Finite Group Databasebeta

← #30774920

Subgroups — #30774920

Order 17152.

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

normal subgroups only

This group has 370 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#22570733C211yesmaximal, characteristic
2#22561929C212yesmaximal
3#22558316C212yesmaximal
4#22558293C211yesmaximal, characteristic
5C67×#1971C211yesmaximal, characteristic
6#16863953—21no—
7C2×#14523983C2212yes—
8#16863805—24no—
9#16863434—21no—
10#16863434C2212yes—
11#16863200—24no—
12C22×Q16×C67C2211yescharacteristic
13C2×#106×C67C2211yescharacteristic
14C2×#103×C67—21no—
15C2×#97×C67C2211yescharacteristic
16#391×C67—21no—
17#14524043—22no—
18#14524035—22no—
19#14524022—42no—
20C4×Dic134—42no—
21#14524002—22no—
22#14523983—22no—
23#14523983—22no—
24C2×C67⋊C16—22no—
25#14523977—22no—
26#14523977D414yes—
27#14523975—22no—
28#14523975—22no—
29C22×Q8×C67—21no—
30C2×Q16×C67—22no—
31C2×Q16×C67—22no—
32C2×Q16×C67—21no—
33C2×Q16×C67Q812yescomplemented
34C22×C8×C67D411yescharacteristic
35C22×C8×C67C2311yescharacteristic
36C2×C4⋊C4×C67—21no—
37C2×C42×C67D411yescharacteristic
38#106×C67—22no—
39#103×C67—42no—
40#97×C67—22no—
41#95×C67—42no—
42C2×Dic134—42no—
43C4⋊Dic67—24no—
44C4⋊Dic67—42no—
45Dic67⋊C4—44no—
46C22⋊Dic67—42no—
47C4×Dic67—42no—
48C67⋊C16—42no—
49C2×Q8×C67—22no—
50C2×Q8×C67—41no—
51C2×Q8×C67—41no—
52C2×Q8×C67—41no—
53C2×Q8×C67—41no—
54C2×Q8×C67—41no—
55C22×C4×C67—41no—
56C22×C4×C67—41no—
57C22×C4×C67D812yes—
58C22×C4×C67C2×D411yescharacteristic
59Q16×C67—24no—
60Q16×C67—42no—
61C2×C8×C67—21no—
62C2×C8×C67—21no—
63C2×C8×C67SD1612yes—
64C2×C8×C67C2×D412yes—
65C2×C8×C67C2×Q811yescharacteristic
66C2×C8×C67#4112yes—
67C2×C8×C67#4111yescharacteristic
68C4⋊C4×C67—41no—
69C4⋊C4×C67—41no—
70C42×C67—21no—
71C42×C67—21no—
72C23×C67C2×D811yescharacteristic
73Q8×C67—42no—
74Q8×C67—42no—
75Q8×C67—81no—
76Q8×C67—81no—
77C2×C4×C67—22no—
78C2×C4×C67—42no—
79C2×C4×C67—42no—
80C2×C4×C67—81no—
81C2×C4×C67—81no—
82C2×C4×C67—82no—
83C2×C4×C67—81no—
84C2×C4×C67#12011yescharacteristic
85C2×C4×C67C22⋊Q811yescharacteristic
86C2×C4×C67#12411yescharacteristic
87C2×C4×C67C2×SD1611yescharacteristic
88C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
89C2×C4×C67#13711yescharacteristic
90C2×Dic67—44no—
91Dic134—84no—
92C8×C67—22no—
93C8×C67C22⋊Q812yes—
94C8×C67#12412yes—
95C22×C67—22no—
96C22×C67#44711yescharacteristic
97C22×C67#47411yescharacteristic
98C22×C67#48611yescharacteristic
99C4×C67—42no—
100C4×C67—82no—
101C4×C67—161no—
102C4×C67#39311yescharacteristic
103C4×C67#44811yescharacteristic
104C4×C67#47611yescharacteristic
105C4×C67#48911yescharacteristic
106Dic67—84no—
107#13368—671nomaximal, Sylow
108C2×C67—21no—
109C2×C67—21no—
110C2×C67#202311yescharacteristic
111C2×C67#219112yes—
112#3162—671no—
113#2141—672no—
114#1971D6711yescharacteristic
115#1386—672no—
116#1377—671no—
117C67#1336811yescharacteristic, complemented, Sylow
118C22×Q16C2×D6711yescharacteristic
119#533—1341no—
120C2×M32—672no—
121#501—1344no—
122#427—672no—
123#427—1341no—
124C2×#106C2×D6711yescharacteristic
125C2×#103—21no—
126C2×#97C2×D6711yescharacteristic
127#391—21no—
128#358—1344no—
129C22×Q8—21no—
130C2×Q16—22no—
131C2×Q16—22no—
132C2×Q16—21no—
133C2×Q16Dic13412yescomplemented
134C22×C8C22⋊D6711yescharacteristic
135C22×C8C22×D6711yescharacteristic
136C4⋊Q8—1342no—
137#125—1342no—
138C22⋊Q8—2682no—
139C4×Q8—2682no—
140#117—1342no—
141C2×C4⋊C4—21no—
142C2×C42C22⋊D6711yescharacteristic
143M32—1342no—
144M32—1342no—
145C2×C16—1342no—
146#107—674no—
147#107—1342no—
148#106—22no—
149#106—1342no—
150#106—1342no—
151#103—42no—
152#97—22no—
153#95—42no—
154C2×Q8—22no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—2682no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4#1176161312yes—
164C22×C4C2×C22⋊D6711yescharacteristic
165Q16—24no—
166Q16—42no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8#1176161612yes—
170C2×C8D4×D6712yes—
171C2×C8C2×Dic13411yescharacteristic
172C2×C8#1176162712yes—
173C2×C8#1176162811yescharacteristic
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—2684no—
177C4⋊C4—2682no—
178C4⋊C4—1344no—
179C22⋊C4—2682no—
180C42—21no—
181C42—21no—
182C42—2682no—
183C16—2682no—
184C23C2×#1176161311yescharacteristic
185Q8—42no—
186Q8—42no—
187Q8—81no—
188Q8—81no—
189Q8—5364no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—82no—
196C2×C4—81no—
197C2×C4—2684no—
198C2×C4#1452401111yescharacteristic
199C2×C4#1452402211yescharacteristic
200C2×C4#1452403011yescharacteristic
201C2×C4C2×#1176161611yescharacteristic
202C2×C4#1452406811yescharacteristic
203C2×C4#1452407311yescharacteristic
204C8—22no—
205C8#1452402012yes—
206C8#1452402912yes—
207C22—22no—
208C22#1686352911yescharacteristic
209C22#1686367611yescharacteristic
210C22#1686374211yescharacteristic
211C4—42no—
212C4—82no—
213C4—161no—
214C4—5364no—
215C4#1686330811yescharacteristic
216C4#1686353411yescharacteristic
217C4#1686368611yescharacteristic
218C4#1686375511yescharacteristic
219C2—21no—
220C2—21no—
221C2#2256134511yescharacteristic
222C2#2256215912yes—
subgroupG/H
centerC22#16863529
commutator, cosocleC22×C8×C67C23
FrattiniC22×C8C22×D67
FittingC67×#1971C2
socleC22×C67#447
prime radicalC23×C67C2×D8
hypercenter#1971D67
hypocenterC67#13368
LCSG ▷ C22×C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ #1971

Complements

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

NH ≅ G/Ntype
33. C2×Q16×C67Q8semidirect
117. C67#13368semidirect
133. C2×Q16Dic134semidirect

Maximal subgroups

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

subgroupindex
#225707332
#225619292
#225583162
#225582932
C67×#19712
#1336867