Finite Group Databasebeta

← #30776179

Subgroups — #30776179

Order 17152.

326 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 79 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#22561874C212yesmaximal
2#22558380C214yesmaximal, complemented
3C67×#3135C211yesmaximal, characteristic
4#16863370—22no—
5#16863191—22no—
6#16863189C2214yes—
7#533×C67C411yescharacteristic, complemented
8#532×C67C2211yescharacteristic
9#491×C67C412yescomplemented
10#490×C67C2212yes—
11#489×C67—22no—
12#416×C67—22no—
13C2×#97×C67C411yescharacteristic, complemented
14#14524002—42no—
15#14523958—24no—
16#14523958—42no—
17C2×#41×C67D412yes—
18C22×C8×C67—21no—
19C4⋊Q8×C67—21no—
20C4⋊Q8×C67C2×C411yescharacteristic
21#127×C67C2311yescharacteristic
22#125×C67C2×C412yes—
23#124×C67D412yes—
24C22⋊Q8×C67—41no—
25#121×C67—41no—
26C4×Q8×C67—41no—
27C4×D4×C67—41no—
28#117×C67—21no—
29C2×C42×C67C2×C411yescharacteristic
30#103×C67—24no—
31#102×C67—24no—
32#97×C67—21no—
33#97×C67C2×C412yes—
34C4⋊Dic67—82no—
35C22⋊Dic67—44no—
36C4×Dic67—82no—
37#41×C67—42no—
38#41×C67—42no—
39C2×Q8×C67—41no—
40C2×Q8×C67C22⋊C412yes—
41C2×D4×C67—22no—
42C2×D4×C67—41no—
43C2×D4×C67C2×D412yes—
44C22×C4×C67—21no—
45C22×C4×C67—22no—
46C22×C4×C67C22⋊C411yescharacteristic
47C2×C8×C67—22no—
48C2×C8×C67—41no—
49C2×C8×C67—41no—
50C4⋊C4×C67—22no—
51C4⋊C4×C67—41no—
52C4⋊C4×C67—42no—
53C4⋊C4×C67—41no—
54C22⋊C4×C67—42no—
55C22⋊C4×C67—41no—
56C42×C67—21no—
57C42×C67—41no—
58C42×C67C22⋊C411yescharacteristic
59C42×C67C22×C411yescharacteristic
60C23×C67—22no—
61C23×C67#9911yescharacteristic
62Q8×C67—42no—
63Q8×C67—82no—
64D4×C67—42no—
65D4×C67—82no—
66D4×C67—82no—
67C2×C4×C67—21no—
68C2×C4×C67—21no—
69C2×C4×C67—21no—
70C2×C4×C67—21no—
71C2×C4×C67—22no—
72C2×C4×C67—42no—
73C2×C4×C67—42no—
74C2×C4×C67—42no—
75C2×C4×C67—81no—
76C2×C4×C67#9911yescharacteristic
77C2×C4×C67C2×C22⋊C411yescharacteristic
78C2×Dic67—84no—
79C8×C67—42no—
80C22×C67—21no—
81C22×C67—41no—
82C22×C67—42no—
83C22×C67—82no—
84C22×C67C2×#9911yescharacteristic
85C4×C67—21no—
86C4×C67—41no—
87C4×C67—41no—
88C4×C67—42no—
89C4×C67—82no—
90C4×C67#35411yescharacteristic
91C4×C67#35411yescharacteristic
92Dic67—164no—
93#13594—671nomaximal, Sylow
94C2×C67—21no—
95C2×C67—41no—
96C2×C67—82no—
97C2×C67C2×#35411yescharacteristic
98#3135D6711yescharacteristic
99#2126—672no—
100#1412—674no—
101C67#1359411yescharacteristic, complemented, Sylow
102#533Dic6711yescharacteristic, complemented
103#532C2×D6711yescharacteristic
104#491Dic6712yescomplemented
105#490C2×D6712yes—
106#489—22no—
107#416—22no—
108#410—1342no—
109C2×#97Dic6711yescharacteristic, complemented
110#354—1342no—
111#353—674no—
112C2×#41C22⋊D6712yes—
113C22×C8—21no—
114C4⋊Q8—21no—
115C4⋊Q8C2×Dic6711yescharacteristic
116#127C22×D6711yescharacteristic
117#125C2×Dic6712yes—
118#124C22⋊D6712yes—
119C22⋊Q8—41no—
120#121—41no—
121C4×Q8—41no—
122C4×D4—41no—
123#117—21no—
124#117—2682no—
125C2×C42C2×Dic6711yescharacteristic
126#103—24no—
127#102—24no—
128#99—2682no—
129#99—1344no—
130#97—21no—
131#97C2×Dic6712yes—
132#41—42no—
133#41—42no—
134C2×Q8—41no—
135C2×Q8C22⋊Dic6712yes—
136C2×D4—22no—
137C2×D4—41no—
138C2×D4C2×C22⋊D6712yes—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4C22⋊Dic6711yescharacteristic
142C2×C8—22no—
143C2×C8—41no—
144C2×C8—41no—
145C4⋊C4—22no—
146C4⋊C4—41no—
147C4⋊C4—42no—
148C4⋊C4—41no—
149C4⋊C4—5362no—
150C22⋊C4—42no—
151C22⋊C4—41no—
152C22⋊C4—2684no—
153C42—21no—
154C42—41no—
155C42—5362no—
156C42C22⋊Dic6711yescharacteristic
157C42C22×Dic6711yescharacteristic
158C23—22no—
159C23#1452395811yescharacteristic
160Q8—42no—
161Q8—82no—
162D4—42no—
163D4—82no—
164D4—82no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—81no—
174C2×C4—5364no—
175C2×C4#1452395811yescharacteristic
176C2×C4C2×C22⋊Dic6711yescharacteristic
177C8—42no—
178C22—21no—
179C22—41no—
180C22—42no—
181C22—82no—
182C22C2×#1452395811yescharacteristic
183C4—21no—
184C4—41no—
185C4—41no—
186C4—42no—
187C4—82no—
188C4—10724no—
189C4#1686319111yescharacteristic
190C4#1686319111yescharacteristic
191C2—21no—
192C2—41no—
193C2—82no—
194C2C2×#1686319111yescharacteristic
subgroupG/H
centerC4#16863191
commutatorC42×C67C22×C4
Frattini#127C22×D67
FittingC67×#3135C2
socleC2×C67C2×#354
prime radical#532×C67C22
cosocle#127×C67C23
hypercenter#3135D67
hypocenterC67#13594
LCSG ▷ C42×C67 ▷ C2×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #3135

Complements

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

NH ≅ G/Ntype
2. #22558380C2semidirect
7. #533×C67C4semidirect
9. #491×C67C4semidirect
13. C2×#97×C67C4semidirect
101. C67#13594semidirect
102. #533Dic67semidirect
104. #491Dic67semidirect
109. C2×#97Dic67semidirect

Maximal subgroups

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

subgroupindex
#225618742
#225583802
C67×#31352
#1359467