Finite Group Databasebeta

← #30755497

Subgroups — #30755497

Order 17152.

466 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C2×#16863775C211yesmaximal, characteristic
2#22561198C211yesmaximal, characteristic
3#22558964C211yesmaximal, characteristic
4#22558891C211yesmaximal, characteristic
5#22558890C211yesmaximal, characteristic
6#22558496C211yesmaximal, characteristic
7C67×#1997C211yesmaximal, characteristic
8C22×#11761618C414yes—
9C2×#14524043C2211yescharacteristic
10#16863775—21no—
11#16863775—21no—
12#16863775—22no—
13#16863449C2211yescharacteristic
14#16863420C2211yescharacteristic
15C2×C4×C67⋊C8C2211yescharacteristic
16#16863164—22no—
17#16863146—22no—
18#16863145—22no—
19#16863144—22no—
20#16863118—21no—
21#16863118—21no—
22C2×C4⋊Q8×C67C2211yescharacteristic
23#423×C67C2211yescharacteristic
24C4×M16×C67C2211yescharacteristic
25#343×C67—22no—
26C22×Dic134C2×C412yes—
27C2×#11761618—24no—
28C2×#11761618—24no—
29C2×#11761618—48no—
30C2×#11761612—21no—
31C22×C67⋊C8C2×C412yes—
32#14524043—21no—
33#14524043—21no—
34#14524043—21no—
35#14524043—21no—
36C2×C4⋊Dic67—21no—
37C2×C4⋊Dic67—21no—
38C4⋊(C67⋊C8)—21no—
39C4⋊(C67⋊C8)—21no—
40C4⋊(C67⋊C8)D412yes—
41C22⋊(C67⋊C8)—21no—
42C4×C67⋊C8—21no—
43C4×C67⋊C8D411yescharacteristic
44C4×C67⋊C8D411yescharacteristic
45C4×C67⋊C8D412yes—
46C22×Q8×C67C2×C412yes—
47C2×M16×C67—21no—
48C2×M16×C67D412yes—
49C4⋊Q8×C67—21no—
50C4⋊Q8×C67—21no—
51C4⋊Q8×C67—21no—
52C4⋊Q8×C67—21no—
53C2×C4⋊C4×C67—21no—
54C2×C4⋊C4×C67—21no—
55C2×C42×C67C2311yescharacteristic
56C4⋊C8×C67—21no—
57C4⋊C8×C67—21no—
58#97×C67—21no—
59C4×C8×C67—21no—
60C2×Dic134—22no—
61C2×Dic134—22no—
62C2×Dic134—44no—
63C22×Dic67—22no—
64#11761618—48no—
65#11761618—48no—
66#11761612—42no—
67C2×C67⋊C8—24no—
68C2×C67⋊C8—22no—
69C2×C67⋊C8—41no—
70C2×C67⋊C8C22⋊C412yes—
71C2×C67⋊C8C22⋊C412yes—
72C4⋊Dic67—41no—
73C4⋊Dic67—41no—
74C4⋊Dic67—41no—
75C4⋊Dic67—41no—
76C2×Q8×C67—22no—
77C2×Q8×C67—22no—
78C2×Q8×C67—44no—
79C22×C4×C67—22no—
80C22×C4×C67C22×C411yescharacteristic
81C22×C4×C67C2×D411yescharacteristic
82C22×C4×C67#4111yescharacteristic
83M16×C67—22no—
84M16×C67—22no—
85M16×C67—42no—
86C2×C8×C67—22no—
87C2×C8×C67—41no—
88C4⋊C4×C67—41no—
89C4⋊C4×C67—41no—
90C4⋊C4×C67—41no—
91C4⋊C4×C67—41no—
92C42×C67C2×D411yescharacteristic
93C42×C67C2×D411yescharacteristic
94C42×C67C2×D411yescharacteristic
95C42×C67#4111yescharacteristic
96C23×C67C4×D411yescharacteristic
97Q8×C67—44no—
98Q8×C67—82no—
99C2×C4×C67—22no—
100C2×C4×C67—22no—
101C2×C4×C67—42no—
102C2×C4×C67—82no—
103C2×C4×C67C2×C22⋊C411yescharacteristic
104C2×C4×C67C4×D411yescharacteristic
105C2×C4×C67#12111yescharacteristic
106C2×C4×C67#12111yescharacteristic
107C2×C4×C67#12411yescharacteristic
108C2×C4×C67#12711yescharacteristic
109C2×C4×C67#13712yes—
110C2×C4×C67#13712yes—
111C2×Dic67—42no—
112C2×Dic67—82no—
113Dic134—44no—
114Dic134—82no—
115C8×C67—42no—
116C8×C67—81no—
117C67⋊C8—22no—
118C67⋊C8—42no—
119C67⋊C8—81no—
120C67⋊C8#10014yes—
121C22×C67—21no—
122C22×C67—21no—
123C22×C67#39011yescharacteristic
124C22×C67#44112yes—
125C4×C67—21no—
126C4×C67—21no—
127C4×C67—82no—
128C4×C67C2×#10012yes—
129C4×C67#47012yes—
130C4×C67#47312yes—
131C4×C67#49711yescharacteristic
132C4×C67#49711yescharacteristic
133Dic67—82no—
134#9807—671nomaximal, Sylow
135C2×C67—22no—
136C2×C67#196511yescharacteristic
137C2×C67#197111yescharacteristic
138C2×C67#197911yescharacteristic
139C2×#494—671no—
140#1997—671no—
141#1997D6711yescharacteristic
142#1566—671no—
143#1548—671no—
144#1548—671no—
145#1451—671no—
146C67#980711yescharacteristic, complemented, Sylow
147C22×Q16—674no—
148C2×C4⋊Q8—671no—
149C2×C4⋊Q8C2×D6711yescharacteristic
150#494—1341no—
151#494—1341no—
152#494—1342no—
153#431—671no—
154#423—671no—
155#423C2×D6711yescharacteristic
156C4×M16C2×D6711yescharacteristic
157C2×C4×C8—671no—
158#343—22no—
159#343—1342no—
160#334—1342no—
161#333—1342no—
162#333—1342no—
163#322—1341no—
164#322—1341no—
165C22×Q8—672no—
166C22×Q8C4×D6712yes—
167C2×Q16—2688no—
168C2×Q16—1344no—
169C2×Q16—1344no—
170C2×M16—21no—
171C2×M16—1341no—
172C2×M16C22⋊D6712yes—
173C22×C8—672no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—1341no—
179C4⋊Q8—1341no—
180C4⋊Q8—1341no—
181C4⋊Q8—1341no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—1341no—
185C2×C4⋊C4—1341no—
186C2×C42C22×D6711yescharacteristic
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—672no—
190C4⋊C8—1341no—
191C4⋊C8—1341no—
192C22⋊C8—1341no—
193#97—21no—
194C4×C8—21no—
195C4×C8—671no—
196C4×C8—671no—
197C4×C8—672no—
198C4×C8—1341no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—44no—
202C2×Q8—2684no—
203C2×Q8—1342no—
204C2×Q8—1342no—
205C22×C4—22no—
206C22×C4—1342no—
207C22×C4C2×C4×D6711yescharacteristic
208C22×C4C2×C22⋊D6711yescharacteristic
209C22×C4#1176162811yescharacteristic
210Q16—2688no—
211Q16—2688no—
212M16—22no—
213M16—22no—
214M16—42no—
215M16—2682no—
216C2×C8—22no—
217C2×C8—41no—
218C2×C8—2681no—
219C2×C8—672no—
220C2×C8—672no—
221C2×C8—1342no—
222C2×C8—1344no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—2681no—
228C4⋊C4—2681no—
229C4⋊C4—2681no—
230C4⋊C4—2681no—
231C42D4×D6711yescharacteristic
232C42D4×D6711yescharacteristic
233C42D4×D6711yescharacteristic
234C42#1176162711yescharacteristic
235C23C4×C22⋊D6711yescharacteristic
236Q8—44no—
237Q8—82no—
238Q8—2684no—
239Q8—5362no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—82no—
244C2×C4—2682no—
245C2×C4—5362no—
246C2×C4C22⋊C4×D6711yescharacteristic
247C2×C4(C22⋊D67)⋊C411yescharacteristic
248C2×C4#1452401511yescharacteristic
249C2×C4#1452401611yescharacteristic
250C2×C4#1452402911yescharacteristic
251C2×C4#1452403911yescharacteristic
252C2×C4#1452407312yes—
253C2×C4#1452407412yes—
254C8—42no—
255C8—81no—
256C8—2682no—
257C8—5361no—
258C8—674no—
259C8—1342no—
260C22—21no—
261C22—21no—
262C22#1686329611yescharacteristic
263C22#1686350112yes—
264C4—21no—
265C4—21no—
266C4—82no—
267C4—5362no—
268C4#100×D6712yes—
269C4#1686365212yes—
270C4#1686367012yes—
271C4#1686378811yescharacteristic
272C4#1686379111yescharacteristic
273C2—22no—
274C2#2256104711yescharacteristic
275C2#2256107911yescharacteristic
276C2#2256112111yescharacteristic
subgroupG/H
centerC22#16863296
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1997C2
socleC22×C67#390
prime radicalC23×C67C4×D4
cosocleC2×C42×C67C23
hypercenter#1997D67
hypocenterC67#9807
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
146. C67#9807semidirect

Maximal subgroups

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

subgroupindex
C2×#168637752
#225611982
#225589642
#225588912
#225588902
#225584962
C67×#19972
#980767