Finite Group Databasebeta

← #30759959

Subgroups — #30759959

Order 17152.

402 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 99 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#22571554C211yesmaximal, characteristic
2#22561186C212yesmaximal
3#22558897C212yesmaximal
4#22558084C211yesmaximal, characteristic
5C67×#1842C211yesmaximal, characteristic
6#16863953C2212yes—
7#16863692—22no—
8#16863637—22no—
9#16863607—22no—
10#16863420C2212yes—
11#16863156—22no—
12#16863156—22no—
13#16863144—22no—
14#16863143—22no—
15C2×C4⋊C8×C67—22no—
16C2×C4⋊C8×C67C2211yescharacteristic
17C2×C4⋊C8×C67C2211yescharacteristic
18C43×C67C2211yescharacteristic
19C2×Dic268—22no—
20C2×#11761612—22no—
21#14524043D412yes—
22#14524043D412yes—
23#14524035—22no—
24#14524022—22no—
25#14524022—22no—
26C4×Dic134—22no—
27C4×Dic134—22no—
28#14524002—22no—
29#14523977—22no—
30C4⋊(C67⋊C8)—22no—
31C4⋊(C67⋊C8)—22no—
32#14523968—24no—
33C22×C8×C67—21no—
34C22×C8×C67D412yes—
35C2×C42×C67—21no—
36C2×C42×C67—21no—
37C2×C42×C67D411yescharacteristic
38C2×C42×C67D411yescharacteristic
39C2×C42×C67C2311yescharacteristic
40C4⋊C8×C67—22no—
41C4⋊C8×C67—22no—
42C4⋊C8×C67—44no—
43C2×Dic134—22no—
44C2×Dic134—22no—
45Dic268—82no—
46#11761612—42no—
47#11761612—42no—
48C2×C67⋊C8—42no—
49C4⋊Dic67—22no—
50C4⋊Dic67—22no—
51C4⋊Dic67—22no—
52C4⋊Dic67—22no—
53Dic67⋊C4—42no—
54Dic67⋊C4—42no—
55C22⋊Dic67—42no—
56C4×Dic67—42no—
57C22×C4×C67—21no—
58C22×C4×C67—21no—
59C22×C4×C67C2×D411yescharacteristic
60C22×C4×C67C2×D411yescharacteristic
61C22×C4×C67#4111yescharacteristic
62C2×C8×C67—22no—
63C2×C8×C67—42no—
64C2×C8×C67—42no—
65C2×C8×C67—41no—
66C42×C67—22no—
67C42×C67—22no—
68C42×C67—21no—
69C42×C67—22no—
70C42×C67—42no—
71C42×C67Q1612yescomplemented
72C42×C67C2×D412yes—
73C42×C67#4112yes—
74C23×C67#12111yescharacteristic
75C2×C4×C67—22no—
76C2×C4×C67—22no—
77C2×C4×C67—22no—
78C2×C4×C67—21no—
79C2×C4×C67—21no—
80C2×C4×C67—22no—
81C2×C4×C67—41no—
82C2×C4×C67—41no—
83C2×C4×C67—41no—
84C2×C4×C67#12011yescharacteristic
85C2×C4×C67#12112yes—
86C2×C4×C67#12312yes—
87C2×C4×C67#12411yescharacteristic
88C2×C4×C67C2×Q1611yescharacteristic
89C2×C4×C67#13511yescharacteristic
90C2×C4×C67#13712yes—
91C2×Dic67—42no—
92C2×Dic67—42no—
93Dic134—42no—
94Dic134—42no—
95Dic134—82no—
96C8×C67—42no—
97C8×C67—81no—
98C67⋊C8—82no—
99C22×C67—21no—
100C22×C67—21no—
101C22×C67#39411yescharacteristic
102C22×C67#46711yescharacteristic
103C22×C67#47011yescharacteristic
104C4×C67—24no—
105C4×C67—22no—
106C4×C67—21no—
107C4×C67—42no—
108C4×C67#45311yescharacteristic
109C4×C67#45611yescharacteristic
110C4×C67#46212yes—
111C4×C67#46412yes—
112Dic67—82no—
113Dic67—82no—
114#10522—671nomaximal, Sylow
115C2×C67—21no—
116C2×C67—21no—
117C2×C67#203911yescharacteristic
118C2×C67#205011yescharacteristic
119C2×C67#205611yescharacteristic
120#3249—671no—
121#1994—672no—
122#1842D6711yescharacteristic
123#1550—672no—
124#1292—671no—
125C67#1052211yescharacteristic, complemented, Sylow
126#533—672no—
127#477—1342no—
128#467—1342no—
129#462—1342no—
130#423—672no—
131C2×C4⋊C8—22no—
132C2×C4⋊C8C2×D6711yescharacteristic
133C2×C4⋊C8C2×D6711yescharacteristic
134C43C2×D6711yescharacteristic
135#339—1342no—
136#339—1342no—
137#333—1342no—
138#332—1342no—
139C2×Q16—1342no—
140C2×M16—1342no—
141C22×C8—21no—
142C22×C8C22⋊D6712yes—
143C4⋊Q8—672no—
144C4⋊Q8—672no—
145#125—1342no—
146C22⋊Q8—1342no—
147C22⋊Q8—1342no—
148C4×Q8—1342no—
149C4×Q8—1342no—
150#117—1342no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42D26811yescharacteristic
154C2×C42D26811yescharacteristic
155C2×C42C22×D6711yescharacteristic
156#107—1342no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—44no—
160C4⋊C8—1342no—
161C4⋊C8—1342no—
162#103—1344no—
163C2×Q8—1342no—
164C2×Q8—1342no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4C2×C22⋊D6711yescharacteristic
168C22×C4C2×D26811yescharacteristic
169C22×C4#1176162811yescharacteristic
170Q16—5362no—
171M16—2682no—
172M16—2682no—
173C2×C8—22no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—41no—
177C2×C8—2682no—
178C4⋊C4—2682no—
179C4⋊C4—2682no—
180C4⋊C4—1342no—
181C4⋊C4—1342no—
182C4⋊C4—1342no—
183C4⋊C4—1342no—
184C22⋊C4—2682no—
185C42—22no—
186C42—22no—
187C42—21no—
188C42—22no—
189C42—42no—
190C42—2682no—
191C42Dic26812yescomplemented
192C42D4×D6712yes—
193C42#1176162912yes—
194C23#1452401711yescharacteristic
195Q8—2682no—
196Q8—2682no—
197Q8—5362no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—2682no—
208C2×C4—2682no—
209C2×C4#1452401111yescharacteristic
210C2×C4#1452401812yes—
211C2×C4#1452402812yes—
212C2×C4#1452403211yescharacteristic
213C2×C4C2×Dic26811yescharacteristic
214C2×C4#1452406511yescharacteristic
215C2×C4#1452407612yes—
216C8—42no—
217C8—81no—
218C8—5362no—
219C22—21no—
220C22—21no—
221C22#1686331311yescharacteristic
222C22#1686363711yescharacteristic
223C22#1686365411yescharacteristic
224C4—24no—
225C4—21no—
226C4—22no—
227C4—42no—
228C4—5362no—
229C4—5362no—
230C4#1686356311yescharacteristic
231C4#1686357811yescharacteristic
232C4#1686360712yes—
233C4#1686361912yes—
234C2—21no—
235C2—21no—
236C2#2256143311yescharacteristic
237C2#2256149611yescharacteristic
238C2#2256153211yescharacteristic
subgroupG/H
centerC22#16863313
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#1842C2
socleC22×C67#394
prime radicalC23×C67#121
hypercenter#1842D67
hypocenterC67#10522
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1842

Complements

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

NH ≅ G/Ntype
71. C42×C67Q16semidirect
125. C67#10522semidirect
191. C42Dic268semidirect

Maximal subgroups

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

subgroupindex
#225715542
#225611862
#225588972
#225580842
C67×#18422
#1052267