharacteristic
30C2×C4×D71D412yes—
31#11765849—22no—
32#11765848—42no—
33C8×D71—42no—
34C4⋊Dic71—22no—
35Dic71⋊C4—22no—
36Dic71⋊C4—22no—
37C4×Dic71—21no—
38C4×Dic71C2×C411yescharacteristic
39C2×Q8×C71C2×C411yescharacteristic
40C2×Q8×C71C2311yescharacteristic
41M16×C71—22no—
42C4⋊C4×C71—22no—
43C42×C71C2×C411yescharacteristic
44C22×D71C22⋊C411yescharacteristic
45Q8×C71—41no—
46Q8×C71—41no—
47C2×C4×C71—21no—
48C2×C4×C71—21no—
49C2×C4×C71C22×C411yescharacteristic
50C2×C4×C71C2×D412yes—
51C2×Dic71—21no—
52C2×Dic71—21no—
53C2×Dic71C22⋊C411yescharacteristic
54C2×Dic71C22⋊C412yes—
55C4×D71—24no—
56C4×D71—22no—
57C4×D71—22no—
58C4×D71—21no—
59C4×D71—21no—
60C4×D71—41no—
61C4×D71—41no—
62Dic142—41no—
63Dic142—42no—
64Dic142—42no—
65Dic142—41no—
66C8×C71—42no—
67C71⋊C8—42no—
68C22×C71C2×C22⋊C411yescharacteristic
69C2×D71—22no—
70C2×D71#9911yescharacteristic
71C2×D71#9911yescharacteristic
72C4×C71—22no—
73C4×C71—22no—
74C4×C71—21no—
75C4×C71—41no—
76Dic71—22no—
77Dic71—21no—
78Dic71—22no—
79Dic71—41no—
80C2×C71—21no—
81C2×C71C2×#9911yescharacteristic
82D71—21no—
83D71#35612yesdirect product factor
84C2×#356—711nomaximal, Sylow
85C71C2×#35611yescharacteristic, complemented, Sylow
86C2×C4⋊Q8—711no—
87C2×#101—712no—
88#356—711no—
89#356—712no—
90#356D7111yescharacteristic, direct product factor
91C22×Q8—711no—
92C22×Q8—711no—
93C2×M16—1422no—
94C4⋊Q8—711no—
95C4⋊Q8—711no—
96C4⋊Q8—711no—
97C4⋊Q8—1422no—
98C4⋊Q8C2×D7111yescharacteristic
99C2×C4⋊C4—1422no—
100C2×C42—711no—
101#101—712no—
102#101—1422no—
103#101C2×D7112yes—
104C2×Q8—2841no—
105C2×Q8—2841no—
106C2×Q8—2841no—
107C2×Q8—2841no—
108C2×Q8—711no—
109C2×Q8—712no—
110C2×Q8—711no—
111C2×Q8—1421no—
112C2×Q8C4×D7111yescharacteristic
113C2×Q8C22×D7111yescharacteristic
114C22×C4—711no—
115C22×C4—712no—
116C22×C4—1421no—
117C22×C4—1421no—
118M16—22no—
119M16—2842no—
120M16—1422no—
121C2×C8—2842no—
122C4⋊C4—22no—
123C4⋊C4—1422no—
124C4⋊C4—1422no—
125C4⋊C4—1422no—
126C42—711no—
127C42—1421no—
128C42C4×D7111yescharacteristic
129C23—711no—
130Q8—41no—
131Q8—41no—
132Q8—2841no—
133Q8—2842no—
134Q8—2841no—
135Q8—2842no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—2841no—
139C2×C4—2841no—
140C2×C4—711no—
141C2×C4—712no—
142C2×C4—1422no—
143C2×C4—1422no—
144C2×C4—1421no—
145C2×C4—1421no—
146C2×C4—1424no—
147C2×C4—1421no—
148C2×C4—1421no—
149C2×C4C2×C4×D7111yescharacteristic
150C2×C4D4×D7112yes—
151C8—42no—
152C8—2842no—
153C22—711no—
154C22—711no—
155C22—1422no—
156C22C22⋊C4×D7111yescharacteristic
157C4—22no—
158C4—22no—
159C4—21no—
160C4—41no—
161C4—2841no—
162C4—1422no—
163C4—1421no—
164C4—1422no—
165C2—21no—
166C2—712no—
167C2—1421no—
168C2#99×D7111yescharacteristic
subgroupG/H
centerC2#99×D71
commutatorC2×C4×C71C22×C4
FrattiniC2×Q8C22×D71
Fitting#356×C71C2
socleC2×C71C2×#99
hypercenter#356D71
hypocenterC71C2×#356
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
3. #17985798C2semidirect
4. #17985797C2semidirect
5. #356×C71C2semidirect
83. D71#356direct (N × H)
85. C71C2×#356semidirect
90. #356D71direct (N × H)