Finite Group Databasebeta

← #30758925

Subgroups — #30758925

Order 17152.

416 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 97 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#22568587C211yesmaximal, characteristic
2#22561186C211yesmaximal, characteristic
3#22558774C212yesmaximal
4#22558068C212yesmaximal
5C67×#1276C211yesmaximal, characteristic
6#16863953C2211yescharacteristic
7C2×#14524043—21no—
8#16863420C2211yescharacteristic
9#16863302—24no—
10#16863262—21no—
11#16863262C2212yes—
12#16863156—21no—
13#16863156—21no—
14#16863134—24no—
15#431×C67C2212yes—
16C43×C67C2211yescharacteristic
17C22×Dic134—22no—
18C2×#11761612—21no—
19#14524043—42no—
20#14524043—42no—
21#14524043Q812yescomplemented
22#14524035—21no—
23#14524022—22no—
24C4×Dic134—22no—
25#14524002—21no—
26C2×C4⋊Dic67—22no—
27C2×C4⋊Dic67—22no—
28C4⋊(C67⋊C8)—22no—
29C67⋊#95—24no—
30C67⋊#95—22no—
31C67⋊#95D414yes—
32C2×C42×C67—22no—
33C2×C42×C67D411yescharacteristic
34C2×C42×C67D411yescharacteristic
35C2×C42×C67C2311yescharacteristic
36C4⋊C8×C67—22no—
37C22⋊C8×C67—22no—
38C4×C8×C67—22no—
39C2×Dic134—22no—
40C2×Dic134—44no—
41C2×Dic134—48no—
42C22×Dic67—24no—
43#11761612—41no—
44#11761612—41no—
45C2×C67⋊C8—41no—
46C4⋊Dic67—22no—
47C4⋊Dic67—22no—
48C4⋊Dic67—44no—
49C4⋊Dic67—42no—
50C4⋊Dic67—42no—
51Dic67⋊C4—42no—
52C22⋊Dic67—41no—
53C4×Dic67—41no—
54C22×C4×C67—21no—
55C22×C4×C67—21no—
56C22×C4×C67C2×D412yes—
57C22×C4×C67C2×D411yescharacteristic
58C2×C8×C67—42no—
59C42×C67—21no—
60C42×C67—21no—
61C42×C67—44no—
62C42×C67SD1612yes—
63C42×C67Q1612yes—
64C42×C67C2×Q811yescharacteristic
65C42×C67#4111yescharacteristic
66C42×C67#4112yes—
67C23×C67#12011yescharacteristic
68C2×C4×C67—24no—
69C2×C4×C67—22no—
70C2×C4×C67—22no—
71C2×C4×C67—42no—
72C2×C4×C67—42no—
73C2×C4×C67C22⋊Q812yes—
74C2×C4×C67C22⋊Q811yescharacteristic
75C2×C4×C67#12412yes—
76C2×C4×C67#12411yescharacteristic
77C2×C4×C67C2×SD1611yescharacteristic
78C2×C4×C67C2×Q1611yescharacteristic
79C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
80C2×C4×C67#13711yescharacteristic
81C2×Dic67—48no—
82C2×Dic67—42no—
83C2×Dic67—44no—
84Dic134—88no—
85Dic134—82no—
86C8×C67—82no—
87C67⋊C8—81no—
88C22×C67—22no—
89C22×C67#39311yescharacteristic
90C22×C67#45011yescharacteristic
91C22×C67#45111yescharacteristic
92C4×C67—24no—
93C4×C67—22no—
94C4×C67—22no—
95C4×C67—42no—
96C4×C67#47511yescharacteristic
97C4×C67#47711yescharacteristic
98C4×C67#48711yescharacteristic
99C4×C67#48911yescharacteristic
100Dic67—84no—
101Dic67—82no—
102#10363—671nomaximal, Sylow
103C2×C67—21no—
104C2×C67—21no—
105C2×C67#202411yescharacteristic
106C2×C67#203112yes—
107#2862—671no—
108#1994—671no—
109#1518—672no—
110#1286—672no—
111#1276D6711yescharacteristic
112C67#1036311yescharacteristic, complemented, Sylow
113#533—671no—
114C2×C4⋊Q8—1341no—
115#431C2×D6712yes—
116#423—671no—
117#391—1344no—
118#382—672no—
119#382—1341no—
120C43C2×D6711yescharacteristic
121#339—1341no—
122#339—1341no—
123#328—1344no—
124C22×Q8—1342no—
125C2×M16—1341no—
126C4⋊Q8—2682no—
127C4⋊Q8—2682no—
128C4⋊Q8—672no—
129#125—1341no—
130C22⋊Q8—1342no—
131C4×Q8—1342no—
132#117—1341no—
133C2×C4⋊C4—1342no—
134C2×C4⋊C4—1342no—
135C2×C42—22no—
136C2×C42D26811yescharacteristic
137C2×C42D26811yescharacteristic
138C2×C42C22×D6711yescharacteristic
139C4⋊C8—22no—
140C4⋊C8—1342no—
141C22⋊C8—22no—
142C4×C8—22no—
143#95—674no—
144#95—1344no—
145#95—1342no—
146C2×Q8—2684no—
147C2×Q8—2688no—
148C2×Q8—1342no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—1344no—
152C22×C4D4×D6712yes—
153C22×C4C2×D26811yescharacteristic
154M16—2681no—
155M16—2681no—
156C2×C8—42no—
157C2×C8—2681no—
158C4⋊C4—2684no—
159C4⋊C4—2682no—
160C4⋊C4—2682no—
161C4⋊C4—2682no—
162C4⋊C4—1342no—
163C4⋊C4—1342no—
164C22⋊C4—2681no—
165C42—21no—
166C42—21no—
167C42—44no—
168C42—2681no—
169C42#1176161712yes—
170C42Dic26812yes—
171C42Q8×D6711yescharacteristic
172C42#1176162711yescharacteristic
173C42#1176162812yes—
174C23#1452401311yescharacteristic
175Q8—5362no—
176Q8—5368no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—2688no—
183C2×C4—2684no—
184C2×C4—2682no—
185C2×C4#1452402112yes—
186C2×C4#1452402311yescharacteristic
187C2×C4#1452402912yes—
188C2×C4#1452403111yescharacteristic
189C2×C4C2×#1176161711yescharacteristic
190C2×C4C2×Dic26811yescharacteristic
191C2×C4#1452407111yescharacteristic
192C2×C4#1452407611yescharacteristic
193C8—82no—
194C8—5361no—
195C22—22no—
196C22#1686330911yescharacteristic
197C22#1686354911yescharacteristic
198C22#1686355411yescharacteristic
199C4—24no—
200C4—22no—
201C4—22no—
202C4—42no—
203C4—5364no—
204C4—5362no—
205C4#1686368211yescharacteristic
206C4#1686369211yescharacteristic
207C4#1686374711yescharacteristic
208C4#1686375611yescharacteristic
209C2—21no—
210C2—21no—
211C2#2256135111yescharacteristic
212C2#2256138212yes—
subgroupG/H
centerC22#16863309
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#1276C2
socleC22×C67#393
prime radicalC23×C67#120
hypercenter#1276D67
hypocenterC67#10363
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
21. #14524043Q8semidirect
112. C67#10363semidirect

Maximal subgroups

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

subgroupindex
#225685872
#225611862
#225587742
#225580682
C67×#12762
#1036367