Subgroups — #30606142

Order 15616.

614 conjugacy classes of subgroups.

The subgroup lattice has 256 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	1	yes	maximal, characteristic
2	C₂³×#11758187	C₂	1	1	yes	maximal, characteristic
3	#21460857	C₂	1	4	yes	maximal, complemented
4	C₂×#16839718	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519851	—	2	2	no	—
6	C₂×#14519851	—	2	2	no	—
7	C₂×#14519851	C₄	1	4	yes	complemented
8	C₂²×#11758187	—	2	2	no	—
9	C₂²×#11758187	—	2	2	no	—
10	C₂²×#11758187	C₄	1	4	yes	complemented
11	C₂²×#11758187	C₂²	1	2	yes	—
12	C₂²×#11758187	C₂²	1	2	yes	—
13	C₂²×#11758194	C₄	1	1	yes	characteristic
14	C₂³×#92418	C₂²	1	1	yes	characteristic
15	#16839718	—	2	2	no	—
16	#16839718	—	2	2	no	—
17	C₂×#14519688	C₄	1	2	yes	—
18	C₂×C₄×#92418	C₂²	1	2	yes	—
19	C₂×#117×C₆₁	C₄	1	1	yes	characteristic
20	#14519851	—	2	8	no	—
21	C₂×#11758187	—	2	8	no	—
22	C₂×#11758187	—	2	2	no	—
23	C₂×#11758187	—	2	2	no	—
24	C₂×#11758187	—	2	2	no	—
25	C₂×#11758187	—	2	2	no	—
26	C₂×#11758187	C₂×C₄	1	4	yes	—
27	C₂×#11758187	C₂×C₄	1	4	yes	—
28	C₂×#11758194	—	2	2	no	—
29	C₂×#11758194	—	2	2	no	—
30	C₂×#11758194	C₈	1	4	yes	—
31	C₂²×#92418	C₂×C₄	1	2	yes	—
32	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
33	C₂²×#92418	C₂×C₄	1	2	yes	—
34	C₂²×#92418	C₂×C₄	1	2	yes	—
35	C₂²×#92418	D₄	1	2	yes	—
36	C₂²×#92418	D₄	1	2	yes	—
37	C₂²×#92418	D₄	1	2	yes	—
38	C₂²×#92418	Q₈	1	2	yes	—
39	C₂²×#92418	C₂³	1	1	yes	characteristic
40	#14519688	—	2	4	no	—
41	C₄×#92418	—	2	2	no	—
42	C₄×#92418	—	2	2	no	—
43	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
44	#117×C₆₁	—	2	2	no	—
45	#117×C₆₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₆₁	C₈	1	2	yes	—
47	C₂×C₂²⋊C₄×C₆₁	C₈	1	2	yes	—
48	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
49	C₂⁴×C₆₁	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
51	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
54	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
57	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
58	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
59	C₄⋊C₄×C₆₁	—	2	4	no	—
60	C₂²⋊C₄×C₆₁	—	2	4	no	—
61	C₄²×C₆₁	—	2	2	no	—
62	C₄²×C₆₁	—	2	2	no	—
63	#11758194	—	2	8	no	—
64	C₂×#92418	—	2	2	no	—
65	C₂×#92418	—	2	2	no	—
66	C₂×#92418	C₄²	1	2	yes	—
67	C₂×#92418	C₂²⋊C₄	1	2	yes	—
68	C₂×#92418	C₂²⋊C₄	1	4	yes	—
69	C₂×#92418	C₂²⋊C₄	1	4	yes	—
70	C₂×#92418	C₄⋊C₄	1	2	yes	—
71	C₂×#92418	C₄⋊C₄	1	4	yes	—
72	C₂×#92418	C₄⋊C₄	1	4	yes	—
73	C₂×#92418	C₂×C₈	1	4	yes	—
74	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
75	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
76	C₂×#92418	C₂×D₄	1	1	yes	characteristic
77	C₂×#92418	C₂×D₄	1	1	yes	characteristic
78	C₂×#92418	C₂×D₄	1	1	yes	characteristic
79	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
80	#11758187	—	2	4	no	—
81	#11758187	—	2	4	no	—
82	C₂³×C₆₁	—	2	4	no	—
83	C₂³×C₆₁	C₄×C₈	1	2	yes	—
84	C₂³×C₆₁	C₂²⋊C₈	1	2	yes	—
85	C₂³×C₆₁	C₄⋊C₈	1	2	yes	—
86	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₆₁	—	2	4	no	—
88	C₂×C₄×C₆₁	—	2	8	no	—
89	C₂×C₄×C₆₁	#95	1	4	yes	—
90	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
91	C₂×C₄×C₆₁	#97	1	4	yes	—
92	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
93	C₂×C₄×C₆₁	C₂²⋊C₈	1	4	yes	—
94	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
95	C₂×C₄×C₆₁	C₄⋊C₈	1	4	yes	—
96	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
102	#92418	—	2	4	no	—
103	#92418	#95	1	4	yes	—
104	#92418	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	#92418	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	#92418	C₂×C₄⋊C₄	1	1	yes	characteristic
107	#92418	C₂×C₄⋊C₄	1	1	yes	characteristic
108	#12545	—	61	1	no	maximal, Sylow
109	C₂²×C₆₁	—	2	4	no	—
110	C₂²×C₆₁	—	2	4	no	—
111	C₂²×C₆₁	—	2	4	no	—
112	C₂²×C₆₁	#336	1	4	yes	—
113	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
114	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
115	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
116	C₂²×C₆₁	#504	1	4	yes	—
117	C₄×C₆₁	—	4	4	no	—
118	C₄×C₆₁	#336	1	4	yes	—
119	C₄×C₆₁	C₂×#95	1	1	yes	characteristic
120	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
121	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
122	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
123	C₂²×M₃₂	—	61	1	no	—
124	C₂³×C₁₆	—	61	1	no	—
125	C₂×#405	—	61	1	no	—
126	#1374	—	61	4	no	—
127	C₂×C₆₁	—	2	4	no	—
128	C₂×C₆₁	C₂×#336	1	1	yes	characteristic
129	C₂×C₆₁	#2108	1	2	yes	—
130	C₂×C₆₁	#2113	1	2	yes	—
131	C₂×C₆₁	#2151	1	2	yes	—
132	C₂²×M₁₆	—	61	1	no	—
133	C₂³×C₈	—	61	1	no	—
134	C₂×#117	#6923	1	1	yes	characteristic
135	C₂×M₃₂	—	61	4	no	—
136	C₂×M₃₂	—	122	2	no	—
137	C₂×M₃₂	—	122	2	no	—
138	C₂²×C₁₆	—	61	2	no	—
139	C₂²×C₁₆	—	61	4	no	—
140	C₂²×C₁₆	—	61	2	no	—
141	C₂²×C₁₆	—	122	2	no	—
142	C₂²×C₁₆	—	122	2	no	—
143	#405	—	122	2	no	—
144	#405	—	122	2	no	—
145	C₂×#97	—	61	2	no	—
146	C₂×C₄×C₈	—	61	2	no	—
147	C₆₁	#12545	1	1	yes	characteristic, complemented, Sylow
148	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
149	C₂×M₁₆	—	61	4	no	—
150	C₂×M₁₆	—	122	2	no	—
151	C₂×M₁₆	—	122	2	no	—
152	C₂²×C₈	—	61	2	no	—
153	C₂²×C₈	—	61	2	no	—
154	C₂²×C₈	—	61	2	no	—
155	C₂²×C₈	—	61	2	no	—
156	C₂²×C₈	—	61	2	no	—
157	C₂²×C₈	—	61	2	no	—
158	C₂²×C₈	—	61	1	no	—
159	C₂²×C₈	—	61	1	no	—
160	C₂²×C₈	—	61	2	no	—
161	#117	—	2	2	no	—
162	#117	—	2	2	no	—
163	C₂×C₄⋊C₄	#92420	1	2	yes	—
164	C₂×C₂²⋊C₄	#92420	1	2	yes	—
165	C₂×C₄²	C₂×#6923	1	2	yes	—
166	M₃₂	—	122	8	no	—
167	C₂×C₁₆	—	61	4	no	—
168	C₂×C₁₆	—	61	4	no	—
169	C₂×C₁₆	—	122	8	no	—
170	C₂×C₁₆	—	122	2	no	—
171	C₂×C₁₆	—	122	2	no	—
172	C₂×C₁₆	—	122	2	no	—
173	C₂×C₁₆	—	122	2	no	—
174	#97	—	122	4	no	—
175	C₄×C₈	—	122	2	no	—
176	C₄×C₈	—	122	2	no	—
177	C₂⁴	C₂×#92420	1	1	yes	characteristic
178	C₂²×C₄	D₆₁⋊C₈	1	2	yes	—
179	C₂²×C₄	#11758213	1	2	yes	—
180	C₂²×C₄	C₄×#6923	1	2	yes	—
181	C₂²×C₄	C₄⋊#6923	1	2	yes	—
182	C₂²×C₄	C₂×#92420	1	4	yes	—
183	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
184	C₂²×C₄	#11758217	1	2	yes	—
185	C₂²×C₄	C₂²⋊#6923	1	2	yes	—
186	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
187	M₁₆	—	122	8	no	—
188	C₂×C₈	—	61	4	no	—
189	C₂×C₈	—	61	4	no	—
190	C₂×C₈	—	61	4	no	—
191	C₂×C₈	—	61	4	no	—
192	C₂×C₈	—	61	1	no	—
193	C₂×C₈	—	61	1	no	—
194	C₂×C₈	—	61	2	no	—
195	C₂×C₈	—	61	1	no	—
196	C₂×C₈	—	61	1	no	—
197	C₂×C₈	—	61	2	no	—
198	C₂×C₈	—	61	1	no	—
199	C₂×C₈	—	61	2	no	—
200	C₂×C₈	—	61	1	no	—
201	C₂×C₈	—	61	4	no	—
202	C₂×C₈	—	122	2	no	—
203	C₂×C₈	—	122	2	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	2	4	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₁₆	—	122	4	no	—
209	C₁₆	—	122	4	no	—
210	C₂³	—	2	4	no	—
211	C₂³	C₄×#92420	1	2	yes	—
212	C₂³	C₂²⋊#92420	1	2	yes	—
213	C₂³	C₄⋊#92420	1	2	yes	—
214	C₂³	C₂²×#92420	1	1	yes	characteristic
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	(C₂×#6923)⋊C₄	1	4	yes	—
218	C₂×C₄	C₄×#92420	1	2	yes	—
219	C₂×C₄	#14519832	1	4	yes	—
220	C₂×C₄	C₂²⋊#92420	1	2	yes	—
221	C₂×C₄	#14519834	1	4	yes	—
222	C₂×C₄	C₄⋊#92420	1	2	yes	—
223	C₂×C₄	#14519844	1	4	yes	—
224	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
225	C₂×C₄	C₂×C₂²⋊#6923	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
227	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
228	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
229	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
230	C₈	—	61	4	no	—
231	C₈	—	61	1	no	—
232	C₈	—	61	1	no	—
233	C₈	—	61	1	no	—
234	C₈	—	61	1	no	—
235	C₈	—	122	4	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	4	no	—
239	C₂²	#16840628	1	4	yes	—
240	C₂²	C₂×C₄×#92420	1	1	yes	characteristic
241	C₂²	C₂×C₂²⋊#92420	1	1	yes	characteristic
242	C₂²	C₂×C₄⋊#92420	1	1	yes	characteristic
243	C₂²	#16840820	1	4	yes	—
244	C₄	—	4	4	no	—
245	C₄	#16840628	1	4	yes	—
246	C₄	C₂×(C₂×#6923)⋊C₄	1	1	yes	characteristic
247	C₄	C₂×#14519832	1	1	yes	characteristic
248	C₄	C₂×#14519834	1	1	yes	characteristic
249	C₄	C₂×#14519844	1	1	yes	characteristic
250	C₂	—	2	4	no	—
251	C₂	C₂×#16840628	1	1	yes	characteristic
252	C₂	#21462028	1	2	yes	—
253	C₂	#21462049	1	2	yes	—
254	C₂	#21462154	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6923
commutator	C₂²×C₆₁	C₂×C₄×C₈
Fitting	C₂×#117×C₆₁	C₄
socle	C₂³×C₆₁	C₂²×C₈
prime radical	C₂⁴×C₆₁	C₂×C₈
cosocle	C₂²×#92418	C₂³
hypercenter	C₂×#117	#6923
hypocenter	C₆₁	#12545
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #21460857	C₂	semidirect
7. C₂×#14519851	C₄	semidirect
10. C₂²×#11758187	C₄	semidirect
147. C₆₁	#12545	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
C₂³×#11758187	2
#21460857	2
C₂×#16839718	2
#12545	61