Finite Group Databasebeta

← C67×#10087 #30689338

Subgroups — C67×#10087 #30689338

Order 17152.

500 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 114 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×#501×C67C212yesmaximal
2C67×#2822C211yesmaximal, characteristic
3C67×#1960C211yesmaximal, characteristic
4C2×#330×C67C212yesmaximal
5C67×#1295C211yesmaximal, characteristic
6C2×C4⋊Q8×C67C2211yescharacteristic
7C2×#125×C67C2212yes—
8#501×C67—24no—
9#501×C67—24no—
10C2×#107×C67—22no—
11C2×#106×C67—22no—
12C2×#103×C67—22no—
13C2×#97×C67C2211yescharacteristic
14C2×#97×C67C2212yes—
15#400×C67—22no—
16#398×C67—22no—
17#398×C67—22no—
18#382×C67C2211yescharacteristic
19#330×C67—24no—
20C22×Q8×C67—21no—
21C22×C8×C67—22no—
22C22×C8×C67—21no—
23C4⋊Q8×C67—22no—
24C4⋊Q8×C67—21no—
25C4⋊Q8×C67D412yes—
26#125×C67—22no—
27#125×C67—22no—
28#125×C67—22no—
29#125×C67—22no—
30C2×C4⋊C4×C67—22no—
31C2×C4⋊C4×C67—22no—
32C2×C4⋊C4×C67—22no—
33C2×C4⋊C4×C67—22no—
34C2×C4⋊C4×C67D412yes—
35C2×C4⋊C4×C67D412yes—
36C2×C4⋊C4×C67D412yes—
37C2×C42×C67C2311yescharacteristic
38#107×C67—28no—
39#106×C67—28no—
40#103×C67—44no—
41#97×C67—22no—
42#97×C67—22no—
43#97×C67D414yes—
44#95×C67—24no—
45#95×C67—22no—
46#95×C67—22no—
47C2×Q8×C67—22no—
48C2×Q8×C67—42no—
49C2×Q8×C67—42no—
50C2×Q8×C67—41no—
51C22×C4×C67—22no—
52C22×C4×C67—22no—
53C22×C4×C67—22no—
54C22×C4×C67—41no—
55C22×C4×C67C2×D412yes—
56C22×C4×C67C2×D411yescharacteristic
57C2×C8×C67—28no—
58C2×C8×C67—24no—
59C2×C8×C67—22no—
60C2×C8×C67—41no—
61C2×C8×C67—41no—
62C4⋊C4×C67—22no—
63C4⋊C4×C67—24no—
64C4⋊C4×C67—22no—
65C4⋊C4×C67—22no—
66C4⋊C4×C67—22no—
67C4⋊C4×C67—42no—
68C4⋊C4×C67—42no—
69C4⋊C4×C67—42no—
70C4⋊C4×C67—42no—
71C4⋊C4×C67—42no—
72C4⋊C4×C67—42no—
73C4⋊C4×C67—82no—
74C42×C67C2×D411yescharacteristic
75C42×C67C2×D412yes—
76C42×C67#4111yescharacteristic
77C23×C67#12011yescharacteristic
78Q8×C67—44no—
79Q8×C67—82no—
80C2×C4×C67—21no—
81C2×C4×C67—21no—
82C2×C4×C67—24no—
83C2×C4×C67—42no—
84C2×C4×C67—42no—
85C2×C4×C67—42no—
86C2×C4×C67—42no—
87C2×C4×C67—42no—
88C2×C4×C67—42no—
89C2×C4×C67—44no—
90C2×C4×C67—44no—
91C2×C4×C67—81no—
92C2×C4×C67—81no—
93C2×C4×C67#12012yes—
94C2×C4×C67#12112yes—
95C2×C4×C67#12111yescharacteristic
96C2×C4×C67#12711yescharacteristic
97C8×C67—28no—
98C8×C67—42no—
99C22×C67#39211yescharacteristic
100C22×C67#45311yescharacteristic
101C22×C67#45411yescharacteristic
102C22×C67#45614yes—
103C4×C67—22no—
104C4×C67—22no—
105C4×C67—21no—
106C4×C67—21no—
107C4×C67—82no—
108C4×C67—82no—
109C4×C67—82no—
110C4×C67—82no—
111#10087C6711yesmaximal, characteristic, direct product factor, Sylow
112C2×C67#166012yes—
113C2×C67#166312yes—
114C2×C67#200512yes—
115C2×C67#200611yescharacteristic
116C2×#501C2×C6712yes—
117#2822C2×C6711yescharacteristic
118#1960C2×C6711yescharacteristic
119C2×#330C2×C6712yes—
120#1295C2×C6711yescharacteristic
121C67#1008711yescharacteristic, direct product factor, Sylow
122C2×C4⋊Q8C22×C6711yescharacteristic
123C2×#125C22×C6712yes—
124#501—24no—
125#501—24no—
126C2×#107—22no—
127C2×#106—22no—
128C2×#103—22no—
129C2×#97C22×C6711yescharacteristic
130C2×#97C22×C6712yes—
131#400—22no—
132#398—22no—
133#398—22no—
134#382C22×C6711yescharacteristic
135#330—24no—
136C22×Q8—21no—
137C22×C8—22no—
138C22×C8—21no—
139C4⋊Q8—22no—
140C4⋊Q8—21no—
141C4⋊Q8D4×C6712yes—
142#125—22no—
143#125—22no—
144#125—22no—
145#125—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4D4×C6712yes—
151C2×C4⋊C4D4×C6712yes—
152C2×C4⋊C4D4×C6712yes—
153C2×C42C23×C6711yescharacteristic
154#107—28no—
155#106—28no—
156#103—44no—
157#97—22no—
158#97—22no—
159#97D4×C6714yes—
160#95—24no—
161#95—22no—
162#95—22no—
163C2×Q8—22no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—41no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—41no—
171C22×C4C2×D4×C6712yes—
172C22×C4C2×D4×C6711yescharacteristic
173C2×C8—28no—
174C2×C8—24no—
175C2×C8—22no—
176C2×C8—41no—
177C2×C8—41no—
178C4⋊C4—22no—
179C4⋊C4—24no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—82no—
190C42C2×D4×C6711yescharacteristic
191C42C2×D4×C6712yes—
192C42#41×C6711yescharacteristic
193C23#120×C6711yescharacteristic
194Q8—44no—
195Q8—82no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—24no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4#120×C6712yes—
210C2×C4#121×C6712yes—
211C2×C4#121×C6711yescharacteristic
212C2×C4#127×C6711yescharacteristic
213C8—28no—
214C8—42no—
215C22#392×C6711yescharacteristic
216C22#453×C6711yescharacteristic
217C22#454×C6711yescharacteristic
218C22#456×C6714yes—
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—82no—
224C4—82no—
225C4—82no—
226C4—82no—
227C2C67×#166012yes—
228C2C67×#166312yes—
229C2C67×#200512yes—
230C2C67×#200611yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C67#120
commutator, Frattini, cosocleC2×C42C23×C67
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C67 ◁ C2×C42×C67 ◁ G

Complements

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

NH ≅ G/Ntype
111. #10087C67direct (N × H)
121. C67#10087direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#501×C672
C67×#28222
C67×#19602
C2×#330×C672
C67×#12952
#1008767