Finite Group Databasebeta

← #30776647

Subgroups — #30776647

Order 17152.

200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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#22562521C211yesmaximal, characteristic
2#22562471C211yesmaximal, characteristic, complemented
3#22562090C211yesmaximal, characteristic, complemented
4#22558406C211yesmaximal, characteristic, complemented
5#22558402C211yesmaximal, characteristic, complemented
6#22558400C211yesmaximal, characteristic, complemented
7C67×#2264C211yesmaximal, characteristic, complemented
8C2×#14523986—21no—
9C2×D1072C2211yescharacteristic
10#16863781—21no—
11#16863513—21no—
12#16863233—21no—
13#16863231—21no—
14#16863230C2211yescharacteristic
15#16863226C2211yescharacteristic
16#16863203—21no—
17C16×Dic67C2211yescharacteristic
18#508×C67C2211yescharacteristic
19C2×Q32×C67C2211yescharacteristic
20Q64×C67—21no—
21SD64×C67—21no—
22M64×C67C2211yescharacteristic
23C2×#11761615—41no—
24C2×D536—21no—
25#14524050—41no—
26#14524032—41no—
27#14523986—42no—
28D1072—21no—
29D1072—21no—
30C2×C67⋊C16D411yescharacteristic
31#14523979—21no—
32#14523970—41no—
33C8×Dic67D411yescharacteristic
34C67⋊C32—21no—
35#135×C67—21no—
36C2×Q16×C67—21no—
37Q32×C67—21no—
38Q32×C67—21no—
39Q32×C67D411yescharacteristic
40SD32×C67—21no—
41D16×C67D411yescharacteristic
42C2×C16×C67C2311yescharacteristic
43C32×C67—21no—
44C2×D268—41no—
45#11761615—81no—
46#11761615—81no—
47D536—41no—
48D536—41no—
49#11761612—41no—
50#11761612—81no—
51C2×C67⋊C8—81no—
52C2×C67⋊C8D811yescharacteristic
53(C2×D67)⋊C4—82no—
54C4×Dic67D811yescharacteristic
55C67⋊C16—21no—
56#41×C67—41no—
57C2×Q8×C67—41no—
58Q16×C67—21no—
59Q16×C67—41no—
60Q16×C67—41no—
61SD16×C67—41no—
62D8×C67—21no—
63C2×C8×C67C2×D411yescharacteristic
64C16×C67C2×D411yescharacteristic
65C16×C67#4111yescharacteristic
66C22×D67—81no—
67Q8×C67—41no—
68Q8×C67—81no—
69Q8×C67—81no—
70D4×C67—41no—
71D4×C67—81no—
72C2×C4×C67—81no—
73C2×C4×C67—81no—
74C2×C4×C67C2×D811yescharacteristic
75C2×Dic67—21no—
76D268—81no—
77D268—81no—
78C8×C67#12111yescharacteristic
79C8×C67AGL(1,ℤ/8)11yescharacteristic
80C67⋊C8—81no—
81C67⋊C8D1611yescharacteristic
82C67⋊C8D1611yescharacteristic
83C22×C67—81no—
84C22×C67#50911yescharacteristic
85C2×D67—162no—
86C2×D67—161no—
87C4×C67—81no—
88C4×C67—161no—
89C4×C67#45911yescharacteristic
90C4×C67C2×D1611yescharacteristic
91Dic67—41no—
92#13691—671nomaximal, Sylow
93C2×C67—21no—
94C2×C67—161no—
95C2×C67#220611yescharacteristic
96D67—321no—
97#2264D6711yescharacteristic, complemented
98#2263—671no—
99#2249—671no—
100#2178—671no—
101#1422—671no—
102#1420—671no—
103#1419—671no—
104C67#1369111yescharacteristic, complemented, Sylow
105#508C2×D6711yescharacteristic
106C2×Q32C2×D6711yescharacteristic
107C2×SD32—1341no—
108C2×D16—671no—
109#495—1341no—
110#443—1341no—
111Q64—21no—
112SD64—21no—
113SD64—1341no—
114D32—1341no—
115M64—671no—
116M64C2×D6711yescharacteristic
117#368—671no—
118#359—1341no—
119C4×C16—671no—
120#135—21no—
121C2×Q16—21no—
122C2×SD16—2681no—
123C2×D8—1341no—
124#131—2681no—
125#124—2681no—
126Q32—21no—
127Q32—21no—
128Q32C22⋊D6711yescharacteristic
129SD32—21no—
130SD32—2682no—
131D16—1341no—
132D16—1341no—
133D16C22⋊D6711yescharacteristic
134C2×C16—671no—
135C2×C16C22×D6711yescharacteristic
136#108—1341no—
137#104—2681no—
138C4×C8—671no—
139C32—21no—
140C32—1341no—
141#41—41no—
142C2×Q8—41no—
143C2×D4—2681no—
144Q16—21no—
145Q16—41no—
146Q16—41no—
147SD16—41no—
148SD16—5361no—
149SD16—5361no—
150D8—21no—
151D8—2681no—
152D8—2681no—
153M16—2681no—
154M16—5361no—
155C2×C8—5361no—
156C2×C8—671no—
157C2×C8D4×D6711yescharacteristic
158C22⋊C4—5362no—
159C42—671no—
160C16—1341no—
161C16C2×C22⋊D6711yescharacteristic
162C16#1176162711yescharacteristic
163C23—5361no—
164Q8—41no—
165Q8—81no—
166Q8—81no—
167D4—41no—
168D4—81no—
169D4—5361no—
170D4—5361no—
171C2×C4—81no—
172C2×C4—81no—
173C2×C4—1341no—
174C2×C4D8×D6711yescharacteristic
175C8—5361no—
176C8—671no—
177C8—671no—
178C8#1452401511yescharacteristic
179C8#1452406711yescharacteristic
180C22—81no—
181C22—10722no—
182C22—10721no—
183C22#1686383011yescharacteristic
184C4—81no—
185C4—2681no—
186C4—161no—
187C4#1686358911yescharacteristic
188C4D16×D6711yescharacteristic
189C2—21no—
190C2—161no—
191C2—21441no—
192C2#2256224211yescharacteristic
subgroupG/H
centerC2#22562242
commutator, cosocleC2×C16×C67C23
FrattiniC2×C16C22×D67
FittingC67×#2264C2
socleC2×C67#2206
prime radical#22562521C2
hypercenter#2264D67
hypocenterC67#13691
LCSG ▷ C2×C16×C67 ▷ C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2264

Complements

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

NH ≅ G/Ntype
2. #22562471C2semidirect
3. #22562090C2semidirect
4. #22558406C2semidirect
5. #22558402C2semidirect
6. #22558400C2semidirect
7. C67×#2264C2semidirect
97. #2264D67semidirect
104. C67#13691semidirect

Maximal subgroups

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

subgroupindex
#225625212
#225624712
#225620902
#225584062
#225584022
#225584002
C67×#22642
#1369167