Subgroups — C₂×#21462162 #30624107

Order 15616.

698 conjugacy classes of subgroups.

The subgroup lattice has 244 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	#21462162	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16840685	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	4	no	—
7	C₂×#14519851	C₂²	1	4	yes	—
8	C₂×#14519851	C₂²	1	8	yes	—
9	C₂²×#11758187	—	2	2	no	—
10	#16840685	C₂²	1	16	yes	complemented
11	C₂²×#11758194	C₄	1	1	yes	characteristic
12	C₂³×#92418	C₂²	1	1	yes	characteristic
13	#16839718	C₄	1	4	yes	—
14	#16839718	C₂²	1	4	yes	—
15	C₂×#14519688	C₄	1	2	yes	—
16	C₂×C₄×#92418	C₂²	1	2	yes	—
17	C₂×#117×C₆₁	C₄	1	1	yes	characteristic
18	#14519851	—	2	8	no	—
19	#14519851	—	2	4	no	—
20	#14519851	C₂³	1	8	yes	—
21	C₂×#11758187	—	2	4	no	—
22	C₂×#11758187	—	2	8	no	—
23	C₂×#11758194	C₈	1	4	yes	—
24	C₂×#11758194	C₈	1	4	yes	—
25	C₂×#11758194	C₂×C₄	1	4	yes	—
26	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
27	C₂²×#92418	C₂×C₄	1	2	yes	—
28	C₂²×#92418	C₂×C₄	1	2	yes	—
29	C₂²×#92418	D₄	1	2	yes	—
30	C₂²×#92418	D₄	1	2	yes	—
31	C₂²×#92418	Q₈	1	2	yes	—
32	C₂²×#92418	Q₈	1	2	yes	—
33	C₂²×#92418	C₂³	1	1	yes	characteristic
34	C₂²×#92418	C₂³	1	2	yes	—
35	#14519688	C₂×C₄	1	8	yes	—
36	C₄×#92418	C₂×C₄	1	4	yes	—
37	C₄×#92418	C₂³	1	4	yes	—
38	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
39	#117×C₆₁	—	2	2	no	—
40	#117×C₆₁	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
42	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
43	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
44	C₂⁴×C₆₁	M₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	—	2	2	no	—
46	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
52	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
54	C₄⋊C₄×C₆₁	—	2	4	no	—
55	C₂²⋊C₄×C₆₁	—	2	4	no	—
56	C₄²×C₆₁	—	2	2	no	—
57	C₄²×C₆₁	C₂²×C₄	1	4	yes	—
58	#11758194	C₂×C₈	1	16	yes	—
59	C₂×#92418	C₄⋊C₄	1	2	yes	—
60	C₂×#92418	C₄⋊C₄	1	4	yes	—
61	C₂×#92418	C₄⋊C₄	1	4	yes	—
62	C₂×#92418	C₄⋊C₄	1	2	yes	—
63	C₂×#92418	C₂×C₈	1	4	yes	—
64	C₂×#92418	C₂×C₈	1	4	yes	—
65	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
66	C₂×#92418	C₂²×C₄	1	2	yes	—
67	C₂×#92418	C₂²×C₄	1	4	yes	—
68	C₂×#92418	C₂×D₄	1	1	yes	characteristic
69	C₂×#92418	C₂×D₄	1	4	yes	—
70	C₂×#92418	C₂×D₄	1	1	yes	characteristic
71	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
72	C₂×#92418	C₂×Q₈	1	4	yes	—
73	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
74	C₂×#92418	C₂⁴	1	1	yes	characteristic
75	#11758187	—	2	8	no	—
76	C₂³×C₆₁	—	4	2	no	—
77	C₂³×C₆₁	C₄⋊C₈	1	2	yes	—
78	C₂³×C₆₁	C₄⋊C₈	1	2	yes	—
79	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
80	C₂³×C₆₁	C₂×M₁₆	1	2	yes	—
81	C₂×C₄×C₆₁	—	2	4	no	—
82	C₂×C₄×C₆₁	—	4	4	no	—
83	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
84	C₂×C₄×C₆₁	C₄⋊C₈	1	4	yes	—
85	C₂×C₄×C₆₁	C₄⋊C₈	1	4	yes	—
86	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
87	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
88	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₆₁	C₂²×C₈	1	4	yes	—
93	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
95	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
96	#92418	C₂×C₄⋊C₄	1	1	yes	characteristic
97	#92418	C₂×C₄⋊C₄	1	1	yes	characteristic
98	#92418	C₂×C₄⋊C₄	1	4	yes	—
99	#92418	C₂²×C₈	1	8	yes	—
100	#92418	C₂²×D₄	1	1	yes	characteristic
101	#92418	C₂²×Q₈	1	1	yes	characteristic
102	C₂×#2153	—	61	1	no	maximal, Sylow
103	C₂²×C₆₁	—	2	2	no	—
104	C₂²×C₆₁	—	2	4	no	—
105	C₂²×C₆₁	—	4	2	no	—
106	C₂²×C₆₁	—	4	2	no	—
107	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
108	C₂²×C₆₁	C₂×C₄⋊C₈	1	4	yes	—
109	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₂²×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
111	C₄×C₆₁	—	4	4	no	—
112	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
113	C₄×C₆₁	C₂×C₄⋊C₈	1	4	yes	—
114	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
115	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
117	C₂²×M₃₂	—	61	2	no	—
118	C₂×#405	—	61	1	no	—
119	#2153	—	61	8	no	—
120	C₂×#364	—	61	4	no	—
121	C₂×C₆₁	—	2	1	no	—
122	C₂×C₆₁	—	2	1	no	—
123	C₂×C₆₁	—	2	4	no	—
124	C₂×C₆₁	#2153	1	2	yes	complemented
125	C₂×C₆₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
126	C₂²×M₁₆	—	61	1	no	—
127	C₂³×C₈	—	61	1	no	—
128	C₂×#117	#6923	1	1	yes	characteristic
129	C₂×M₃₂	—	61	4	no	—
130	C₂×M₃₂	—	61	8	no	—
131	C₂×M₃₂	—	122	2	no	—
132	C₂×M₃₂	—	122	4	no	—
133	C₂²×C₁₆	—	122	2	no	—
134	#405	—	61	4	no	—
135	#405	—	61	4	no	—
136	C₂×#97	—	61	2	no	—
137	C₂×C₄×C₈	—	61	2	no	—
138	#364	—	61	16	no	—
139	C₆₁	C₂×#2153	1	1	yes	characteristic, complemented, Sylow
140	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
141	C₂×M₁₆	—	61	4	no	—
142	C₂×M₁₆	—	61	4	no	—
143	C₂×M₁₆	—	61	4	no	—
144	C₂²×C₈	—	61	2	no	—
145	C₂²×C₈	—	61	2	no	—
146	C₂²×C₈	—	61	2	no	—
147	C₂²×C₈	—	61	2	no	—
148	C₂²×C₈	—	61	1	no	—
149	C₂²×C₈	—	61	1	no	—
150	C₂²×C₈	—	61	2	no	—
151	C₂²×C₈	—	61	2	no	—
152	C₂²×C₈	—	61	2	no	—
153	#117	—	2	2	no	—
154	#117	C₂×#6923	1	4	yes	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₂²⋊C₄	—	2	1	no	—
157	C₂×C₄²	C₂×#6923	1	2	yes	—
158	M₃₂	—	61	8	no	—
159	M₃₂	—	122	8	no	—
160	M₃₂	—	122	4	no	—
161	C₂×C₁₆	—	122	8	no	—
162	C₂×C₁₆	—	122	4	no	—
163	#97	—	61	8	no	—
164	C₄×C₈	—	61	4	no	—
165	C₄×C₈	—	61	4	no	—
166	C₂⁴	#11758217	1	1	yes	characteristic
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	D₆₁⋊C₈	1	2	yes	—
169	C₂²×C₄	D₆₁⋊C₈	1	2	yes	—
170	C₂²×C₄	C₄⋊#6923	1	2	yes	—
171	C₂²×C₄	C₄⋊#6923	1	2	yes	—
172	C₂²×C₄	#11758217	1	1	yes	characteristic
173	C₂²×C₄	#11758217	1	2	yes	—
174	C₂²×C₄	C₂²×#6923	1	2	yes	—
175	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
176	M₁₆	—	61	16	no	—
177	C₂×C₈	—	61	4	no	—
178	C₂×C₈	—	61	1	no	—
179	C₂×C₈	—	61	4	no	—
180	C₂×C₈	—	61	2	no	—
181	C₂×C₈	—	61	4	no	—
182	C₂×C₈	—	61	1	no	—
183	C₂×C₈	—	61	4	no	—
184	C₂×C₈	—	61	1	no	—
185	C₂×C₈	—	61	2	no	—
186	C₂×C₈	—	61	1	no	—
187	C₂×C₈	—	61	4	no	—
188	C₂×C₈	—	61	4	no	—
189	C₂×C₈	—	61	1	no	—
190	C₂×C₈	—	61	1	no	—
191	C₂×C₈	—	61	4	no	—
192	C₂×C₈	—	61	2	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₂²⋊C₄	—	2	4	no	—
195	C₄²	—	2	2	no	—
196	C₄²	C₂²×#6923	1	4	yes	—
197	C₁₆	—	122	8	no	—
198	C₂³	—	4	2	no	—
199	C₂³	#14519844	1	2	yes	—
200	C₂³	#14519844	1	2	yes	—
201	C₂³	C₂×#11758217	1	1	yes	characteristic
202	C₂³	C₂×#11758217	1	2	yes	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	#14519844	1	2	yes	—
206	C₂×C₄	#14519844	1	4	yes	—
207	C₂×C₄	#14519844	1	2	yes	—
208	C₂×C₄	#14519844	1	4	yes	—
209	C₂×C₄	C₂×C₄⋊#6923	1	4	yes	—
210	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
212	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
213	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
214	C₂×C₄	C₂×D₆₁⋊C₈	1	4	yes	—
215	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
216	C₂×C₄	C₂×#11758217	1	2	yes	—
217	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
218	C₈	—	61	4	no	—
219	C₈	—	61	1	no	—
220	C₈	—	61	1	no	—
221	C₈	—	61	1	no	—
222	C₈	—	61	1	no	—
223	C₈	—	61	8	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	C₂×#14519844	1	1	yes	characteristic
229	C₂²	C₂×#14519844	1	4	yes	—
230	C₂²	C₂×#14519844	1	1	yes	characteristic
231	C₂²	C₂²×#11758217	1	1	yes	characteristic
232	C₄	—	4	4	no	—
233	C₄	C₂×#14519844	1	1	yes	characteristic
234	C₄	C₂×#14519844	1	4	yes	—
235	C₄	C₂×#14519844	1	1	yes	characteristic
236	C₄	C₂²×C₄⋊#6923	1	1	yes	characteristic
237	C₄	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
238	C₂	—	2	1	no	—
239	C₂	—	2	1	no	—
240	C₂	—	2	4	no	—
241	C₂	#21462162	1	2	yes	direct product factor
242	C₂	C₂²×#14519844	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#6923
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₂×#117×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₄⋊C₈
prime radical	C₂⁴×C₆₁	M₁₆
cosocle	C₂×#92418	C₂⁴
hypercenter	C₂×#117	#6923
hypocenter	C₆₁	C₂×#2153
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. #21462162	C₂	direct (N × H)
3. C₂×#16840685	C₂	semidirect
10. #16840685	C₂²	semidirect
124. C₂×C₆₁	#2153	semidirect
139. C₆₁	C₂×#2153	semidirect
241. C₂	#21462162	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×#14519851	2
#21462162	2
C₂×#16840685	2
C₂×#16839718	2
C₂×#2153	61

Subgroups — C2×#21462162 #30624107

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21462162 #30624107