Finite Group Databasebeta

← #30776153

Subgroups — #30776153

Order 17152.

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

normal subgroups only

This group has 406 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
1#22561874C212yesmaximal
2#22558374C214yesmaximal, complemented
3C67×#3152C211yesmaximal, characteristic
4#16863370—22no—
5#16863191—22no—
6#16863189C2214yes—
7C2×#137×C67—22no—
8#532×C67C2211yescharacteristic
9C2×C4⋊Q8×C67C411yescharacteristic, complemented
10#497×C67—22no—
11#494×C67C412yescomplemented
12#492×C67C2212yes—
13C4×M16×C67C411yescharacteristic, complemented
14#14524002—42no—
15#14523958—24no—
16#14523958—42no—
17C2×#41×C67D412yes—
18C22×Q8×C67—21no—
19#137×C67—44no—
20#137×C67—44no—
21C2×Q16×C67—24no—
22C2×SD16×C67—24no—
23C2×M16×C67—21no—
24C4⋊Q8×C67—21no—
25C4⋊Q8×C67—41no—
26C4⋊Q8×C67C2×C412yes—
27C4⋊Q8×C67C2×C411yescharacteristic
28#127×C67C2311yescharacteristic
29#124×C67D412yes—
30#121×C67—41no—
31C4×D4×C67—41no—
32C2×C4⋊C4×C67—41no—
33C2×C42×C67C2×C411yescharacteristic
34#97×C67—21no—
35C4×C8×C67C2×C412yes—
36C4⋊Dic67—82no—
37C22⋊Dic67—44no—
38C4×Dic67—82no—
39#41×C67—42no—
40#41×C67—42no—
41C2×Q8×C67—22no—
42C2×Q8×C67—42no—
43C2×Q8×C67—42no—
44C2×Q8×C67—41no—
45C2×Q8×C67C22⋊C412yes—
46C2×D4×C67—22no—
47C2×D4×C67—41no—
48C2×D4×C67C2×D412yes—
49C22×C4×C67—21no—
50C22×C4×C67—22no—
51C22×C4×C67—41no—
52C22×C4×C67C22⋊C411yescharacteristic
53Q16×C67—48no—
54SD16×C67—48no—
55M16×C67—24no—
56C2×C8×C67—22no—
57C4⋊C4×C67—41no—
58C4⋊C4×C67—42no—
59C4⋊C4×C67—42no—
60C22⋊C4×C67—42no—
61C42×C67—21no—
62C42×C67C22⋊C411yescharacteristic
63C42×C67C22×C411yescharacteristic
64C23×C67—22no—
65C23×C67#9911yescharacteristic
66Q8×C67—42no—
67Q8×C67—44no—
68Q8×C67—82no—
69D4×C67—42no—
70D4×C67—82no—
71D4×C67—82no—
72C2×C4×C67—21no—
73C2×C4×C67—22no—
74C2×C4×C67—21no—
75C2×C4×C67—22no—
76C2×C4×C67—42no—
77C2×C4×C67—42no—
78C2×C4×C67—42no—
79C2×C4×C67—42no—
80C2×C4×C67—81no—
81C2×C4×C67—81no—
82C2×C4×C67#9911yescharacteristic
83C2×C4×C67C2×C22⋊C411yescharacteristic
84C2×Dic67—84no—
85C8×C67—24no—
86C22×C67—21no—
87C22×C67—41no—
88C22×C67—42no—
89C22×C67—82no—
90C22×C67C2×#9911yescharacteristic
91C4×C67—22no—
92C4×C67—21no—
93C4×C67—41no—
94C4×C67—42no—
95C4×C67—82no—
96C4×C67#35312yes—
97Dic67—164no—
98#13586—671nomaximal, Sylow
99C2×C67—21no—
100C2×C67—41no—
101C2×C67—82no—
102C2×C67C2×#35311yescharacteristic
103#3152D6711yescharacteristic
104#2126—672no—
105#1409—674no—
106C67#1358611yescharacteristic, complemented, Sylow
107C2×#137—22no—
108#532C2×D6711yescharacteristic
109C2×C4⋊Q8Dic6711yescharacteristic, complemented
110#497—22no—
111#494Dic6712yescomplemented
112#492C2×D6712yes—
113#410—1342no—
114C4×M16Dic6711yescharacteristic, complemented
115#354—1342no—
116#353—674no—
117C2×#41C22⋊D6712yes—
118C22×Q8—21no—
119#137—44no—
120#137—44no—
121C2×Q16—24no—
122C2×SD16—24no—
123C2×M16—21no—
124C4⋊Q8—21no—
125C4⋊Q8—41no—
126C4⋊Q8C2×Dic6712yes—
127C4⋊Q8C2×Dic6711yescharacteristic
128#127C22×D6711yescharacteristic
129#124C22⋊D6712yes—
130#121—41no—
131C4×D4—41no—
132#117—2682no—
133C2×C4⋊C4—41no—
134C2×C42C2×Dic6711yescharacteristic
135#99—2682no—
136#99—1344no—
137#97—21no—
138C4×C8C2×Dic6712yes—
139#41—42no—
140#41—42no—
141C2×Q8—22no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—41no—
145C2×Q8C22⋊Dic6712yes—
146C2×D4—22no—
147C2×D4—41no—
148C2×D4C2×C22⋊D6712yes—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—41no—
152C22×C4C22⋊Dic6711yescharacteristic
153Q16—48no—
154SD16—48no—
155M16—24no—
156C2×C8—22no—
157C4⋊C4—41no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—5362no—
161C22⋊C4—42no—
162C22⋊C4—2684no—
163C42—21no—
164C42—5362no—
165C42C22⋊Dic6711yescharacteristic
166C42C22×Dic6711yescharacteristic
167C23—22no—
168C23#1452395811yescharacteristic
169Q8—42no—
170Q8—44no—
171Q8—82no—
172D4—42no—
173D4—82no—
174D4—82no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—5364no—
186C2×C4#1452395811yescharacteristic
187C2×C4C2×C22⋊Dic6711yescharacteristic
188C8—24no—
189C22—21no—
190C22—41no—
191C22—42no—
192C22—82no—
193C22C2×#1452395811yescharacteristic
194C4—22no—
195C4—21no—
196C4—41no—
197C4—42no—
198C4—82no—
199C4—10724no—
200C4#1686318912yes—
201C2—21no—
202C2—41no—
203C2—82no—
204C2C2×#1686318911yescharacteristic
subgroupG/H
centerC2C2×#16863189
commutatorC42×C67C22×C4
Frattini#127C22×D67
FittingC67×#3152C2
socleC2×C67C2×#353
prime radical#532×C67C22
cosocle#127×C67C23
hypercenter#3152D67
hypocenterC67#13586
LCSG ▷ C42×C67 ▷ C2×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #3152

Complements

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

NH ≅ G/Ntype
2. #22558374C2semidirect
9. C2×C4⋊Q8×C67C4semidirect
11. #494×C67C4semidirect
13. C4×M16×C67C4semidirect
106. C67#13586semidirect
109. C2×C4⋊Q8Dic67semidirect
111. #494Dic67semidirect
114. C4×M16Dic67semidirect

Maximal subgroups

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

subgroupindex
#225618742
#225583742
C67×#31522
#1358667