Finite Group Databasebeta

← #356×D103 #27196823

Subgroups — #356×D103 #27196823

Order 13184.

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×D103C211yesmaximal, characteristic
2#101×D103C212yesmaximal
3#20267590C211yesmaximal, characteristic, complemented
4#20267589C212yesmaximal, complemented
5#356×C103C211yesmaximal, characteristic, complemented
6C2×Q8×D103C411yescharacteristic
7C2×Q8×D103C2211yescharacteristic
8M16×D103—22no—
9#15693021C411yescharacteristic
10#15693020—22no—
11#15693019C411yescharacteristic
12#15693019C2211yescharacteristic
13C4⋊C4×D103—22no—
14C42×D103C411yescharacteristic
15#15692940C2212yes—
16#15692939—22no—
17C4⋊Q8×C103C2211yescharacteristic
18#101×C103C2212yes—
19C2×Dic206—21no—
20C2×Dic206C2×C411yescharacteristic
21C2×Dic206C2×C411yescharacteristic
22C2×Dic206D412yes—
23Q8×D103—41no—
24Q8×D103—41no—
25Q8×D103—41no—
26Q8×D103—41no—
27C2×C4×D103—21no—
28C2×C4×D103—21no—
29C2×C4×D103C2×C411yescharacteristic
30C2×C4×D103D412yes—
31#13110089—22no—
32#13110088—42no—
33C8×D103—42no—
34C4⋊Dic103—22no—
35Dic103⋊C4—22no—
36Dic103⋊C4—22no—
37C4×Dic103—21no—
38C4×Dic103C2×C411yescharacteristic
39C2×Q8×C103C2×C411yescharacteristic
40C2×Q8×C103C2311yescharacteristic
41M16×C103—22no—
42C4⋊C4×C103—22no—
43C42×C103C2×C411yescharacteristic
44C22×D103C22⋊C411yescharacteristic
45Q8×C103—41no—
46Q8×C103—41no—
47C2×C4×C103—21no—
48C2×C4×C103—21no—
49C2×C4×C103C22×C411yescharacteristic
50C2×C4×C103C2×D412yes—
51C2×Dic103—21no—
52C2×Dic103—21no—
53C2×Dic103C22⋊C411yescharacteristic
54C2×Dic103C22⋊C412yes—
55C4×D103—24no—
56C4×D103—22no—
57C4×D103—22no—
58C4×D103—21no—
59C4×D103—21no—
60C4×D103—41no—
61C4×D103—41no—
62Dic206—41no—
63Dic206—42no—
64Dic206—42no—
65Dic206—41no—
66C8×C103—42no—
67C103⋊C8—42no—
68C22×C103C2×C22⋊C411yescharacteristic
69C2×D103—22no—
70C2×D103#9911yescharacteristic
71C2×D103#9911yescharacteristic
72C4×C103—22no—
73C4×C103—22no—
74C4×C103—21no—
75C4×C103—41no—
76Dic103—22no—
77Dic103—21no—
78Dic103—22no—
79Dic103—41no—
80C2×C103—21no—
81C2×C103C2×#9911yescharacteristic
82D103—21no—
83D103#35612yesdirect product factor
84C2×#356—1031nomaximal, Sylow
85C103C2×#35611yescharacteristic, complemented, Sylow
86C2×C4⋊Q8—1031no—
87C2×#101—1032no—
88#356—1031no—
89#356—1032no—
90#356D10311yescharacteristic, direct product factor
91C22×Q8—1031no—
92C22×Q8—1031no—
93C2×M16—2062no—
94C4⋊Q8—1031no—
95C4⋊Q8—1031no—
96C4⋊Q8—1031no—
97C4⋊Q8—2062no—
98C4⋊Q8C2×D10311yescharacteristic
99C2×C4⋊C4—2062no—
100C2×C42—1031no—
101#101—1032no—
102#101—2062no—
103#101C2×D10312yes—
104C2×Q8—1031no—
105C2×Q8—1032no—
106C2×Q8—1031no—
107C2×Q8—4121no—
108C2×Q8—4121no—
109C2×Q8—4121no—
110C2×Q8—4121no—
111C2×Q8—2061no—
112C2×Q8C4×D10311yescharacteristic
113C2×Q8C22×D10311yescharacteristic
114C22×C4—1031no—
115C22×C4—1032no—
116C22×C4—2061no—
117C22×C4—2061no—
118M16—22no—
119M16—4122no—
120M16—2062no—
121C2×C8—4122no—
122C4⋊C4—22no—
123C4⋊C4—2062no—
124C4⋊C4—2062no—
125C4⋊C4—2062no—
126C42—1031no—
127C42—2061no—
128C42C4×D10311yescharacteristic
129C23—1031no—
130Q8—41no—
131Q8—41no—
132Q8—4121no—
133Q8—4122no—
134Q8—4121no—
135Q8—4122no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—1031no—
139C2×C4—1032no—
140C2×C4—4121no—
141C2×C4—4121no—
142C2×C4—2062no—
143C2×C4—2062no—
144C2×C4—2061no—
145C2×C4—2061no—
146C2×C4—2064no—
147C2×C4—2061no—
148C2×C4—2061no—
149C2×C4C2×C4×D10311yescharacteristic
150C2×C4D4×D10312yes—
151C8—42no—
152C8—4122no—
153C22—1031no—
154C22—1031no—
155C22—2062no—
156C22C22⋊C4×D10311yescharacteristic
157C4—22no—
158C4—22no—
159C4—21no—
160C4—41no—
161C4—4121no—
162C4—2062no—
163C4—2061no—
164C4—2062no—
165C2—21no—
166C2—1032no—
167C2—2061no—
168C2#99×D10311yescharacteristic
subgroupG/H
centerC2#99×D103
commutatorC2×C4×C103C22×C4
FrattiniC2×Q8C22×D103
Fitting#356×C103C2
socleC2×C103C2×#99
prime radicalC22×D103C22⋊C4
cosocleC2×Q8×C103C23
hypercenter#356D103
hypocenterC103C2×#356
LCSG ▷ C2×C4×C103 ▷ C22×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
3. #20267590C2semidirect
4. #20267589C2semidirect
5. #356×C103C2semidirect
83. D103#356direct (N × H)
85. C103C2×#356semidirect
90. #356D103direct (N × H)

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1032
#101×D1032
#202675902
#202675892
#356×C1032
C2×#356103