Finite Group Databasebeta

← #31909374

Subgroups — #31909374

Order 18176.

240 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C4×D1136C211yesmaximal, characteristic
2#22587224C211yesmaximal, characteristic, complemented
3#22587134C211yesmaximal, characteristic, complemented
4#22586300C211yesmaximal, characteristic, complemented
5#22586300C211yesmaximal, characteristic, complemented
6#22586259C211yesmaximal, characteristic, complemented
7C71×#1372C211yesmaximal, characteristic, complemented
8C2×D1136C411yescharacteristic
9C4×D568C2211yescharacteristic
10C4×D568C2211yescharacteristic
11#17986076—21no—
12#17986064—21no—
13#17985825C411yescharacteristic
14#17985801C411yescharacteristic
15#17985801C411yescharacteristic
16C4×C71⋊C16C2211yescharacteristic
17#17985726—21no—
18#17985726—21no—
19C8×C71⋊C8C2211yescharacteristic
20C4×C16×C71C2211yescharacteristic
21#335×C71C2211yescharacteristic
22#335×C71C2211yescharacteristic
23C2×D568C2×C411yescharacteristic
24C2×D568C2×C411yescharacteristic
25C2×#11765848—41no—
26C2×C8×D71—41no—
27C4×D284—21no—
28C4×D284—21no—
29D1136—22no—
30C2×C71⋊C16—21no—
31#14556062C2×C411yescharacteristic
32#14556062C2×C411yescharacteristic
33#14556050—21no—
34#14556050—21no—
35(C2×D71)⋊C8—41no—
36(C2×D71)⋊C8—41no—
37C4×C71⋊C8D411yescharacteristic
38C4×C71⋊C8D411yescharacteristic
39C2×C16×C71C2×C411yescharacteristic
40C2×C16×C71C2×C411yescharacteristic
41C4⋊C8×C71—21no—
42C4⋊C8×C71—21no—
43C4×C8×C71C2311yescharacteristic
44C2×D284—21no—
45C2×D284—21no—
46C2×C4×D71—41no—
47C2×C4×D71—41no—
48D568—21no—
49D568—21no—
50D568—21no—
51D568M1612yescomplemented
52#11765848—82no—
53C2×C71⋊C8—21no—
54C2×C71⋊C8D811yescharacteristic
55C2×C71⋊C8D811yescharacteristic
56C8×D71—82no—
57C4⋊Dic71—21no—
58C4⋊Dic71—21no—
59(C2×D71)⋊C4—41no—
60(C2×D71)⋊C4—41no—
61C71⋊C16—41no—
62C2×C8×C71—41no—
63C2×C8×C71—41no—
64C2×C8×C71C22×C411yescharacteristic
65C2×C8×C71#4111yescharacteristic
66C42×C71C2×D411yescharacteristic
67C16×C71—21no—
68C16×C71—21no—
69C22×D71—41no—
70C22×D71—41no—
71C2×C4×C71C4×D411yescharacteristic
72C2×C4×C71C2×D811yescharacteristic
73C2×C4×C71#13511yescharacteristic
74C2×Dic71—41no—
75C2×Dic71—41no—
76D284—22no—
77D284—41no—
78D284—41no—
79D284—41no—
80C4×D71—81no—
81C4×D71—81no—
82C4×D71—81no—
83C4×D71—81no—
84C8×C71—21no—
85C8×C71—81no—
86C8×C71—81no—
87C8×C71C2×M1611yescharacteristic
88C8×C71#13111yescharacteristic
89C71⋊C8—22no—
90C71⋊C8—21no—
91C71⋊C8—21no—
92C22×C71C4×D811yescharacteristic
93C2×D71—42no—
94C2×D71—81no—
95C2×D71—81no—
96C2×D71—81no—
97C2×D71—81no—
98C2×D71—81no—
99C4×C71—21no—
100C4×C71#43511yescharacteristic
101C4×C71#44411yescharacteristic
102C4×C71#50911yescharacteristic
103C4×C71#50911yescharacteristic
104Dic71—81no—
105Dic71—81no—
106#12073—711nomaximal, Sylow
107C2×C71#158111yescharacteristic
108C2×C71#217711yescharacteristic
109C2×C71#217811yescharacteristic
110D71—82no—
111D71—161no—
112C4×D16—711no—
113#1589—711no—
114C8×D8—711no—
115#1372D7111yescharacteristic, complemented
116#1331—711no—
117#1331—711no—
118#1312—711no—
119C71#1207311yescharacteristic, complemented, Sylow
120C2×D16—711no—
121C4×D8—711no—
122C4×D8—711no—
123#436—1421no—
124C8×D4—1421no—
125#366—711no—
126#357—711no—
127#357—711no—
128C4×C16—711no—
129C4×C16C2×D7111yescharacteristic
130#335C2×D7111yescharacteristic
131#335C2×D7111yescharacteristic
132D4⋊C8—1421no—
133D4⋊C8—1421no—
134C82—711no—
135C2×D8—711no—
136C2×D8—711no—
137C2×M16—2841no—
138C22×C8—2841no—
139C4×D4—1421no—
140C4×D4—1421no—
141D16—1422no—
142C2×C16—1421no—
143C2×C16C4×D7111yescharacteristic
144C2×C16C4×D7111yescharacteristic
145#107—711no—
146#107—711no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149#102—1421no—
150#102—1421no—
151C22⋊C8—2841no—
152C22⋊C8—2841no—
153C4×C8—711no—
154C4×C8—711no—
155C4×C8C22×D7111yescharacteristic
156C2×D4—1421no—
157C2×D4—1421no—
158C22×C4—2841no—
159C22×C4—2841no—
160D8—712no—
161D8—1421no—
162D8—1421no—
163D8—1421no—
164M16—5682no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—5682no—
168C2×C8—711no—
169C2×C8—711no—
170C2×C8—1421no—
171C2×C8C2×C4×D7111yescharacteristic
172C2×C8#1176586611yescharacteristic
173C4⋊C4—1421no—
174C4⋊C4—1421no—
175C22⋊C4—2841no—
176C22⋊C4—2841no—
177C42D4×D7111yescharacteristic
178C16—21no—
179C16—21no—
180C16—2841no—
181C23—2841no—
182C23—2841no—
183D4—2841no—
184D4—2841no—
185D4—2841no—
186D4—1422no—
187C2×C4—2841no—
188C2×C4—2841no—
189C2×C4—5681no—
190C2×C4—5681no—
191C2×C4—5681no—
192C2×C4—5681no—
193C2×C4#1455609011yescharacteristic
194C2×C4D8×D7111yescharacteristic
195C2×C4#1455615111yescharacteristic
196C8—21no—
197C8—81no—
198C8—81no—
199C8—1422no—
200C8—1421no—
201C8—1421no—
202C8M16×D7111yescharacteristic
203C8#1455613411yescharacteristic
204C22—2842no—
205C22—5681no—
206C22—5681no—
207C22—5681no—
208C22—5681no—
209C22—5681no—
210C22#1798608111yescharacteristic
211C4—21no—
212C4—5681no—
213C4—5681no—
214C4#1798606911yescharacteristic
215C4#1798612011yescharacteristic
216C4#1798643611yescharacteristic
217C4#1798643611yescharacteristic
218C2—5682no—
219C2—11361no—
220C2#2258716811yescharacteristic
221C2#2259021511yescharacteristic
222C2#2259022011yescharacteristic
subgroupG/H
centerC2×C4D8×D71
commutatorC2×C8×C71C22×C4
FrattiniC4×C8C22×D71
FittingC71×#1372C2
socleC22×C71C4×D8
prime radicalC2×D1136C4
cosocleC4×C8×C71C23
hypercenter#1372D71
hypocenterC71#12073
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ #1372

Complements

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

NH ≅ G/Ntype
2. #22587224C2semidirect
3. #22587134C2semidirect
4. #22586300C2semidirect
5. #22586300C2semidirect
6. #22586259C2semidirect
7. C71×#1372C2semidirect
51. D568M16semidirect
115. #1372D71semidirect
119. C71#12073semidirect

Maximal subgroups

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

subgroupindex
C4×D11362
#225872242
#225871342
#225863002
#225862592
C71×#13722
#1207371