Subgroups — #72225317

Order 1024.

290 conjugacy classes of subgroups.

The subgroup lattice has 209 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#1398	C₂	1	1	yes	maximal, characteristic
2	#129101	C₂	1	1	yes	maximal, characteristic
3	#129090	C₂	1	1	yes	maximal, characteristic
4	C₂×#1398	—	2	1	no	—
5	C₂×#1398	C₄	1	2	yes	—
6	C₄×M₆₄	—	2	1	no	—
7	C₄×#364	—	2	1	no	—
8	C₄×#364	—	2	1	no	—
9	C₄×#346	C₂²	1	1	yes	characteristic
10	C₂×#1373	C₄	1	1	yes	characteristic
11	C₂×#1373	C₄	1	1	yes	characteristic
12	C₂×#1369	C₄	1	2	yes	—
13	#7161	—	2	1	no	—
14	#7160	—	2	1	no	—
15	C₂×M₆₄	—	2	2	no	—
16	C₂×#364	—	2	2	no	—
17	C₂×#364	—	2	1	no	—
18	C₂×#364	—	2	1	no	—
19	C₄×M₃₂	—	4	1	no	—
20	C₄×M₃₂	—	4	1	no	—
21	C₂×#346	C₂×C₄	1	1	yes	characteristic
22	C₂×#346	C₂×C₄	1	1	yes	characteristic
23	C₂×#346	D₄	1	1	yes	characteristic
24	C₂×#346	Q₈	1	1	yes	characteristic
25	C₂×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₆	D₄	1	1	yes	characteristic
27	C₄²×C₈	D₄	1	1	yes	characteristic
28	#1398	—	2	4	no	—
29	#1398	—	2	2	no	—
30	#1398	C₈	1	4	yes	—
31	#1397	—	2	2	no	—
32	C₄×C₃₂	—	2	2	no	—
33	#1376	—	4	1	no	—
34	#1376	—	4	1	no	—
35	#1373	—	2	1	no	—
36	#1373	—	2	1	no	—
37	#1373	C₈	1	2	yes	—
38	#1373	C₈	1	2	yes	—
39	#1369	—	2	2	no	—
40	#1369	C₈	1	4	yes	—
41	C₂×M₃₂	—	4	2	no	—
42	C₂×M₃₂	—	4	1	no	—
43	C₂×M₃₂	—	4	1	no	—
44	C₂²×C₁₆	C₄²	1	1	yes	characteristic
45	C₂²×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₆	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₆	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₈	D₈	1	1	yes	characteristic
52	C₂×C₄×C₈	SD₁₆	1	1	yes	characteristic
53	C₂×C₄×C₈	SD₁₆	1	1	yes	characteristic
54	C₄³	Q₁₆	1	1	yes	characteristic
55	M₆₄	—	2	4	no	—
56	M₆₄	—	2	4	no	—
57	C₂×C₃₂	—	2	4	no	—
58	#364	—	2	4	no	—
59	#364	—	2	2	no	—
60	#364	—	2	2	no	—
61	#364	—	4	1	no	—
62	#364	—	4	1	no	—
63	#364	—	4	2	no	—
64	#346	—	2	2	no	—
65	#346	—	2	2	no	—
66	#346	—	8	1	no	—
67	#346	—	8	1	no	—
68	#346	C₂×C₈	1	2	yes	—
69	#346	C₂×C₈	1	2	yes	—
70	#346	M₁₆	1	2	yes	—
71	#346	M₁₆	1	2	yes	—
72	C₄×C₁₆	—	2	1	no	—
73	C₄×C₁₆	—	2	1	no	—
74	C₄×C₁₆	—	8	1	no	—
75	C₄×C₁₆	—	8	1	no	—
76	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
77	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
78	C₄×C₁₆	M₁₆	1	1	yes	characteristic
79	C₄×C₁₆	M₁₆	1	1	yes	characteristic
80	C₂²×C₈	#95	1	1	yes	characteristic
81	C₂²×C₈	#102	1	1	yes	characteristic
82	C₂²×C₈	#102	1	1	yes	characteristic
83	C₂²×C₈	#106	1	1	yes	characteristic
84	C₂×C₄²	—	4	1	no	—
85	C₂×C₄²	#103	1	1	yes	characteristic
86	C₂×C₄²	#103	1	1	yes	characteristic
87	C₂×C₄²	#107	1	1	yes	characteristic
88	M₃₂	—	4	4	no	—
89	M₃₂	—	4	2	no	—
90	M₃₂	—	4	2	no	—
91	M₃₂	—	8	2	no	—
92	M₃₂	—	8	1	no	—
93	M₃₂	—	8	1	no	—
94	C₂×C₁₆	—	2	1	no	—
95	C₂×C₁₆	—	2	1	no	—
96	C₂×C₁₆	—	2	1	no	—
97	C₂×C₁₆	—	2	1	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	—	8	2	no	—
101	C₂×C₁₆	—	8	1	no	—
102	C₂×C₁₆	—	8	1	no	—
103	C₂×C₁₆	C₄×C₈	1	1	yes	characteristic
104	C₂×C₁₆	C₄×C₈	1	1	yes	characteristic
105	C₂×C₁₆	C₄×C₈	1	1	yes	characteristic
106	C₂×C₁₆	#97	1	1	yes	characteristic
107	C₂×C₁₆	#97	1	1	yes	characteristic
108	C₂×C₁₆	#97	1	1	yes	characteristic
109	C₂×C₁₆	C₂²⋊C₈	1	2	yes	—
110	C₂×C₁₆	C₄⋊C₈	1	1	yes	characteristic
111	C₂×C₁₆	C₄⋊C₈	1	1	yes	characteristic
112	C₂×C₁₆	C₄⋊C₈	1	2	yes	—
113	C₄×C₈	—	2	1	no	—
114	C₄×C₈	—	2	1	no	—
115	C₄×C₈	—	2	1	no	—
116	C₄×C₈	—	2	1	no	—
117	C₄×C₈	—	4	1	no	—
118	C₄×C₈	C₂²⋊C₈	1	1	yes	characteristic
119	C₄×C₈	C₂²⋊C₈	1	1	yes	characteristic
120	C₄×C₈	C₂²⋊C₈	1	2	yes	—
121	C₄×C₈	#104	1	1	yes	characteristic
122	C₄×C₈	#104	1	1	yes	characteristic
123	C₄×C₈	#104	1	2	yes	—
124	C₄×C₈	C₄⋊C₈	1	2	yes	—
125	C₄×C₈	#108	1	2	yes	—
126	C₃₂	—	2	8	no	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	#340	1	1	yes	characteristic
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	D₄⋊C₈	1	2	yes	—
136	C₂×C₈	D₄⋊C₈	1	1	yes	characteristic
137	C₂×C₈	D₄⋊C₈	1	1	yes	characteristic
138	C₂×C₈	#334	1	2	yes	—
139	C₂×C₈	#336	1	1	yes	characteristic
140	C₂×C₈	#336	1	1	yes	characteristic
141	C₂×C₈	#336	1	1	yes	characteristic
142	C₂×C₈	#339	1	1	yes	characteristic
143	C₂×C₈	#339	1	1	yes	characteristic
144	C₂×C₈	#341	1	1	yes	characteristic
145	C₂×C₈	#360	1	1	yes	characteristic
146	C₂×C₈	#360	1	1	yes	characteristic
147	C₂×C₈	#361	1	1	yes	characteristic
148	C₂×C₈	#362	1	1	yes	characteristic
149	C₄²	—	2	1	no	—
150	C₄²	—	2	1	no	—
151	C₄²	—	2	1	no	—
152	C₄²	—	8	2	no	—
153	C₄²	Q₈⋊C₈	1	1	yes	characteristic
154	C₄²	Q₈⋊C₈	1	2	yes	—
155	C₄²	Q₈⋊C₈	1	1	yes	characteristic
156	C₄²	#335	1	2	yes	—
157	C₁₆	—	2	2	no	—
158	C₁₆	—	2	2	no	—
159	C₁₆	—	2	1	no	—
160	C₁₆	—	2	1	no	—
161	C₁₆	—	8	4	no	—
162	C₁₆	—	8	2	no	—
163	C₁₆	—	8	2	no	—
164	C₁₆	C₈²	1	1	yes	characteristic
165	C₁₆	#322	1	1	yes	characteristic
166	C₁₆	#322	1	1	yes	characteristic
167	C₁₆	#322	1	1	yes	characteristic
168	C₂³	#1388	1	1	yes	characteristic
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	8	1	no	—
175	C₂×C₄	—	8	1	no	—
176	C₂×C₄	#1277	1	1	yes	characteristic
177	C₂×C₄	#1279	1	1	yes	characteristic
178	C₂×C₄	#1279	1	1	yes	characteristic
179	C₂×C₄	#1386	1	1	yes	characteristic
180	C₂×C₄	#1387	1	1	yes	characteristic
181	C₂×C₄	#1389	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	—	2	2	no	—
184	C₈	#1273	1	1	yes	characteristic
185	C₈	#1276	1	1	yes	characteristic
186	C₈	#1278	1	1	yes	characteristic
187	C₈	#1278	1	1	yes	characteristic
188	C₈	#1333	1	1	yes	characteristic
189	C₈	#1333	1	1	yes	characteristic
190	C₈	#1334	1	1	yes	characteristic
191	C₈	#1334	1	1	yes	characteristic
192	C₂²	—	4	1	no	—
193	C₂²	#7158	1	1	yes	characteristic
194	C₂²	#7176	1	1	yes	characteristic
195	C₂²	#7177	1	1	yes	characteristic
196	C₄	—	2	2	no	—
197	C₄	—	4	1	no	—
198	C₄	—	4	2	no	—
199	C₄	—	4	2	no	—
200	C₄	#7015	1	1	yes	characteristic
201	C₄	#7159	1	1	yes	characteristic
202	C₄	#7175	1	1	yes	characteristic
203	C₄	#7178	1	1	yes	characteristic
204	C₂	—	2	1	no	—
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	#93270	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#1333
commutator	C₁₆	C₈²
Frattini, cosocle	C₄×#346	C₂²
socle	C₂	#93270
prime radical	C₂³	#1388
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ C₄×#346 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
C₄×#1398	2
#129101	2
#129090	2