Subgroups — C₂×C₄×#14519713 #29678109

Order 15616.

1086 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14519713	C₂	1	2	yes	maximal
2	C₂×C₄×#11758194	C₂	1	2	yes	maximal
3	C₄×#14519713	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16839527	C₂	1	2	yes	maximal
5	C₂²×C₄×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758194	C₂²	1	4	yes	—
7	C₂×#14519713	C₄	1	16	yes	direct product factor
8	C₂×#14519713	C₂²	1	8	yes	—
9	C₄×#11758194	—	2	4	no	—
10	C₄×#11758194	C₂²	1	8	yes	—
11	C₂×#14519688	—	2	2	no	—
12	C₂×C₄×#92418	—	2	2	no	—
13	#16839527	C₂²	1	8	yes	—
14	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₆₁	C₄	1	4	yes	—
18	C₂×C₄×C₈×C₆₁	C₄	1	4	yes	—
19	C₂×C₄×C₈×C₆₁	C₂²	1	4	yes	—
20	C₂×#11758194	—	2	4	no	—
21	C₂×#11758194	—	2	8	no	—
22	C₂×#11758194	C₂×C₄	1	16	yes	—
23	C₂×#11758194	C₂³	1	8	yes	—
24	C₂²×#92418	—	2	4	no	—
25	#14519713	C₂×C₄	1	32	yes	direct product factor
26	#14519688	—	2	8	no	—
27	C₄×#92418	—	2	8	no	—
28	C₂³×C₄×C₆₁	Q₈	1	2	yes	—
29	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₆₁	—	2	4	no	—
31	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₆₁	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
38	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₁	D₄	1	2	yes	—
42	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₁	Q₈	1	2	yes	—
44	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
45	C₄×C₈×C₆₁	C₂×C₄	1	16	yes	—
46	C₂⁴×C₆₁	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	—	2	8	no	—
48	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
54	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	C₂×Q₈	1	4	yes	—
57	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	#41	1	4	yes	—
59	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
60	C₂×C₈×C₆₁	—	2	4	no	—
61	C₂×C₈×C₆₁	—	2	4	no	—
62	C₂×C₈×C₆₁	—	2	4	no	—
63	C₂×C₈×C₆₁	C₄²	1	8	yes	—
64	C₂×C₈×C₆₁	C₄²	1	8	yes	—
65	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
66	C₂×C₈×C₆₁	C₂²×C₄	1	8	yes	—
67	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
68	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
69	C₄²×C₆₁	C₄⋊C₄	1	4	yes	—
70	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
71	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
72	C₄²×C₆₁	C₂×D₄	1	2	yes	—
73	C₄²×C₆₁	C₂×D₄	1	2	yes	—
74	#11758194	—	2	16	no	—
75	#11758194	—	2	8	no	—
76	#11758194	C₂²×C₄	1	16	yes	—
77	C₂×#92418	—	2	16	no	—
78	C₂×#92418	—	2	8	no	—
79	C₂³×C₆₁	—	2	4	no	—
80	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂³×C₆₁	C₄×Q₈	1	4	yes	—
83	C₂³×C₆₁	C₂²×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₆₁	—	2	4	no	—
85	C₂×C₄×C₆₁	—	2	8	no	—
86	C₂×C₄×C₆₁	—	2	8	no	—
87	C₂×C₄×C₆₁	#108	1	8	yes	complemented
88	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
89	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
90	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
91	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
94	C₂×C₄×C₆₁	#117	1	2	yes	—
95	C₂×C₄×C₆₁	#117	1	2	yes	—
96	C₂×C₄×C₆₁	C₄×D₄	1	4	yes	—
97	C₂×C₄×C₆₁	C₄×D₄	1	4	yes	—
98	C₂×C₄×C₆₁	C₄×Q₈	1	4	yes	—
99	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂³×C₄	1	2	yes	—
101	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
103	C₈×C₆₁	—	2	4	no	—
104	C₈×C₆₁	C₂×C₄²	1	8	yes	—
105	#92418	—	2	16	no	—
106	C₂×C₄×#108	—	61	1	no	maximal, Sylow
107	C₂²×C₆₁	—	2	2	no	—
108	C₂²×C₆₁	—	2	2	no	—
109	C₂²×C₆₁	—	2	8	no	—
110	C₂²×C₆₁	C₄×C₄⋊C₄	1	2	yes	—
111	C₂²×C₆₁	C₄×C₄⋊C₄	1	2	yes	—
112	C₂²×C₆₁	C₂×#108	1	4	yes	—
113	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₆₁	C₂×C₄×Q₈	1	2	yes	—
115	C₄×C₆₁	—	2	4	no	—
116	C₄×C₆₁	C₄×C₄⋊C₄	1	2	yes	—
117	C₄×C₆₁	C₄×C₄⋊C₄	1	2	yes	—
118	C₄×C₆₁	C₂×#108	1	8	yes	complemented
119	C₄×C₆₁	C₂²×C₄²	1	1	yes	characteristic
120	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
121	C₄×C₆₁	C₂×C₄×D₄	1	2	yes	—
122	C₂²×#108	—	61	2	no	—
123	C₂×C₄×M₁₆	—	61	2	no	—
124	C₂²×C₄×C₈	D₆₁	1	1	yes	characteristic
125	C₄×#108	—	61	8	no	—
126	C₂×#341	—	61	2	no	—
127	C₂×C₆₁	—	2	2	no	—
128	C₂×C₆₁	—	2	2	no	—
129	C₂×C₆₁	C₄×#108	1	4	yes	complemented
130	C₂×C₆₁	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₆₁	C₂²×#108	1	2	yes	—
132	C₂²×M₁₆	—	61	4	no	—
133	C₂³×C₈	C₂×D₆₁	1	1	yes	characteristic
134	C₂³×C₈	C₂×D₆₁	1	1	yes	characteristic
135	C₂²×C₄²	C₂×D₆₁	1	1	yes	characteristic
136	C₂×#108	—	61	16	no	—
137	C₂×#108	—	61	8	no	—
138	C₄×M₁₆	—	61	8	no	—
139	C₄×M₁₆	—	122	4	no	—
140	C₂×#97	—	122	2	no	—
141	C₂×C₄×C₈	—	122	2	no	—
142	C₂×C₄×C₈	Dic₆₁	1	4	yes	—
143	C₂×C₄×C₈	Dic₆₁	1	4	yes	—
144	C₂×C₄×C₈	C₂×D₆₁	1	4	yes	—
145	#341	—	61	8	no	—
146	C₆₁	C₂×C₄×#108	1	1	yes	characteristic, complemented, Sylow
147	C₂³×C₄	Dic₁₂₂	1	2	yes	—
148	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
149	C₂×M₁₆	—	61	16	no	—
150	C₂×M₁₆	—	61	8	no	—
151	C₂×M₁₆	—	122	4	no	—
152	C₂×M₁₆	—	122	8	no	—
153	C₂²×C₈	—	2	4	no	—
154	C₂²×C₈	—	122	4	no	—
155	C₂²×C₈	C₄×D₆₁	1	8	yes	—
156	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
157	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
158	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
159	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
160	C₂²×C₈	C₂²×D₆₁	1	2	yes	—
161	C₂²×C₈	C₂²×D₆₁	1	2	yes	—
162	C₂×C₄²	Dic₁₂₂	1	2	yes	—
163	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
164	C₂×C₄²	D₂₄₄	1	2	yes	—
165	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
166	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
167	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
168	C₂×C₄²	C₂²×D₆₁	1	2	yes	—
169	#108	—	61	32	no	—
170	#97	—	122	8	no	—
171	C₄×C₈	—	122	8	no	—
172	C₄×C₈	C₂×Dic₆₁	1	16	yes	—
173	C₂⁴	C₂×Dic₁₂₂	1	1	yes	characteristic
174	C₂²×C₄	—	2	8	no	—
175	C₂²×C₄	C₄⋊Dic₆₁	1	2	yes	—
176	C₂²×C₄	C₄⋊Dic₆₁	1	4	yes	—
177	C₂²×C₄	C₄⋊Dic₆₁	1	2	yes	—
178	C₂²×C₄	C₂×Dic₁₂₂	1	4	yes	—
179	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₄×D₆₁	1	4	yes	—
181	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
183	C₂²×C₄	#11758222	1	4	yes	—
184	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
185	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
186	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
187	M₁₆	—	61	16	no	—
188	M₁₆	—	122	8	no	—
189	M₁₆	—	122	16	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	122	8	no	—
194	C₂×C₈	—	122	16	no	—
195	C₂×C₈	C₄×Dic₆₁	1	8	yes	—
196	C₂×C₈	C₄×Dic₆₁	1	8	yes	—
197	C₂×C₈	C₂×C₄×D₆₁	1	8	yes	—
198	C₂×C₈	C₂²×Dic₆₁	1	4	yes	—
199	C₂×C₈	C₂²×Dic₆₁	1	4	yes	—
200	C₄²	C₄⋊Dic₆₁	1	2	yes	—
201	C₄²	C₄⋊Dic₆₁	1	4	yes	—
202	C₄²	C₄⋊Dic₆₁	1	2	yes	—
203	C₄²	C₂×D₂₄₄	1	2	yes	—
204	C₄²	C₂×D₂₄₄	1	2	yes	—
205	C₄²	C₂²×Dic₆₁	1	4	yes	—
206	C₂³	—	2	4	no	—
207	C₂³	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
208	C₂³	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
209	C₂³	C₄×Dic₁₂₂	1	4	yes	—
210	C₂³	C₂²×Dic₁₂₂	1	1	yes	characteristic
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	2	8	no	—
214	C₂×C₄	#14519713	1	8	yes	direct product factor
215	C₂×C₄	C₂×C₄×Dic₆₁	1	2	yes	—
216	C₂×C₄	C₂×C₄×Dic₆₁	1	2	yes	—
217	C₂×C₄	C₂×C₄⋊Dic₆₁	1	4	yes	—
218	C₂×C₄	C₂×C₄⋊Dic₆₁	1	4	yes	—
219	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
220	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
221	C₂×C₄	#14519736	1	2	yes	—
222	C₂×C₄	#14519736	1	2	yes	—
223	C₂×C₄	C₄×D₂₄₄	1	4	yes	—
224	C₂×C₄	C₄×D₂₄₄	1	4	yes	—
225	C₂×C₄	C₄×Dic₁₂₂	1	4	yes	—
226	C₂×C₄	C₂²×C₄×D₆₁	1	2	yes	—
227	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
228	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
229	C₂×C₄	C₂×#11758222	1	2	yes	—
230	C₈	—	2	4	no	—
231	C₈	—	122	16	no	—
232	C₈	C₂×C₄×Dic₆₁	1	8	yes	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	8	no	—
236	C₂²	C₄×C₄⋊Dic₆₁	1	2	yes	—
237	C₂²	C₄×C₄⋊Dic₆₁	1	2	yes	—
238	C₂²	C₂×#14519713	1	4	yes	—
239	C₂²	C₂²×C₄⋊Dic₆₁	1	1	yes	characteristic
240	C₂²	C₂×C₄×Dic₁₂₂	1	2	yes	—
241	C₄	—	2	4	no	—
242	C₄	C₄×C₄⋊Dic₆₁	1	2	yes	—
243	C₄	C₄×C₄⋊Dic₆₁	1	2	yes	—
244	C₄	C₂×#14519713	1	8	yes	direct product factor
245	C₄	C₂²×C₄×Dic₆₁	1	1	yes	characteristic
246	C₄	C₂×#14519736	1	1	yes	characteristic
247	C₄	C₂×C₄×D₂₄₄	1	2	yes	—
248	C₂	—	2	2	no	—
249	C₂	—	2	2	no	—
250	C₂	C₄×#14519713	1	4	yes	direct product factor
251	C₂	C₂×C₄×C₄⋊Dic₆₁	1	1	yes	characteristic
252	C₂	C₂²×#14519713	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	D₂₄₄
commutator	C₄×C₆₁	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₂²×C₄×C₈×C₆₁	C₂
socle	C₂³×C₆₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₁	C₂×Q₈
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	C₂²×C₄×C₈	D₆₁
hypocenter	C₆₁	C₂×C₄×#108
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₄×#14519713	C₂	direct (N × H)
7. C₂×#14519713	C₄	direct (N × H)
25. #14519713	C₂×C₄	direct (N × H)
87. C₂×C₄×C₆₁	#108	semidirect
118. C₄×C₆₁	C₂×#108	semidirect
129. C₂×C₆₁	C₄×#108	semidirect
146. C₆₁	C₂×C₄×#108	semidirect
214. C₂×C₄	#14519713	direct (N × H)
244. C₄	C₂×#14519713	direct (N × H)
250. C₂	C₄×#14519713	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14519713	2
C₂×C₄×#11758194	2
C₄×#14519713	2
C₂×#16839527	2
C₂²×C₄×C₈×C₆₁	2
C₂×C₄×#108	61

Subgroups — C2×C4×#14519713 #29678109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#14519713 #29678109