Finite Group Databasebeta

← #30750341

Subgroups — #30750341

Order 17152.

412 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 111 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#22569276C211yesmaximal, characteristic
2#22561038C211yesmaximal, characteristic
3#22558998C212yesmaximal, complemented
4#22558041C212yesmaximal, complemented
5C67×#1543C211yesmaximal, characteristic
6#16863474—22no—
7D4×C67⋊C8—22no—
8#16863449C2212yes—
9#16863420C2211yescharacteristic
10C2×#14523970—22no—
11#16863363—22no—
12C42×Dic67C2211yescharacteristic
13#16863149—22no—
14#16863147—22no—
15#532×C67C2211yescharacteristic
16#431×C67C2212yes—
17#333×C67C411yescharacteristic, complemented
18#333×C67C411yescharacteristic, complemented
19#331×C67C412yescomplemented
20C2×#11761612—22no—
21C2×#11761612—21no—
22C22×C67⋊C8—22no—
23C2×C4×Dic67—21no—
24C2×C4×Dic67—22no—
25C4⋊(C67⋊C8)—22no—
26C4⋊(C67⋊C8)D412yes—
27C4⋊(C67⋊C8)D412yes—
28#14523970—44no—
29C22⋊(C67⋊C8)—24no—
30C22⋊(C67⋊C8)—22no—
31C4×C67⋊C8—22no—
32C2×#41×C67D412yes—
33C4⋊Q8×C67C2×C411yescharacteristic
34#127×C67C2×C411yescharacteristic
35#124×C67Q812yes—
36#121×C67—22no—
37C4×D4×C67—22no—
38C2×C42×C67C2311yescharacteristic
39C4⋊C8×C67C2×C412yes—
40C4⋊C8×C67C2×C412yes—
41C22⋊C8×C67—22no—
42C4×C8×C67—22no—
43C22×Dic67—41no—
44#11761612—42no—
45#11761612—82no—
46C2×C67⋊C8—22no—
47C2×C67⋊C8—22no—
48C2×C67⋊C8—41no—
49C2×C67⋊C8—82no—
50C4×Dic67—22no—
51C4×Dic67—24no—
52C4×Dic67—42no—
53C4×Dic67—41no—
54#41×C67—42no—
55#41×C67—42no—
56C2×Q8×C67C4⋊C412yes—
57C2×D4×C67—22no—
58C2×D4×C67—22no—
59C2×D4×C67C4⋊C412yes—
60C22×C4×C67—22no—
61C22×C4×C67C2×D411yescharacteristic
62C22×C4×C67#4112yes—
63C2×C8×C67—22no—
64C2×C8×C67—22no—
65C4⋊C4×C67—22no—
66C22⋊C4×C67—24no—
67C42×C67C22×C411yescharacteristic
68C42×C67C2×D412yes—
69C42×C67C2×Q811yescharacteristic
70C23×C67—22no—
71C23×C67#12311yescharacteristic
72Q8×C67—42no—
73D4×C67—42no—
74D4×C67—42no—
75D4×C67—42no—
76D4×C67—82no—
77C2×C4×C67—22no—
78C2×C4×C67—22no—
79C2×C4×C67—21no—
80C2×C4×C67—21no—
81C2×C4×C67—22no—
82C2×C4×C67—42no—
83C2×C4×C67—42no—
84C2×C4×C67C2×C4⋊C411yescharacteristic
85C2×C4×C67C4×D412yes—
86C2×C4×C67#12011yescharacteristic
87C2×C4×C67C22⋊Q812yes—
88C2×Dic67—42no—
89C2×Dic67—44no—
90C8×C67—42no—
91C8×C67—42no—
92C67⋊C8—42no—
93C67⋊C8—42no—
94C67⋊C8—81no—
95C22×C67—21no—
96C22×C67—22no—
97C22×C67—42no—
98C22×C67—44no—
99C22×C67#38511yescharacteristic
100C4×C67—21no—
101C4×C67—22no—
102C4×C67—42no—
103C4×C67#44312yes—
104C4×C67#44412yes—
105C4×C67#45311yescharacteristic
106C4×C67#45511yescharacteristic
107Dic67—44no—
108#8935—671nomaximal, Sylow
109C2×C67—41no—
110C2×C67—82no—
111C2×C67#185611yescharacteristic
112C2×C67#186612yes—
113#2979—671no—
114#1962—671no—
115#1574—672no—
116#1543D6711yescharacteristic
117#1276—672no—
118C67#893511yescharacteristic, complemented, Sylow
119#532C2×D6711yescharacteristic
120#436—1342no—
121C8×D4—1342no—
122#431—672no—
123#431C2×D6712yes—
124#423—671no—
125C2×#104—1342no—
126#408—1342no—
127C43—671no—
128#335—1342no—
129#334—1342no—
130#333Dic6711yescharacteristic, complemented
131#333Dic6711yescharacteristic, complemented
132#331Dic6712yescomplemented
133C2×#41D26812yes—
134C2×M16—1341no—
135C2×M16—1342no—
136C22×C8—1342no—
137C4⋊Q8C2×Dic6711yescharacteristic
138#127C2×Dic6711yescharacteristic
139#124Dic13412yes—
140#121—22no—
141C4×D4—22no—
142C2×C42—1341no—
143C2×C42—1342no—
144C2×C42C22×D6711yescharacteristic
145C4⋊C8—672no—
146C4⋊C8—672no—
147C4⋊C8—1342no—
148C4⋊C8C2×Dic6712yes—
149C4⋊C8C2×Dic6712yes—
150#104—2684no—
151C22⋊C8—22no—
152C22⋊C8—1344no—
153C22⋊C8—1342no—
154C4×C8—22no—
155C4×C8—1342no—
156#41—42no—
157#41—42no—
158C2×Q8C4⋊Dic6712yes—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4C4⋊Dic6712yes—
162C22×C4—22no—
163C22×C4—2681no—
164C22×C4C2×D26811yescharacteristic
165C22×C4#1176162712yes—
166M16—2682no—
167M16—5362no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—2681no—
171C2×C8—5362no—
172C2×C8—1342no—
173C2×C8—1342no—
174C4⋊C4—22no—
175C22⋊C4—24no—
176C42—2682no—
177C42—2681no—
178C42—1342no—
179C42—1344no—
180C42C22×Dic6711yescharacteristic
181C42D4×D6712yes—
182C42C2×Dic13411yescharacteristic
183C23—22no—
184C23#1452402711yescharacteristic
185Q8—42no—
186D4—42no—
187D4—42no—
188D4—42no—
189D4—82no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—2682no—
198C2×C4—2684no—
199C2×C4C2×C4⋊Dic6711yescharacteristic
200C2×C4D4×Dic6712yes—
201C2×C4#1452401311yescharacteristic
202C2×C4#1452402012yes—
203C8—42no—
204C8—42no—
205C8—2682no—
206C8—2682no—
207C8—5361no—
208C22—21no—
209C22—22no—
210C22—42no—
211C22—44no—
212C22#1686327111yescharacteristic
213C4—21no—
214C4—22no—
215C4—42no—
216C4—2684no—
217C4#1686351312yes—
218C4#1686351812yes—
219C4#1686356611yescharacteristic
220C4#1686357611yescharacteristic
221C2—41no—
222C2—82no—
223C2#2256042111yescharacteristic
224C2#2256048612yes—
subgroupG/H
centerC2×C4#14524013
commutatorC42×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1543C2
socleC22×C67#385
prime radical#532×C67C22
cosocleC2×C42×C67C23
hypercenter#1543D67
hypocenterC67#8935
LCSG ▷ C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1543

Complements

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

NH ≅ G/Ntype
3. #22558998C2semidirect
4. #22558041C2semidirect
17. #333×C67C4semidirect
18. #333×C67C4semidirect
19. #331×C67C4semidirect
118. C67#8935semidirect
130. #333Dic67semidirect
131. #333Dic67semidirect
132. #331Dic67semidirect

Maximal subgroups

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

subgroupindex
#225692762
#225610382
#225589982
#225580412
C67×#15432
#893567