Finite Group Databasebeta

← #31918236

Subgroups — #31918236

Order 18176.

220 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C2×#17985838C211yesmaximal, characteristic, complemented
2C2×#17985837C211yesmaximal, characteristic, complemented
3#22590588C211yesmaximal, characteristic
4#22590198C211yesmaximal, characteristic, complemented
5Dic71⋊C32C211yesmaximal, characteristic, complemented
6#22586517C211yesmaximal, characteristic, complemented
7C71×#1415C211yesmaximal, characteristic, complemented
8C2×Dic568C2211yescharacteristic
9C2×D1136C2211yescharacteristic
10C2×#14556069—21no—
11#17986374—21no—
12#17986093—21no—
13#17985838—22no—
14#17985837—22no—
15C2×C71⋊C32C2211yescharacteristic
16#17985802—21no—
17C16×Dic71C2211yescharacteristic
18C2×D16×C71C2211yescharacteristic
19C2×C32×C71C2211yescharacteristic
20#366×C71C2211yescharacteristic
21C2×Dic284—21no—
22C2×D568—21no—
23C2×#11765850—41no—
24#14556124—41no—
25#14556093—41no—
26Dic568—21no—
27Dic568D412yescomplemented
28D1136—21no—
29D1136—21no—
30#14556069—42no—
31C2×C71⋊C16D411yescharacteristic
32#14556051—41no—
33C8×Dic71D411yescharacteristic
34C71⋊C32—21no—
35C2×D8×C71—21no—
36D16×C71—21no—
37D16×C71—21no—
38C2×C16×C71C2311yescharacteristic
39#107×C71—21no—
40C32×C71—21no—
41C2×Dic142—41no—
42C2×D284—41no—
43C2×C22⋊D71—81no—
44C2×C22⋊D71—81no—
45Dic284—22no—
46Dic284—41no—
47D568—41no—
48D568—41no—
49#11765850—82no—
50C2×C71⋊C8D811yescharacteristic
51Dic71⋊C4—81no—
52C4×Dic71—81no—
53C4×Dic71D811yescharacteristic
54C71⋊C16—21no—
55C2×D4×C71—41no—
56D8×C71—41no—
57D8×C71—41no—
58C2×C8×C71C2×D411yescharacteristic
59C4⋊C4×C71—41no—
60C16×C71C2×D411yescharacteristic
61C16×C71#4111yescharacteristic
62C23×C71—81no—
63C22×D71—81no—
64D4×C71—81no—
65D4×C71—81no—
66C2×C4×C71—81no—
67C2×C4×C71C2×D811yescharacteristic
68C22⋊D71—162no—
69C22⋊D71—162no—
70C2×Dic71—81no—
71C2×Dic71D1611yescharacteristic
72C2×Dic71D1611yescharacteristic
73D284—81no—
74D284—81no—
75Dic142—42no—
76Dic142—81no—
77C8×C71#12111yescharacteristic
78C8×C71AGL(1,ℤ/8)11yescharacteristic
79C71⋊C8—21no—
80C22×C71—161no—
81C22×C71—161no—
82C22×C71—161no—
83C22×C71C2×D1611yescharacteristic
84C2×D71—161no—
85C2×D71—161no—
86C2×D71—161no—
87C4×C71—161no—
88C4×C71#45911yescharacteristic
89C4×C71#50911yescharacteristic
90Dic71—21no—
91Dic71—82no—
92Dic71SD6412yescomplemented
93#13686—711nomaximal, Sylow
94C2×C71—321no—
95C2×C71#220611yescharacteristic
96C2×C71C2×SD6411yescharacteristic
97C2×C71#226311yescharacteristic
98D71—321no—
99C2×SD64—711no—
100C2×SD64—711no—
101#2246—711no—
102C4×Q32—711no—
103C4⋊C32—711no—
104#1415—711no—
105#1415D7111yescharacteristic, complemented
106C71#1368611yescharacteristic, complemented, Sylow
107C2×Q32—711no—
108C2×D16—711no—
109C2×D16—1421no—
110C2×D16C2×D7111yescharacteristic
111#493—1421no—
112C4×Q16—1421no—
113SD64—1422no—
114SD64—1422no—
115C2×C32—711no—
116C2×C32C2×D7111yescharacteristic
117#366C2×D7111yescharacteristic
118#358—1421no—
119C4×C16—711no—
120C2×Q16—1421no—
121C2×D8—21no—
122C2×D8—2841no—
123C2×D8—1421no—
124#127—2841no—
125C4×Q8—2841no—
126Q32—712no—
127Q32—1421no—
128D16—21no—
129D16—21no—
130D16—2842no—
131D16—1421no—
132D16—1421no—
133C2×C16—711no—
134C2×C16C22×D7111yescharacteristic
135#107—21no—
136#103—2841no—
137C4×C8—711no—
138C32—21no—
139C32—1421no—
140C2×Q8—2841no—
141C2×D4—41no—
142C2×D4—2841no—
143C2×D4—5681no—
144C2×D4—5681no—
145Q16—2841no—
146Q16—1422no—
147D8—41no—
148D8—41no—
149D8—2841no—
150D8—2841no—
151D8—5682no—
152C2×C8—711no—
153C2×C8D4×D7111yescharacteristic
154C4⋊C4—41no—
155C4⋊C4—5681no—
156C42—5681no—
157C42—711no—
158C16—1421no—
159C16D4×D7111yescharacteristic
160C16#1176586511yescharacteristic
161C23—81no—
162C23—5681no—
163Q8—2842no—
164Q8—5681no—
165D4—81no—
166D4—81no—
167D4—5681no—
168D4—5681no—
169D4—11362no—
170D4—11362no—
171C2×C4—81no—
172C2×C4—5681no—
173C2×C4—711no—
174C2×C4—711no—
175C2×C4D8×D7111yescharacteristic
176C8—1421no—
177C8#1455610111yescharacteristic
178C8#1455615511yescharacteristic
179C22—161no—
180C22—161no—
181C22—161no—
182C22—11361no—
183C22—11361no—
184C22—11361no—
185C22D16×D7111yescharacteristic
186C4—161no—
187C4—5682no—
188C4—712no—
189C4—1421no—
190C4#1798619311yescharacteristic
191C4#1798643611yescharacteristic
192C2—321no—
193C2—22721no—
194C2#2259037411yescharacteristic
195C2SD64×D7111yescharacteristic
196C2#2259065111yescharacteristic
subgroupG/H
centerC22D16×D71
commutator, cosocleC2×C16×C71C23
FrattiniC2×C16C22×D71
FittingC71×#1415C2
socleC22×C71C2×D16
prime radical#22590588C2
hypercenter#1415D71
hypocenterC71#13686
LCSG ▷ C2×C16×C71 ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1415

Complements

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

NH ≅ G/Ntype
1. C2×#17985838C2semidirect
2. C2×#17985837C2semidirect
4. #22590198C2semidirect
5. Dic71⋊C32C2semidirect
6. #22586517C2semidirect
7. C71×#1415C2semidirect
27. Dic568D4semidirect
92. Dic71SD64semidirect
105. #1415D71semidirect
106. C71#13686semidirect

Maximal subgroups

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

subgroupindex
C2×#179858382
C2×#179858372
#225905882
#225901982
Dic71⋊C322
#225865172
C71×#14152
#1368671