Finite Group Databasebeta

← #356×D199 #38770529

Subgroups — #356×D199 #38770529

Order 25472.

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×D199C211yesmaximal, characteristic
2#101×D199C212yesmaximal
3#27163817C211yesmaximal, characteristic, complemented
4#27163816C212yesmaximal, complemented
5#356×C199C211yesmaximal, characteristic, complemented
6C2×Q8×D199C411yescharacteristic
7C2×Q8×D199C2211yescharacteristic
8M16×D199—22no—
9#20263955C411yescharacteristic
10#20263954—22no—
11#20263953C411yescharacteristic
12#20263953C2211yescharacteristic
13C4⋊C4×D199—22no—
14C42×D199C411yescharacteristic
15#20263874C2212yes—
16#20263873—22no—
17C4⋊Q8×C199C2211yescharacteristic
18#101×C199C2212yes—
19C2×Dic398—21no—
20C2×Dic398C2×C411yescharacteristic
21C2×Dic398C2×C411yescharacteristic
22C2×Dic398D412yes—
23Q8×D199—41no—
24Q8×D199—41no—
25Q8×D199—41no—
26Q8×D199—41no—
27C2×C4×D199—21no—
28C2×C4×D199—21no—
29C2×C4×D199C2×C411yescharacteristic
30C2×C4×D199D412yes—
31#15691870—22no—
32#15691869—42no—
33C8×D199—42no—
34C4⋊Dic199—22no—
35Dic199⋊C4—22no—
36Dic199⋊C4—22no—
37C4×Dic199—21no—
38C4×Dic199C2×C411yescharacteristic
39C2×Q8×C199C2×C411yescharacteristic
40C2×Q8×C199C2311yescharacteristic
41M16×C199—22no—
42C4⋊C4×C199—22no—
43C42×C199C2×C411yescharacteristic
44C22×D199C22⋊C411yescharacteristic
45Dic398—41no—
46Dic398—42no—
47Dic398—42no—
48Dic398—41no—
49C2×Dic199—21no—
50C2×Dic199—21no—
51C2×Dic199C22⋊C411yescharacteristic
52C2×Dic199C22⋊C412yes—
53C4×D199—24no—
54C4×D199—22no—
55C4×D199—22no—
56C4×D199—21no—
57C4×D199—21no—
58C4×D199—41no—
59C4×D199—41no—
60C199⋊C8—42no—
61Q8×C199—41no—
62Q8×C199—41no—
63C2×C4×C199—21no—
64C2×C4×C199—21no—
65C2×C4×C199C22×C411yescharacteristic
66C2×C4×C199C2×D412yes—
67C8×C199—42no—
68C22×C199C2×C22⋊C411yescharacteristic
69C2×D199—22no—
70C2×D199#9911yescharacteristic
71C2×D199#9911yescharacteristic
72C4×C199—22no—
73C4×C199—22no—
74C4×C199—21no—
75C4×C199—41no—
76Dic199—22no—
77Dic199—22no—
78Dic199—21no—
79Dic199—41no—
80C2×C199—21no—
81C2×C199C2×#9911yescharacteristic
82D199—21no—
83D199#35612yesdirect product factor
84C199C2×#35611yescharacteristic, complemented, Sylow
85C2×#356—1991nomaximal, Sylow
86C2×C4⋊Q8—1991no—
87C2×#101—1992no—
88#356—1991no—
89#356—1992no—
90#356D19911yescharacteristic, direct product factor
91C22×Q8—1991no—
92C22×Q8—1991no—
93C2×M16—3982no—
94C4⋊Q8—3982no—
95C4⋊Q8—1991no—
96C4⋊Q8—1991no—
97C4⋊Q8—1991no—
98C4⋊Q8C2×D19911yescharacteristic
99C2×C4⋊C4—3982no—
100C2×C42—1991no—
101#101—3982no—
102#101—1992no—
103#101C2×D19912yes—
104C2×Q8—7961no—
105C2×Q8—7961no—
106C2×Q8—7961no—
107C2×Q8—7961no—
108C2×Q8—3981no—
109C2×Q8—1991no—
110C2×Q8—1991no—
111C2×Q8—1992no—
112C2×Q8C4×D19911yescharacteristic
113C2×Q8C22×D19911yescharacteristic
114C22×C4—3981no—
115C22×C4—3981no—
116C22×C4—1991no—
117C22×C4—1992no—
118M16—22no—
119M16—7962no—
120M16—3982no—
121C2×C8—7962no—
122C4⋊C4—22no—
123C4⋊C4—3982no—
124C4⋊C4—3982no—
125C4⋊C4—3982no—
126C42—3981no—
127C42—1991no—
128C42C4×D19911yescharacteristic
129C23—1991no—
130Q8—41no—
131Q8—41no—
132Q8—7961no—
133Q8—7962no—
134Q8—7961no—
135Q8—7962no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—7961no—
139C2×C4—7961no—
140C2×C4—3984no—
141C2×C4—3982no—
142C2×C4—3982no—
143C2×C4—3981no—
144C2×C4—3981no—
145C2×C4—3981no—
146C2×C4—3981no—
147C2×C4—1991no—
148C2×C4—1992no—
149C2×C4C2×C4×D19911yescharacteristic
150C2×C4D4×D19912yes—
151C8—42no—
152C8—7962no—
153C22—3982no—
154C22—1991no—
155C22—1991no—
156C22C22⋊C4×D19911yescharacteristic
157C4—22no—
158C4—22no—
159C4—21no—
160C4—41no—
161C4—7961no—
162C4—3982no—
163C4—3981no—
164C4—3982no—
165C2—21no—
166C2—3981no—
167C2—1992no—
168C2#99×D19911yescharacteristic
subgroupG/H
centerC2#99×D199
commutatorC2×C4×C199C22×C4
FrattiniC2×Q8C22×D199
Fitting#356×C199C2
socleC2×C199C2×#99
prime radicalC22×D199C22⋊C4
cosocleC2×Q8×C199C23
hypercenter#356D199
hypocenterC199C2×#356
LCSG ▷ C2×C4×C199 ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
3. #27163817C2semidirect
4. #27163816C2semidirect
5. #356×C199C2semidirect
83. D199#356direct (N × H)
84. C199C2×#356semidirect
90. #356D199direct (N × H)

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1992
#101×D1992
#271638172
#271638162
#356×C1992
C2×#356199