Finite Group Databasebeta

Subgroups — #72322808

Order 1024.

252 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#513074	C₂	1	1	yes	maximal, characteristic
2	#153425	C₂	1	1	yes	maximal, characteristic, complemented
3	#151057	C₂	1	1	yes	maximal, characteristic, complemented
4	#94614	C₂	1	4	yes	maximal, complemented
5	#33740	—	2	1	no	—
6	C₂×M₁₂₈	—	2	1	no	—
7	#13732	C₄	1	1	yes	characteristic
8	#13725	C₄	1	1	yes	characteristic
9	#13723	—	2	1	no	—
10	#13697	C₄	1	2	yes	—
11	#13694	—	2	1	no	—
12	#13654	C₂²	1	2	yes	—
13	C₄×D₃₂	—	2	1	no	—
14	#13549	C₂²	1	1	yes	characteristic
15	#12027	—	2	1	no	—
16	#7511	—	2	1	no	—
17	#7511	C₂²	1	2	yes	—
18	#7464	C₂²	1	2	yes	—
19	#7462	—	2	1	no	—
20	#3176	—	4	1	no	—
21	C₂×D₃₂	—	2	1	no	—
22	C₂×D₃₂	C₂×C₄	1	2	yes	—
23	C₂×M₆₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃₂	D₄	1	1	yes	characteristic
25	#2253	—	2	1	no	—
26	#2246	—	2	1	no	—
27	#2244	—	4	1	no	—
28	#2218	—	4	1	no	—
29	#2215	—	2	2	no	—
30	#2177	—	4	1	no	—
31	C₄×D₁₆	—	2	2	no	—
32	#2166	—	4	1	no	—
33	#2108	D₄	1	1	yes	characteristic
34	M₁₂₈	—	2	2	no	—
35	M₁₂₈	—	2	2	no	—
36	C₂×C₆₄	—	2	2	no	—
37	#1424	C₂×C₄	1	2	yes	—
38	#1423	—	2	1	no	—
39	#1415	—	2	2	no	—
40	#1415	—	2	2	no	—
41	#1397	D₄	1	1	yes	characteristic
42	#1397	C₂³	1	1	yes	characteristic
43	C₄×C₃₂	C₂×C₄	1	1	yes	characteristic
44	C₄×C₃₂	D₄	1	1	yes	characteristic
45	#1373	—	2	2	no	—
46	#1369	—	4	1	no	—
47	#540	—	8	1	no	—
48	C₂×D₁₆	—	2	2	no	—
49	C₂×D₁₆	—	4	1	no	—
50	C₂×M₃₂	SD₁₆	1	1	yes	characteristic
51	C₂²×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
52	#499	—	4	1	no	—
53	#493	—	4	1	no	—
54	#490	—	8	1	no	—
55	#469	—	8	1	no	—
56	#466	—	4	2	no	—
57	#442	—	8	1	no	—
58	C₄×D₈	—	4	2	no	—
59	#405	D₈	1	1	yes	characteristic
60	D₃₂	—	2	4	no	—
61	D₃₂	—	4	1	no	—
62	D₃₂	—	4	1	no	—
63	M₆₄	—	2	2	no	—
64	C₂×C₃₂	—	2	1	no	—
65	C₂×C₃₂	—	2	1	no	—
66	C₂×C₃₂	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₃₂	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₃₂	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃₂	C₂×D₄	1	1	yes	characteristic
70	#367	—	4	1	no	—
71	#366	—	2	2	no	—
72	#364	—	4	2	no	—
73	#357	—	4	2	no	—
74	#357	—	4	2	no	—
75	#349	—	8	1	no	—
76	#347	—	8	1	no	—
77	#346	D₈	1	1	yes	characteristic
78	#346	SD₁₆	1	1	yes	characteristic
79	C₄×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
80	C₄×C₁₆	C₂×D₄	1	1	yes	characteristic
81	C₆₄	—	2	4	no	—
82	C₂×D₈	—	4	2	no	—
83	C₂×D₈	—	8	1	no	—
84	C₂×M₁₆	D₁₆	1	1	yes	characteristic
85	C₂²×C₈	#102	1	1	yes	characteristic
86	#127	—	8	1	no	—
87	#125	—	8	1	no	—
88	#123	—	16	2	no	—
89	#121	—	8	2	no	—
90	#121	—	16	2	no	—
91	C₄×D₄	—	8	2	no	—
92	#117	SD₃₂	1	1	yes	characteristic
93	D₁₆	—	4	2	no	—
94	D₁₆	—	4	2	no	—
95	D₁₆	—	8	2	no	—
96	M₃₂	—	2	2	no	—
97	M₃₂	—	8	2	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	#102	1	1	yes	characteristic
101	C₂×C₁₆	#102	1	1	yes	characteristic
102	C₂×C₁₆	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₁₆	C₂×SD₁₆	1	1	yes	characteristic
104	#107	—	4	2	no	—
105	#106	—	8	1	no	—
106	#102	—	8	2	no	—
107	#102	—	8	2	no	—
108	#97	D₁₆	1	1	yes	characteristic
109	#97	SD₃₂	1	1	yes	characteristic
110	C₄×C₈	#102	1	1	yes	characteristic
111	C₄×C₈	C₂×D₈	1	1	yes	characteristic
112	C₃₂	—	2	1	no	—
113	C₃₂	—	2	1	no	—
114	C₃₂	#104	1	1	yes	characteristic
115	C₃₂	#104	1	1	yes	characteristic
116	C₃₂	#131	1	2	yes	—
117	C₂×D₄	—	8	2	no	—
118	C₂×D₄	—	16	2	no	—
119	C₂×D₄	—	16	2	no	—
120	C₂²×C₄	—	16	2	no	—
121	C₂²×C₄	#357	1	1	yes	characteristic
122	D₈	—	8	2	no	—
123	D₈	—	8	2	no	—
124	D₈	—	16	2	no	—
125	M₁₆	—	4	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	#357	1	1	yes	characteristic
129	C₂×C₈	#357	1	1	yes	characteristic
130	C₂×C₈	C₂×#102	1	1	yes	characteristic
131	C₂×C₈	C₂×D₁₆	1	1	yes	characteristic
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	8	2	no	—
134	C₄⋊C₄	—	16	2	no	—
135	C₂²⋊C₄	—	2	1	no	—
136	C₂²⋊C₄	—	16	2	no	—
137	C₂²⋊C₄	—	16	2	no	—
138	C₄²	#357	1	1	yes	characteristic
139	C₄²	C₂×SD₃₂	1	1	yes	characteristic
140	C₁₆	—	2	1	no	—
141	C₁₆	—	2	1	no	—
142	C₁₆	—	16	2	no	—
143	C₁₆	#359	1	2	yes	—
144	C₁₆	#408	1	1	yes	characteristic
145	C₁₆	C₂×#104	1	1	yes	characteristic
146	C₂³	—	16	2	no	—
147	C₂³	#1419	1	1	yes	characteristic
148	D₄	—	16	2	no	—
149	D₄	—	16	2	no	—
150	D₄	—	64	2	no	—
151	D₄	—	64	2	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	16	2	no	—
157	C₂×C₄	—	64	2	no	—
158	C₂×C₄	#1419	1	1	yes	characteristic
159	C₂×C₄	C₂×#357	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	#1492	1	1	yes	characteristic
164	C₈	C₂×#359	1	1	yes	characteristic
165	C₂²	—	2	1	no	—
166	C₂²	—	4	1	no	—
167	C₂²	—	32	2	no	—
168	C₂²	—	32	2	no	—
169	C₂²	—	32	2	no	—
170	C₂²	C₂×#1419	1	1	yes	characteristic
171	C₄	—	2	1	no	—
172	C₄	—	4	1	no	—
173	C₄	—	4	1	no	—
174	C₄	—	32	2	no	—
175	C₄	#11671	1	1	yes	characteristic
176	C₄	#13622	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	4	1	no	—
179	C₂	—	64	2	no	—
180	C₂	#150158	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#150158
commutator	C₂×C₃₂	C₂²×C₄
Frattini, cosocle	#1397	C₂³
prime radical	#513074	C₂
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ #2108 ◁ #13549 ◁ G
LECS	G ▷ #1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #153425	C₂	semidirect
3. #151057	C₂	semidirect
4. #94614	C₂	semidirect

Maximal subgroups

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

subgroup	index
#513074	2
#153425	2
#151057	2
#94614	2