Finite Group Databasebeta

← #30758912

Subgroups — #30758912

Order 17152.

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

normal subgroups only

This group has 392 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#22568484C211yesmaximal, characteristic
2#22561183C212yesmaximal
3#22558091C212yesmaximal
4#22558090C211yesmaximal, characteristic
5C67×#1960C211yesmaximal, characteristic
6C2×#14523954C2212yes—
7#16863330—22no—
8#16863329—22no—
9#16863328—22no—
10#16863262C2212yes—
11#16863158—22no—
12#16863158—22no—
13C2×C4⋊Q8×C67C2211yescharacteristic
14C2×#103×C67—22no—
15C2×#97×C67C2211yescharacteristic
16#382×C67C2211yescharacteristic
17C22×C67⋊C8—22no—
18C2×C4⋊Dic67—22no—
19C2×C4⋊Dic67D412yes—
20C2×C4⋊Dic67D412yes—
21C2×Dic67⋊C4—22no—
22C2×Dic67⋊C4—22no—
23#14523954—22no—
24#14523954—22no—
25C67⋊#95—22no—
26C67⋊#95—22no—
27C67⋊#95—22no—
28C22×Q8×C67—21no—
29C22×C8×C67—21no—
30C4⋊Q8×C67—22no—
31C4⋊Q8×C67—21no—
32C4⋊Q8×C67Q812yes—
33C2×C4⋊C4×C67—22no—
34C2×C4⋊C4×C67D412yes—
35C2×C42×C67C2311yescharacteristic
36#103×C67—44no—
37#97×C67—22no—
38#95×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—
47Dic67⋊C4—82no—
48Dic67⋊C4—82no—
49C2×Q8×C67—22no—
50C2×Q8×C67—42no—
51C2×Q8×C67—42no—
52C2×Q8×C67—41no—
53C22×C4×C67—22no—
54C22×C4×C67—41no—
55C22×C4×C67C2×D412yes—
56C22×C4×C67C2×D411yescharacteristic
57C2×C8×C67—22no—
58C2×C8×C67—41no—
59C2×C8×C67—41no—
60C4⋊C4×C67—24no—
61C4⋊C4×C67—42no—
62C4⋊C4×C67—42no—
63C42×C67C2×Q811yescharacteristic
64C42×C67#4112yes—
65C42×C67#4111yescharacteristic
66C23×C67#12011yescharacteristic
67Q8×C67—44no—
68Q8×C67—82no—
69C2×C4×C67—21no—
70C2×C4×C67—21no—
71C2×C4×C67—22no—
72C2×C4×C67—22no—
73C2×C4×C67—42no—
74C2×C4×C67—42no—
75C2×C4×C67—42no—
76C2×C4×C67—42no—
77C2×C4×C67—81no—
78C2×C4×C67—81no—
79C2×C4×C67C22⋊Q812yes—
80C2×C4×C67C22⋊Q811yescharacteristic
81C2×C4×C67#12412yes—
82C2×C4×C67#12411yescharacteristic
83C2×Dic67—42no—
84C2×Dic67—42no—
85C2×Dic67—42no—
86C2×Dic67—42no—
87C2×Dic67—42no—
88C2×Dic67—42no—
89C8×C67—42no—
90C67⋊C8—44no—
91C22×C67#39311yescharacteristic
92C22×C67#45311yescharacteristic
93C22×C67#45411yescharacteristic
94C22×C67#45512yes—
95C22×C67#45612yes—
96C4×C67—21no—
97C4×C67—21no—
98C4×C67—22no—
99C4×C67—22no—
100C4×C67—82no—
101C4×C67—82no—
102Dic67—82no—
103Dic67—82no—
104#10361—671nomaximal, Sylow
105C2×C67#166112yes—
106C2×C67#166211yescharacteristic
107C2×C67#166311yescharacteristic
108C2×C67#202711yescharacteristic
109C2×C67#202812yes—
110#2845—671no—
111#1993—672no—
112#1960D6711yescharacteristic
113#1295—672no—
114#1295—671no—
115C67#1036111yescharacteristic, complemented, Sylow
116C2×C4⋊Q8C2×D6711yescharacteristic
117C2×#103—22no—
118C2×#97—672no—
119C2×#97C2×D6711yescharacteristic
120#398—1342no—
121#398—1342no—
122#398—1342no—
123#382—672no—
124#382C2×D6711yescharacteristic
125#340—1342no—
126#340—1342no—
127C22×Q8—21no—
128C22×C8—21no—
129C22×C8—1342no—
130C4⋊Q8—22no—
131C4⋊Q8—21no—
132C4⋊Q8Dic13412yes—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—672no—
135C2×C4⋊C4—672no—
136C2×C4⋊C4—1342no—
137C2×C4⋊C4—1342no—
138C2×C4⋊C4—1342no—
139C2×C4⋊C4C22⋊D6712yes—
140C2×C42C22×D6711yescharacteristic
141#103—44no—
142#97—22no—
143#97—1342no—
144#97—1342no—
145#95—22no—
146#95—1342no—
147#95—1342no—
148#95—1342no—
149C2×Q8—22no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—41no—
153C22×C4—22no—
154C22×C4—41no—
155C22×C4—1342no—
156C22×C4—1342no—
157C22×C4D4×D6712yes—
158C22×C4C2×C22⋊D6711yescharacteristic
159C2×C8—22no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—2682no—
163C2×C8—2682no—
164C2×C8—1344no—
165C4⋊C4—24no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—2682no—
169C4⋊C4—2682no—
170C4⋊C4—5362no—
171C4⋊C4—5362no—
172C4⋊C4—5362no—
173C42C2×Dic13411yescharacteristic
174C42#1176162712yes—
175C42#1176162811yescharacteristic
176C23#1452401111yescharacteristic
177Q8—44no—
178Q8—82no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—2682no—
190C2×C4—2682no—
191C2×C4—2682no—
192C2×C4—2682no—
193C2×C4—2682no—
194C2×C4—2682no—
195C2×C4#1452402012yes—
196C2×C4#1452402211yescharacteristic
197C2×C4#1452402912yes—
198C2×C4#1452403011yescharacteristic
199C8—42no—
200C8—2684no—
201C22#1686330811yescharacteristic
202C22#1686356311yescharacteristic
203C22#1686356811yescharacteristic
204C22#1686357212yes—
205C22#1686357712yes—
206C4—21no—
207C4—21no—
208C4—22no—
209C4—22no—
210C4—82no—
211C4—82no—
212C4—5362no—
213C4—5362no—
214C2#2255952312yes—
215C2#2255953011yescharacteristic
216C2#2255953511yescharacteristic
217C2#2256136511yescharacteristic
218C2#2256136812yes—
subgroupG/H
centerC23#14524011
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#1960C2
socle, prime radicalC23×C67#120
hypercenter#1960D67
hypocenterC67#10361
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
115. C67#10361semidirect

Maximal subgroups

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

subgroupindex
#225684842
#225611832
#225580912
#225580902
C67×#19602
#1036167