Finite Group Databasebeta

← #30745141

Subgroups — #30745141

Order 17152.

350 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 87 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#22561772C212yesmaximal
2#22561747C211yesmaximal, characteristic
3#22558035C211yesmaximal, characteristic
4#22558032C212yesmaximal
5C67×#2809C211yesmaximal, characteristic
6C2×#14523976—21no—
7C2×#14523967—21no—
8#16863266C2211yescharacteristic
9#16863158—22no—
10#16863158—22no—
11#16863151C2211yescharacteristic
12#16863151C2212yes—
13#400×C67—21no—
14#398×C67—22no—
15#391×C67—21no—
16#391×C67C2211yescharacteristic
17#389×C67C2212yes—
18#387×C67—22no—
19C22×C67⋊C8—21no—
20C22×C67⋊C8—22no—
21C2×C4⋊Dic67—21no—
22C2×C4⋊Dic67—22no—
23#14523976—42no—
24#14523967—42no—
25C22×Q8×C67D411yescharacteristic
26C2×C4⋊C4×C67—22no—
27C2×C4⋊C4×C67—21no—
28C2×C4⋊C4×C67—21no—
29C2×C4⋊C4×C67D412yes—
30C2×C4⋊C4×C67D412yes—
31C2×C4⋊C4×C67D411yescharacteristic
32C2×C42×C67—21no—
33C2×C42×C67C2311yescharacteristic
34#95×C67—21no—
35#95×C67—22no—
36#95×C67—22no—
37#95×C67—22no—
38#95×C67—22no—
39C22×Dic67—41no—
40C2×C67⋊C8—22no—
41C2×C67⋊C8—41no—
42C2×C67⋊C8—41no—
43C2×C67⋊C8—42no—
44C2×C67⋊C8—42no—
45C2×C67⋊C8—42no—
46C4⋊Dic67—41no—
47C4⋊Dic67—41no—
48C4⋊Dic67—44no—
49C2×Q8×C67—21no—
50C2×Q8×C67—41no—
51C2×Q8×C67—41no—
52C2×Q8×C67Q1612yes—
53C22×C4×C67—21no—
54C22×C4×C67—22no—
55C22×C4×C67—22no—
56C22×C4×C67—21no—
57C22×C4×C67C2×D411yescharacteristic
58C22×C4×C67C2×D412yes—
59C4⋊C4×C67—22no—
60C4⋊C4×C67—22no—
61C4⋊C4×C67—42no—
62C4⋊C4×C67—41no—
63C4⋊C4×C67—42no—
64C4⋊C4×C67—42no—
65C4⋊C4×C67—81no—
66C4⋊C4×C67—81no—
67C42×C67—41no—
68C42×C67—81no—
69C23×C67#12011yescharacteristic
70Q8×C67—42no—
71Q8×C67—81no—
72C2×C4×C67—21no—
73C2×C4×C67—22no—
74C2×C4×C67—22no—
75C2×C4×C67—21no—
76C2×C4×C67—22no—
77C2×C4×C67—41no—
78C2×C4×C67—41no—
79C2×C4×C67—42no—
80C2×C4×C67—42no—
81C2×C4×C67—42no—
82C2×C4×C67—41no—
83C2×C4×C67—42no—
84C2×C4×C67—42no—
85C2×C4×C67—42no—
86C2×C4×C67—42no—
87C2×C4×C67C2×Q1611yescharacteristic
88C2×C4×C67#13512yes—
89C2×C4×C67#13712yes—
90C2×C4×C67#13711yescharacteristic
91C2×Dic67—81no—
92C2×Dic67—82no—
93C67⋊C8—42no—
94C67⋊C8—82no—
95C22×C67#45111yescharacteristic
96C22×C67#45212yes—
97C22×C67#45311yescharacteristic
98C22×C67#45412yes—
99C22×C67#45811yescharacteristic
100C4×C67—21no—
101C4×C67—21no—
102C4×C67—22no—
103C4×C67—22no—
104C4×C67—42no—
105C4×C67—82no—
106C4×C67—82no—
107C4×C67—81no—
108Dic67—161no—
109#7831—671nomaximal, Sylow
110C2×C67#161411yescharacteristic
111C2×C67#161512yes—
112C2×C67#163411yescharacteristic
113C2×C67#163612yes—
114C2×C67#165711yescharacteristic
115#2809D6711yescharacteristic
116#2100—672no—
117#2095—671no—
118#1272—671no—
119#1271—672no—
120C67#783111yescharacteristic, complemented, Sylow
121C2×#107—1341no—
122C2×#103—1341no—
123#400—21no—
124#398—22no—
125#391—21no—
126#391C2×D6711yescharacteristic
127#389C2×D6712yes—
128#387—22no—
129#384—671no—
130#340—1342no—
131#340—1342no—
132#336—671no—
133#336—672no—
134C22×Q8C22⋊D6711yescharacteristic
135C22×C8—1342no—
136C22×C8—1341no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—1342no—
141C2×C4⋊C4—1341no—
142C2×C4⋊C4C22⋊D6712yes—
143C2×C4⋊C4C22⋊D6712yes—
144C2×C4⋊C4C22⋊D6711yescharacteristic
145C2×C42—21no—
146C2×C42C22×D6711yescharacteristic
147#107—2682no—
148#103—2682no—
149#95—21no—
150#95—22no—
151#95—22no—
152#95—22no—
153#95—22no—
154C2×Q8—21no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8#1176161812yes—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—2681no—
163C22×C4C2×C22⋊D6711yescharacteristic
164C22×C4C2×C22⋊D6712yes—
165C2×C8—2681no—
166C2×C8—2681no—
167C2×C8—2682no—
168C2×C8—2682no—
169C2×C8—2682no—
170C2×C8—1342no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—81no—
178C4⋊C4—81no—
179C4⋊C4—2681no—
180C4⋊C4—2681no—
181C4⋊C4—2684no—
182C42—41no—
183C42—81no—
184C23#1452401211yescharacteristic
185Q8—42no—
186Q8—81no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—5361no—
203C2×C4—5362no—
204C2×C4C2×#1176161811yescharacteristic
205C2×C4#1452406312yes—
206C2×C4#1452407312yes—
207C2×C4#1452407411yescharacteristic
208C8—2682no—
209C8—5362no—
210C22#1686355211yescharacteristic
211C22#1686355812yes—
212C22#1686356411yescharacteristic
213C22#1686356912yes—
214C22#1686358611yescharacteristic
215C4—21no—
216C4—21no—
217C4—22no—
218C4—22no—
219C4—42no—
220C4—82no—
221C4—81no—
222C4—82no—
223C4—10721no—
224C2#2255923411yescharacteristic
225C2#2255924012yes—
226C2#2255935411yescharacteristic
227C2#2255936712yes—
228C2#2255950711yescharacteristic
subgroupG/H
centerC23#14524012
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC67×#2809C2
socle, prime radicalC23×C67#120
hypercenter#2809D67
hypocenterC67#7831
LCSG ▷ C2×C42×C67 ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2809

Complements

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

NH ≅ G/Ntype
120. C67#7831semidirect

Maximal subgroups

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

subgroupindex
#225617722
#225617472
#225580352
#225580322
C67×#28092
#783167