Finite Group Databasebeta

← #356×D179 #37510679

Subgroups — #356×D179 #37510679

Order 22912.

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×D179C211yesmaximal, characteristic
2#101×D179C212yesmaximal
3#26038238C211yesmaximal, characteristic, complemented
4#26038237C212yesmaximal, complemented
5#356×C179C211yesmaximal, characteristic, complemented
6C2×Q8×D179C411yescharacteristic
7C2×Q8×D179C2211yescharacteristic
8M16×D179—22no—
9#19153113C411yescharacteristic
10#19153112—22no—
11#19153111C411yescharacteristic
12#19153111C2211yescharacteristic
13C4⋊C4×D179—22no—
14C42×D179C411yescharacteristic
15#19153032C2212yes—
16#19153031—22no—
17C4⋊Q8×C179C2211yescharacteristic
18#101×C179C2212yes—
19C2×Dic358—21no—
20C2×Dic358C2×C411yescharacteristic
21C2×Dic358C2×C411yescharacteristic
22C2×Dic358D412yes—
23Q8×D179—41no—
24Q8×D179—41no—
25Q8×D179—41no—
26Q8×D179—41no—
27C2×C4×D179—21no—
28C2×C4×D179—21no—
29C2×C4×D179C2×C411yescharacteristic
30C2×C4×D179D412yes—
31#15668243—22no—
32#15668242—42no—
33C8×D179—42no—
34C4⋊Dic179—22no—
35Dic179⋊C4—22no—
36Dic179⋊C4—22no—
37C4×Dic179—21no—
38C4×Dic179C2×C411yescharacteristic
39C2×Q8×C179C2×C411yescharacteristic
40C2×Q8×C179C2311yescharacteristic
41M16×C179—22no—
42C4⋊C4×C179—22no—
43C42×C179C2×C411yescharacteristic
44C22×D179C22⋊C411yescharacteristic
45Dic358—41no—
46Dic358—42no—
47Dic358—42no—
48Dic358—41no—
49C2×Dic179—21no—
50C2×Dic179—21no—
51C2×Dic179C22⋊C411yescharacteristic
52C2×Dic179C22⋊C412yes—
53C4×D179—24no—
54C4×D179—22no—
55C4×D179—22no—
56C4×D179—21no—
57C4×D179—21no—
58C4×D179—41no—
59C4×D179—41no—
60C179⋊C8—42no—
61Q8×C179—41no—
62Q8×C179—41no—
63C2×C4×C179—21no—
64C2×C4×C179—21no—
65C2×C4×C179C22×C411yescharacteristic
66C2×C4×C179C2×D412yes—
67C8×C179—42no—
68C22×C179C2×C22⋊C411yescharacteristic
69C2×D179—22no—
70C2×D179#9911yescharacteristic
71C2×D179#9911yescharacteristic
72C4×C179—22no—
73C4×C179—22no—
74C4×C179—21no—
75C4×C179—41no—
76Dic179—22no—
77Dic179—22no—
78Dic179—21no—
79Dic179—41no—
80C2×C179—21no—
81C2×C179C2×#9911yescharacteristic
82D179—21no—
83D179#35612yesdirect product factor
84C179C2×#35611yescharacteristic, complemented, Sylow
85C2×#356—1791nomaximal, Sylow
86C2×C4⋊Q8—1791no—
87C2×#101—1792no—
88#356—1791no—
89#356—1792no—
90#356D17911yescharacteristic, direct product factor
91C22×Q8—1791no—
92C22×Q8—1791no—
93C2×M16—3582no—
94C4⋊Q8—3582no—
95C4⋊Q8—1791no—
96C4⋊Q8—1791no—
97C4⋊Q8—1791no—
98C4⋊Q8C2×D17911yescharacteristic
99C2×C4⋊C4—3582no—
100C2×C42—1791no—
101#101—3582no—
102#101—1792no—
103#101C2×D17912yes—
104C2×Q8—3581no—
105C2×Q8—1791no—
106C2×Q8—1791no—
107C2×Q8—1792no—
108C2×Q8—7161no—
109C2×Q8—7161no—
110C2×Q8—7161no—
111C2×Q8—7161no—
112C2×Q8C4×D17911yescharacteristic
113C2×Q8C22×D17911yescharacteristic
114C22×C4—3581no—
115C22×C4—3581no—
116C22×C4—1791no—
117C22×C4—1792no—
118M16—22no—
119M16—3582no—
120M16—7162no—
121C2×C8—7162no—
122C4⋊C4—22no—
123C4⋊C4—3582no—
124C4⋊C4—3582no—
125C4⋊C4—3582no—
126C42—3581no—
127C42—1791no—
128C42C4×D17911yescharacteristic
129C23—1791no—
130Q8—41no—
131Q8—41no—
132Q8—7162no—
133Q8—7161no—
134Q8—7161no—
135Q8—7162no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—3584no—
139C2×C4—3582no—
140C2×C4—3582no—
141C2×C4—3581no—
142C2×C4—3581no—
143C2×C4—3581no—
144C2×C4—3581no—
145C2×C4—1791no—
146C2×C4—1792no—
147C2×C4—7161no—
148C2×C4—7161no—
149C2×C4C2×C4×D17911yescharacteristic
150C2×C4D4×D17912yes—
151C8—42no—
152C8—7162no—
153C22—3582no—
154C22—1791no—
155C22—1791no—
156C22C22⋊C4×D17911yescharacteristic
157C4—22no—
158C4—22no—
159C4—21no—
160C4—41no—
161C4—3582no—
162C4—3581no—
163C4—3582no—
164C4—7161no—
165C2—21no—
166C2—3581no—
167C2—1792no—
168C2#99×D17911yescharacteristic
subgroupG/H
centerC2#99×D179
commutatorC2×C4×C179C22×C4
FrattiniC2×Q8C22×D179
Fitting#356×C179C2
socleC2×C179C2×#99
prime radicalC22×D179C22⋊C4
cosocleC2×Q8×C179C23
hypercenter#356D179
hypocenterC179C2×#356
LCSG ▷ C2×C4×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
3. #26038238C2semidirect
4. #26038237C2semidirect
5. #356×C179C2semidirect
83. D179#356direct (N × H)
84. C179C2×#356semidirect
90. #356D179direct (N × H)

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1792
#101×D1792
#260382382
#260382372
#356×C1792
C2×#356179