Finite Group Databasebeta

← #34112809

Subgroups — #34112809

Order 20096.

336 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 125 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
1Q8×Dic314C211yesmaximal, characteristic
2#23757998C211yesmaximal, characteristic
3#23757623C211yesmaximal, characteristic
4#23757622C212yesmaximal
5#23757569C211yesmaximal, characteristic
6#23757566C212yesmaximal
7#23757481C212yesmaximal
8#23757397C211yesmaximal, characteristic
9Q16×Dic157C212yesmaximal
10C4×#14578944C211yesmaximal, characteristic
11#462×C157C211yesmaximal, characteristic
12C2×#14578944C2211yescharacteristic
13#19097771—21no—
14#19097771C2211yescharacteristic
15#19097770—22no—
16#19097770C2212yes—
17#19097769C2212yes—
18#19097762—21no—
19#19097761—22no—
20Q8×Dic157C2212yes—
21Q8×Dic157C2212yes—
22C4×Dic314—21no—
23C4×Dic314C2211yescharacteristic
24C4×Dic314C2211yescharacteristic
25#19097736—21no—
26#19097705C2212yes—
27#19097704C2211yescharacteristic
28#19097700C2212yes—
29#19097696C2212yes—
30#19097695C2211yescharacteristic
31#19097695C2212yes—
32#19097695C2212yes—
33#19097694C2211yescharacteristic
34C4×#11929434C2211yescharacteristic
35C8×Dic157C2212yes—
36C2×Q16×C157C2212yes—
37C4⋊Q8×C157C2211yescharacteristic
38C4×Q8×C157C2211yescharacteristic
39C4⋊C8×C157C2211yescharacteristic
40#103×C157C2212yes—
41C2×Dic314—22no—
42C2×Dic314—21no—
43C2×Dic314D411yescharacteristic
44C2×Dic314C2311yescharacteristic
45#14578944—22no—
46C2×#11929434C2311yescharacteristic
47C2×#11929434C2311yescharacteristic
48C4⋊Dic157—22no—
49C4⋊Dic157—22no—
50C4⋊Dic157—21no—
51C4⋊Dic157D411yescharacteristic
52C4⋊Dic157C2312yes—
53C4⋊Dic157C2311yescharacteristic
54#14578933—22no—
55#14578933—22no—
56#14578933—22no—
57#14578933—21no—
58#14578933D412yes—
59C4×Dic157—22no—
60C4×Dic157—22no—
61C4×Dic157—21no—
62C4×Dic157C2312yes—
63C2×Q8×C157C2312yes—
64C2×Q8×C157C2311yescharacteristic
65Q16×C157—24no—
66C2×C8×C157C2312yes—
67C4⋊C4×C157—21no—
68C4⋊C4×C157—21no—
69C4⋊C4×C157—21no—
70C4⋊C4×C157C2311yescharacteristic
71C42×C157—21no—
72C42×C157C2311yescharacteristic
73Dic314—21no—
74Dic314—21no—
75Dic314—82no—
76Dic314—81no—
77Dic314Q1614yescomplemented
78C2×Dic157—22no—
79C2×Dic157—21no—
80C2×Dic157—21no—
81C2×Dic157C2×D412yes—
82C2×Dic157C2×D412yes—
83#11929434—21no—
84#11929434—21no—
85Q8×C157—21no—
86Q8×C157—22no—
87Q8×C157—22no—
88Q8×C157#4112yes—
89C2×C4×C157—21no—
90C2×C4×C157—21no—
91C2×C4×C157—22no—
92C2×C4×C157C2×D411yescharacteristic
93C2×C4×C157C2×D411yescharacteristic
94C2×C4×C157C2411yescharacteristic
95C8×C157—22no—
96C22×C157C22×D411yescharacteristic
97C4×C157—22no—
98C4×C157—21no—
99C4×C157—42no—
100C4×C157—41no—
101C4×C157C2×Q1612yes—
102C4×C157C2×#4111yescharacteristic
103C4×C1572−1+411yescharacteristic
104Dic157—22no—
105Dic157—41no—
106Dic157—41no—
107Dic157—42no—
108Dic157C2×Q1614yes—
109C2×C157#54811yescharacteristic
110C2×C157C22×Q1611yescharacteristic
111C2×C157#57711yescharacteristic
112C157#336311yescharacteristic, complemented, Sylow
113#3363—1571nomaximal, Sylow
114Q82—1571no—
115#557—1571no—
116#487—1571no—
117#487—1572no—
118#477—1571no—
119#477—1572no—
120#462—1572no—
121#462D15711yescharacteristic
122C8×Q8—1571no—
123C4×Q16—1572no—
124C4×Q16—1571no—
125C2×Q16—1571no—
126C2×Q16C2×D15712yes—
127C4⋊Q8—3141no—
128C4⋊Q8—3142no—
129C4⋊Q8—1572no—
130C4⋊Q8—1572no—
131C4⋊Q8—1571no—
132C4⋊Q8C2×D15711yescharacteristic
133#125—3142no—
134#125—3141no—
135C4×Q8—3141no—
136C4×Q8—3141no—
137C4×Q8—1572no—
138C4×Q8—1572no—
139C4×Q8—1571no—
140C4×Q8—1571no—
141C4×Q8C2×D15711yescharacteristic
142#107—1572no—
143#107—1571no—
144C4⋊C8—1572no—
145C4⋊C8C2×D15711yescharacteristic
146#103—1571no—
147#103—1572no—
148#103—1572no—
149#103—1572no—
150#103—1571no—
151#103C2×D15712yes—
152C4×C8—1572no—
153C4×C8—1571no—
154C2×Q8—3142no—
155C2×Q8—3141no—
156C2×Q8—1571no—
157C2×Q8—1571no—
158C2×Q8C22×D15712yes—
159C2×Q8C22×D15711yescharacteristic
160Q16—24no—
161Q16—3142no—
162C2×C8—1571no—
163C2×C8—1571no—
164C2×C8C22×D15712yes—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—3142no—
169C4⋊C4—3142no—
170C4⋊C4—3142no—
171C4⋊C4—3142no—
172C4⋊C4—3142no—
173C4⋊C4—3141no—
174C4⋊C4—3141no—
175C4⋊C4—1572no—
176C4⋊C4—1571no—
177C4⋊C4—1572no—
178C4⋊C4—1571no—
179C4⋊C4C22×D15711yescharacteristic
180C42—21no—
181C42—3142no—
182C42—3142no—
183C42—3141no—
184C42—1572no—
185C42C22×D15711yescharacteristic
186Q8—21no—
187Q8—22no—
188Q8—22no—
189Q8—3141no—
190Q8—3141no—
191Q8—1574no—
192Q8—12561no—
193Q8—12562no—
194Q8#1457895312yes—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—3141no—
199C2×C4—3141no—
200C2×C4—3142no—
201C2×C4—1572no—
202C2×C4—1572no—
203C2×C4D4×D15711yescharacteristic
204C2×C4D4×D15711yescharacteristic
205C2×C4C23×D15711yescharacteristic
206C8—22no—
207C8—3141no—
208C8—3141no—
209C22C2×D4×D15711yescharacteristic
210C4—21no—
211C4—22no—
212C4—41no—
213C4—42no—
214C4—3142no—
215C4—6282no—
216C4—6281no—
217C4—6281no—
218C4—1574no—
219C4Q16×D15712yes—
220C4C2×#1457895311yescharacteristic
221C4#1909781811yescharacteristic
222C2#2375792911yescharacteristic
223C2C2×Q16×D15711yescharacteristic
224C2#2375808211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#462×C157C2
socle, prime radicalC22×C157C22×D4
hypercenter#462D157
hypocenterC157#3363
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
77. Dic314Q16semidirect
112. C157#3363semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic3142
#237579982
#237576232
#237576222
#237575692
#237575662
#237574812
#237573972
Q16×Dic1572
C4×#145789442
#462×C1572
#3363157