Subgroups — #30606153

Order 15616.

586 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14519851	C₂	1	2	yes	maximal
2	#21460860	C₂	1	4	yes	maximal, complemented
3	C₂×#16839718	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519851	—	2	4	no	—
5	C₂×#14519851	—	2	4	no	—
6	C₂×#14519851	C₄	1	4	yes	complemented
7	C₂×#14519851	C₂²	1	4	yes	—
8	C₂²×#11758187	C₄	1	4	yes	complemented
9	C₂²×#11758194	C₄	1	1	yes	characteristic
10	C₂³×#92418	C₂²	1	1	yes	characteristic
11	#16839718	—	2	4	no	—
12	C₂×#14519688	C₂²	1	2	yes	—
13	C₂×C₄×#92418	C₄	1	2	yes	—
14	C₂×#117×C₆₁	C₄	1	1	yes	characteristic
15	#14519851	—	2	8	no	—
16	#14519851	—	2	4	no	—
17	#14519851	—	2	4	no	—
18	C₂×#11758187	—	2	8	no	—
19	C₂×#11758187	C₂×C₄	1	8	yes	—
20	C₂×#11758194	—	2	2	no	—
21	C₂×#11758194	—	2	2	no	—
22	C₂×#11758194	C₈	1	4	yes	—
23	C₂²×#92418	C₂×C₄	1	2	yes	—
24	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
25	C₂²×#92418	C₂×C₄	1	2	yes	—
26	C₂²×#92418	C₂×C₄	1	2	yes	—
27	C₂²×#92418	D₄	1	2	yes	—
28	C₂²×#92418	D₄	1	4	yes	—
29	C₂²×#92418	Q₈	1	2	yes	—
30	C₂²×#92418	C₂³	1	1	yes	characteristic
31	#14519688	—	2	4	no	—
32	C₄×#92418	—	2	4	no	—
33	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
34	#117×C₆₁	—	2	2	no	—
35	#117×C₆₁	C₈	1	4	yes	—
36	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
37	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
38	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
39	C₂⁴×C₆₁	M₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	—	2	2	no	—
41	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
42	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
48	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
49	C₄⋊C₄×C₆₁	—	2	4	no	—
50	C₂²⋊C₄×C₆₁	—	2	4	no	—
51	C₄²×C₆₁	—	2	2	no	—
52	C₄²×C₆₁	C₂×C₈	1	4	yes	—
53	#11758194	—	2	8	no	—
54	C₂×#92418	—	2	2	no	—
55	C₂×#92418	—	2	2	no	—
56	C₂×#92418	C₄²	1	1	yes	characteristic
57	C₂×#92418	C₄²	1	1	yes	characteristic
58	C₂×#92418	C₂²⋊C₄	1	2	yes	—
59	C₂×#92418	C₂²⋊C₄	1	4	yes	—
60	C₂×#92418	C₂²⋊C₄	1	4	yes	—
61	C₂×#92418	C₄⋊C₄	1	2	yes	—
62	C₂×#92418	C₄⋊C₄	1	4	yes	—
63	C₂×#92418	C₄⋊C₄	1	4	yes	—
64	C₂×#92418	C₂×C₈	1	4	yes	—
65	C₂×#92418	C₂²×C₄	1	2	yes	—
66	C₂×#92418	C₂×D₄	1	1	yes	characteristic
67	C₂×#92418	C₂×D₄	1	2	yes	—
68	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
69	#11758187	—	2	8	no	—
70	C₂³×C₆₁	—	4	2	no	—
71	C₂³×C₆₁	#97	1	2	yes	—
72	C₂³×C₆₁	C₂²⋊C₈	1	2	yes	—
73	C₂³×C₆₁	C₄⋊C₈	1	2	yes	—
74	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₆₁	—	2	4	no	—
76	C₂×C₄×C₆₁	—	4	4	no	—
77	C₂×C₄×C₆₁	#95	1	4	yes	—
78	C₂×C₄×C₆₁	C₄×C₈	1	4	yes	—
79	C₂×C₄×C₆₁	#97	1	2	yes	—
80	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
81	C₂×C₄×C₆₁	C₂²⋊C₈	1	4	yes	—
82	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
83	C₂×C₄×C₆₁	C₄⋊C₈	1	4	yes	—
84	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
90	#92418	—	2	4	no	—
91	#92418	#95	1	2	yes	—
92	#92418	#95	1	2	yes	—
93	#92418	C₂×C₂²⋊C₄	1	2	yes	—
94	#92418	C₂×C₄⋊C₄	1	2	yes	—
95	#12547	—	61	1	no	maximal, Sylow
96	C₂²×C₆₁	—	2	2	no	—
97	C₂²×C₆₁	—	2	4	no	—
98	C₂²×C₆₁	—	4	2	no	—
99	C₂²×C₆₁	—	4	2	no	—
100	C₂²×C₆₁	#336	1	4	yes	—
101	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
102	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
103	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
104	C₄×C₆₁	—	4	4	no	—
105	C₄×C₆₁	#336	1	4	yes	—
106	C₄×C₆₁	C₂×#95	1	1	yes	characteristic
107	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
108	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
109	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₂²×M₃₂	—	61	2	no	—
111	C₂×#405	—	61	1	no	—
112	#1375	—	61	4	no	—
113	C₂×C₆₁	—	2	1	no	—
114	C₂×C₆₁	—	2	4	no	—
115	C₂×C₆₁	—	2	1	no	—
116	C₂×C₆₁	C₂×#336	1	1	yes	characteristic
117	C₂×C₆₁	#2110	1	2	yes	—
118	C₂²×M₁₆	—	61	1	no	—
119	C₂³×C₈	—	61	1	no	—
120	C₂×#117	#6923	1	1	yes	characteristic
121	C₂×M₃₂	—	61	4	no	—
122	C₂×M₃₂	—	61	4	no	—
123	C₂×M₃₂	—	122	4	no	—
124	C₂×M₃₂	—	122	4	no	—
125	C₂²×C₁₆	—	61	4	no	—
126	#405	—	122	4	no	—
127	C₂×#97	—	61	2	no	—
128	C₂×C₄×C₈	—	61	2	no	—
129	C₆₁	#12547	1	1	yes	characteristic, complemented, Sylow
130	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
131	C₂×M₁₆	—	61	4	no	—
132	C₂×M₁₆	—	122	2	no	—
133	C₂×M₁₆	—	122	2	no	—
134	C₂²×C₈	—	61	2	no	—
135	C₂²×C₈	—	61	2	no	—
136	C₂²×C₈	—	61	2	no	—
137	C₂²×C₈	—	61	2	no	—
138	C₂²×C₈	—	61	4	no	—
139	C₂²×C₈	—	61	1	no	—
140	C₂²×C₈	—	61	1	no	—
141	C₂²×C₈	—	61	2	no	—
142	#117	—	2	2	no	—
143	#117	#92420	1	4	yes	—
144	C₂×C₄⋊C₄	—	2	1	no	—
145	C₂×C₂²⋊C₄	—	2	1	no	—
146	C₂×C₄²	C₂×#6923	1	2	yes	—
147	M₃₂	—	122	8	no	—
148	M₃₂	—	122	4	no	—
149	M₃₂	—	122	4	no	—
150	C₂×C₁₆	—	61	8	no	—
151	C₂×C₁₆	—	122	8	no	—
152	#97	—	122	4	no	—
153	C₄×C₈	—	122	4	no	—
154	C₂⁴	#11758217	1	1	yes	characteristic
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	D₆₁⋊C₈	1	2	yes	—
157	C₂²×C₄	#11758213	1	2	yes	—
158	C₂²×C₄	C₄×#6923	1	2	yes	—
159	C₂²×C₄	C₄⋊#6923	1	2	yes	—
160	C₂²×C₄	C₂×#92420	1	2	yes	—
161	C₂²×C₄	#11758217	1	1	yes	characteristic
162	C₂²×C₄	C₂²⋊#6923	1	2	yes	—
163	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
164	M₁₆	—	122	8	no	—
165	C₂×C₈	—	61	4	no	—
166	C₂×C₈	—	61	4	no	—
167	C₂×C₈	—	61	4	no	—
168	C₂×C₈	—	61	4	no	—
169	C₂×C₈	—	61	2	no	—
170	C₂×C₈	—	61	1	no	—
171	C₂×C₈	—	61	1	no	—
172	C₂×C₈	—	61	2	no	—
173	C₂×C₈	—	61	2	no	—
174	C₂×C₈	—	61	1	no	—
175	C₂×C₈	—	61	2	no	—
176	C₂×C₈	—	61	1	no	—
177	C₂×C₈	—	61	4	no	—
178	C₂×C₈	—	122	2	no	—
179	C₂×C₈	—	122	2	no	—
180	C₄⋊C₄	—	2	4	no	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₄²	—	2	2	no	—
183	C₄²	C₂×#92420	1	4	yes	—
184	C₁₆	—	122	8	no	—
185	C₂³	—	4	2	no	—
186	C₂³	#14519832	1	2	yes	—
187	C₂³	C₂²⋊#92420	1	2	yes	—
188	C₂³	#14519844	1	2	yes	—
189	C₂³	C₂×#11758217	1	1	yes	characteristic
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	(C₂×#6923)⋊C₄	1	4	yes	—
193	C₂×C₄	C₄×#92420	1	4	yes	—
194	C₂×C₄	#14519832	1	2	yes	—
195	C₂×C₄	C₂²⋊#92420	1	2	yes	—
196	C₂×C₄	#14519834	1	4	yes	—
197	C₂×C₄	C₄⋊#92420	1	4	yes	—
198	C₂×C₄	#14519844	1	2	yes	—
199	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
200	C₂×C₄	C₂×C₂²⋊#6923	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
202	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
203	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
204	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
205	C₈	—	61	2	no	—
206	C₈	—	61	2	no	—
207	C₈	—	61	2	no	—
208	C₈	—	61	2	no	—
209	C₈	—	122	4	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	#16840628	1	4	yes	—
215	C₂²	C₂×#14519832	1	1	yes	characteristic
216	C₂²	C₂×C₂²⋊#92420	1	1	yes	characteristic
217	C₂²	C₂×#14519844	1	1	yes	characteristic
218	C₄	—	4	4	no	—
219	C₄	#16840628	1	4	yes	—
220	C₄	C₂×(C₂×#6923)⋊C₄	1	1	yes	characteristic
221	C₄	C₂×C₄×#92420	1	1	yes	characteristic
222	C₄	C₂×#14519834	1	1	yes	characteristic
223	C₄	C₂×C₄⋊#92420	1	1	yes	characteristic
224	C₂	—	2	1	no	—
225	C₂	—	2	1	no	—
226	C₂	—	2	4	no	—
227	C₂	C₂×#16840628	1	1	yes	characteristic
228	C₂	#21462040	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#6923
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×#117×C₆₁	C₄
socle	C₂²×C₆₁	C₂×#97
prime radical	C₂⁴×C₆₁	M₁₆
cosocle	C₂²×#92418	C₂³
hypercenter	C₂×#117	#6923
hypocenter	C₆₁	#12547
LCS	G ▷ C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #21460860	C₂	semidirect
6. C₂×#14519851	C₄	semidirect
8. C₂²×#11758187	C₄	semidirect
129. C₆₁	#12547	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14519851	2
#21460860	2
C₂×#16839718	2
#12547	61