Finite Group Databasebeta

← #30765176

Subgroups — #30765176

Order 17152.

296 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 91 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#22569142C211yesmaximal, characteristic
2#22562005C212yesmaximal
3#22558151C211yesmaximal, characteristic
4#22558150C212yesmaximal
5C67×#1323C211yesmaximal, characteristic
6#16863949C411yescharacteristic
7#16863525—22no—
8#16863520—22no—
9#16863422C2211yescharacteristic
10#16863419C411yescharacteristic
11#16863354C411yescharacteristic
12#16863352C411yescharacteristic
13C2×C8×Dic67C2211yescharacteristic
14Dic67⋊C16—22no—
15C22⋊(C67⋊C16)C2212yes—
16C16×Dic67—22no—
17#424×C67C2211yescharacteristic
18C22⋊C16×C67C2212yes—
19C2×#11761612C2×C411yescharacteristic
20C22×C67⋊C8C2×C411yescharacteristic
21#14524042C812yes—
22#14524034C812yes—
23#14524020—22no—
24C4×Dic134—22no—
25#14524002C2×C411yescharacteristic
26C2×C4×Dic67C2×C411yescharacteristic
27C2×C67⋊C16—22no—
28Dic67⋊C8—22no—
29Dic67⋊C8—22no—
30#14523954—21no—
31#14523953—21no—
32C4×C67⋊C8—21no—
33C8×Dic67—21no—
34C8×Dic67D412yes—
35C8×Dic67D412yes—
36C2×M16×C67C2×C411yescharacteristic
37C22×C8×C67C2311yescharacteristic
38#117×C67C2×C411yescharacteristic
39C2×C16×C67—22no—
40C4⋊C8×C67—22no—
41C2×Dic134—22no—
42C22×Dic67M1612yes—
43#11761612—41no—
44C2×C67⋊C8—22no—
45C2×C67⋊C8—21no—
46C2×C67⋊C8C22⋊C411yescharacteristic
47C2×C67⋊C8C22⋊C411yescharacteristic
48C4⋊Dic67—22no—
49C4⋊Dic67C2×C812yes—
50Dic67⋊C4—22no—
51Dic67⋊C4—22no—
52C22⋊Dic67—21no—
53C4×Dic67—21no—
54C4×Dic67C22⋊C412yes—
55C4×Dic67C2×C811yescharacteristic
56C4×Dic67C2×C811yescharacteristic
57C67⋊C16—42no—
58C22×C4×C67C22×C411yescharacteristic
59M16×C67—41no—
60C2×C8×C67—21no—
61C2×C8×C67—41no—
62C2×C8×C67C2×D412yes—
63C4⋊C4×C67C2×C812yes—
64C22⋊C4×C67—21no—
65C42×C67—21no—
66C16×C67—42no—
67C23×C67C2×M1611yescharacteristic
68C2×C4×C67—21no—
69C2×C4×C67—21no—
70C2×C4×C67—22no—
71C2×C4×C67C2×C22⋊C411yescharacteristic
72C2×C4×C67C22×C811yescharacteristic
73C2×Dic67—24no—
74C2×Dic67—22no—
75C2×Dic67C22⋊C812yes—
76C2×Dic67C22⋊C812yes—
77Dic134—44no—
78C8×C67—22no—
79C8×C67—41no—
80C67⋊C8—21no—
81C67⋊C8—41no—
82C67⋊C8#10112yes—
83C22×C67—22no—
84C22×C67—21no—
85C22×C67C2×C22⋊C811yescharacteristic
86C4×C67—21no—
87C4×C67—42no—
88C4×C67#41011yescharacteristic
89C4×C67C2×#10111yescharacteristic
90Dic67—22no—
91Dic67—42no—
92Dic67#35014yes—
93#11496—671nomaximal, Sylow
94C2×C67—41no—
95C2×C67#146111yescharacteristic
96C2×C67C2×#35012yes—
97#2960—671no—
98#2162—672no—
99#1323—671no—
100#1323—672no—
101#1323D6711yescharacteristic
102C67#1149611yescharacteristic, complemented, Sylow
103#533—671no—
104#446—1342no—
105C8×Q8—1342no—
106#424—671no—
107#424C2×D6711yescharacteristic
108#423—671no—
109#405—671no—
110#405—671no—
111C2×C4×C8—671no—
112C4⋊C16—1342no—
113C22⋊C16—672no—
114C22⋊C16C2×D6712yes—
115C4×C16—1342no—
116C2×M16—671no—
117C2×M16C4×D6711yescharacteristic
118C22×C8—671no—
119C22×C8C22×D6711yescharacteristic
120C4⋊Q8—672no—
121#125—672no—
122C22⋊Q8—1342no—
123C4×Q8—1342no—
124#117—671no—
125#117C4×D6711yescharacteristic
126C2×C42—671no—
127C2×C16—22no—
128C2×C16—1342no—
129C4⋊C8—22no—
130C4⋊C8—1342no—
131C4⋊C8—1342no—
132#97—1341no—
133#97—1341no—
134C4×C8—672no—
135C4×C8—672no—
136C4×C8—1341no—
137C4×C8—1341no—
138C2×Q8—1342no—
139C22×C4—672no—
140C22×C4C2×C4×D6711yescharacteristic
141M16—41no—
142M16—2681no—
143C2×C8—21no—
144C2×C8—41no—
145C2×C8—671no—
146C2×C8—671no—
147C2×C8—1341no—
148C2×C8—1342no—
149C2×C8D4×D6712yes—
150C4⋊C4—672no—
151C4⋊C4—1342no—
152C4⋊C4—1342no—
153C4⋊C4—1342no—
154C4⋊C4C8×D6712yes—
155C22⋊C4—21no—
156C22⋊C4—1341no—
157C42—21no—
158C42—672no—
159C42—671no—
160C42—671no—
161C42—1341no—
162C16—42no—
163C16—2682no—
164C23M16×D6711yescharacteristic
165Q8—2684no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—672no—
170C2×C4—672no—
171C2×C4—1342no—
172C2×C4—1344no—
173C2×C4C22⋊C4×D6711yescharacteristic
174C2×C4C2×C8×D6711yescharacteristic
175C8—22no—
176C8—41no—
177C8—2681no—
178C8—672no—
179C8—1341no—
180C22—22no—
181C22—21no—
182C22C22⋊C8×D6711yescharacteristic
183C4—21no—
184C4—42no—
185C4—2682no—
186C4—674no—
187C4—1342no—
188C4#1686336811yescharacteristic
189C4#101×D6711yescharacteristic
190C2—41no—
191C2#2255853111yescharacteristic
192C2#350×D6712yes—
subgroupG/H
centerC2×C4C22⋊C4×D67
commutatorC2×C4×C67C22×C8
FrattiniC22×C8C22×D67
FittingC67×#1323C2
socleC22×C67C2×C22⋊C8
prime radicalC23×C67C2×M16
cosocleC22×C8×C67C23
hypercenter#1323D67
hypocenterC67#11496
LCSG ▷ C2×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1323

Complements

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

NH ≅ G/Ntype
102. C67#11496semidirect

Maximal subgroups

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

subgroupindex
#225691422
#225620052
#225581512
#225581502
C67×#13232
#1149667