Finite Group Databasebeta

← #30765340

Subgroups — #30765340

Order 17152.

262 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 71 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#22569146C211yesmaximal, characteristic
2#22562022C212yesmaximal
3#22558161C212yesmaximal
4#22558160C211yesmaximal, characteristic
5C67×#1327C211yesmaximal, characteristic
6#16863953C411yescharacteristic
7#16863525—22no—
8#16863525—22no—
9#16863424C411yescharacteristic
10#16863424C2211yescharacteristic
11#16863354C411yescharacteristic
12#16863354C2211yescharacteristic
13C2×#14523954C411yescharacteristic
14#16863216—22no—
15#16863178C2212yes—
16#16863173—22no—
17#423×C67C2211yescharacteristic
18#349×C67C2212yes—
19C22×C67⋊C8C2×C411yescharacteristic
20C22×C67⋊C8C2×C411yescharacteristic
21#14524043—21no—
22#14524035—21no—
23#14524022—22no—
24C4×Dic134—22no—
25#14524002C2×C411yescharacteristic
26#14524002C2×C411yescharacteristic
27#14523983—22no—
28Dic67⋊C8—22no—
29Dic67⋊C8—22no—
30#14523954—21no—
31#14523954—21no—
32#14523953—21no—
33#14523953D412yes—
34C8×Dic67—21no—
35C8×Dic67D412yes—
36C2×M16×C67C2×C411yescharacteristic
37C2×M16×C67C2311yescharacteristic
38C2×C42×C67C2×C411yescharacteristic
39M32×C67—22no—
40C4⋊C8×C67—22no—
41C2×Dic134—22no—
42C2×C67⋊C8—22no—
43C2×C67⋊C8—21no—
44C2×C67⋊C8—41no—
45C2×C67⋊C8C22⋊C411yescharacteristic
46C2×C67⋊C8C22⋊C411yescharacteristic
47C4⋊Dic67—22no—
48C4⋊Dic67—21no—
49C4⋊Dic67—21no—
50Dic67⋊C4—24no—
51C22⋊Dic67—21no—
52C22⋊Dic67M1612yes—
53C4×Dic67—21no—
54C4×Dic67C22⋊C412yes—
55C67⋊C16—42no—
56C22×C4×C67—21no—
57C22×C4×C67C22×C411yescharacteristic
58M16×C67—41no—
59M16×C67—41no—
60C2×C8×C67—21no—
61C2×C8×C67C2×D412yes—
62C42×C67—21no—
63C42×C67M1611yescharacteristic
64C42×C67M1611yescharacteristic
65C16×C67—42no—
66C23×C67C2×M1611yescharacteristic
67C2×C4×C67—21no—
68C2×C4×C67—21no—
69C2×C4×C67—21no—
70C2×C4×C67—21no—
71C2×C4×C67—81no—
72C2×C4×C67C2×C22⋊C411yescharacteristic
73C2×C4×C67C2×M1611yescharacteristic
74C2×Dic67—22no—
75C2×Dic67—22no—
76Dic134—82no—
77C8×C67—22no—
78C8×C67—41no—
79C67⋊C8—21no—
80C67⋊C8—41no—
81C67⋊C8#10112yes—
82C22×C67—21no—
83C22×C67—41no—
84C22×C67#40711yescharacteristic
85C4×C67—21no—
86C4×C67—41no—
87C4×C67—41no—
88C4×C67#41011yescharacteristic
89C4×C67C2×#10111yescharacteristic
90Dic67—42no—
91Dic67—42no—
92#11540—671nomaximal, Sylow
93C2×C67—21no—
94C2×C67—41no—
95C2×C67#146711yescharacteristic
96#2961—671no—
97#2166—672no—
98#1328—672no—
99#1327—671no—
100#1327D6711yescharacteristic
101C67#1154011yescharacteristic, complemented, Sylow
102#533—671no—
103#446—1342no—
104#446—1342no—
105#424—671no—
106#424—671no—
107#423C2×D6711yescharacteristic
108#405—671no—
109#405—671no—
110C2×#97—671no—
111#364—1342no—
112#349—672no—
113#349C2×D6712yes—
114#347—1342no—
115C2×M16C4×D6711yescharacteristic
116C2×M16C22×D6711yescharacteristic
117C22×C8—671no—
118C22×C8—671no—
119C4⋊Q8—1341no—
120#125—1341no—
121C22⋊Q8—1342no—
122C4×Q8—1342no—
123#117—671no—
124#117—671no—
125C2×C42C4×D6711yescharacteristic
126M32—22no—
127M32—1342no—
128C4⋊C8—22no—
129C4⋊C8—1342no—
130C4⋊C8—1342no—
131#97—672no—
132#97—1341no—
133#97—1341no—
134#97—1341no—
135C4×C8—672no—
136C4×C8—1341no—
137C2×Q8—1342no—
138C22×C4—21no—
139C22×C4C2×C4×D6711yescharacteristic
140M16—41no—
141M16—41no—
142C2×C8—21no—
143C2×C8—2681no—
144C2×C8—671no—
145C2×C8—671no—
146C2×C8—1341no—
147C2×C8—1342no—
148C2×C8D4×D6712yes—
149C4⋊C4—1344no—
150C4⋊C4—1342no—
151C4⋊C4—1341no—
152C4⋊C4—1341no—
153C22⋊C4—672no—
154C22⋊C4—1341no—
155C42—21no—
156C42—672no—
157C42—1341no—
158C42#1176161111yescharacteristic
159C42#1176161111yescharacteristic
160C16—42no—
161C16—2682no—
162C23M16×D6711yescharacteristic
163Q8—5362no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—81no—
169C2×C4—1342no—
170C2×C4—1342no—
171C2×C4C22⋊C4×D6711yescharacteristic
172C2×C4C2×#1176161111yescharacteristic
173C8—22no—
174C8—41no—
175C8—2681no—
176C8—672no—
177C8—1341no—
178C22—21no—
179C22—41no—
180C22#1686335811yescharacteristic
181C4—21no—
182C4—41no—
183C4—41no—
184C4—2682no—
185C4—2682no—
186C4#1686336811yescharacteristic
187C4#101×D6711yescharacteristic
188C2—21no—
189C2—41no—
190C2#2255855111yescharacteristic
subgroupG/H
centerC4#16863368
commutatorC22×C4×C67C22×C4
FrattiniC2×M16C22×D67
FittingC67×#1327C2
socleC2×C67#1467
prime radicalC23×C67C2×M16
cosocleC2×M16×C67C23
hypercenter#1327D67
hypocenterC67#11540
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1327

Complements

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

NH ≅ G/Ntype
101. C67#11540semidirect

Maximal subgroups

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

subgroupindex
#225691462
#225620222
#225581612
#225581602
C67×#13272
#1154067