Finite Group Databasebeta

← C67×#10584 #30689835

Subgroups — C67×#10584 #30689835

Order 17152.

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

normal subgroups only

This group has 374 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
1C67×#2844C211yesmaximal, characteristic
2C67×#1847C211yesmaximal, characteristic
3C67×#1836C212yesmaximal
4C67×#1295C212yesmaximal
5C67×#1295C211yesmaximal, characteristic
6C2×#107×C67—21no—
7C2×#106×C67—21no—
8C2×#97×C67C2212yes—
9C2×#97×C67C2211yescharacteristic
10#400×C67—22no—
11#400×C67—21no—
12#395×C67—22no—
13#395×C67—21no—
14#384×C67C2211yescharacteristic
15#382×C67C2212yes—
16#382×C67C2211yescharacteristic
17#340×C67—22no—
18#340×C67—22no—
19C22×Q8×C67—21no—
20C22×C8×C67—21no—
21C22×C8×C67—22no—
22C2×C4⋊C4×C67—22no—
23C2×C4⋊C4×C67—21no—
24C2×C4⋊C4×C67—21no—
25C2×C4⋊C4×C67—22no—
26C2×C4⋊C4×C67D412yes—
27C2×C4⋊C4×C67D412yes—
28C2×C4⋊C4×C67D411yescharacteristic
29C2×C4⋊C4×C67D411yescharacteristic
30C2×C42×C67C2311yescharacteristic
31#107×C67—42no—
32#106×C67—42no—
33#97×C67—21no—
34#97×C67—22no—
35#97×C67Q812yescomplemented
36#97×C67Q814yescomplemented
37#95×C67—21no—
38#95×C67—22no—
39#95×C67—21no—
40#95×C67—22no—
41#95×C67—22no—
42#95×C67—22no—
43C2×Q8×C67—81no—
44C2×Q8×C67—82no—
45C22×C4×C67—21no—
46C22×C4×C67—22no—
47C22×C4×C67—21no—
48C22×C4×C67—22no—
49C22×C4×C67—41no—
50C22×C4×C67C2×D411yescharacteristic
51C22×C4×C67C2×D412yes—
52C2×C8×C67—22no—
53C2×C8×C67—22no—
54C2×C8×C67—22no—
55C2×C8×C67—24no—
56C2×C8×C67—42no—
57C2×C8×C67—42no—
58C4⋊C4×C67—41no—
59C4⋊C4×C67—41no—
60C4⋊C4×C67—42no—
61C4⋊C4×C67—42no—
62C4⋊C4×C67—41no—
63C4⋊C4×C67—41no—
64C4⋊C4×C67—44no—
65C4⋊C4×C67—81no—
66C4⋊C4×C67—82no—
67C42×C67C2×Q811yescharacteristic
68C42×C67C2×Q812yes—
69C42×C67#4111yescharacteristic
70C23×C67#12011yescharacteristic
71Q8×C67—84no—
72C2×C4×C67—21no—
73C2×C4×C67—22no—
74C2×C4×C67—22no—
75C2×C4×C67—21no—
76C2×C4×C67—41no—
77C2×C4×C67—41no—
78C2×C4×C67—42no—
79C2×C4×C67—42no—
80C2×C4×C67—42no—
81C2×C4×C67—42no—
82C2×C4×C67—42no—
83C2×C4×C67—42no—
84C2×C4×C67—42no—
85C2×C4×C67—42no—
86C2×C4×C67—81no—
87C2×C4×C67—82no—
88C2×C4×C67C22⋊Q811yescharacteristic
89C2×C4×C67C22⋊Q812yes—
90C2×C4×C67C4⋊Q811yescharacteristic
91C2×C4×C67C4⋊Q812yes—
92C8×C67—24no—
93C8×C67—44no—
94C22×C67#39511yescharacteristic
95C22×C67#45311yescharacteristic
96C22×C67#45412yes—
97C22×C67#45512yes—
98C22×C67#45611yescharacteristic
99C4×C67—21no—
100C4×C67—21no—
101C4×C67—22no—
102C4×C67—22no—
103C4×C67—81no—
104C4×C67—81no—
105C4×C67—82no—
106C4×C67—82no—
107C4×C67—161no—
108#10584C6711yesmaximal, characteristic, direct product factor, Sylow
109C2×C67#165711yescharacteristic
110C2×C67#165911yescharacteristic
111C2×C67#166412yes—
112C2×C67#206011yescharacteristic
113C2×C67#206112yes—
114#2844C2×C6711yescharacteristic
115#1847C2×C6711yescharacteristic
116#1836C2×C6712yes—
117#1295C2×C6712yes—
118#1295C2×C6711yescharacteristic
119C67#1058411yescharacteristic, direct product factor, Sylow
120C2×#107—21no—
121C2×#106—21no—
122C2×#97C22×C6712yes—
123C2×#97C22×C6711yescharacteristic
124#400—22no—
125#400—21no—
126#395—22no—
127#395—21no—
128#384C22×C6711yescharacteristic
129#382C22×C6712yes—
130#382C22×C6711yescharacteristic
131#340—22no—
132#340—22no—
133C22×Q8—21no—
134C22×C8—21no—
135C22×C8—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4D4×C6712yes—
141C2×C4⋊C4D4×C6712yes—
142C2×C4⋊C4D4×C6711yescharacteristic
143C2×C4⋊C4D4×C6711yescharacteristic
144C2×C42C23×C6711yescharacteristic
145#107—42no—
146#106—42no—
147#97—21no—
148#97—22no—
149#97Q8×C6712yescomplemented
150#97Q8×C6714yescomplemented
151#95—22no—
152#95—21no—
153#95—21no—
154#95—22no—
155#95—22no—
156#95—22no—
157C2×Q8—81no—
158C2×Q8—82no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—41no—
164C22×C4C2×D4×C6711yescharacteristic
165C22×C4C2×D4×C6712yes—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—24no—
170C2×C8—42no—
171C2×C8—42no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—44no—
179C4⋊C4—81no—
180C4⋊C4—82no—
181C42C2×Q8×C6711yescharacteristic
182C42C2×Q8×C6712yes—
183C42#41×C6711yescharacteristic
184C23#120×C6711yescharacteristic
185Q8—84no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—81no—
201C2×C4—82no—
202C2×C4C22⋊Q8×C6711yescharacteristic
203C2×C4C22⋊Q8×C6712yes—
204C2×C4C4⋊Q8×C6711yescharacteristic
205C2×C4C4⋊Q8×C6712yes—
206C8—24no—
207C8—44no—
208C22#395×C6711yescharacteristic
209C22#453×C6711yescharacteristic
210C22#454×C6712yes—
211C22#455×C6712yes—
212C22#456×C6711yescharacteristic
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—81no—
218C4—81no—
219C4—82no—
220C4—82no—
221C4—161no—
222C2C67×#165711yescharacteristic
223C2C67×#165911yescharacteristic
224C2C67×#166412yes—
225C2C67×#206011yescharacteristic
226C2C67×#206112yes—
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
35. #97×C67Q8semidirect
36. #97×C67Q8semidirect
108. #10584C67direct (N × H)
119. C67#10584direct (N × H)
149. #97Q8×C67semidirect
150. #97Q8×C67semidirect

Maximal subgroups

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

subgroupindex
C67×#28442
C67×#18472
C67×#18362
C67×#12952
#1058467