Finite Group Databasebeta

← #31918130

Subgroups — #31918130

Order 18176.

284 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 63 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#22590657C212yesmaximal, complemented
2#22590448C211yesmaximal, characteristic, complemented
3#22586527C212yesmaximal, complemented
4#22586480C211yesmaximal, characteristic, complemented
5C71×#3414C211yesmaximal, characteristic
6C2×Dic568C2211yescharacteristic, complemented
7#17986240—21no—
8#17985839—22no—
9#17985837—22no—
10#17985833C2212yescomplemented
11#17985827C2211yescharacteristic, complemented
12#17985804—21no—
13C2×#135×C71—21no—
14#510×C71—22no—
15#509×C71—22no—
16#508×C71—21no—
17#508×C71C2212yes—
18C2×M32×C71C2211yescharacteristic
19C2×Dic284—21no—
20#14556107—41no—
21Dic568—22no—
22#14556062—21no—
23#14556053—41no—
24C71⋊C32—22no—
25C2×#41×C71—41no—
26#135×C71—22no—
27#135×C71—22no—
28#135×C71—41no—
29#135×C71—41no—
30C2×Q16×C71—21no—
31C2×SD16×C71—41no—
32C2×D8×C71—21no—
33C22×C8×C71D411yescharacteristic
34Q32×C71—21no—
35Q32×C71D412yes—
36SD32×C71—22no—
37SD32×C71—22no—
38D16×C71—21no—
39D16×C71D412yes—
40M32×C71—22no—
41C2×C16×C71D411yescharacteristic
42C2×C16×C71C2311yescharacteristic
43C2×Dic142—41no—
44Dic284—41no—
45Dic284—41no—
46C4⋊Dic71—41no—
47Dic71⋊C4—81no—
48C22⋊Dic71—81no—
49#41×C71—42no—
50#41×C71—42no—
51#41×C71—81no—
52#41×C71—81no—
53C2×Q8×C71—41no—
54C2×D4×C71—41no—
55C2×D4×C71—81no—
56C22×C4×C71—81no—
57C22×C4×C71D811yescharacteristic
58Q16×C71—22no—
59Q16×C71—41no—
60SD16×C71—42no—
61SD16×C71—42no—
62D8×C71—22no—
63D8×C71—41no—
64C2×C8×C71—21no—
65C2×C8×C71—21no—
66C2×C8×C71D811yescharacteristic
67C2×C8×C71C2×D411yescharacteristic
68C16×C71—21no—
69C16×C71C2×D412yes—
70C23×C71—81no—
71C23×C71D1611yescharacteristic
72Q8×C71—42no—
73Q8×C71—81no—
74D4×C71—42no—
75D4×C71—81no—
76D4×C71—82no—
77D4×C71—82no—
78C2×C4×C71—21no—
79C2×C4×C71—21no—
80C2×C4×C71—82no—
81C2×C4×C71—81no—
82C2×C4×C71—82no—
83C2×C4×C71D1611yescharacteristic
84C2×C4×C71C2×D811yescharacteristic
85C2×Dic71—81no—
86C2×Dic71—81no—
87Dic142—81no—
88Dic142—81no—
89C8×C71—21no—
90C8×C71#12011yescharacteristic
91C8×C71AGL(1,ℤ/8)11yescharacteristic
92C22×C71—21no—
93C22×C71—41no—
94C22×C71—82no—
95C22×C71—162no—
96C22×C71C2×D1611yescharacteristic
97C4×C71—21no—
98C4×C71—82no—
99C4×C71#44711yescharacteristic
100C4×C71#50911yescharacteristic
101Dic71—161no—
102Dic71—161no—
103#13668—711nomaximal, Sylow
104C2×C71—21no—
105C2×C71—41no—
106C2×C71—162no—
107C2×C71#218411yescharacteristic
108#3414D7111yescharacteristic
109#2264—712no—
110#2219—711no—
111#1418—712no—
112#1400—711no—
113C71#1366811yescharacteristic, complemented, Sylow
114C2×#135—21no—
115#510—22no—
116#509—22no—
117#508—21no—
118#508C2×D7112yes—
119C2×Q32—711no—
120C2×M32C2×D7111yescharacteristic
121#467—1421no—
122Q64—1422no—
123SD64—1422no—
124M64—712no—
125#367—711no—
126#358—1421no—
127C2×#41—41no—
128#135—22no—
129#135—22no—
130#135—41no—
131#135—41no—
132C2×Q16—21no—
133C2×Q16—1421no—
134C2×SD16—41no—
135C2×D8—21no—
136C22×C8C22⋊D7111yescharacteristic
137C22⋊Q8—2841no—
138Q32—21no—
139Q32—1422no—
140Q32C22⋊D7112yes—
141SD32—22no—
142SD32—22no—
143D16—21no—
144D16C22⋊D7112yes—
145M32—22no—
146C2×C16C22⋊D7111yescharacteristic
147C2×C16C22×D7111yescharacteristic
148#107—1421no—
149#103—2841no—
150C32—1422no—
151#41—42no—
152#41—42no—
153#41—81no—
154#41—81no—
155C2×Q8—41no—
156C2×Q8—2841no—
157C2×D4—41no—
158C2×D4—81no—
159C22×C4—81no—
160C22×C4#1176585011yescharacteristic
161Q16—22no—
162Q16—41no—
163Q16—2841no—
164Q16—2841no—
165SD16—42no—
166SD16—42no—
167D8—22no—
168D8—41no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8#1176585011yescharacteristic
172C2×C8C2×C22⋊D7111yescharacteristic
173C4⋊C4—2841no—
174C4⋊C4—5681no—
175C22⋊C4—5681no—
176C16—21no—
177C16C2×C22⋊D7112yes—
178C23—81no—
179C23#1455606911yescharacteristic
180Q8—42no—
181Q8—81no—
182Q8—5681no—
183Q8—5681no—
184D4—42no—
185D4—81no—
186D4—82no—
187D4—82no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—82no—
191C2×C4—81no—
192C2×C4—82no—
193C2×C4—5681no—
194C2×C4—5681no—
195C2×C4#1455606911yescharacteristic
196C2×C4C2×#1176585011yescharacteristic
197C8—21no—
198C8#1455609711yescharacteristic
199C8#1455615411yescharacteristic
200C22—21no—
201C22—41no—
202C22—82no—
203C22—162no—
204C22C2×#1455606911yescharacteristic
205C4—21no—
206C4—82no—
207C4—11361no—
208C4—11361no—
209C4#1798613311yescharacteristic
210C4#1798643511yescharacteristic
211C2—21no—
212C2—41no—
213C2—162no—
214C2#2259025211yescharacteristic
subgroupG/H
centerC2#22590252
commutator, cosocleC2×C16×C71C23
FrattiniC2×C16C22×D71
Fitting, prime radicalC71×#3414C2
socleC2×C71#2184
hypercenter#3414D71
hypocenterC71#13668
LCSG ▷ C2×C16×C71 ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ C2×M32 ◁ #3414

Complements

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

NH ≅ G/Ntype
1. #22590657C2semidirect
2. #22590448C2semidirect
3. #22586527C2semidirect
4. #22586480C2semidirect
6. C2×Dic568C22semidirect
10. #17985833C22semidirect
11. #17985827C22semidirect
113. C71#13668semidirect

Maximal subgroups

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

subgroupindex
#225906572
#225904482
#225865272
#225864802
C71×#34142
#1366871