Finite Group Databasebeta

← #30758924

Subgroups — #30758924

Order 17152.

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

normal subgroups only

This group has 416 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#22568587C211yesmaximal, characteristic
2#22558773C212yesmaximal
3#22558069C212yesmaximal
4#22558042C211yesmaximal, characteristic
5C67×#1994C211yesmaximal, characteristic
6C2×#14524043—21no—
7#16863449C2212yes—
8#16863302—24no—
9#16863262—21no—
10#16863262C2212yes—
11#16863133—24no—
12#533×C67C2211yescharacteristic
13#423×C67C2211yescharacteristic
14C43×C67C2211yescharacteristic
15#339×C67—21no—
16#339×C67—21no—
17C22×Dic134—22no—
18#14524043—42no—
19#14524043—42no—
20C2×C4⋊Dic67—22no—
21C2×C4⋊Dic67—22no—
22C4⋊(C67⋊C8)—22no—
23C22⋊(C67⋊C8)—22no—
24C4×C67⋊C8—22no—
25C67⋊#95—24no—
26C67⋊#95—22no—
27C67⋊#95D414yes—
28C2×M16×C67—21no—
29C4⋊Q8×C67Q812yescomplemented
30#125×C67—21no—
31C22⋊Q8×C67—22no—
32C4×Q8×C67—22no—
33#117×C67—21no—
34C2×C42×C67—22no—
35C2×C42×C67D411yescharacteristic
36C2×C42×C67D411yescharacteristic
37C2×C42×C67C2311yescharacteristic
38C4⋊C8×C67—22no—
39C2×Dic134—44no—
40C2×Dic134—48no—
41C22×Dic67—24no—
42C2×C67⋊C8—42no—
43C4⋊Dic67—44no—
44C4⋊Dic67—42no—
45C4⋊Dic67—42no—
46C2×Q8×C67—22no—
47C22×C4×C67—21no—
48C22×C4×C67—21no—
49C22×C4×C67C2×D412yes—
50C22×C4×C67C2×D411yescharacteristic
51M16×C67—41no—
52M16×C67—41no—
53C2×C8×C67—41no—
54C4⋊C4×C67—22no—
55C4⋊C4×C67—22no—
56C4⋊C4×C67—42no—
57C22⋊C4×C67—41no—
58C42×C67—21no—
59C42×C67—21no—
60C42×C67—44no—
61C42×C67—41no—
62C42×C67SD1612yes—
63C42×C67Q1612yes—
64C42×C67C2×Q811yescharacteristic
65C42×C67#4112yes—
66C42×C67#4111yescharacteristic
67C23×C67#12011yescharacteristic
68Q8×C67—82no—
69C2×C4×C67—24no—
70C2×C4×C67—22no—
71C2×C4×C67—22no—
72C2×C4×C67—42no—
73C2×C4×C67—42no—
74C2×C4×C67—42no—
75C2×C4×C67C22⋊Q812yes—
76C2×C4×C67C22⋊Q811yescharacteristic
77C2×C4×C67#12412yes—
78C2×C4×C67#12411yescharacteristic
79C2×C4×C67C2×SD1611yescharacteristic
80C2×C4×C67C2×Q1611yescharacteristic
81C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
82C2×C4×C67#13711yescharacteristic
83C2×Dic67—48no—
84C2×Dic67—44no—
85Dic134—88no—
86C8×C67—81no—
87C67⋊C8—82no—
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—82no—
97C4×C67#47511yescharacteristic
98C4×C67#47711yescharacteristic
99C4×C67#48711yescharacteristic
100C4×C67#48911yescharacteristic
101Dic67—84no—
102#10363—671nomaximal, Sylow
103C2×C67—21no—
104C2×C67—21no—
105C2×C67#202411yescharacteristic
106C2×C67#203112yes—
107#2862—671no—
108#1994D6711yescharacteristic
109#1518—672no—
110#1286—672no—
111#1276—671no—
112C67#1036311yescharacteristic, complemented, Sylow
113#533C2×D6711yescharacteristic
114C2×C4⋊Q8—1341no—
115#431—672no—
116#423C2×D6711yescharacteristic
117#391—1344no—
118#382—672no—
119#382—1341no—
120C43C2×D6711yescharacteristic
121#339—21no—
122#339—21no—
123#328—1344no—
124C22×Q8—1342no—
125C2×M16—21no—
126C4⋊Q8—2682no—
127C4⋊Q8—2682no—
128C4⋊Q8Dic13412yescomplemented
129#125—21no—
130C22⋊Q8—22no—
131C4×Q8—22no—
132#117—21no—
133C2×C4⋊C4—1342no—
134C2×C4⋊C4—1342no—
135C2×C42—22no—
136C2×C42C22⋊D6711yescharacteristic
137C2×C42C22⋊D6711yescharacteristic
138C2×C42C22×D6711yescharacteristic
139C4⋊C8—22no—
140C4⋊C8—1342no—
141C22⋊C8—1342no—
142C4×C8—1342no—
143#95—674no—
144#95—1344no—
145#95—1342no—
146C2×Q8—22no—
147C2×Q8—2684no—
148C2×Q8—2688no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—1344no—
152C22×C4D4×D6712yes—
153C22×C4C2×C22⋊D6711yescharacteristic
154M16—41no—
155M16—41no—
156C2×C8—41no—
157C2×C8—2682no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—42no—
161C4⋊C4—2684no—
162C4⋊C4—2682no—
163C4⋊C4—2682no—
164C22⋊C4—41no—
165C42—21no—
166C42—21no—
167C42—44no—
168C42—41no—
169C42#1176161512yes—
170C42#1176161812yes—
171C42C2×Dic13411yescharacteristic
172C42#1176162712yes—
173C42#1176162811yescharacteristic
174C23#1452401111yescharacteristic
175Q8—82no—
176Q8—5368no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—2688no—
184C2×C4—2684no—
185C2×C4#1452402012yes—
186C2×C4#1452402211yescharacteristic
187C2×C4#1452402912yes—
188C2×C4#1452403011yescharacteristic
189C2×C4C2×#1176161511yescharacteristic
190C2×C4C2×#1176161811yescharacteristic
191C2×C4#1452406811yescharacteristic
192C2×C4#1452407311yescharacteristic
193C8—81no—
194C8—5362no—
195C22—22no—
196C22#1686330811yescharacteristic
197C22#1686354611yescharacteristic
198C22#1686355111yescharacteristic
199C4—24no—
200C4—22no—
201C4—22no—
202C4—42no—
203C4—82no—
204C4—5364no—
205C4#1686368111yescharacteristic
206C4#1686369111yescharacteristic
207C4#1686374611yescharacteristic
208C4#1686375511yescharacteristic
209C2—21no—
210C2—21no—
211C2#2256135011yescharacteristic
212C2#2256138112yes—
subgroupG/H
centerC22#16863308
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#1994C2
socleC22×C67#393
prime radicalC23×C67#120
hypercenter#1994D67
hypocenterC67#10363
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1994

Complements

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

NH ≅ G/Ntype
29. C4⋊Q8×C67Q8semidirect
112. C67#10363semidirect
128. C4⋊Q8Dic134semidirect

Maximal subgroups

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

subgroupindex
#225685872
#225587732
#225580692
#225580422
C67×#19942
#1036367