Subgroups — #30606724

Order 15616.

318 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462151	C₂	1	1	yes	maximal, characteristic
2	C₂×#16840654	C₂	1	2	yes	maximal
3	C₄×#14519851	C₂	1	1	yes	maximal, characteristic
4	#21460860	C₂	1	2	yes	maximal
5	#21445688	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519851	C₂²	1	1	yes	characteristic
7	C₂×#14519851	C₂²	1	2	yes	—
8	C₂×#14519851	C₂²	1	1	yes	characteristic
9	C₄⋊#11758187	C₄	1	2	yes	complemented
10	C₄⋊#11758187	C₄	1	2	yes	complemented
11	#16840654	—	2	4	no	—
12	#16840650	C₄	1	2	yes	complemented
13	C₄×#11758187	C₄	1	2	yes	complemented
14	C₂×C₄⋊#92418	C₄	1	1	yes	characteristic
15	C₂×#14519688	C₂²	1	2	yes	—
16	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
17	#16839517	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₆₁	C₄	1	1	yes	characteristic
19	#14519851	—	2	2	no	—
20	#14519851	—	2	1	no	—
21	#14519851	—	2	1	no	—
22	#14519851	—	2	2	no	—
23	#14519851	D₄	1	2	yes	—
24	#14519851	Q₈	1	2	yes	—
25	C₂×#11758187	C₂×C₄	1	2	yes	—
26	C₂×#11758187	C₂×C₄	1	4	yes	—
27	C₂×#11758187	C₂×C₄	1	2	yes	—
28	C₂²×#92418	C₂×C₄	1	2	yes	—
29	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
30	C₂²×#92418	C₂³	1	1	yes	characteristic
31	C₄⋊#92418	—	2	1	no	—
32	C₄⋊#92418	—	2	1	no	—
33	#14519688	—	2	2	no	—
34	#14519688	—	2	2	no	—
35	C₄×#92418	C₂×C₄	1	1	yes	characteristic
36	C₄×#92418	C₂×C₄	1	1	yes	characteristic
37	C₄×#92418	C₂×C₄	1	1	yes	characteristic
38	C₄×#92418	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
40	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
42	#95×C₆₁	—	2	1	no	—
43	C₂²×C₄×C₆₁	—	2	2	no	—
44	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₄⋊C₄×C₆₁	—	4	2	no	—
48	C₄²×C₆₁	—	2	2	no	—
49	C₄²×C₆₁	—	2	2	no	—
50	C₄²×C₆₁	C₄²	1	1	yes	characteristic
51	C₄²×C₆₁	C₄²	1	1	yes	characteristic
52	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
53	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
54	C₂×#92418	—	2	1	no	—
55	C₂×#92418	—	2	1	no	—
56	C₂×#92418	—	2	2	no	—
57	C₂×#92418	—	2	2	no	—
58	C₂×#92418	—	2	2	no	—
59	C₂×#92418	C₄²	1	1	yes	characteristic
60	C₂×#92418	C₄²	1	1	yes	characteristic
61	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
62	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
63	C₂×#92418	C₂×D₄	1	1	yes	characteristic
64	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
65	C₂×#92418	#41	1	1	yes	characteristic
66	C₂×#92418	#41	1	1	yes	characteristic
67	#11758187	—	2	2	no	—
68	#11758187	—	2	4	no	—
69	#11758187	C₄⋊C₄	1	4	yes	complemented
70	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₆₁	—	2	2	no	—
72	C₂×C₄×C₆₁	—	4	1	no	—
73	C₂×C₄×C₆₁	—	4	2	no	—
74	C₂×C₄×C₆₁	—	4	2	no	—
75	C₂×C₄×C₆₁	—	4	2	no	—
76	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
77	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
78	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
79	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
85	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
86	#92418	—	2	1	no	—
87	#92418	—	2	1	no	—
88	#92418	—	4	2	no	—
89	#92418	C₂×C₄⋊C₄	1	1	yes	characteristic
90	#92418	#117	1	1	yes	characteristic
91	#92418	C₄×D₄	1	1	yes	characteristic
92	#92418	C₄×Q₈	1	1	yes	characteristic
93	#12666	—	61	1	no	maximal, Sylow
94	C₂²×C₆₁	—	2	1	no	—
95	C₂²×C₆₁	—	2	1	no	—
96	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
97	C₂²×C₆₁	#436	1	1	yes	characteristic
98	C₂²×C₆₁	#446	1	1	yes	characteristic
99	C₄×C₆₁	—	2	1	no	—
100	C₄×C₆₁	—	2	1	no	—
101	C₄×C₆₁	—	2	1	no	—
102	C₄×C₆₁	—	8	2	no	—
103	C₄×C₆₁	#347	1	2	yes	complemented
104	C₄×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
105	C₄×C₆₁	#405	1	1	yes	characteristic
106	C₄×C₆₁	#436	1	1	yes	characteristic
107	C₄×C₆₁	#446	1	1	yes	characteristic
108	#2150	—	61	1	no	—
109	C₂×#347	—	61	2	no	—
110	C₄×M₃₂	—	61	1	no	—
111	#1770	—	61	1	no	—
112	#1375	—	61	2	no	—
113	C₂×C₆₁	—	2	1	no	—
114	C₂×C₆₁	—	2	1	no	—
115	C₂×C₆₁	#1770	1	1	yes	characteristic
116	C₂×C₆₁	C₂×#347	1	1	yes	characteristic
117	C₂×C₆₁	#2110	1	1	yes	characteristic
118	C₂×M₃₂	—	61	1	no	—
119	C₂×M₃₂	—	61	1	no	—
120	C₂×M₃₂	—	61	2	no	—
121	C₂×C₄⋊C₈	—	61	1	no	—
122	C₂×#97	—	61	2	no	—
123	C₂×C₄×C₈	—	61	1	no	—
124	C₄×C₄⋊C₄	#6923	1	1	yes	characteristic
125	C₄⋊C₁₆	—	61	2	no	—
126	C₄⋊C₁₆	—	61	2	no	—
127	#347	—	122	4	no	—
128	#346	—	61	2	no	—
129	C₄×C₁₆	—	61	2	no	—
130	#336	—	61	2	no	—
131	C₆₁	#12666	1	1	yes	characteristic, complemented, Sylow
132	C₂²×C₈	—	61	1	no	—
133	C₂²×C₈	—	61	1	no	—
134	C₂²×C₈	—	61	2	no	—
135	C₂×C₄⋊C₄	—	2	1	no	—
136	C₂×C₄²	C₂×#6923	1	2	yes	—
137	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
138	M₃₂	—	61	2	no	—
139	M₃₂	—	61	2	no	—
140	M₃₂	—	122	2	no	—
141	M₃₂	—	122	1	no	—
142	M₃₂	—	122	1	no	—
143	M₃₂	—	122	2	no	—
144	C₂×C₁₆	—	61	2	no	—
145	C₂×C₁₆	—	61	2	no	—
146	C₂×C₁₆	—	61	4	no	—
147	C₄⋊C₈	—	122	1	no	—
148	C₄⋊C₈	—	122	1	no	—
149	#97	—	122	2	no	—
150	#97	—	122	2	no	—
151	C₄×C₈	—	61	1	no	—
152	C₄×C₈	—	61	1	no	—
153	C₄×C₈	—	61	1	no	—
154	C₄×C₈	—	61	1	no	—
155	#95	—	2	1	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	#11758217	1	1	yes	characteristic
158	C₂²×C₄	#11758217	1	1	yes	characteristic
159	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
160	C₂×C₈	—	61	1	no	—
161	C₂×C₈	—	61	1	no	—
162	C₂×C₈	—	61	1	no	—
163	C₂×C₈	—	61	1	no	—
164	C₂×C₈	—	61	1	no	—
165	C₂×C₈	—	61	1	no	—
166	C₂×C₈	—	61	1	no	—
167	C₂×C₈	—	61	1	no	—
168	C₂×C₈	—	122	1	no	—
169	C₂×C₈	—	122	1	no	—
170	C₂×C₈	—	122	2	no	—
171	C₂×C₈	—	122	2	no	—
172	C₂×C₈	—	122	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	#11758213	1	1	yes	characteristic
177	C₄²	#11758213	1	1	yes	characteristic
178	C₄²	C₄×#6923	1	1	yes	characteristic
179	C₄²	C₄×#6923	1	1	yes	characteristic
180	C₁₆	—	61	4	no	—
181	C₁₆	—	122	2	no	—
182	C₁₆	—	122	4	no	—
183	C₂³	C₂×#11758217	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	#14519832	1	1	yes	characteristic
190	C₂×C₄	#14519832	1	1	yes	characteristic
191	C₂×C₄	#14519832	1	1	yes	characteristic
192	C₂×C₄	#14519832	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
194	C₂×C₄	D₄×#6923	1	1	yes	characteristic
195	C₂×C₄	Q₈×#6923	1	1	yes	characteristic
196	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
197	C₂×C₄	#14519864	1	1	yes	characteristic
198	C₂×C₄	#14519865	1	1	yes	characteristic
199	C₈	—	61	1	no	—
200	C₈	—	61	1	no	—
201	C₈	—	61	1	no	—
202	C₈	—	61	1	no	—
203	C₈	—	122	1	no	—
204	C₈	—	122	1	no	—
205	C₈	—	244	2	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	C₂×#14519832	1	1	yes	characteristic
209	C₂²	#16840797	1	1	yes	characteristic
210	C₂²	#16840813	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	8	2	no	—
215	C₄	#16840654	1	2	yes	complemented
216	C₄	C₄⋊C₄×#6923	1	1	yes	characteristic
217	C₄	#16840727	1	1	yes	characteristic
218	C₄	#16840796	1	1	yes	characteristic
219	C₄	#16840812	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#21461615	1	1	yes	characteristic
223	C₂	C₂×#16840654	1	1	yes	characteristic
224	C₂	#21462040	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#6923
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₄×C₄⋊C₄×C₆₁	C₄
socle	C₂²×C₆₁	C₂×#97
prime radical	C₂³×C₆₁	C₂×M₁₆
cosocle	C₂²×#92418	C₂³
hypercenter	C₄×C₄⋊C₄	#6923
hypocenter	C₆₁	#12666
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. C₄⋊#11758187	C₄	semidirect
10. C₄⋊#11758187	C₄	semidirect
12. #16840650	C₄	semidirect
13. C₄×#11758187	C₄	semidirect
69. #11758187	C₄⋊C₄	semidirect
103. C₄×C₆₁	#347	semidirect
131. C₆₁	#12666	semidirect
215. C₄	#16840654	semidirect

Maximal subgroups

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

subgroup	index
#21462151	2
C₂×#16840654	2
C₄×#14519851	2
#21460860	2
#21445688	2
#12666	61