Subgroups — C₂²×C₂²⋊#63216 #17978167

Order 4352.

1470 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#63216	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊#63216	C₂	1	12	yes	maximal, direct product factor
3	C₂⁴×#3622	C₂	1	1	yes	maximal, characteristic
4	C₂²×#63216	—	2	8	no	—
5	C₂²×#63216	C₂²	1	12	yes	complemented
6	C₂²⋊#63216	C₂²	1	16	yes	direct product factor
7	C₂³×#3622	—	2	8	no	—
8	C₂³×#3622	C₄	1	6	yes	—
9	C₂³×#3622	C₄	1	1	yes	characteristic
10	C₂³×#3622	C₂²	1	6	yes	—
11	C₂³×#3622	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₁₇	C₄	1	1	yes	characteristic
13	C₂⁵×C₁₇	C₈	1	1	yes	characteristic
14	C₂³×C₄×C₁₇	—	2	4	no	—
15	C₂³×C₄×C₁₇	—	2	4	no	—
16	C₂³×C₄×C₁₇	C₈	1	1	yes	characteristic
17	C₂³×C₄×C₁₇	C₈	1	6	yes	—
18	C₂³×C₄×C₁₇	C₂×C₄	1	6	yes	—
19	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×#3622	—	2	24	no	—
21	C₂²×#3622	—	2	24	no	—
22	C₂²×#3622	—	2	4	no	—
23	C₂²×#3622	C₂×C₄	1	12	yes	—
24	C₂²×#3622	C₂×C₄	1	6	yes	—
25	C₂²×#3622	C₂×C₄	1	3	yes	—
26	C₂²×#3622	D₄	1	8	yes	—
27	C₂²×#3622	C₂³	1	4	yes	—
28	C₂²×#3622	C₂³	1	3	yes	—
29	C₂×#63216	—	2	24	no	—
30	C₂×#63216	C₂³	1	8	yes	complemented
31	C₂⁴×C₁₇	—	2	4	no	—
32	C₂⁴×C₁₇	—	2	4	no	—
33	C₂⁴×C₁₇	C₁₆	1	8	yes	complemented
34	C₂⁴×C₁₇	C₂×C₈	1	6	yes	—
35	C₂⁴×C₁₇	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₇	—	2	4	no	—
37	C₂²×C₄×C₁₇	—	2	24	no	—
38	C₂²×C₄×C₁₇	—	2	12	no	—
39	C₂²×C₄×C₁₇	—	2	12	no	—
40	C₂²×C₄×C₁₇	C₂²⋊C₄	1	4	yes	—
41	C₂²×C₄×C₁₇	C₂×C₈	1	6	yes	—
42	C₂²×C₄×C₁₇	C₂×C₈	1	12	yes	—
43	C₂²×C₄×C₁₇	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₁₇	M₁₆	1	4	yes	—
45	C₂²×C₄×C₁₇	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
47	C₂×#3622	—	2	12	no	—
48	C₂×#3622	—	2	24	no	—
49	C₂×#3622	—	2	8	no	—
50	C₂×#3622	C₂²⋊C₄	1	12	yes	—
51	C₂×#3622	C₂²×C₄	1	4	yes	—
52	C₂×#3622	C₂²×C₄	1	3	yes	—
53	C₂×#3622	C₂×D₄	1	12	yes	—
54	C₂×#3622	C₂⁴	1	1	yes	characteristic
55	#63216	—	2	8	no	—
56	C₂²×C₂²⋊C₁₆	—	17	1	no	maximal, Sylow
57	C₂³×C₁₇	—	2	32	no	—
58	C₂³×C₁₇	—	2	12	no	—
59	C₂³×C₁₇	—	2	12	no	—
60	C₂³×C₁₇	C₂²⋊C₈	1	4	yes	—
61	C₂³×C₁₇	C₂×C₁₆	1	4	yes	—
62	C₂³×C₁₇	C₂×C₁₆	1	24	yes	complemented
63	C₂³×C₁₇	M₃₂	1	4	yes	—
64	C₂³×C₁₇	C₂²×C₈	1	4	yes	—
65	C₂³×C₁₇	C₂²×C₈	1	3	yes	—
66	C₂×C₄×C₁₇	—	2	12	no	—
67	C₂×C₄×C₁₇	—	2	4	no	—
68	C₂×C₄×C₁₇	—	2	4	no	—
69	C₂×C₄×C₁₇	—	2	24	no	—
70	C₂×C₄×C₁₇	C₂²⋊C₈	1	12	yes	—
71	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	6	yes	—
72	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
73	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
74	C₂×C₄×C₁₇	C₂×M₁₆	1	6	yes	—
75	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
76	#3622	—	2	4	no	—
77	#3622	C₂×C₂²⋊C₄	1	6	yes	—
78	#3622	C₂²×D₄	1	2	yes	—
79	C₂⁴×C₈	—	17	1	no	—
80	C₂³×C₁₆	—	17	2	no	—
81	C₂×C₂²⋊C₁₆	—	17	12	no	—
82	C₂²×C₁₇	—	2	4	no	—
83	C₂²×C₁₇	—	2	48	no	—
84	C₂²×C₁₇	—	2	4	no	—
85	C₂²×C₁₇	C₂²⋊C₁₆	1	16	yes	complemented
86	C₂²×C₁₇	C₂×C₂²⋊C₈	1	6	yes	—
87	C₂²×C₁₇	C₂²×C₁₆	1	6	yes	—
88	C₂²×C₁₇	C₂²×C₁₆	1	8	yes	complemented
89	C₂²×C₁₇	C₂×M₃₂	1	6	yes	—
90	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
91	C₄×C₁₇	—	2	4	no	—
92	C₄×C₁₇	C₂×C₂²⋊C₈	1	6	yes	—
93	C₄×C₁₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
94	C₄×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
95	C₂⁴×C₄	#595	1	1	yes	characteristic
96	C₂³×C₈	—	17	1	no	—
97	C₂³×C₈	—	17	6	no	—
98	C₂³×C₈	—	17	6	no	—
99	C₂³×C₈	—	17	1	no	—
100	C₂³×C₈	—	34	8	no	—
101	C₂²×C₁₆	—	17	12	no	—
102	C₂²×C₁₆	—	34	8	no	—
103	C₂²⋊C₁₆	—	17	16	no	—
104	C₂×C₁₇	—	2	8	no	—
105	C₂×C₁₇	C₂×C₂²⋊C₁₆	1	12	yes	complemented
106	C₂×C₁₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
107	C₂×C₁₇	C₂³×C₁₆	1	1	yes	characteristic
108	C₂×C₁₇	C₂²×M₃₂	1	1	yes	characteristic
109	C₂⁵	#3624	1	1	yes	characteristic
110	C₂³×C₄	—	2	4	no	—
111	C₂³×C₄	—	2	4	no	—
112	C₂³×C₄	#3624	1	1	yes	characteristic
113	C₂³×C₄	#3624	1	6	yes	—
114	C₂³×C₄	C₂×#595	1	6	yes	—
115	C₂³×C₄	C₂×#595	1	1	yes	characteristic
116	C₂²×C₈	—	17	8	no	—
117	C₂²×C₈	—	17	3	no	—
118	C₂²×C₈	—	17	3	no	—
119	C₂²×C₈	—	17	4	no	—
120	C₂²×C₈	—	17	12	no	—
121	C₂²×C₈	—	17	6	no	—
122	C₂²×C₈	—	34	24	no	—
123	C₂²×C₈	—	34	24	no	—
124	C₂²×C₈	—	34	4	no	—
125	C₂×C₁₆	—	17	8	no	—
126	C₂×C₁₆	—	34	24	no	—
127	C₁₇	C₂²×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
128	C₂⁴	—	2	4	no	—
129	C₂⁴	—	2	4	no	—
130	C₂⁴	#63216	1	8	yes	complemented
131	C₂⁴	C₂×#3624	1	6	yes	—
132	C₂⁴	C₂×#3624	1	1	yes	characteristic
133	C₂²×C₄	—	2	4	no	—
134	C₂²×C₄	—	2	12	no	—
135	C₂²×C₄	—	2	12	no	—
136	C₂²×C₄	—	2	24	no	—
137	C₂²×C₄	C₂×#3624	1	6	yes	—
138	C₂²×C₄	C₂×#3624	1	12	yes	—
139	C₂²×C₄	C₂×#3624	1	3	yes	—
140	C₂²×C₄	#63247	1	4	yes	—
141	C₂²×C₄	C₂²⋊#595	1	4	yes	—
142	C₂²×C₄	C₂²×#595	1	4	yes	—
143	C₂²×C₄	C₂²×#595	1	3	yes	—
144	C₂×C₈	—	17	12	no	—
145	C₂×C₈	—	17	12	no	—
146	C₂×C₈	—	17	1	no	—
147	C₂×C₈	—	17	3	no	—
148	C₂×C₈	—	17	4	no	—
149	C₂×C₈	—	34	8	no	—
150	C₂×C₈	—	34	24	no	—
151	C₂×C₈	—	34	12	no	—
152	C₁₆	—	34	8	no	—
153	C₂³	—	2	12	no	—
154	C₂³	—	2	12	no	—
155	C₂³	—	2	32	no	—
156	C₂³	C₂×#63216	1	4	yes	—
157	C₂³	C₂×#63216	1	24	yes	complemented
158	C₂³	#10587538	1	4	yes	—
159	C₂³	C₂²⋊#3624	1	4	yes	—
160	C₂³	C₂²×#3624	1	4	yes	—
161	C₂³	C₂²×#3624	1	3	yes	—
162	C₂×C₄	—	2	12	no	—
163	C₂×C₄	—	2	24	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	C₂²⋊#3624	1	12	yes	—
167	C₂×C₄	C₂²×#3624	1	4	yes	—
168	C₂×C₄	C₂²×#3624	1	3	yes	—
169	C₂×C₄	C₂×#63247	1	6	yes	—
170	C₂×C₄	C₂×C₂²⋊#595	1	6	yes	—
171	C₂×C₄	C₂³×#595	1	1	yes	characteristic
172	C₈	—	17	2	no	—
173	C₈	—	17	6	no	—
174	C₈	—	34	4	no	—
175	C₂²	—	2	4	no	—
176	C₂²	—	2	48	no	—
177	C₂²	—	2	4	no	—
178	C₂²	C₂²⋊#63216	1	16	yes	direct product factor
179	C₂²	C₂×C₂²⋊#3624	1	6	yes	—
180	C₂²	C₂²×#63216	1	6	yes	—
181	C₂²	C₂²×#63216	1	8	yes	complemented
182	C₂²	C₂×#10587538	1	6	yes	—
183	C₂²	C₂³×#3624	1	1	yes	characteristic
184	C₄	—	2	4	no	—
185	C₄	C₂×C₂²⋊#3624	1	6	yes	—
186	C₄	C₂²×C₂²⋊#595	1	1	yes	characteristic
187	C₄	C₂²×#63247	1	1	yes	characteristic
188	C₂	—	2	8	no	—
189	C₂	C₂×C₂²⋊#63216	1	12	yes	direct product factor
190	C₂	C₂²×C₂²⋊#3624	1	1	yes	characteristic
191	C₂	C₂³×#63216	1	1	yes	characteristic
192	C₂	C₂²×#10587538	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×#595
commutator	C₂×C₁₇	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×#595
Fitting	C₂⁴×C₄×C₁₇	C₄
socle	C₂⁴×C₁₇	C₂×C₈
prime radical	C₂⁵×C₁₇	C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	C₂⁴×C₄	#595
hypocenter	C₁₇	C₂²×C₂²⋊C₁₆
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#63216	C₂	semidirect
2. C₂×C₂²⋊#63216	C₂	direct (N × H)
5. C₂²×#63216	C₂²	semidirect
6. C₂²⋊#63216	C₂²	direct (N × H)
30. C₂×#63216	C₂³	semidirect
33. C₂⁴×C₁₇	C₁₆	semidirect
62. C₂³×C₁₇	C₂×C₁₆	semidirect
85. C₂²×C₁₇	C₂²⋊C₁₆	semidirect
88. C₂²×C₁₇	C₂²×C₁₆	semidirect
105. C₂×C₁₇	C₂×C₂²⋊C₁₆	semidirect
127. C₁₇	C₂²×C₂²⋊C₁₆	semidirect
130. C₂⁴	#63216	semidirect
157. C₂³	C₂×#63216	semidirect
178. C₂²	C₂²⋊#63216	direct (N × H)
181. C₂²	C₂²×#63216	semidirect
189. C₂	C₂×C₂²⋊#63216	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#63216	2
C₂×C₂²⋊#63216	2
C₂⁴×#3622	2
C₂²×C₂²⋊C₁₆	17

Subgroups — C22×C22⋊#63216 #17978167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂²⋊#63216 #17978167