Subgroups — C₂²×C₄×#63242 #17979400

Order 4352.

4638 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#63242	C₂	1	2	yes	maximal
2	C₂×C₄×#63242	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄×#3624	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄²×D₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63242	C₄	1	32	yes	direct product factor
6	C₂²×#63242	C₂²	1	24	yes	—
7	C₂³×#3624	C₂²	1	4	yes	—
8	C₄×#63242	C₂²	1	64	yes	direct product factor
9	C₂×C₄×#3624	C₂²	1	48	yes	complemented
10	C₂³×C₄×D₁₇	C₄	1	2	yes	—
11	C₂³×C₄×D₁₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄×Dic₁₇	C₄	1	1	yes	characteristic
13	C₂²×C₄×Dic₁₇	C₂²	1	2	yes	—
14	C₂×C₄²×D₁₇	C₄	1	12	yes	—
15	C₂×C₄²×D₁₇	C₂²	1	12	yes	—
16	C₂²×C₄²×C₁₇	C₄	1	1	yes	characteristic
17	C₂⁴×D₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₁₇	C₂×C₄	1	2	yes	—
20	C₂³×Dic₁₇	C₂×C₄	1	2	yes	—
21	C₂³×Dic₁₇	C₂³	1	1	yes	characteristic
22	C₂²×C₄×D₁₇	C₈	1	32	yes	complemented
23	C₂²×C₄×D₁₇	C₂×C₄	1	6	yes	—
24	C₂²×C₄×D₁₇	C₂×C₄	1	16	yes	—
25	C₂²×C₄×D₁₇	C₂×C₄	1	24	yes	—
26	C₂²×C₄×D₁₇	C₂³	1	6	yes	—
27	C₂²×#3624	C₂×C₄	1	32	yes	complemented
28	C₂²×#3624	C₂³	1	24	yes	—
29	C₂×#63242	C₂×C₄	1	192	yes	direct product factor
30	C₂×#63242	C₂³	1	32	yes	—
31	C₂×C₄²×C₁₇	C₂×C₄	1	12	yes	—
32	C₂×C₄×Dic₁₇	C₂×C₄	1	12	yes	—
33	C₂×C₄×Dic₁₇	C₂×C₄	1	12	yes	—
34	C₂×C₄×Dic₁₇	C₂³	1	12	yes	—
35	C₄²×D₁₇	C₂×C₄	1	48	yes	—
36	C₄²×D₁₇	C₂³	1	16	yes	—
37	C₄×#3624	C₂³	1	64	yes	complemented
38	C₂⁴×C₁₇	C₂²×C₄	1	1	yes	characteristic
39	C₂³×D₁₇	C₄²	1	8	yes	—
40	C₂³×D₁₇	C₂×C₈	1	16	yes	—
41	C₂³×D₁₇	C₂²×C₄	1	6	yes	—
42	C₂²×C₄×C₁₇	C₄²	1	8	yes	—
43	C₂²×C₄×C₁₇	C₂×C₈	1	16	yes	complemented
44	C₂²×C₄×C₁₇	C₂²×C₄	1	6	yes	—
45	C₂²×C₄×C₁₇	C₂²×C₄	1	12	yes	—
46	C₂²×Dic₁₇	C₂×C₈	1	16	yes	complemented
47	C₂²×Dic₁₇	C₂²×C₄	1	3	yes	—
48	C₂²×Dic₁₇	C₂²×C₄	1	8	yes	—
49	C₂²×Dic₁₇	C₂²×C₄	1	12	yes	—
50	C₂²×Dic₁₇	C₂⁴	1	3	yes	—
51	C₂×C₄×D₁₇	C₄²	1	48	yes	—
52	C₂×C₄×D₁₇	C₂×C₈	1	192	yes	complemented
53	C₂×C₄×D₁₇	C₂²×C₄	1	12	yes	—
54	C₂×C₄×D₁₇	C₂²×C₄	1	48	yes	—
55	C₂×C₄×D₁₇	C₂²×C₄	1	32	yes	—
56	C₂×C₄×D₁₇	C₂⁴	1	4	yes	—
57	C₂×#3624	C₂²×C₄	1	96	yes	complemented
58	C₂×#3624	C₂⁴	1	16	yes	—
59	#63242	C₂²×C₄	1	128	yes	direct product factor
60	C₄²×C₁₇	C₂²×C₄	1	16	yes	—
61	C₄×Dic₁₇	C₂²×C₄	1	24	yes	—
62	C₄×Dic₁₇	C₂²×C₄	1	16	yes	—
63	C₄×Dic₁₇	C₂⁴	1	8	yes	—
64	C₂³×C₄×C₈	—	17	1	no	maximal, Sylow
65	C₂³×C₁₇	C₂×C₄²	1	4	yes	—
66	C₂³×C₁₇	C₂²×C₈	1	8	yes	—
67	C₂³×C₁₇	C₂³×C₄	1	3	yes	—
68	C₂²×D₁₇	C₄×C₈	1	64	yes	complemented
69	C₂²×D₁₇	C₂×C₄²	1	24	yes	—
70	C₂²×D₁₇	C₂²×C₈	1	48	yes	—
71	C₂²×D₁₇	C₂³×C₄	1	4	yes	—
72	C₂×C₄×C₁₇	C₂×C₄²	1	24	yes	—
73	C₂×C₄×C₁₇	C₂²×C₈	1	48	yes	complemented
74	C₂×C₄×C₁₇	C₂³×C₄	1	8	yes	—
75	C₂×C₄×C₁₇	C₂³×C₄	1	4	yes	—
76	C₂×Dic₁₇	C₂×C₄²	1	12	yes	—
77	C₂×Dic₁₇	C₂²×C₈	1	48	yes	complemented
78	C₂×Dic₁₇	C₂³×C₄	1	8	yes	—
79	C₂×Dic₁₇	C₂³×C₄	1	12	yes	—
80	C₂×Dic₁₇	C₂³×C₄	1	3	yes	—
81	C₂×Dic₁₇	C₂⁵	1	1	yes	characteristic
82	C₄×D₁₇	C₂×C₄²	1	48	yes	—
83	C₄×D₁₇	C₂²×C₈	1	128	yes	complemented
84	C₄×D₁₇	C₂³×C₄	1	16	yes	—
85	#3624	C₂³×C₄	1	32	yes	complemented
86	C₂⁴×C₈	—	17	2	no	—
87	C₂³×C₄²	—	17	1	no	—
88	C₂²×C₄×C₈	—	17	4	no	—
89	C₂²×C₄×C₈	—	17	24	no	—
90	C₂²×C₁₇	C₂×C₄×C₈	1	16	yes	complemented
91	C₂²×C₁₇	C₂²×C₄²	1	6	yes	—
92	C₂²×C₁₇	C₂³×C₈	1	12	yes	—
93	C₂²×C₁₇	C₂⁴×C₄	1	1	yes	characteristic
94	C₂×D₁₇	C₂×C₄×C₈	1	96	yes	complemented
95	C₂×D₁₇	C₂²×C₄²	1	8	yes	—
96	C₂×D₁₇	C₂³×C₈	1	16	yes	—
97	C₄×C₁₇	C₂²×C₄²	1	8	yes	—
98	C₄×C₁₇	C₂³×C₈	1	16	yes	complemented
99	Dic₁₇	C₂²×C₄²	1	6	yes	—
100	Dic₁₇	C₂³×C₈	1	16	yes	complemented
101	Dic₁₇	C₂⁴×C₄	1	2	yes	—
102	C₂⁴×C₄	—	17	2	no	—
103	C₂⁴×C₄	—	17	1	no	—
104	C₂³×C₈	—	17	4	no	—
105	C₂³×C₈	—	17	24	no	—
106	C₂³×C₈	—	17	32	no	—
107	C₂²×C₄²	—	17	2	no	—
108	C₂²×C₄²	—	17	1	no	—
109	C₂²×C₄²	—	17	12	no	—
110	C₂²×C₄²	—	17	12	no	—
111	C₂²×C₄²	#595	1	1	yes	characteristic
112	C₂×C₄×C₈	—	17	48	no	—
113	C₂×C₄×C₈	—	17	64	no	—
114	C₂×C₁₇	C₂²×C₄×C₈	1	12	yes	complemented
115	C₂×C₁₇	C₂³×C₄²	1	1	yes	characteristic
116	C₂×C₁₇	C₂⁴×C₈	1	2	yes	—
117	D₁₇	C₂²×C₄×C₈	1	16	yes	complemented
118	C₂⁵	—	17	1	no	—
119	C₂³×C₄	—	17	2	no	—
120	C₂³×C₄	—	17	1	no	—
121	C₂³×C₄	—	17	32	no	—
122	C₂³×C₄	—	17	16	no	—
123	C₂³×C₄	—	17	24	no	—
124	C₂³×C₄	—	17	6	no	—
125	C₂³×C₄	—	17	6	no	—
126	C₂³×C₄	C₂×#595	1	2	yes	—
127	C₂³×C₄	C₂×#595	1	1	yes	characteristic
128	C₂²×C₈	—	17	192	no	—
129	C₂²×C₈	—	17	32	no	—
130	C₂²×C₈	—	17	32	no	—
131	C₂²×C₈	—	17	24	no	—
132	C₂×C₄²	—	17	12	no	—
133	C₂×C₄²	—	17	12	no	—
134	C₂×C₄²	—	17	12	no	—
135	C₂×C₄²	—	17	16	no	—
136	C₂×C₄²	—	17	48	no	—
137	C₂×C₄²	C₂×#595	1	12	yes	—
138	C₄×C₈	—	17	64	no	—
139	C₁₇	C₂³×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
140	C₂⁴	—	17	16	no	—
141	C₂⁴	—	17	8	no	—
142	C₂⁴	—	17	6	no	—
143	C₂⁴	C₂²×#595	1	1	yes	characteristic
144	C₂²×C₄	—	17	16	no	—
145	C₂²×C₄	—	17	8	no	—
146	C₂²×C₄	—	17	3	no	—
147	C₂²×C₄	—	17	12	no	—
148	C₂²×C₄	—	17	3	no	—
149	C₂²×C₄	—	17	192	no	—
150	C₂²×C₄	—	17	48	no	—
151	C₂²×C₄	—	17	48	no	—
152	C₂²×C₄	—	17	12	no	—
153	C₂²×C₄	—	17	32	no	—
154	C₂²×C₄	—	17	4	no	—
155	C₂²×C₄	#63242	1	16	yes	direct product factor
156	C₂²×C₄	C₄×#595	1	8	yes	—
157	C₂²×C₄	C₂²×#595	1	12	yes	—
158	C₂²×C₄	C₂²×#595	1	6	yes	—
159	C₂×C₈	—	17	128	no	—
160	C₂×C₈	—	17	96	no	—
161	C₂×C₈	—	17	16	no	—
162	C₄²	—	17	8	no	—
163	C₄²	—	17	16	no	—
164	C₄²	—	17	24	no	—
165	C₄²	C₂²×#595	1	16	yes	—
166	C₂³	—	17	64	no	—
167	C₂³	—	17	24	no	—
168	C₂³	—	17	48	no	—
169	C₂³	—	17	4	no	—
170	C₂³	C₂×#63242	1	8	yes	—
171	C₂³	C₂×C₄×#595	1	4	yes	—
172	C₂³	C₂³×#595	1	3	yes	—
173	C₂×C₄	—	17	12	no	—
174	C₂×C₄	—	17	12	no	—
175	C₂×C₄	—	17	48	no	—
176	C₂×C₄	—	17	1	no	—
177	C₂×C₄	—	17	3	no	—
178	C₂×C₄	—	17	8	no	—
179	C₂×C₄	—	17	16	no	—
180	C₂×C₄	—	17	48	no	—
181	C₂×C₄	—	17	128	no	—
182	C₂×C₄	C₂×#63242	1	48	yes	direct product factor
183	C₂×C₄	C₂×C₄×#595	1	24	yes	—
184	C₂×C₄	C₂³×#595	1	4	yes	—
185	C₂×C₄	C₂³×#595	1	8	yes	—
186	C₈	—	17	32	no	—
187	C₂²	—	17	8	no	—
188	C₂²	—	17	16	no	—
189	C₂²	—	17	96	no	—
190	C₂²	C₄×#63242	1	16	yes	direct product factor
191	C₂²	C₂²×C₄×#595	1	6	yes	—
192	C₂²	C₂²×#63242	1	12	yes	—
193	C₂²	C₂⁴×#595	1	1	yes	characteristic
194	C₄	—	17	2	no	—
195	C₄	—	17	6	no	—
196	C₄	—	17	16	no	—
197	C₄	C₂²×C₄×#595	1	8	yes	—
198	C₄	C₂²×#63242	1	16	yes	direct product factor
199	C₂	—	17	16	no	—
200	C₂	C₂×C₄×#63242	1	12	yes	direct product factor
201	C₂	C₂³×C₄×#595	1	1	yes	characteristic
202	C₂	C₂³×#63242	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₄²	#595
commutator, hypocenter	C₁₇	C₂³×C₄×C₈
Frattini	C₂²	C₂⁴×#595
Fitting	C₂²×C₄²×C₁₇	C₄
socle	C₂⁴×C₁₇	C₂²×C₄
prime radical	C₂⁴×D₁₇	C₂×C₄
cosocle	C₂×Dic₁₇	C₂⁵
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#63242	C₂	direct (N × H)
3. C₂²×C₄×#3624	C₂	semidirect
5. C₂²×#63242	C₄	direct (N × H)
8. C₄×#63242	C₂²	direct (N × H)
9. C₂×C₄×#3624	C₂²	semidirect
22. C₂²×C₄×D₁₇	C₈	semidirect
27. C₂²×#3624	C₂×C₄	semidirect
29. C₂×#63242	C₂×C₄	direct (N × H)
37. C₄×#3624	C₂³	semidirect
43. C₂²×C₄×C₁₇	C₂×C₈	semidirect
46. C₂²×Dic₁₇	C₂×C₈	semidirect
52. C₂×C₄×D₁₇	C₂×C₈	semidirect
57. C₂×#3624	C₂²×C₄	semidirect
59. #63242	C₂²×C₄	direct (N × H)
68. C₂²×D₁₇	C₄×C₈	semidirect
73. C₂×C₄×C₁₇	C₂²×C₈	semidirect
77. C₂×Dic₁₇	C₂²×C₈	semidirect
83. C₄×D₁₇	C₂²×C₈	semidirect
85. #3624	C₂³×C₄	semidirect
90. C₂²×C₁₇	C₂×C₄×C₈	semidirect
94. C₂×D₁₇	C₂×C₄×C₈	semidirect
98. C₄×C₁₇	C₂³×C₈	semidirect
100. Dic₁₇	C₂³×C₈	semidirect
114. C₂×C₁₇	C₂²×C₄×C₈	semidirect
117. D₁₇	C₂²×C₄×C₈	semidirect
139. C₁₇	C₂³×C₄×C₈	semidirect
155. C₂²×C₄	#63242	direct (N × H)
182. C₂×C₄	C₂×#63242	direct (N × H)
190. C₂²	C₄×#63242	direct (N × H)
198. C₄	C₂²×#63242	direct (N × H)
200. C₂	C₂×C₄×#63242	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×#63242	2
C₂×C₄×#63242	2
C₂²×C₄×#3624	2
C₂²×C₄²×D₁₇	2
C₂³×C₄×C₈	17

Subgroups — C22×C4×#63242 #17979400

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×#63242 #17979400