Finite Group Databasebeta

← #30776194

Subgroups — #30776194

Order 17152.

312 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C2×#16863195C212yesmaximal
2#22558382C214yesmaximal, complemented
3C67×#3139C211yesmaximal, characteristic
4C2×#14523962—22no—
5#16863195—22no—
6#16863195C2214yes—
7#533×C67C411yescharacteristic
8C2×C4⋊Q8×C67C2211yescharacteristic
9#491×C67C412yes—
10#491×C67C2212yes—
11#487×C67—22no—
12#419×C67—22no—
13C4×M16×C67C411yescharacteristic
14C2×#11761612—42no—
15#14523962—24no—
16#14523962—42no—
17C22×Q8×C67D412yes—
18C2×M16×C67—21no—
19C4⋊Q8×C67—21no—
20C4⋊Q8×C67—41no—
21C4⋊Q8×C67C2×C411yescharacteristic
22C4⋊Q8×C67D412yes—
23C4⋊Q8×C67C2311yescharacteristic
24#125×C67C2×C412yes—
25C22⋊Q8×C67—41no—
26C4×Q8×C67—41no—
27#117×C67—21no—
28C2×C4⋊C4×C67—41no—
29C2×C42×C67C2×C411yescharacteristic
30#103×C67—24no—
31#103×C67—24no—
32#97×C67C2×C412yes—
33C4×C8×C67—21no—
34#11761612—44no—
35#11761612—82no—
36C2×C67⋊C8—82no—
37C2×Q8×C67—22no—
38C2×Q8×C67—41no—
39C2×Q8×C67—42no—
40C2×Q8×C67—42no—
41C2×Q8×C67C22⋊C412yes—
42C2×Q8×C67C2×D412yes—
43C22×C4×C67—21no—
44C22×C4×C67—22no—
45C22×C4×C67C22⋊C411yescharacteristic
46M16×C67—42no—
47C2×C8×C67—22no—
48C4⋊C4×C67—22no—
49C4⋊C4×C67—41no—
50C4⋊C4×C67—42no—
51C4⋊C4×C67—41no—
52C4⋊C4×C67—42no—
53C4⋊C4×C67—41no—
54C22⋊C4×C67—41no—
55C42×C67—21no—
56C42×C67—41no—
57C42×C67C22⋊C411yescharacteristic
58C42×C67C22×C411yescharacteristic
59C23×C67#9911yescharacteristic
60Q8×C67—42no—
61Q8×C67—42no—
62Q8×C67—82no—
63Q8×C67—82no—
64C2×C4×C67—22no—
65C2×C4×C67—21no—
66C2×C4×C67—21no—
67C2×C4×C67—22no—
68C2×C4×C67—22no—
69C2×C4×C67—42no—
70C2×C4×C67—42no—
71C2×C4×C67—42no—
72C2×C4×C67—42no—
73C2×C4×C67#9911yescharacteristic
74C2×C4×C67C2×C22⋊C411yescharacteristic
75C8×C67—42no—
76C67⋊C8—84no—
77C22×C67—41no—
78C22×C67#35412yes—
79C22×C67C2×#9911yescharacteristic
80C4×C67—22no—
81C4×C67—21no—
82C4×C67—21no—
83C4×C67—41no—
84C4×C67—42no—
85C4×C67—42no—
86C4×C67—82no—
87#13599—671nomaximal, Sylow
88C2×C67—21no—
89C2×C67—22no—
90C2×C67C2×#35411yescharacteristic
91#3139D6711yescharacteristic
92C2×#356—672no—
93#1413—674no—
94C67#1359911yescharacteristic, complemented, Sylow
95#533Dic6711yescharacteristic
96C2×C4⋊Q8C2×D6711yescharacteristic
97#491Dic6712yes—
98#491C2×D6712yes—
99#487—22no—
100#419—22no—
101C2×#101—1342no—
102C4×M16Dic6711yescharacteristic
103#356—674no—
104#356—1342no—
105C22×Q8C22⋊D6712yes—
106C2×M16—21no—
107C2×M16—2682no—
108C4⋊Q8—21no—
109C4⋊Q8—41no—
110C4⋊Q8C2×Dic6711yescharacteristic
111C4⋊Q8C22⋊D6712yes—
112C4⋊Q8C22×D6711yescharacteristic
113#125C2×Dic6712yes—
114C22⋊Q8—41no—
115C4×Q8—41no—
116#117—21no—
117C2×C4⋊C4—41no—
118C2×C42C2×Dic6711yescharacteristic
119#103—24no—
120#103—24no—
121#101—2682no—
122#101—1344no—
123#97C2×Dic6712yes—
124C4×C8—21no—
125C2×Q8—22no—
126C2×Q8—41no—
127C2×Q8—42no—
128C2×Q8—42no—
129C2×Q8C22⋊Dic6712yes—
130C2×Q8C2×C22⋊D6712yes—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4C22⋊Dic6711yescharacteristic
134M16—42no—
135M16—2684no—
136M16—5362no—
137C2×C8—22no—
138C2×C8—5362no—
139C4⋊C4—22no—
140C4⋊C4—41no—
141C4⋊C4—42no—
142C4⋊C4—41no—
143C4⋊C4—42no—
144C4⋊C4—41no—
145C22⋊C4—41no—
146C42—21no—
147C42—41no—
148C42C22⋊Dic6711yescharacteristic
149C42C22×Dic6711yescharacteristic
150C23#1452395811yescharacteristic
151Q8—42no—
152Q8—42no—
153Q8—82no—
154Q8—82no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4#1452395811yescharacteristic
165C2×C4C2×C22⋊Dic6711yescharacteristic
166C8—42no—
167C8—5364no—
168C22—41no—
169C22#1686319112yes—
170C22C2×#1452395811yescharacteristic
171C4—22no—
172C4—21no—
173C4—21no—
174C4—41no—
175C4—42no—
176C4—42no—
177C4—82no—
178C2—21no—
179C2—22no—
180C2C2×#1686319111yescharacteristic
subgroupG/H
centerC2C2×#16863191
commutatorC42×C67C22×C4
FrattiniC4⋊Q8C22×D67
FittingC67×#3139C2
socleC2×C67C2×#354
prime radicalC23×C67#99
cosocleC4⋊Q8×C67C23
hypercenter#3139D67
hypocenterC67#13599
LCSG ▷ C42×C67 ▷ C2×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ #3139

Complements

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

NH ≅ G/Ntype
2. #22558382C2semidirect
94. C67#13599semidirect

Maximal subgroups

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

subgroupindex
C2×#168631952
#225583822
C67×#31392
#1359967