Finite Group Databasebeta

← #31779402

Subgroups — #31779402

Order 17536.

212 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 61 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#22577959C211yesmaximal, characteristic
2#22577458C211yesmaximal, characteristic
3#97×D137C211yesmaximal, characteristic
4#22577145C211yesmaximal, characteristic, complemented
5#22577144C211yesmaximal, characteristic, complemented
6#22577144C211yesmaximal, characteristic, complemented
7#330×C137C211yesmaximal, characteristic, complemented
8C2×C8×D137—21no—
9#17984087C411yescharacteristic
10#17984087C411yescharacteristic
11#17984080C2211yescharacteristic
12#17984066—21no—
13#17984066—21no—
14#17984053—21no—
15#17984053—21no—
16#17984046C411yescharacteristic
17#17984046C411yescharacteristic
18C42×D137C2211yescharacteristic
19#17984017—21no—
20#17984000—21no—
21#17983999C2211yescharacteristic
22#17983999C2211yescharacteristic
23#17983998—21no—
24#125×C137C2211yescharacteristic
25#97×C137C2211yescharacteristic
26#97×C137C2211yescharacteristic
27C2×Dic274—21no—
28C2×Dic274—21no—
29C2×C4×D137C2×C411yescharacteristic
30C2×C4×D137D411yescharacteristic
31C2×C4×D137D411yescharacteristic
32C2×#11764040—21no—
33C2×#11764040—21no—
34C8×D137—24no—
35C4⋊Dic137C2×C411yescharacteristic
36C4⋊Dic137C2×C411yescharacteristic
37#14555489—21no—
38#14555489—21no—
39#14555489—21no—
40#14555489—21no—
41#14555487—21no—
42#14555487—21no—
43C4×Dic137—21no—
44C4×Dic137—21no—
45C4×Dic137C2×C411yescharacteristic
46C4×Dic137D411yescharacteristic
47C4×Dic137D411yescharacteristic
48C2×C8×C137—21no—
49C2×C8×C137—21no—
50C4⋊C4×C137—21no—
51C4⋊C4×C137—21no—
52C4⋊C4×C137C2×C411yescharacteristic
53C4⋊C4×C137C2×C411yescharacteristic
54C42×C137C2311yescharacteristic
55C22×D137C22⋊C411yescharacteristic
56Dic274—42no—
57Dic274—42no—
58C2×Dic137—21no—
59C2×Dic137—21no—
60C2×Dic137C22⋊C411yescharacteristic
61C2×Dic137C22⋊C411yescharacteristic
62C2×Dic137C22⋊C411yescharacteristic
63C4×D137—22no—
64C4×D137—22no—
65C4×D137—41no—
66#11764040—22no—
67#11764040—41no—
68C2×C4×C137—21no—
69C2×C4×C137—21no—
70C2×C4×C137C22×C411yescharacteristic
71C2×C4×C137C2×D411yescharacteristic
72C2×C4×C137C2×D411yescharacteristic
73C8×C137—22no—
74C8×C137—41no—
75C22×C137C2×C22⋊C411yescharacteristic
76C2×D137—21no—
77C2×D137—21no—
78C2×D137#10112yes—
79C4×C137—21no—
80C4×C137—21no—
81C4×C137—21no—
82C4×C137—41no—
83C4×C137—41no—
84Dic137—21no—
85Dic137—21no—
86Dic137—41no—
87Dic137—41no—
88Dic137#10412yes—
89C2×C137C2×#10111yescharacteristic
90C2×C137C2×#10411yescharacteristic
91C2×C137#42111yescharacteristic
92D137—22no—
93C137#153111yescharacteristic, complemented, Sylow
94#1531—1371nomaximal, Sylow
95#533—1371no—
96#432—1371no—
97C2×#97—1371no—
98#330—1371no—
99#330—1371no—
100#330—1371no—
101#330D13711yescharacteristic, complemented
102C22×C8—2741no—
103C4⋊Q8—1371no—
104C4⋊Q8—1371no—
105#125—1371no—
106#125C2×D13711yescharacteristic
107C22⋊Q8—2741no—
108C22⋊Q8—2741no—
109C4×Q8—2741no—
110C4×Q8—2741no—
111#117—1371no—
112#117—1371no—
113C2×C42—1371no—
114C4⋊C8—2741no—
115C22⋊C8—2741no—
116#97—2741no—
117#97—1371no—
118#97—1371no—
119#97C2×D13711yescharacteristic
120#97C2×D13711yescharacteristic
121C2×Q8—2741no—
122C2×Q8—2741no—
123C22×C4—1371no—
124C22×C4—1371no—
125C22×C4—1371no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—2744no—
129C2×C8—2741no—
130C2×C8—2741no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—2741no—
134C4⋊C4—2741no—
135C4⋊C4—2741no—
136C4⋊C4—2741no—
137C4⋊C4—1371no—
138C4⋊C4—1371no—
139C4⋊C4C4×D13711yescharacteristic
140C4⋊C4C4×D13711yescharacteristic
141C22⋊C4—2741no—
142C22⋊C4—2741no—
143C42—2741no—
144C42—2741no—
145C42—1371no—
146C42—1371no—
147C42—1371no—
148C42C22×D13711yescharacteristic
149C23—1371no—
150Q8—5482no—
151Q8—5482no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—2742no—
155C2×C4—2742no—
156C2×C4—2741no—
157C2×C4—2741no—
158C2×C4—5481no—
159C2×C4—1371no—
160C2×C4—1371no—
161C2×C4—1371no—
162C2×C4C2×C4×D13711yescharacteristic
163C2×C4D4×D13711yescharacteristic
164C2×C4D4×D13711yescharacteristic
165C8—22no—
166C8—41no—
167C8—2742no—
168C8—5481no—
169C22—2741no—
170C22—2741no—
171C22—1372no—
172C22C22⋊C4×D13711yescharacteristic
173C4—21no—
174C4—21no—
175C4—21no—
176C4—41no—
177C4—41no—
178C4—2741no—
179C4—2741no—
180C4—5481no—
181C4—5481no—
182C4—1372no—
183C2—2742no—
184C2#101×D13711yescharacteristic
185C2#104×D13711yescharacteristic
186C2#2257741811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D137
commutatorC2×C4×C137C22×C4
FrattiniC42C22×D137
Fitting#330×C137C2
socleC22×C137C2×C22⋊C4
prime radicalC22×D137C22⋊C4
cosocleC42×C137C23
hypercenter#330D137
hypocenterC137#1531
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
4. #22577145C2semidirect
5. #22577144C2semidirect
6. #22577144C2semidirect
7. #330×C137C2semidirect
93. C137#1531semidirect
101. #330D137semidirect

Maximal subgroups

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

subgroupindex
#225779592
#225774582
#97×D1372
#225771452
#225771442
#330×C1372
#1531137