Finite Group Databasebeta

← #30759322

Subgroups — #30759322

Order 17152.

386 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 93 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#22561179C212yesmaximal
2#22560342C211yesmaximal, characteristic
3C2×#16863138C212yesmaximal
4#22558090C211yesmaximal, characteristic
5C67×#2822C211yesmaximal, characteristic
6C2×#14523954C2212yes—
7C2×#14523954C2211yescharacteristic
8#16863266C2211yescharacteristic
9#16863158—22no—
10#16863158—22no—
11#16863158—22no—
12#16863138—24no—
13C2×#125×C67C2212yes—
14#400×C67—22no—
15#398×C67—22no—
16#398×C67—22no—
17#382×C67C2211yescharacteristic
18C22×C67⋊C8—22no—
19C22×C67⋊C8—21no—
20C2×C4⋊Dic67—22no—
21C2×C4⋊Dic67—21no—
22#14523954—22no—
23#14523954—21no—
24#14523954—22no—
25#14523954D412yes—
26#125×C67—22no—
27#125×C67—22no—
28#125×C67—22no—
29#125×C67—22no—
30C2×C4⋊C4×C67—22no—
31C2×C4⋊C4×C67—22no—
32C2×C4⋊C4×C67—22no—
33C2×C4⋊C4×C67D412yes—
34C2×C4⋊C4×C67D412yes—
35C2×C4⋊C4×C67D412yes—
36C2×C42×C67C2311yescharacteristic
37#95×C67—24no—
38#95×C67—22no—
39#95×C67—22no—
40C22×Dic67—41no—
41C2×C67⋊C8—24no—
42C2×C67⋊C8—22no—
43C2×C67⋊C8—42no—
44C2×C67⋊C8—41no—
45C2×C67⋊C8—41no—
46C2×C67⋊C8—42no—
47C4⋊Dic67—42no—
48C4⋊Dic67—44no—
49C22×C4×C67—22no—
50C22×C4×C67—22no—
51C22×C4×C67—22no—
52C22×C4×C67C2×D412yes—
53C22×C4×C67C2×D411yescharacteristic
54C4⋊C4×C67—22no—
55C4⋊C4×C67—22no—
56C4⋊C4×C67—22no—
57C4⋊C4×C67—22no—
58C4⋊C4×C67—42no—
59C4⋊C4×C67—42no—
60C4⋊C4×C67—42no—
61C4⋊C4×C67—42no—
62C4⋊C4×C67—42no—
63C4⋊C4×C67—82no—
64C42×C67C2×D411yescharacteristic
65C42×C67#4111yescharacteristic
66C42×C67#4112yes—
67C23×C67#12011yescharacteristic
68C2×C4×C67—22no—
69C2×C4×C67—21no—
70C2×C4×C67—21no—
71C2×C4×C67—22no—
72C2×C4×C67—42no—
73C2×C4×C67—42no—
74C2×C4×C67—42no—
75C2×C4×C67—42no—
76C2×C4×C67—42no—
77C2×C4×C67—42no—
78C2×C4×C67—42no—
79C2×C4×C67—42no—
80C2×C4×C67—42no—
81C2×C4×C67#12111yescharacteristic
82C2×C4×C67#12112yes—
83C2×C4×C67#12312yes—
84C2×C4×C67#12411yescharacteristic
85C2×Dic67—81no—
86C2×Dic67—82no—
87C67⋊C8—44no—
88C67⋊C8—42no—
89C22×C67#39411yescharacteristic
90C22×C67#45411yescharacteristic
91C22×C67#45412yes—
92C22×C67#45511yescharacteristic
93C22×C67#45612yes—
94C4×C67—22no—
95C4×C67—22no—
96C4×C67—21no—
97C4×C67—21no—
98C4×C67—82no—
99C4×C67—82no—
100C4×C67—82no—
101Dic67—161no—
102#10428—671nomaximal, Sylow
103C2×C67#165811yescharacteristic
104C2×C67#166311yescharacteristic
105C2×C67#166412yes—
106C2×C67#203812yes—
107C2×C67#204011yescharacteristic
108#2822D6711yescharacteristic
109#1992—672no—
110#1836—671no—
111C2×#330—672no—
112#1295—671no—
113C67#1042811yescharacteristic, complemented, Sylow
114C2×#125C2×D6712yes—
115C2×#97—671no—
116C2×#97—672no—
117#400—22no—
118#398—22no—
119#398—22no—
120#384—671no—
121#382C2×D6711yescharacteristic
122#340—1342no—
123#340—1342no—
124#340—1342no—
125#330—1344no—
126C22×C8—1341no—
127C22×C8—1342no—
128#125—22no—
129#125—22no—
130#125—22no—
131#125—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—1341no—
136C2×C4⋊C4—1342no—
137C2×C4⋊C4C22⋊D6712yes—
138C2×C4⋊C4C22⋊D6712yes—
139C2×C4⋊C4C22⋊D6712yes—
140C2×C42C22×D6711yescharacteristic
141#97—672no—
142#97—1342no—
143#97—1342no—
144#97—1341no—
145#95—24no—
146#95—22no—
147#95—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—2681no—
152C22×C4C2×C22⋊D6711yescharacteristic
153C22×C4C2×C22⋊D6712yes—
154C2×C8—2682no—
155C2×C8—2681no—
156C2×C8—2681no—
157C2×C8—2682no—
158C2×C8—1344no—
159C2×C8—1342no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—82no—
170C4⋊C4—2684no—
171C4⋊C4—2682no—
172C42D4×D6711yescharacteristic
173C42#1176162711yescharacteristic
174C42#1176162712yes—
175C23#1452401211yescharacteristic
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—5361no—
190C2×C4—5362no—
191C2×C4#1452401411yescharacteristic
192C2×C4#1452401512yes—
193C2×C4#1452402412yes—
194C2×C4#1452403011yescharacteristic
195C8—2684no—
196C8—2682no—
197C22#1686331011yescharacteristic
198C22#1686356911yescharacteristic
199C22#1686356912yes—
200C22#1686357411yescharacteristic
201C22#1686357912yes—
202C4—22no—
203C4—22no—
204C4—21no—
205C4—21no—
206C4—82no—
207C4—82no—
208C4—82no—
209C4—10721no—
210C2#2255951111yescharacteristic
211C2#2255953711yescharacteristic
212C2#2255954312yes—
213C2#2256142312yes—
214C2#2256143511yescharacteristic
subgroupG/H
centerC23#14524012
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#2822C2
socle, prime radicalC23×C67#120
hypercenter#2822D67
hypocenterC67#10428
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2822

Complements

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

NH ≅ G/Ntype
113. C67#10428semidirect

Maximal subgroups

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

subgroupindex
#225611792
#225603422
C2×#168631382
#225580902
C67×#28222
#1042867