Subgroups — #17978790

Order 4352.

570 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#63216	C₂	1	2	yes	maximal, complemented
2	D₄×#63216	C₂	1	6	yes	maximal, complemented
3	#14545141	C₂	1	3	yes	maximal, complemented
4	C₂×C₄⋊#63216	C₂	1	3	yes	maximal, complemented
5	#41×#3622	C₂	1	1	yes	maximal, characteristic
6	C₂²×#63216	C₂²	1	6	yes	—
7	C₄⋊#63216	C₂²	1	6	yes	—
8	C₄⋊#63216	C₂²	1	3	yes	—
9	C₄⋊#63216	C₂²	1	1	yes	characteristic
10	C₂²⋊#63216	C₂²	1	6	yes	—
11	C₄×#63216	C₂²	1	6	yes	—
12	Q₈×#3622	C₄	1	1	yes	characteristic
13	Q₈×#3622	C₂²	1	1	yes	characteristic
14	D₄×#3622	C₄	1	3	yes	—
15	D₄×#3622	C₂²	1	3	yes	—
16	#11762862	C₄	1	3	yes	—
17	C₂×C₄×#3622	C₂²	1	3	yes	—
18	C₄×#41×C₁₇	C₄	1	1	yes	characteristic
19	C₂×#41×C₁₇	C₈	1	1	yes	characteristic
20	C₄×Q₈×C₁₇	C₈	1	1	yes	characteristic
21	C₄×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₄×D₄×C₁₇	C₈	1	3	yes	—
23	C₄×D₄×C₁₇	C₂×C₄	1	3	yes	—
24	C₁₇×#117	C₈	1	3	yes	—
25	C₂×C₄²×C₁₇	C₂×C₄	1	3	yes	—
26	C₂²×#3622	C₂×C₄	1	3	yes	—
27	C₂²×#3622	C₂³	1	3	yes	—
28	C₂×#63216	—	4	6	no	—
29	C₂×#63216	C₂³	1	6	yes	—
30	C₂×#63216	C₂³	1	2	yes	—
31	C₂²⋊#3622	C₂×C₄	1	6	yes	—
32	C₄⋊#3622	C₂×C₄	1	3	yes	—
33	C₄⋊#3622	C₂×C₄	1	3	yes	—
34	C₄×#3622	C₂×C₄	1	3	yes	—
35	C₄×#3622	C₂×C₄	1	3	yes	—
36	C₄×#3622	C₂³	1	1	yes	characteristic
37	C₄×#3622	C₂³	1	3	yes	—
38	#41×C₁₇	C₁₆	1	8	yes	complemented
39	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
40	C₂×D₄×C₁₇	C₂×C₈	1	3	yes	—
41	C₂²×C₄×C₁₇	C₂×C₈	1	3	yes	—
42	C₂²×C₄×C₁₇	C₂×C₈	1	3	yes	—
43	C₂²×C₄×C₁₇	C₂²×C₄	1	3	yes	—
44	C₄⋊C₄×C₁₇	C₂×C₈	1	3	yes	—
45	C₄⋊C₄×C₁₇	C₂×C₈	1	3	yes	—
46	C₂²⋊C₄×C₁₇	C₂×C₈	1	6	yes	—
47	C₄²×C₁₇	C₂×C₈	1	3	yes	—
48	C₄²×C₁₇	C₂×C₈	1	3	yes	—
49	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₁₇	C₂²×C₄	1	3	yes	—
51	C₂×#3622	—	2	3	no	—
52	C₂×#3622	—	2	3	no	—
53	C₂×#3622	—	4	3	no	—
54	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
55	C₂×#3622	C₂²×C₄	1	3	yes	—
56	C₂×#3622	C₂²×C₄	1	3	yes	—
57	C₂×#3622	C₂⁴	1	1	yes	characteristic
58	#63216	—	2	6	no	—
59	#63216	—	2	2	no	—
60	#33502	—	17	1	no	maximal, Sylow
61	C₂³×C₁₇	C₂²×C₈	1	3	yes	—
62	Q₈×C₁₇	C₂×C₁₆	1	4	yes	complemented
63	D₄×C₁₇	C₂×C₁₆	1	12	yes	complemented
64	C₂×C₄×C₁₇	—	2	3	no	—
65	C₂×C₄×C₁₇	—	2	3	no	—
66	C₂×C₄×C₁₇	—	4	3	no	—
67	C₂×C₄×C₁₇	C₂×C₁₆	1	12	yes	complemented
68	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
70	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
71	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
73	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
74	#3622	—	2	3	no	—
75	#3622	—	2	1	no	—
76	#3622	—	2	3	no	—
77	#3622	H(5,𝔽₂)	1	1	yes	characteristic
78	#3622	2₋¹⁺⁴	1	1	yes	characteristic
79	C₈×#41	—	17	1	no	—
80	Q₈×C₁₆	—	17	2	no	—
81	D₄×C₁₆	—	17	6	no	—
82	#2162	—	17	3	no	—
83	C₂×C₄⋊C₁₆	—	17	3	no	—
84	C₂²×C₁₇	—	2	3	no	—
85	C₂²×C₁₇	—	2	3	no	—
86	C₂²×C₁₇	C₂²×C₁₆	1	6	yes	—
87	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
88	C₄×C₁₇	—	2	1	no	—
89	C₄×C₁₇	—	2	3	no	—
90	C₄×C₁₇	—	2	3	no	—
91	C₄×C₁₇	C₂²×C₁₆	1	2	yes	—
92	C₄×C₁₇	C₂²×C₁₆	1	6	yes	—
93	C₄×C₁₇	#520	1	1	yes	characteristic
94	C₄×C₁₇	#568	1	1	yes	characteristic
95	C₄×#41	#595	1	1	yes	characteristic
96	C₂²×C₁₆	—	17	6	no	—
97	C₈×Q₈	—	17	1	no	—
98	C₈×Q₈	—	17	1	no	—
99	C₈×D₄	—	17	3	no	—
100	C₈×D₄	—	17	3	no	—
101	#431	—	17	3	no	—
102	C₂×C₄×C₈	—	17	3	no	—
103	C₄⋊C₁₆	—	17	3	no	—
104	C₄⋊C₁₆	—	17	6	no	—
105	C₄⋊C₁₆	—	17	1	no	—
106	C₂²⋊C₁₆	—	17	6	no	—
107	C₄×C₁₆	—	17	6	no	—
108	C₂×C₁₇	—	2	6	no	—
109	C₂×C₁₇	#2988	1	1	yes	characteristic
110	C₂×C₁₇	C₂³×C₁₆	1	1	yes	characteristic
111	C₂×C₁₇	#3407	1	1	yes	characteristic
112	C₂×#41	#3624	1	1	yes	characteristic
113	C₂²×C₈	—	17	3	no	—
114	C₂²×C₈	—	17	3	no	—
115	C₄×Q₈	#3624	1	1	yes	characteristic
116	C₄×Q₈	C₂×#595	1	1	yes	characteristic
117	C₄×D₄	#3624	1	3	yes	—
118	C₄×D₄	C₂×#595	1	3	yes	—
119	#117	#3624	1	3	yes	—
120	C₂×C₄²	C₂×#595	1	3	yes	—
121	C₂×C₁₆	—	17	2	no	—
122	C₂×C₁₆	—	17	6	no	—
123	C₂×C₁₆	—	68	6	no	—
124	C₄⋊C₈	—	17	3	no	—
125	C₄⋊C₈	—	17	3	no	—
126	C₂²⋊C₈	—	17	6	no	—
127	C₄×C₈	—	17	3	no	—
128	C₄×C₈	—	17	3	no	—
129	C₄×C₈	—	17	3	no	—
130	C₄×C₈	—	17	1	no	—
131	C₁₇	#33502	1	1	yes	characteristic, complemented, Sylow
132	#41	#63216	1	8	yes	complemented
133	C₂×Q₈	C₂×#3624	1	1	yes	characteristic
134	C₂×D₄	C₂×#3624	1	3	yes	—
135	C₂²×C₄	C₂×#3624	1	3	yes	—
136	C₂²×C₄	C₂×#3624	1	3	yes	—
137	C₂²×C₄	C₂²×#595	1	3	yes	—
138	C₂×C₈	—	17	1	no	—
139	C₂×C₈	—	17	1	no	—
140	C₂×C₈	—	17	3	no	—
141	C₂×C₈	—	17	3	no	—
142	C₂×C₈	—	34	3	no	—
143	C₂×C₈	—	34	3	no	—
144	C₂×C₈	—	68	3	no	—
145	C₄⋊C₄	C₂×#3624	1	3	yes	—
146	C₄⋊C₄	C₂×#3624	1	3	yes	—
147	C₂²⋊C₄	C₂×#3624	1	6	yes	—
148	C₄²	C₂×#3624	1	3	yes	—
149	C₄²	C₂×#3624	1	3	yes	—
150	C₄²	C₂²×#595	1	1	yes	characteristic
151	C₄²	C₂²×#595	1	3	yes	—
152	C₁₆	—	34	6	no	—
153	C₁₆	—	34	2	no	—
154	C₂³	C₂²×#3624	1	3	yes	—
155	Q₈	C₂×#63216	1	4	yes	complemented
156	D₄	C₂×#63216	1	12	yes	complemented
157	C₂×C₄	—	2	3	no	—
158	C₂×C₄	—	2	3	no	—
159	C₂×C₄	—	4	3	no	—
160	C₂×C₄	C₂×#63216	1	12	yes	complemented
161	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
162	C₂×C₄	C₂²×#3624	1	3	yes	—
163	C₂×C₄	C₂²×#3624	1	3	yes	—
164	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
165	C₂×C₄	C₂²×#3624	1	3	yes	—
166	C₂×C₄	C₂³×#595	1	1	yes	characteristic
167	C₈	—	17	1	no	—
168	C₈	—	17	1	no	—
169	C₈	—	34	3	no	—
170	C₈	—	34	1	no	—
171	C₈	—	34	3	no	—
172	C₂²	—	2	3	no	—
173	C₂²	—	2	3	no	—
174	C₂²	C₂²×#63216	1	6	yes	—
175	C₂²	C₂³×#3624	1	1	yes	characteristic
176	C₄	—	2	3	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	3	no	—
179	C₄	C₂²×#63216	1	2	yes	—
180	C₄	C₂²×#63216	1	6	yes	—
181	C₄	#11763878	1	1	yes	characteristic
182	C₄	#11763888	1	1	yes	characteristic
183	C₂	—	2	6	no	—
184	C₂	#14545699	1	1	yes	characteristic
185	C₂	C₂³×#63216	1	1	yes	characteristic
186	C₂	#14545751	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#595
commutator	C₂×C₁₇	C₂³×C₁₆
Fitting	C₄×#41×C₁₇	C₄
socle	C₂²×C₁₇	C₂³×C₈
prime radical	C₂×#41×C₁₇	C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	C₄×#41	#595
hypocenter	C₁₇	#33502
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×#63216	C₂	semidirect
2. D₄×#63216	C₂	semidirect
3. #14545141	C₂	semidirect
4. C₂×C₄⋊#63216	C₂	semidirect
38. #41×C₁₇	C₁₆	semidirect
62. Q₈×C₁₇	C₂×C₁₆	semidirect
63. D₄×C₁₇	C₂×C₁₆	semidirect
67. C₂×C₄×C₁₇	C₂×C₁₆	semidirect
131. C₁₇	#33502	semidirect
132. #41	#63216	semidirect
155. Q₈	C₂×#63216	semidirect
156. D₄	C₂×#63216	semidirect
160. C₂×C₄	C₂×#63216	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#63216	2
D₄×#63216	2
#14545141	2
C₂×C₄⋊#63216	2
#41×#3622	2
#33502	17