Finite Group Databasebeta

← #31907996

Subgroups — #31907996

Order 18176.

316 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 75 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#22598990C211yesmaximal, characteristic
2#22588095C211yesmaximal, characteristic, complemented
3#22586332C212yesmaximal, complemented
4#22586326C212yesmaximal, complemented
5C71×#2166C211yesmaximal, characteristic, complemented
6#17986604—21no—
7#17986401C411yescharacteristic, complemented
8#17986376C411yescharacteristic, complemented
9#17986364—21no—
10#17986252C2212yes—
11#17986235—21no—
12#17986112C412yescomplemented
13C4×D568—21no—
14#17985954—21no—
15C71⋊#338C2211yescharacteristic
16#17985756C2211yescharacteristic
17#17985755—21no—
18#405×C71C2211yescharacteristic
19#364×C71—21no—
20#349×C71C2212yes—
21#347×C71—21no—
22C2×D568—21no—
23C2×D568C2×C412yes—
24C2×#11765849—21no—
25#14556125—21no—
26#14556120—21no—
27#14556113—42no—
28#14556102—22no—
29#14556101—42no—
30C4×D284—22no—
31#14556087—21no—
32#14556062—21no—
33#14556060C2×C412yes—
34#14556050—22no—
35#14556050—22no—
36#14556038—22no—
37C8×Dic71—22no—
38C2×M16×C71C2311yescharacteristic
39C22×C8×C71D411yescharacteristic
40#117×C71D411yescharacteristic
41M32×C71—22no—
42#97×C71C2×C411yescharacteristic
43#97×C71D411yescharacteristic
44C4×C8×C71C2×C411yescharacteristic
45C4×C8×C71D411yescharacteristic
46C2×D284—22no—
47C2×D284—42no—
48C2×C22⋊D71—42no—
49C2×C4×D71—42no—
50D568—42no—
51D568—41no—
52D568—41no—
53#11765849—41no—
54#11765849—41no—
55C2×C71⋊C8—22no—
56C4⋊Dic71—22no—
57C4⋊Dic71—41no—
58Dic71⋊C4—42no—
59C22⋊Dic71—41no—
60(C2×D71)⋊C4—42no—
61(C2×D71)⋊C4—42no—
62C4×Dic71—22no—
63C22×C4×C71C2×D411yescharacteristic
64M16×C71—41no—
65C2×C8×C71—21no—
66C2×C8×C71—21no—
67C2×C8×C71C22⋊C411yescharacteristic
68C2×C8×C71C22⋊C411yescharacteristic
69C2×C8×C71C22×C411yescharacteristic
70C2×C8×C71C2×D411yescharacteristic
71C4⋊C4×C71—21no—
72C22⋊C4×C71—21no—
73C42×C71C22⋊C411yescharacteristic
74C42×C71C22⋊C411yescharacteristic
75C16×C71—42no—
76C23×C71AGL(1,ℤ/8)11yescharacteristic
77C22×D71—42no—
78C2×C4×C71—21no—
79C2×C4×C71—21no—
80C2×C4×C71—21no—
81C2×C4×C71—21no—
82C2×C4×C71C2×C22⋊C411yescharacteristic
83C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
84C22⋊D71—162no—
85C2×Dic71—42no—
86C2×Dic71—42no—
87D284—42no—
88D284—42no—
89D284—162no—
90C4×D71—162no—
91C8×C71—21no—
92C8×C71—21no—
93C8×C71—21no—
94C8×C71#10411yescharacteristic
95C8×C71#10411yescharacteristic
96C71⋊C8—42no—
97C22×C71—21no—
98C22×C71—41no—
99C22×C71#41811yescharacteristic
100C2×D71—82no—
101C2×D71—82no—
102C2×D71—82no—
103C4×C71—21no—
104C4×C71—41no—
105C4×C71—41no—
106C4×C71#41011yescharacteristic
107C4×C71C2×#10411yescharacteristic
108Dic71—82no—
109Dic71—82no—
110#11770—711nomaximal, Sylow
111C2×C71—21no—
112C2×C71—41no—
113C2×C71#150511yescharacteristic
114D71—162no—
115#3176—711no—
116#2166D7111yescharacteristic, complemented
117#1747—711no—
118#1341—712no—
119#1339—712no—
120C71#1177011yescharacteristic, complemented, Sylow
121#540—1421no—
122#499—711no—
123#493—711no—
124#490—1421no—
125#469—712no—
126#466—1421no—
127#442—712no—
128C4×D8—1421no—
129#405—1421no—
130#405C2×D7111yescharacteristic
131#364—21no—
132#349C2×D7112yes—
133#347—21no—
134#338—711no—
135H(3,ℤ/4)—711no—
136H(3,ℤ/4)—1421no—
137C2×D8—712no—
138C2×D8—1421no—
139C2×M16—1421no—
140C2×M16C22×D7111yescharacteristic
141C22×C8D28411yescharacteristic
142#127—1421no—
143#125—1421no—
144#123—2842no—
145#121—2842no—
146#121—1422no—
147C4×D4—1422no—
148#117—1421no—
149#117D28411yescharacteristic
150M32—22no—
151#107—1421no—
152#106—712no—
153#102—1422no—
154#102—1422no—
155#97—1422no—
156#97C4×D7111yescharacteristic
157#97C22⋊D7111yescharacteristic
158C4×C8—1422no—
159C4×C8C4×D7111yescharacteristic
160C4×C8C22⋊D7111yescharacteristic
161C2×D4—2842no—
162C2×D4—2842no—
163C2×D4—1422no—
164C22×C4—2842no—
165C22×C4C2×D28411yescharacteristic
166D8—2842no—
167D8—2841no—
168D8—2841no—
169M16—41no—
170M16—2841no—
171M16—2841no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—1422no—
175C2×C8(C2×D71)⋊C411yescharacteristic
176C2×C8(C2×D71)⋊C411yescharacteristic
177C2×C8C2×C4×D7111yescharacteristic
178C2×C8C2×C22⋊D7111yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—2842no—
181C4⋊C4—2841no—
182C4⋊C4—1422no—
183C22⋊C4—21no—
184C22⋊C4—2842no—
185C22⋊C4—2841no—
186C22⋊C4—2842no—
187C42—1422no—
188C42(C2×D71)⋊C411yescharacteristic
189C42(C2×D71)⋊C411yescharacteristic
190C16—42no—
191C23—2842no—
192C23#1455615611yescharacteristic
193D4—2842no—
194D4—2842no—
195D4—11362no—
196D4—11362no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—2842no—
202C2×C4—2842no—
203C2×C4—11362no—
204C2×C4C2×(C2×D71)⋊C411yescharacteristic
205C2×C4#1455615611yescharacteristic
206C8—21no—
207C8—21no—
208C8—21no—
209C8—2842no—
210C8#1455605411yescharacteristic
211C8#1455605411yescharacteristic
212C22—21no—
213C22—41no—
214C22—5682no—
215C22—5682no—
216C22—5682no—
217C22#1798600511yescharacteristic
218C4—21no—
219C4—41no—
220C4—41no—
221C4—5682no—
222C4—5682no—
223C4#1798597211yescharacteristic
224C4C2×#1455605411yescharacteristic
225C2—21no—
226C2—41no—
227C2—11362no—
228C2#2258684911yescharacteristic
subgroupG/H
centerC2#22586849
commutatorC2×C8×C71C22×C4
FrattiniC2×M16C22×D71
FittingC71×#2166C2
socleC2×C71#1505
prime radical#22598990C2
cosocleC2×M16×C71C23
hypercenter#2166D71
hypocenterC71#11770
LCSG ▷ C2×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2166

Complements

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

NH ≅ G/Ntype
2. #22588095C2semidirect
3. #22586332C2semidirect
4. #22586326C2semidirect
5. C71×#2166C2semidirect
7. #17986401C4semidirect
8. #17986376C4semidirect
12. #17986112C4semidirect
116. #2166D71semidirect
120. C71#11770semidirect

Maximal subgroups

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

subgroupindex
#225989902
#225880952
#225863322
#225863262
C71×#21662
#1177071