Subgroups — #30606161

Order 15616.

586 conjugacy classes of subgroups.

The subgroup lattice has 222 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	#21460863	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	2	no	—
6	C₂×#14519851	C₄	1	8	yes	complemented
7	C₂×#14519851	C₂²	1	4	yes	—
8	C₂²×#11758187	—	2	2	no	—
9	C₂²×#11758194	C₄	1	1	yes	characteristic
10	C₂³×#92418	C₂²	1	1	yes	characteristic
11	#16839718	—	2	4	no	—
12	C₂×#14519688	C₄	1	2	yes	—
13	C₂×C₄×#92418	C₂²	1	2	yes	—
14	C₂×#117×C₆₁	C₄	1	1	yes	characteristic
15	#14519851	—	2	8	no	—
16	#14519851	—	2	4	no	—
17	#14519851	C₂×C₄	1	8	yes	—
18	C₂×#11758187	—	2	8	no	—
19	C₂×#11758187	—	2	4	no	—
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	4	yes	—
26	C₂²×#92418	D₄	1	2	yes	—
27	C₂²×#92418	D₄	1	4	yes	—
28	C₂²×#92418	Q₈	1	2	yes	—
29	C₂²×#92418	C₂³	1	1	yes	characteristic
30	#14519688	—	2	4	no	—
31	C₄×#92418	—	2	4	no	—
32	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
33	#117×C₆₁	—	2	2	no	—
34	#117×C₆₁	C₈	1	4	yes	—
35	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
36	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
37	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
38	C₂⁴×C₆₁	M₁₆	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	—	2	2	no	—
40	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
41	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
47	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
48	C₄⋊C₄×C₆₁	—	2	4	no	—
49	C₂²⋊C₄×C₆₁	—	2	4	no	—
50	C₄²×C₆₁	—	2	2	no	—
51	C₄²×C₆₁	C₂×C₈	1	4	yes	—
52	#11758194	—	2	8	no	—
53	C₂×#92418	—	2	2	no	—
54	C₂×#92418	—	2	2	no	—
55	C₂×#92418	C₄²	1	2	yes	—
56	C₂×#92418	C₂²⋊C₄	1	2	yes	—
57	C₂×#92418	C₂²⋊C₄	1	4	yes	—
58	C₂×#92418	C₂²⋊C₄	1	4	yes	—
59	C₂×#92418	C₄⋊C₄	1	2	yes	—
60	C₂×#92418	C₄⋊C₄	1	8	yes	—
61	C₂×#92418	C₂×C₈	1	4	yes	—
62	C₂×#92418	C₂²×C₄	1	2	yes	—
63	C₂×#92418	C₂×D₄	1	1	yes	characteristic
64	C₂×#92418	C₂×D₄	1	2	yes	—
65	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
66	#11758187	—	2	8	no	—
67	C₂³×C₆₁	—	4	2	no	—
68	C₂³×C₆₁	#97	1	2	yes	—
69	C₂³×C₆₁	C₂²⋊C₈	1	2	yes	—
70	C₂³×C₆₁	C₄⋊C₈	1	2	yes	—
71	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₆₁	—	2	4	no	—
73	C₂×C₄×C₆₁	—	4	4	no	—
74	C₂×C₄×C₆₁	#95	1	4	yes	—
75	C₂×C₄×C₆₁	C₄×C₈	1	4	yes	—
76	C₂×C₄×C₆₁	#97	1	2	yes	—
77	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
78	C₂×C₄×C₆₁	C₂²⋊C₈	1	4	yes	—
79	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
80	C₂×C₄×C₆₁	C₄⋊C₈	1	4	yes	—
81	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
87	#92418	—	2	4	no	—
88	#92418	#95	1	4	yes	—
89	#92418	C₂×C₂²⋊C₄	1	2	yes	—
90	#92418	C₂×C₄⋊C₄	1	2	yes	—
91	#12549	—	61	1	no	maximal, Sylow
92	C₂²×C₆₁	—	2	2	no	—
93	C₂²×C₆₁	—	2	4	no	—
94	C₂²×C₆₁	—	4	2	no	—
95	C₂²×C₆₁	—	4	2	no	—
96	C₂²×C₆₁	#336	1	4	yes	—
97	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
98	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
99	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₄×C₆₁	—	4	4	no	—
101	C₄×C₆₁	#336	1	4	yes	—
102	C₄×C₆₁	C₂×#95	1	1	yes	characteristic
103	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
104	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
105	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
106	C₂²×M₃₂	—	61	2	no	—
107	C₂×#405	—	61	1	no	—
108	#1376	—	61	4	no	—
109	C₂×C₆₁	—	2	1	no	—
110	C₂×C₆₁	—	2	4	no	—
111	C₂×C₆₁	—	2	1	no	—
112	C₂×C₆₁	C₂×#336	1	1	yes	characteristic
113	C₂×C₆₁	#2153	1	2	yes	—
114	C₂²×M₁₆	—	61	1	no	—
115	C₂³×C₈	—	61	1	no	—
116	C₂×#117	#6923	1	1	yes	characteristic
117	C₂×M₃₂	—	61	4	no	—
118	C₂×M₃₂	—	61	8	no	—
119	C₂×M₃₂	—	122	2	no	—
120	C₂×M₃₂	—	122	4	no	—
121	C₂²×C₁₆	—	122	2	no	—
122	#405	—	122	4	no	—
123	C₂×#97	—	61	2	no	—
124	C₂×C₄×C₈	—	61	2	no	—
125	C₆₁	#12549	1	1	yes	characteristic, complemented, Sylow
126	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
127	C₂×M₁₆	—	61	4	no	—
128	C₂×M₁₆	—	122	2	no	—
129	C₂×M₁₆	—	122	2	no	—
130	C₂²×C₈	—	61	4	no	—
131	C₂²×C₈	—	61	2	no	—
132	C₂²×C₈	—	61	2	no	—
133	C₂²×C₈	—	61	4	no	—
134	C₂²×C₈	—	61	1	no	—
135	C₂²×C₈	—	61	1	no	—
136	C₂²×C₈	—	61	2	no	—
137	#117	—	2	2	no	—
138	#117	#92420	1	4	yes	—
139	C₂×C₄⋊C₄	—	2	1	no	—
140	C₂×C₂²⋊C₄	—	2	1	no	—
141	C₂×C₄²	C₂×#6923	1	2	yes	—
142	M₃₂	—	61	8	no	—
143	M₃₂	—	122	8	no	—
144	M₃₂	—	122	4	no	—
145	C₂×C₁₆	—	122	8	no	—
146	C₂×C₁₆	—	122	4	no	—
147	#97	—	122	4	no	—
148	C₄×C₈	—	122	4	no	—
149	C₂⁴	#11758217	1	1	yes	characteristic
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	D₆₁⋊C₈	1	2	yes	—
152	C₂²×C₄	#11758213	1	2	yes	—
153	C₂²×C₄	C₄×#6923	1	2	yes	—
154	C₂²×C₄	C₄⋊#6923	1	2	yes	—
155	C₂²×C₄	C₂×#92420	1	2	yes	—
156	C₂²×C₄	#11758217	1	1	yes	characteristic
157	C₂²×C₄	C₂²⋊#6923	1	2	yes	—
158	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
159	M₁₆	—	122	8	no	—
160	C₂×C₈	—	61	4	no	—
161	C₂×C₈	—	61	4	no	—
162	C₂×C₈	—	61	8	no	—
163	C₂×C₈	—	61	2	no	—
164	C₂×C₈	—	61	2	no	—
165	C₂×C₈	—	61	2	no	—
166	C₂×C₈	—	61	2	no	—
167	C₂×C₈	—	61	1	no	—
168	C₂×C₈	—	61	2	no	—
169	C₂×C₈	—	61	1	no	—
170	C₂×C₈	—	61	4	no	—
171	C₂×C₈	—	122	2	no	—
172	C₂×C₈	—	122	2	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₂²⋊C₄	—	2	4	no	—
175	C₄²	—	2	2	no	—
176	C₄²	C₂×#92420	1	4	yes	—
177	C₁₆	—	122	8	no	—
178	C₂³	—	4	2	no	—
179	C₂³	#14519832	1	2	yes	—
180	C₂³	C₂²⋊#92420	1	2	yes	—
181	C₂³	#14519844	1	2	yes	—
182	C₂³	C₂×#11758217	1	1	yes	characteristic
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	(C₂×#6923)⋊C₄	1	4	yes	—
186	C₂×C₄	C₄×#92420	1	4	yes	—
187	C₂×C₄	#14519832	1	2	yes	—
188	C₂×C₄	C₂²⋊#92420	1	2	yes	—
189	C₂×C₄	#14519834	1	4	yes	—
190	C₂×C₄	C₄⋊#92420	1	4	yes	—
191	C₂×C₄	#14519844	1	2	yes	—
192	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
193	C₂×C₄	C₂×C₂²⋊#6923	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
195	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
196	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
197	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
198	C₈	—	61	4	no	—
199	C₈	—	61	2	no	—
200	C₈	—	61	2	no	—
201	C₈	—	122	4	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	#16840628	1	4	yes	—
207	C₂²	C₂×#14519832	1	1	yes	characteristic
208	C₂²	C₂×C₂²⋊#92420	1	1	yes	characteristic
209	C₂²	C₂×#14519844	1	1	yes	characteristic
210	C₄	—	4	4	no	—
211	C₄	#16840628	1	4	yes	—
212	C₄	C₂×(C₂×#6923)⋊C₄	1	1	yes	characteristic
213	C₄	C₂×C₄×#92420	1	1	yes	characteristic
214	C₄	C₂×#14519834	1	1	yes	characteristic
215	C₄	C₂×C₄⋊#92420	1	1	yes	characteristic
216	C₂	—	2	1	no	—
217	C₂	—	2	1	no	—
218	C₂	—	2	4	no	—
219	C₂	C₂×#16840628	1	1	yes	characteristic
220	C₂	#21462162	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×C₄⋊C₈
prime radical	C₂⁴×C₆₁	M₁₆
cosocle	C₂²×#92418	C₂³
hypercenter	C₂×#117	#6923
hypocenter	C₆₁	#12549
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #21460863	C₂	semidirect
6. C₂×#14519851	C₄	semidirect
125. C₆₁	#12549	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
#21460863	2
C₂×#16839718	2
#12549	61