Subgroups — #30624495

Order 15616.

532 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11758187	C₂	1	2	yes	maximal, complemented
2	#21462203	C₂	1	6	yes	maximal, complemented
3	#21462176	C₂	1	3	yes	maximal, complemented
4	#21462151	C₂	1	3	yes	maximal, complemented
5	#41×#92418	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519851	C₂²	1	6	yes	—
7	C₄⋊#11758187	C₂²	1	3	yes	—
8	C₄⋊#11758187	C₂²	1	6	yes	—
9	C₄⋊#11758187	C₂²	1	1	yes	characteristic
10	C₂²⋊#11758187	C₂²	1	6	yes	—
11	C₄×#11758187	C₂²	1	6	yes	—
12	Q₈×#92418	C₄	1	1	yes	characteristic
13	Q₈×#92418	C₂²	1	1	yes	characteristic
14	D₄×#92418	C₄	1	3	yes	—
15	D₄×#92418	C₂²	1	3	yes	—
16	#16839815	C₄	1	3	yes	—
17	C₂×C₄×#92418	C₂²	1	3	yes	—
18	C₄×#41×C₆₁	C₄	1	1	yes	characteristic
19	#14519851	—	4	6	no	—
20	C₂×#11758187	C₂³	1	6	yes	—
21	C₂×#11758187	C₂³	1	2	yes	—
22	C₂²×#92418	C₂×C₄	1	3	yes	—
23	C₂²×#92418	C₂³	1	3	yes	—
24	C₄⋊#92418	C₂×C₄	1	3	yes	—
25	C₄⋊#92418	C₂×C₄	1	3	yes	—
26	C₂²⋊#92418	C₂×C₄	1	6	yes	—
27	C₄×#92418	C₂×C₄	1	3	yes	—
28	C₄×#92418	C₂×C₄	1	3	yes	—
29	C₄×#92418	C₂³	1	1	yes	characteristic
30	C₄×#92418	C₂³	1	3	yes	—
31	C₂×#41×C₆₁	C₈	1	1	yes	characteristic
32	C₄×Q₈×C₆₁	C₈	1	1	yes	characteristic
33	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₄×D₄×C₆₁	C₈	1	3	yes	—
35	C₄×D₄×C₆₁	C₂×C₄	1	3	yes	—
36	#117×C₆₁	C₈	1	3	yes	—
37	C₂×C₄²×C₆₁	C₂×C₄	1	3	yes	—
38	#41×C₆₁	—	2	4	no	—
39	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
40	C₂×D₄×C₆₁	C₂×C₈	1	3	yes	—
41	C₂²×C₄×C₆₁	C₂×C₈	1	3	yes	—
42	C₂²×C₄×C₆₁	C₂×C₈	1	3	yes	—
43	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
44	C₄⋊C₄×C₆₁	C₂×C₈	1	3	yes	—
45	C₄⋊C₄×C₆₁	C₂×C₈	1	3	yes	—
46	C₂²⋊C₄×C₆₁	C₂×C₈	1	6	yes	—
47	C₄²×C₆₁	C₂×C₈	1	3	yes	—
48	C₄²×C₆₁	C₂×C₈	1	3	yes	—
49	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₆₁	C₂²×C₄	1	3	yes	—
51	C₂×#92418	—	2	3	no	—
52	C₂×#92418	—	2	3	no	—
53	C₂×#92418	—	4	3	no	—
54	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
55	C₂×#92418	C₂²×C₄	1	3	yes	—
56	C₂×#92418	C₂²×C₄	1	3	yes	—
57	C₂×#92418	C₂⁴	1	1	yes	characteristic
58	#11758187	—	2	6	no	—
59	#11758187	—	2	2	no	—
60	C₂³×C₆₁	C₂²×C₈	1	3	yes	—
61	Q₈×C₆₁	M₃₂	1	4	yes	complemented
62	D₄×C₆₁	—	2	6	no	—
63	C₂×C₄×C₆₁	—	2	3	no	—
64	C₂×C₄×C₆₁	—	2	3	no	—
65	C₂×C₄×C₆₁	—	2	6	no	—
66	C₂×C₄×C₆₁	—	2	6	no	—
67	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₂²×C₈	1	3	yes	—
69	C₂×C₄×C₆₁	C₂²×C₈	1	3	yes	—
70	C₂×C₄×C₆₁	C₂²×C₈	1	3	yes	—
71	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
73	#92418	—	2	3	no	—
74	#92418	—	2	1	no	—
75	#92418	—	2	3	no	—
76	#92418	H(5,𝔽₂)	1	1	yes	characteristic
77	#92418	2₋¹⁺⁴	1	1	yes	characteristic
78	#33505	—	61	1	no	maximal, Sylow
79	C₂²×C₆₁	—	2	3	no	—
80	C₂²×C₆₁	—	2	3	no	—
81	C₂²×C₆₁	—	2	3	no	—
82	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
83	C₄×C₆₁	—	2	1	no	—
84	C₄×C₆₁	—	2	3	no	—
85	C₄×C₆₁	—	2	3	no	—
86	C₄×C₆₁	C₂×M₃₂	1	6	yes	—
87	C₄×C₆₁	#504	1	2	yes	—
88	C₄×C₆₁	#520	1	1	yes	characteristic
89	C₄×C₆₁	#568	1	1	yes	characteristic
90	C₈×#41	—	61	1	no	—
91	Q₈×C₁₆	—	61	2	no	—
92	#2169	—	61	6	no	—
93	#2162	—	61	3	no	—
94	#2150	—	61	3	no	—
95	C₂×C₆₁	—	4	3	no	—
96	C₂×C₆₁	#2988	1	1	yes	characteristic
97	C₂×C₆₁	C₂²×M₃₂	1	1	yes	characteristic
98	C₂×C₆₁	C₂×#504	1	1	yes	characteristic
99	C₄×#41	#6923	1	1	yes	characteristic
100	C₂×M₃₂	—	61	6	no	—
101	C₈×Q₈	—	61	1	no	—
102	C₈×Q₈	—	61	1	no	—
103	C₈×D₄	—	61	3	no	—
104	C₈×D₄	—	61	3	no	—
105	#431	—	61	3	no	—
106	C₂×C₄×C₈	—	61	3	no	—
107	C₄⋊C₁₆	—	61	3	no	—
108	C₄⋊C₁₆	—	61	6	no	—
109	C₄⋊C₁₆	—	61	1	no	—
110	C₂²⋊C₁₆	—	61	6	no	—
111	C₄×C₁₆	—	61	6	no	—
112	C₆₁	#33505	1	1	yes	characteristic, complemented, Sylow
113	C₂×#41	#92420	1	1	yes	characteristic
114	C₂²×C₈	—	61	3	no	—
115	C₂²×C₈	—	61	3	no	—
116	C₄×Q₈	#92420	1	1	yes	characteristic
117	C₄×Q₈	C₂×#6923	1	1	yes	characteristic
118	C₄×D₄	#92420	1	3	yes	—
119	C₄×D₄	C₂×#6923	1	3	yes	—
120	#117	#92420	1	3	yes	—
121	C₂×C₄²	C₂×#6923	1	3	yes	—
122	M₃₂	—	244	6	no	—
123	C₂×C₁₆	—	61	2	no	—
124	C₂×C₁₆	—	61	6	no	—
125	C₄⋊C₈	—	61	3	no	—
126	C₄⋊C₈	—	61	3	no	—
127	C₂²⋊C₈	—	61	6	no	—
128	C₄×C₈	—	61	3	no	—
129	C₄×C₈	—	61	1	no	—
130	C₄×C₈	—	61	3	no	—
131	C₄×C₈	—	61	3	no	—
132	#41	—	2	4	no	—
133	C₂×Q₈	C₂×#92420	1	1	yes	characteristic
134	C₂×D₄	C₂×#92420	1	3	yes	—
135	C₂²×C₄	C₂×#92420	1	3	yes	—
136	C₂²×C₄	C₂×#92420	1	3	yes	—
137	C₂²×C₄	C₂²×#6923	1	3	yes	—
138	C₂×C₈	—	61	1	no	—
139	C₂×C₈	—	61	1	no	—
140	C₂×C₈	—	61	3	no	—
141	C₂×C₈	—	61	3	no	—
142	C₂×C₈	—	122	3	no	—
143	C₂×C₈	—	122	3	no	—
144	C₂×C₈	—	244	3	no	—
145	C₄⋊C₄	C₂×#92420	1	3	yes	—
146	C₄⋊C₄	C₂×#92420	1	3	yes	—
147	C₂²⋊C₄	C₂×#92420	1	6	yes	—
148	C₄²	C₂×#92420	1	3	yes	—
149	C₄²	C₂×#92420	1	3	yes	—
150	C₄²	C₂²×#6923	1	1	yes	characteristic
151	C₄²	C₂²×#6923	1	3	yes	—
152	C₁₆	—	122	2	no	—
153	C₁₆	—	122	6	no	—
154	C₂³	C₂²×#92420	1	3	yes	—
155	Q₈	#14519851	1	4	yes	complemented
156	D₄	—	2	6	no	—
157	C₂×C₄	—	2	6	no	—
158	C₂×C₄	—	2	6	no	—
159	C₂×C₄	—	2	3	no	—
160	C₂×C₄	—	2	3	no	—
161	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
162	C₂×C₄	C₂²×#92420	1	3	yes	—
163	C₂×C₄	C₂²×#92420	1	3	yes	—
164	C₂×C₄	C₂²×#92420	1	3	yes	—
165	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
166	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
167	C₈	—	61	1	no	—
168	C₈	—	61	1	no	—
169	C₈	—	122	3	no	—
170	C₈	—	122	3	no	—
171	C₈	—	122	1	no	—
172	C₂²	—	2	3	no	—
173	C₂²	—	2	3	no	—
174	C₂²	—	2	3	no	—
175	C₂²	C₂³×#92420	1	1	yes	characteristic
176	C₄	—	2	3	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	3	no	—
179	C₄	C₂×#14519851	1	6	yes	—
180	C₄	#16840820	1	2	yes	—
181	C₄	#16840831	1	1	yes	characteristic
182	C₄	#16840841	1	1	yes	characteristic
183	C₂	—	4	3	no	—
184	C₂	#21462734	1	1	yes	characteristic
185	C₂	C₂²×#14519851	1	1	yes	characteristic
186	C₂	C₂×#16840820	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₄×#41×C₆₁	C₄
prime radical	C₂×#41×C₆₁	C₈
cosocle	C₂×#92418	C₂⁴
hypercenter	C₄×#41	#6923
hypocenter	C₆₁	#33505
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×#11758187	C₂	semidirect
2. #21462203	C₂	semidirect
3. #21462176	C₂	semidirect
4. #21462151	C₂	semidirect
61. Q₈×C₆₁	M₃₂	semidirect
112. C₆₁	#33505	semidirect
155. Q₈	#14519851	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11758187	2
#21462203	2
#21462176	2
#21462151	2
#41×#92418	2
#33505	61