Subgroups — C₂×#21443872 #29591276

Order 15616.

318 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840164	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11758185	C₂	1	1	yes	maximal, characteristic
3	#21443872	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₆₁×#334	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519769	—	2	1	no	—
6	C₂²×#11758185	—	2	1	no	—
7	#16840164	—	2	1	no	—
8	#16840164	—	2	1	no	—
9	#16840164	C₄	1	2	yes	—
10	#16840164	C₂²	1	2	yes	—
11	C₂×#14519711	C₄	1	1	yes	characteristic
12	C₂×#14519711	C₄	1	1	yes	characteristic
13	C₂×#14519709	—	2	1	no	—
14	C₄⋊#11758185	—	2	1	no	—
15	C₄⋊#11758185	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
17	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#334	—	2	1	no	—
19	C₆₁×#334	C₂²	1	2	yes	—
20	#14519769	—	2	2	no	—
21	#14519769	—	2	2	no	—
22	#14519769	—	4	1	no	—
23	#14519769	—	4	1	no	—
24	C₂×C₄⋊Dic₆₁	—	2	1	no	—
25	C₂×C₄⋊Dic₆₁	—	2	1	no	—
26	C₂×#11758196	—	4	1	no	—
27	C₂×#11758196	—	4	1	no	—
28	C₂×#11758185	—	2	2	no	—
29	C₂×#11758185	—	4	2	no	—
30	#14519711	—	2	1	no	—
31	#14519711	—	2	1	no	—
32	#14519711	C₂×C₄	1	2	yes	—
33	#14519711	C₂×C₄	1	2	yes	—
34	#14519709	—	2	2	no	—
35	#14519709	—	2	2	no	—
36	C₂²×C₈×C₆₁	—	4	1	no	—
37	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
40	C₄⋊C₈×C₆₁	—	2	2	no	—
41	C₄⋊C₈×C₆₁	—	4	1	no	—
42	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
45	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
49	C₂²×Dic₆₁	—	4	1	no	—
50	C₂²×Dic₆₁	—	4	1	no	—
51	C₂×C₈×C₆₁	—	2	2	no	—
52	C₂×C₈×C₆₁	—	4	2	no	—
53	C₂×C₈×C₆₁	—	8	2	no	—
54	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
57	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
58	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₆₁	D₈	1	1	yes	characteristic
62	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
63	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
64	C₄⋊Dic₆₁	—	2	2	no	—
65	C₄⋊Dic₆₁	—	2	2	no	—
66	C₄⋊Dic₆₁	—	4	1	no	—
67	C₄⋊Dic₆₁	—	4	1	no	—
68	#11758196	—	4	2	no	—
69	#11758196	—	4	2	no	—
70	#11758196	—	4	2	no	—
71	#11758196	—	4	2	no	—
72	#11758185	—	4	2	no	—
73	C₂³×C₆₁	#102	1	1	yes	characteristic
74	C₂×C₄×C₆₁	—	2	2	no	—
75	C₂×C₄×C₆₁	—	2	2	no	—
76	C₂×C₄×C₆₁	#101	1	2	yes	—
77	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
78	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
79	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
80	C₂×C₄×C₆₁	#103	1	2	yes	—
81	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×Dic₆₁	—	4	2	no	—
85	C₂×Dic₆₁	—	4	2	no	—
86	C₂×Dic₆₁	—	8	2	no	—
87	C₂×Dic₆₁	—	8	2	no	—
88	C₈×C₆₁	—	2	1	no	—
89	C₈×C₆₁	—	2	1	no	—
90	C₈×C₆₁	—	8	2	no	—
91	C₈×C₆₁	#103	1	2	yes	—
92	C₈×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
93	C₈×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×#1363	—	61	1	no	maximal, Sylow
95	C₂²×C₆₁	#332	1	2	yes	—
96	C₂²×C₆₁	#359	1	2	yes	—
97	C₂²×C₆₁	#360	1	2	yes	—
98	C₂²×C₆₁	C₂×#102	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	—	2	1	no	—
103	C₄×C₆₁	#332	1	2	yes	—
104	C₄×C₆₁	C₂×#101	1	1	yes	characteristic
105	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
106	Dic₆₁	—	8	2	no	—
107	Dic₆₁	—	8	2	no	—
108	C₂×#499	—	61	1	no	—
109	C₂×C₄⋊C₁₆	—	61	1	no	—
110	C₂×#334	D₆₁	1	1	yes	characteristic
111	#1363	—	61	4	no	—
112	C₂×C₆₁	#1363	1	4	yes	complemented
113	C₂×C₆₁	C₂×#332	1	1	yes	characteristic
114	C₂×C₆₁	C₂×#359	1	1	yes	characteristic
115	C₂×C₆₁	C₂×#360	1	1	yes	characteristic
116	C₂×#125	—	122	1	no	—
117	C₂²×C₁₆	—	122	1	no	—
118	#499	—	61	2	no	—
119	#499	—	61	2	no	—
120	#499	—	122	1	no	—
121	#499	—	122	1	no	—
122	C₂×#107	—	61	1	no	—
123	C₂×#107	—	61	1	no	—
124	C₂×#106	—	122	1	no	—
125	C₂×C₄⋊C₈	—	2	1	no	—
126	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
127	C₄⋊C₁₆	—	61	2	no	—
128	C₄⋊C₁₆	—	122	1	no	—
129	#334	—	2	1	no	—
130	#334	C₂×D₆₁	1	2	yes	—
131	C₆₁	C₂×#1363	1	1	yes	characteristic, complemented, Sylow
132	C₂²×C₈	—	4	1	no	—
133	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
134	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
135	#125	—	122	2	no	—
136	#125	—	122	2	no	—
137	#125	—	244	1	no	—
138	#125	—	244	1	no	—
139	C₂×C₄⋊C₄	—	122	1	no	—
140	C₂×C₄⋊C₄	—	122	1	no	—
141	C₂×C₄⋊C₄	—	244	1	no	—
142	C₂×C₄⋊C₄	—	244	1	no	—
143	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
144	C₂×C₁₆	—	122	2	no	—
145	C₂×C₁₆	—	244	2	no	—
146	#107	—	61	2	no	—
147	#107	—	61	2	no	—
148	#107	—	122	1	no	—
149	#107	—	122	1	no	—
150	#106	—	122	2	no	—
151	#106	—	122	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	4	1	no	—
154	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
155	C₄×C₈	D₂₄₄	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
157	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
158	C₂²×C₄	—	244	1	no	—
159	C₂²×C₄	—	244	1	no	—
160	C₂²×C₄	#11758198	1	1	yes	characteristic
161	C₂²×C₄	#11758199	1	1	yes	characteristic
162	C₂²×C₄	#11758201	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	8	2	no	—
166	C₂×C₈	#11758190	1	2	yes	—
167	C₂×C₈	Dic₂₄₄	1	2	yes	—
168	C₂×C₈	#11758198	1	1	yes	characteristic
169	C₂×C₈	#11758198	1	1	yes	characteristic
170	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
171	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
172	C₄⋊C₄	—	122	2	no	—
173	C₄⋊C₄	—	122	2	no	—
174	C₄⋊C₄	—	244	1	no	—
175	C₄⋊C₄	—	244	1	no	—
176	C₄⋊C₄	—	244	2	no	—
177	C₄⋊C₄	—	244	2	no	—
178	C₄⋊C₄	—	244	2	no	—
179	C₄⋊C₄	—	244	2	no	—
180	C₄²	#11758198	1	1	yes	characteristic
181	C₄²	#11758199	1	1	yes	characteristic
182	C₄²	#11758201	1	1	yes	characteristic
183	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
184	C₁₆	—	244	2	no	—
185	C₂³	#14519697	1	1	yes	characteristic
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	488	2	no	—
189	C₂×C₄	—	488	2	no	—
190	C₂×C₄	—	244	2	no	—
191	C₂×C₄	—	244	2	no	—
192	C₂×C₄	#14519695	1	2	yes	—
193	C₂×C₄	#14519697	1	1	yes	characteristic
194	C₂×C₄	#14519697	1	1	yes	characteristic
195	C₂×C₄	#14519697	1	1	yes	characteristic
196	C₂×C₄	#14519702	1	2	yes	—
197	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
198	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
199	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	2	1	no	—
202	C₈	—	8	2	no	—
203	C₈	#14519702	1	2	yes	—
204	C₈	C₂×#11758190	1	1	yes	characteristic
205	C₈	C₂×Dic₂₄₄	1	1	yes	characteristic
206	C₂²	#16839509	1	2	yes	—
207	C₂²	#16839568	1	2	yes	—
208	C₂²	#16839571	1	2	yes	—
209	C₂²	C₂×#14519697	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	488	2	no	—
215	C₄	—	488	2	no	—
216	C₄	#16839509	1	2	yes	—
217	C₄	C₂×#14519695	1	1	yes	characteristic
218	C₄	C₂×#14519702	1	1	yes	characteristic
219	C₂	#21443872	1	4	yes	direct product factor
220	C₂	C₂×#16839509	1	1	yes	characteristic
221	C₂	C₂×#16839568	1	1	yes	characteristic
222	C₂	C₂×#16839571	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519697
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×#334	C₂
socle, prime radical	C₂³×C₆₁	#102
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	C₂×#334	D₆₁
hypocenter	C₆₁	C₂×#1363
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
3. #21443872	C₂	direct (N × H)
112. C₂×C₆₁	#1363	semidirect
131. C₆₁	C₂×#1363	semidirect
219. C₂	#21443872	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16840164	2
C₂×C₄⋊#11758185	2
#21443872	2
C₂×C₆₁×#334	2
C₂×#1363	61

Subgroups — C2×#21443872 #29591276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21443872 #29591276