Finite Group Databasebeta

← C67×#10936 #30690187

Subgroups — C67×#10936 #30690187

Order 17152.

1258 conjugacy classes of subgroups; only the 302 normal subgroups are shown.

normal subgroups only

This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 302 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
1C2×#381×C67C213yesmaximal
2C67×#2273C211yesmaximal, characteristic
3C2×#342×C67C213yesmaximal
4C24×C4×C67C2211yescharacteristic
5C22×C22⋊C4×C67C2213yes—
6C22×C22⋊C4×C67C2213yes—
7#381×C67—26no—
8#381×C67C4112yescomplemented
9C4×C22⋊C4×C67—26no—
10C2×#95×C67—26no—
11#342×C67—26no—
12C25×C67C2311yescharacteristic
13C23×C4×C67—23no—
14C23×C4×C67—23no—
15C23×C4×C67—26no—
16C23×C4×C67—23no—
17C23×C4×C67C2×C416yes—
18C23×C4×C67D416yes—
19C2×C22⋊C4×C67—26no—
20C2×C22⋊C4×C67—26no—
21C2×C22⋊C4×C67—26no—
22C2×C22⋊C4×C67—26no—
23C2×C22⋊C4×C67—23no—
24C2×C22⋊C4×C67C2×C416yes—
25C2×C22⋊C4×C67C2×C416yes—
26C2×C22⋊C4×C67D416yes—
27C2×C42×C67—24no—
28#95×C67—24no—
29#95×C67—224no—
30C24×C67—44no—
31C24×C67C22×C413yes—
32C24×C67C2×D413yes—
33C24×C67C2×D413yes—
34C24×C67#4116yes—
35C22×C4×C67—26no—
36C22×C4×C67—26no—
37C22×C4×C67—212no—
38C22×C4×C67—212no—
39C22×C4×C67—26no—
40C22×C4×C67—26no—
41C22×C4×C67—26no—
42C22×C4×C67—41no—
43C22×C4×C67—46no—
44C22×C4×C67—43no—
45C22×C4×C67—43no—
46C22×C4×C67—42no—
47C22×C4×C67—41no—
48C22×C4×C67—412no—
49C22×C4×C67—412no—
50C22×C4×C67—46no—
51C22×C4×C67C4214yescomplemented
52C22×C4×C67C22⋊C4112yes—
53C22⋊C4×C67—212no—
54C22⋊C4×C67—212no—
55C22⋊C4×C67—212no—
56C22⋊C4×C67—43no—
57C22⋊C4×C67—43no—
58C42×C67—84no—
59C23×C67—212no—
60C23×C67—46no—
61C23×C67—46no—
62C23×C67—412no—
63C23×C67C2×C4211yescharacteristic
64C23×C67C2×C22⋊C413yes—
65C23×C67#11713yes—
66C23×C67C4×D416yes—
67C23×C67C4×D416yes—
68C23×C67#12011yescharacteristic
69C23×C67#12116yes—
70C23×C67#12313yes—
71C23×C67#12413yes—
72C23×C67#12612yes—
73C23×C67#12711yescharacteristic
74C2×C4×C67—212no—
75C2×C4×C67—212no—
76C2×C4×C67—46no—
77C2×C4×C67—412no—
78C2×C4×C67—46no—
79C2×C4×C67—412no—
80C2×C4×C67—412no—
81C2×C4×C67—412no—
82C2×C4×C67—46no—
83C2×C4×C67—86no—
84C2×C4×C67—812no—
85C22×C67—26no—
86C22×C67—212no—
87C22×C67—26no—
88C22×C67—212no—
89C22×C67—41no—
90C22×C67—43no—
91C22×C67—46no—
92C22×C67—43no—
93C22×C67—42no—
94C22×C67—41no—
95C22×C67C4×C22⋊C413yes—
96C22×C67#38613yes—
97C22×C67#38816yes—
98C22×C67#39013yes—
99C22×C67#45814yes—
100C4×C67—412no—
101C4×C67—44no—
102C4×C67—86no—
103#10936C6711yesmaximal, characteristic, direct product factor, Sylow
104C2×C67—23no—
105C2×C67—23no—
106C2×C67—26no—
107C2×C67#143711yescharacteristic
108C2×C67#189016yes—
109C2×#381C2×C6713yes—
110#2273C2×C6711yescharacteristic
111C2×#342C2×C6713yes—
112C67#1093611yescharacteristic, direct product factor, Sylow
113C24×C4C22×C6711yescharacteristic
114C22×C22⋊C4C22×C6713yes—
115C22×C22⋊C4C22×C6713yes—
116#381—26no—
117#381C4×C67112yescomplemented
118C4×C22⋊C4—26no—
119C2×#95—26no—
120#342—26no—
121C25C23×C6711yescharacteristic
122C23×C4—23no—
123C23×C4—23no—
124C23×C4—26no—
125C23×C4—23no—
126C23×C4C2×C4×C6716yes—
127C23×C4D4×C6716yes—
128C2×C22⋊C4—26no—
129C2×C22⋊C4—26no—
130C2×C22⋊C4—26no—
131C2×C22⋊C4—26no—
132C2×C22⋊C4—23no—
133C2×C22⋊C4C2×C4×C6716yes—
134C2×C22⋊C4C2×C4×C6716yes—
135C2×C22⋊C4D4×C6716yes—
136C2×C42—24no—
137#95—24no—
138#95—224no—
139C24—44no—
140C24C22×C4×C6713yes—
141C24C2×D4×C6713yes—
142C24C2×D4×C6713yes—
143C24#41×C6716yes—
144C22×C4—26no—
145C22×C4—26no—
146C22×C4—212no—
147C22×C4—212no—
148C22×C4—26no—
149C22×C4—26no—
150C22×C4—26no—
151C22×C4—41no—
152C22×C4—46no—
153C22×C4—43no—
154C22×C4—43no—
155C22×C4—42no—
156C22×C4—41no—
157C22×C4—412no—
158C22×C4—412no—
159C22×C4—46no—
160C22×C4C42×C6714yescomplemented
161C22×C4C22⋊C4×C67112yes—
162C22⋊C4—212no—
163C22⋊C4—212no—
164C22⋊C4—212no—
165C22⋊C4—43no—
166C22⋊C4—43no—
167C42—84no—
168C23—212no—
169C23—46no—
170C23—46no—
171C23—412no—
172C23C2×C42×C6711yescharacteristic
173C23C2×C22⋊C4×C6713yes—
174C23#117×C6713yes—
175C23C4×D4×C6716yes—
176C23C4×D4×C6716yes—
177C23#120×C6711yescharacteristic
178C23#121×C6716yes—
179C23#123×C6713yes—
180C23#124×C6713yes—
181C23#126×C6712yes—
182C23#127×C6711yescharacteristic
183C2×C4—212no—
184C2×C4—212no—
185C2×C4—412no—
186C2×C4—46no—
187C2×C4—412no—
188C2×C4—46no—
189C2×C4—412no—
190C2×C4—412no—
191C2×C4—46no—
192C2×C4—86no—
193C2×C4—812no—
194C22—26no—
195C22—212no—
196C22—26no—
197C22—212no—
198C22—41no—
199C22—43no—
200C22—46no—
201C22—43no—
202C22—42no—
203C22—41no—
204C22C4×C22⋊C4×C6713yes—
205C22#386×C6713yes—
206C22#388×C6716yes—
207C22#390×C6713yes—
208C22#458×C6714yes—
209C4—412no—
210C4—44no—
211C4—86no—
212C2—23no—
213C2—23no—
214C2—26no—
215C2C67×#143711yescharacteristic
216C2C67×#189016yes—
subgroupG/H
center, socleC23×C67#120
commutatorC23C2×C42×C67
Frattini, cosocleC25C23×C67
prime radicalC25×C67C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C67 ◁ C25×C67 ◁ G

Complements

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

NH ≅ G/Ntype
8. #381×C67C4semidirect
51. C22×C4×C67C42semidirect
103. #10936C67direct (N × H)
112. C67#10936direct (N × H)
117. #381C4×C67semidirect
160. C22×C4C42×C67semidirect

Maximal subgroups

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

subgroupindex
C2×#381×C672
C67×#22732
C2×#342×C672
#1093667