Finite Group Databasebeta

← #30758903

Subgroups — #30758903

Order 17152.

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

normal subgroups only

This group has 400 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#22561181C212yesmaximal
2#22561028C211yesmaximal, characteristic
3#22558090C212yesmaximal
4#22558090C211yesmaximal, characteristic
5C67×#2844C211yesmaximal, characteristic
6C2×#14524043C2211yescharacteristic
7C2×#14523966—21no—
8C2×#14523966—21no—
9C2×#14523954C2211yescharacteristic
10C2×#14523954C2212yes—
11#16863158—22no—
12#16863158—22no—
13#400×C67—21no—
14#400×C67—22no—
15#395×C67—21no—
16#395×C67—22no—
17#382×C67C2211yescharacteristic
18#382×C67C2212yes—
19C22×Dic134—21no—
20C22×C67⋊C8—21no—
21C22×C67⋊C8—22no—
22#14524043—22no—
23#14524043—21no—
24#14524043Q812yescomplemented
25C2×C4⋊Dic67—22no—
26#14523966—42no—
27#14523966—42no—
28#14523954—22no—
29#14523954—22no—
30#14523954—22no—
31C22×Q8×C67—21no—
32C2×C4⋊C4×C67—21no—
33C2×C4⋊C4×C67—22no—
34C2×C4⋊C4×C67D411yescharacteristic
35C2×C4⋊C4×C67D411yescharacteristic
36C2×C4⋊C4×C67D412yes—
37C2×C4⋊C4×C67D412yes—
38C2×C42×C67C2311yescharacteristic
39#95×C67—21no—
40#95×C67—22no—
41#95×C67—21no—
42#95×C67—22no—
43#95×C67—22no—
44#95×C67—22no—
45C2×Dic134—22no—
46C2×Dic134—41no—
47C2×Dic134—41no—
48C2×Dic134—41no—
49C2×Dic134—41no—
50C2×Dic134—41no—
51C22×Dic67—41no—
52C2×C67⋊C8—22no—
53C2×C67⋊C8—24no—
54C2×C67⋊C8—41no—
55C2×C67⋊C8—42no—
56C2×C67⋊C8—42no—
57C2×C67⋊C8—41no—
58C4⋊Dic67—24no—
59C4⋊Dic67—42no—
60C2×Q8×C67—81no—
61C2×Q8×C67—82no—
62C22×C4×C67—21no—
63C22×C4×C67—22no—
64C22×C4×C67—21no—
65C22×C4×C67—22no—
66C22×C4×C67C2×D412yes—
67C22×C4×C67C2×D411yescharacteristic
68C4⋊C4×C67—42no—
69C4⋊C4×C67—42no—
70C4⋊C4×C67—41no—
71C4⋊C4×C67—41no—
72C4⋊C4×C67—81no—
73C4⋊C4×C67—82no—
74C42×C67C2×Q811yescharacteristic
75C42×C67#4112yes—
76C42×C67#4111yescharacteristic
77C23×C67#12011yescharacteristic
78Q8×C67—84no—
79C2×C4×C67—21no—
80C2×C4×C67—21no—
81C2×C4×C67—22no—
82C2×C4×C67—22no—
83C2×C4×C67—41no—
84C2×C4×C67—41no—
85C2×C4×C67—42no—
86C2×C4×C67—42no—
87C2×C4×C67—42no—
88C2×C4×C67—42no—
89C2×C4×C67—42no—
90C2×C4×C67—42no—
91C2×C4×C67—42no—
92C2×C4×C67—42no—
93C2×C4×C67C22⋊Q812yes—
94C2×C4×C67C22⋊Q811yescharacteristic
95C2×C4×C67#12412yes—
96C2×C4×C67#12411yescharacteristic
97C2×Dic67—42no—
98C2×Dic67—81no—
99C2×Dic67—81no—
100Dic134—42no—
101Dic134—42no—
102Dic134—81no—
103Dic134—81no—
104C67⋊C8—42no—
105C67⋊C8—44no—
106C22×C67#39311yescharacteristic
107C22×C67#45311yescharacteristic
108C22×C67#45411yescharacteristic
109C22×C67#45512yes—
110C22×C67#45612yes—
111C4×C67—21no—
112C4×C67—21no—
113C4×C67—22no—
114C4×C67—22no—
115C4×C67—81no—
116C4×C67—82no—
117C4×C67—81no—
118C4×C67—82no—
119Dic67—82no—
120#10359—671nomaximal, Sylow
121C2×C67#165911yescharacteristic
122C2×C67#166011yescharacteristic
123C2×C67#166412yes—
124C2×C67#202711yescharacteristic
125C2×C67#202812yes—
126#2844D6711yescharacteristic
127#1993—672no—
128#1960—671no—
129#1295—671no—
130#1295—672no—
131C67#1035911yescharacteristic, complemented, Sylow
132C2×C4⋊Q8—671no—
133C2×#103—1341no—
134C2×#103—1341no—
135C2×#97—671no—
136C2×#97—672no—
137#400—22no—
138#400—21no—
139#395—22no—
140#395—21no—
141#382C2×D6711yescharacteristic
142#382C2×D6712yes—
143#340—1342no—
144#340—1342no—
145C22×Q8—21no—
146C22×Q8—1341no—
147C22×C8—1342no—
148C22×C8—1341no—
149C4⋊Q8—672no—
150C4⋊Q8—1341no—
151C4⋊Q8—1342no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—1342no—
155C2×C4⋊C4C22⋊D6712yes—
156C2×C4⋊C4C22⋊D6712yes—
157C2×C4⋊C4C22⋊D6711yescharacteristic
158C2×C4⋊C4C22⋊D6711yescharacteristic
159C2×C42C22×D6711yescharacteristic
160#103—2682no—
161#103—2682no—
162#97—1342no—
163#97—1342no—
164#97—1342no—
165#95—22no—
166#95—21no—
167#95—21no—
168#95—22no—
169#95—22no—
170#95—22no—
171C2×Q8—81no—
172C2×Q8—82no—
173C2×Q8—2681no—
174C2×Q8—2681no—
175C2×Q8—2681no—
176C2×Q8—2681no—
177C2×Q8—2681no—
178C2×Q8—1342no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—2681no—
184C22×C4C2×C22⋊D6712yes—
185C22×C4C2×C22⋊D6711yescharacteristic
186C2×C8—2681no—
187C2×C8—2682no—
188C2×C8—2682no—
189C2×C8—2681no—
190C2×C8—1342no—
191C2×C8—1344no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—81no—
197C4⋊C4—82no—
198C4⋊C4—2682no—
199C4⋊C4—1344no—
200C42Q8×D6711yescharacteristic
201C42#1176162912yes—
202C42#1176162911yescharacteristic
203C23#1452401211yescharacteristic
204Q8—84no—
205Q8—2682no—
206Q8—2682no—
207Q8—5361no—
208Q8—5361no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—2682no—
224C2×C4—5361no—
225C2×C4—5361no—
226C2×C4#1452401912yes—
227C2×C4#1452401911yescharacteristic
228C2×C4#1452403212yes—
229C2×C4#1452403211yescharacteristic
230C8—2682no—
231C8—2684no—
232C22#1686330711yescharacteristic
233C22#1686356411yescharacteristic
234C22#1686356911yescharacteristic
235C22#1686357412yes—
236C22#1686357912yes—
237C4—21no—
238C4—21no—
239C4—22no—
240C4—22no—
241C4—81no—
242C4—82no—
243C4—81no—
244C4—82no—
245C4—5362no—
246C2#2255951511yescharacteristic
247C2#2255952011yescharacteristic
248C2#2255954312yes—
249C2#2256136211yescharacteristic
250C2#2256136712yes—
subgroupG/H
centerC23#14524012
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#2844C2
socle, prime radicalC23×C67#120
hypercenter#2844D67
hypocenterC67#10359
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2844

Complements

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

NH ≅ G/Ntype
24. #14524043Q8semidirect
131. C67#10359semidirect

Maximal subgroups

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

subgroupindex
#225611812
#225610282
#225580902
C67×#28442
#1035967