Finite Group Databasebeta

← #30750214

Subgroups — #30750214

Order 17152.

396 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 111 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#22563395C211yesmaximal, characteristic
2#22560344C212yesmaximal
3C2×#16863138C211yesmaximal, characteristic
4#22558091C212yesmaximal
5C67×#1992C211yesmaximal, characteristic
6C2×#14523954C2212yes—
7#16863295—22no—
8#16863282—22no—
9#16863282—22no—
10#16863262C2212yes—
11#16863158—22no—
12#16863158—22no—
13#16863138C414yescomplemented
14C2×#125×C67C2211yescharacteristic
15C2×#97×C67C2211yescharacteristic
16#384×C67C2211yescharacteristic
17#340×C67—22no—
18C22×C67⋊C8—22no—
19C2×C4⋊Dic67—22no—
20C2×C4⋊Dic67D412yes—
21C2×C4⋊Dic67D412yes—
22C2×C4×Dic67—22no—
23#14523954—22no—
24#14523954C2×C414yes—
25C67⋊#95—22no—
26C67⋊#95—22no—
27C67⋊#95—22no—
28C67⋊#95—22no—
29C22×C8×C67—21no—
30#125×C67—22no—
31#125×C67C2×C412yes—
32#125×C67Q812yes—
33C2×C4⋊C4×C67—22no—
34C2×C4⋊C4×C67—21no—
35C2×C4⋊C4×C67—22no—
36C2×C4⋊C4×C67D412yes—
37C2×C42×C67C2311yescharacteristic
38#97×C67—22no—
39C22×Dic67—22no—
40C22×Dic67—22no—
41C2×C67⋊C8—24no—
42C2×C67⋊C8—42no—
43C2×C67⋊C8—42no—
44C4⋊Dic67—42no—
45C4⋊Dic67—42no—
46C4⋊Dic67—82no—
47C4×Dic67—82no—
48C22×C4×C67—22no—
49C22×C4×C67—41no—
50C22×C4×C67C2×D412yes—
51C22×C4×C67C2×D411yescharacteristic
52C2×C8×C67—22no—
53C2×C8×C67—41no—
54C2×C8×C67—41no—
55C4⋊C4×C67—24no—
56C4⋊C4×C67—24no—
57C4⋊C4×C67—22no—
58C4⋊C4×C67—42no—
59C4⋊C4×C67—44no—
60C4⋊C4×C67C4⋊C414yes—
61C42×C67C22×C411yescharacteristic
62C42×C67C2×Q811yescharacteristic
63C42×C67#4112yes—
64C23×C67#12011yescharacteristic
65C2×C4×C67—21no—
66C2×C4×C67—22no—
67C2×C4×C67—22no—
68C2×C4×C67—21no—
69C2×C4×C67—24no—
70C2×C4×C67—42no—
71C2×C4×C67—42no—
72C2×C4×C67—82no—
73C2×C4×C67—81no—
74C2×C4×C67C2×C4⋊C411yescharacteristic
75C2×C4×C67C4×D412yes—
76C2×C4×C67C22⋊Q812yes—
77C2×C4×C67#12311yescharacteristic
78C2×Dic67—42no—
79C2×Dic67—42no—
80C2×Dic67—42no—
81C2×Dic67—42no—
82C2×Dic67—42no—
83C2×Dic67—42no—
84C8×C67—42no—
85C67⋊C8—44no—
86C22×C67#38511yescharacteristic
87C22×C67#45412yes—
88C22×C67#45411yescharacteristic
89C22×C67#45512yes—
90C22×C67#45611yescharacteristic
91C4×C67—21no—
92C4×C67—21no—
93C4×C67—22no—
94C4×C67—22no—
95C4×C67—44no—
96C4×C67—161no—
97Dic67—82no—
98Dic67—82no—
99#8915—671nomaximal, Sylow
100C2×C67#165811yescharacteristic
101C2×C67#166211yescharacteristic
102C2×C67#166412yes—
103C2×C67#185711yescharacteristic
104C2×C67#186012yes—
105#2382—671no—
106#1992D6711yescharacteristic
107#1836—672no—
108C2×#330—671no—
109#1295—672no—
110C67#891511yescharacteristic, complemented, Sylow
111C2×#125C2×D6711yescharacteristic
112C2×#97—672no—
113C2×#97C2×D6711yescharacteristic
114#389—1342no—
115#387—1342no—
116#387—1342no—
117#384C2×D6711yescharacteristic
118#382—672no—
119#340—22no—
120#340—1342no—
121#340—1342no—
122#330—674no—
123C22×C8—21no—
124C22×C8—1342no—
125#125—22no—
126#125Dic13412yes—
127#125C4×D6712yes—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—672no—
132C2×C4⋊C4—672no—
133C2×C4⋊C4—1342no—
134C2×C4⋊C4C22⋊D6712yes—
135C2×C42—1342no—
136C2×C42C22×D6711yescharacteristic
137#97—22no—
138#97—674no—
139#97—1342no—
140#95—1342no—
141#95—1342no—
142#95—1342no—
143#95—1342no—
144C22×C4—22no—
145C22×C4—41no—
146C22×C4—1342no—
147C22×C4—1342no—
148C22×C4D4×D6712yes—
149C22×C4C2×C22⋊D6711yescharacteristic
150C2×C8—22no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—2682no—
154C2×C8—2682no—
155C2×C8—1344no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—22no—
159C4⋊C4—44no—
160C4⋊C4—42no—
161C4⋊C4—2682no—
162C4⋊C4—2682no—
163C4⋊C4—5362no—
164C4⋊C4Dic67⋊C414yes—
165C42—5362no—
166C42C2×C4×D6711yescharacteristic
167C42C2×Dic13411yescharacteristic
168C42#1176162712yes—
169C23#1452401111yescharacteristic
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—24no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—82no—
178C2×C4—81no—
179C2×C4—2682no—
180C2×C4—2682no—
181C2×C4—2682no—
182C2×C4—2682no—
183C2×C4—2682no—
184C2×C4—2682no—
185C2×C4C2×Dic67⋊C411yescharacteristic
186C2×C4(C22⋊D67)⋊C412yes—
187C2×C4#1452402012yes—
188C2×C4#1452402411yescharacteristic
189C8—42no—
190C8—2684no—
191C22#1686326811yescharacteristic
192C22#1686356712yes—
193C22#1686356811yescharacteristic
194C22#1686357212yes—
195C22#1686357811yescharacteristic
196C4—21no—
197C4—21no—
198C4—22no—
199C4—22no—
200C4—44no—
201C4—161no—
202C4—5362no—
203C4—5362no—
204C2#2255951011yescharacteristic
205C2#2255953011yescharacteristic
206C2#2255954112yes—
207C2#2256042311yescharacteristic
208C2#2256044012yes—
subgroupG/H
centerC23#14524011
commutatorC42×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1992C2
socle, prime radicalC23×C67#120
cosocleC2×C42×C67C23
hypercenter#1992D67
hypocenterC67#8915
LCSG ▷ C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1992

Complements

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

NH ≅ G/Ntype
13. #16863138C4semidirect
110. C67#8915semidirect

Maximal subgroups

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

subgroupindex
#225633952
#225603442
C2×#168631382
#225580912
C67×#19922
#891567