Subgroups — Q₈×#10587538 #17978670

Order 4352.

530 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545199	C₂	1	6	yes	maximal, complemented
2	Q₈×#63216	C₂	1	2	yes	maximal, complemented
3	#14545116	C₂	1	3	yes	maximal
4	C₄×#10587538	C₂	1	3	yes	maximal
5	C₂×Q₈×#3622	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587538	C₂²	1	3	yes	—
7	C₂×#10587538	C₂²	1	1	yes	characteristic
8	C₄⋊#63216	C₂²	1	6	yes	—
9	C₄⋊#63216	C₂²	1	6	yes	—
10	#11763697	C₂²	1	6	yes	—
11	C₄×#63216	C₂²	1	6	yes	—
12	Q₈×#3622	C₄	1	3	yes	—
13	Q₈×#3622	C₄	1	1	yes	characteristic
14	Q₈×#3622	C₂²	1	1	yes	characteristic
15	Q₈×#3622	C₂²	1	3	yes	—
16	C₂×C₄⋊#3622	C₄	1	3	yes	—
17	C₂×C₄×#3622	C₂²	1	3	yes	—
18	C₂×C₄×Q₈×C₁₇	C₄	1	1	yes	characteristic
19	C₂²×Q₈×C₁₇	C₈	1	1	yes	characteristic
20	C₄×Q₈×C₁₇	C₈	1	3	yes	—
21	C₄×Q₈×C₁₇	C₈	1	1	yes	characteristic
22	C₄×Q₈×C₁₇	C₂×C₄	1	3	yes	—
23	C₄×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₁₇	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	1	yes	characteristic
28	#10587538	—	2	6	no	—
29	#10587538	Q₈	1	4	yes	direct product factor
30	C₂×#63216	C₂³	1	6	yes	—
31	C₂×#63216	C₂³	1	2	yes	—
32	C₄⋊#3622	C₂×C₄	1	3	yes	—
33	C₄⋊#3622	C₂×C₄	1	6	yes	—
34	C₄⋊#3622	C₂×C₄	1	3	yes	—
35	C₄×#3622	C₂×C₄	1	3	yes	—
36	C₄×#3622	C₂×C₄	1	3	yes	—
37	C₄×#3622	C₂³	1	3	yes	—
38	C₄×#3622	C₂³	1	3	yes	—
39	C₂×Q₈×C₁₇	—	2	4	no	—
40	C₂×Q₈×C₁₇	C₂×C₈	1	3	yes	—
41	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	C₂×C₈	1	3	yes	—
43	C₂²×C₄×C₁₇	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
45	C₄⋊C₄×C₁₇	C₂×C₈	1	6	yes	—
46	C₄⋊C₄×C₁₇	C₂×C₈	1	3	yes	—
47	C₄⋊C₄×C₁₇	C₂×C₈	1	3	yes	—
48	C₄²×C₁₇	C₂×C₈	1	3	yes	—
49	C₄²×C₁₇	C₂×C₈	1	3	yes	—
50	C₄²×C₁₇	C₂²×C₄	1	3	yes	—
51	C₄²×C₁₇	C₂²×C₄	1	3	yes	—
52	C₂×#3622	—	2	6	no	—
53	C₂×#3622	C₂²×C₄	1	3	yes	—
54	C₂×#3622	C₂²×C₄	1	3	yes	—
55	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
56	C₂×#3622	C₂×Q₈	1	2	yes	—
57	C₂×#3622	#41	1	2	yes	—
58	C₂×#3622	C₂⁴	1	1	yes	characteristic
59	#63216	—	2	6	no	—
60	#63216	C₂×Q₈	1	4	yes	complemented
61	Q₈×M₃₂	—	17	1	no	maximal, Sylow
62	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
63	Q₈×C₁₇	—	2	6	no	—
64	Q₈×C₁₇	M₃₂	1	4	yes	complemented
65	C₂×C₄×C₁₇	—	2	6	no	—
66	C₂×C₄×C₁₇	—	4	3	no	—
67	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
68	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
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	#131	1	2	yes	—
74	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
75	#3622	—	2	3	no	—
76	#3622	—	2	3	no	—
77	#3622	C₄×Q₈	1	2	yes	—
78	#3622	C₂²×Q₈	1	1	yes	characteristic
79	#3622	C₂×#41	1	1	yes	characteristic
80	C₂×C₈×Q₈	—	17	1	no	—
81	#2183	—	17	6	no	—
82	Q₈×C₁₆	—	17	2	no	—
83	#2150	—	17	3	no	—
84	C₄×M₃₂	—	17	3	no	—
85	C₂²×C₁₇	—	2	1	no	—
86	C₂²×C₁₇	—	2	1	no	—
87	C₂²×C₁₇	C₈×Q₈	1	2	yes	—
88	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
89	C₄×C₁₇	—	2	3	no	—
90	C₄×C₁₇	—	2	3	no	—
91	C₄×C₁₇	—	2	3	no	—
92	C₄×C₁₇	C₈×Q₈	1	2	yes	—
93	C₄×C₁₇	C₂×M₃₂	1	6	yes	—
94	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
95	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
96	C₂×C₄×Q₈	#595	1	1	yes	characteristic
97	C₂×M₃₂	—	17	1	no	—
98	C₂×M₃₂	—	17	3	no	—
99	C₈×Q₈	—	17	1	no	—
100	C₈×Q₈	—	17	3	no	—
101	C₈×Q₈	—	17	3	no	—
102	C₈×Q₈	—	17	1	no	—
103	C₂×C₄⋊C₈	—	17	3	no	—
104	C₂×C₄×C₈	—	17	3	no	—
105	C₄⋊C₁₆	—	17	6	no	—
106	C₄⋊C₁₆	—	17	6	no	—
107	#346	—	17	6	no	—
108	C₄×C₁₆	—	17	6	no	—
109	C₂×C₁₇	—	2	2	no	—
110	C₂×C₁₇	C₂×C₈×Q₈	1	1	yes	characteristic
111	C₂×C₁₇	C₂²×M₃₂	1	1	yes	characteristic
112	C₂×C₁₇	#3407	1	1	yes	characteristic
113	C₂²×Q₈	#3624	1	1	yes	characteristic
114	C₂²×C₈	—	17	1	no	—
115	C₂²×C₈	—	17	3	no	—
116	C₄×Q₈	#3624	1	3	yes	—
117	C₄×Q₈	#3624	1	1	yes	characteristic
118	C₄×Q₈	C₂×#595	1	3	yes	—
119	C₄×Q₈	C₂×#595	1	1	yes	characteristic
120	C₂×C₄⋊C₄	#3624	1	3	yes	—
121	C₂×C₄²	C₂×#595	1	3	yes	—
122	M₃₂	—	17	4	no	—
123	M₃₂	—	34	6	no	—
124	C₂×C₁₆	—	17	2	no	—
125	C₂×C₁₆	—	17	6	no	—
126	C₄⋊C₈	—	17	3	no	—
127	C₄⋊C₈	—	17	6	no	—
128	C₄⋊C₈	—	17	3	no	—
129	C₄×C₈	—	17	3	no	—
130	C₄×C₈	—	17	3	no	—
131	C₄×C₈	—	17	3	no	—
132	C₄×C₈	—	17	3	no	—
133	C₁₇	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
134	C₂×Q₈	—	2	4	no	—
135	C₂×Q₈	C₂×#3624	1	3	yes	—
136	C₂×Q₈	C₂×#3624	1	1	yes	characteristic
137	C₂²×C₄	C₂×#3624	1	3	yes	—
138	C₂²×C₄	C₂×#3624	1	3	yes	—
139	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
140	C₂×C₈	—	17	2	no	—
141	C₂×C₈	—	17	2	no	—
142	C₂×C₈	—	17	1	no	—
143	C₂×C₈	—	17	1	no	—
144	C₂×C₈	—	17	3	no	—
145	C₂×C₈	—	17	3	no	—
146	C₂×C₈	—	34	6	no	—
147	C₄⋊C₄	C₂×#3624	1	6	yes	—
148	C₄⋊C₄	C₂×#3624	1	3	yes	—
149	C₄⋊C₄	C₂×#3624	1	3	yes	—
150	C₄²	C₂×#3624	1	3	yes	—
151	C₄²	C₂×#3624	1	3	yes	—
152	C₄²	C₂²×#595	1	3	yes	—
153	C₄²	C₂²×#595	1	3	yes	—
154	C₁₆	—	17	4	no	—
155	C₁₆	—	34	6	no	—
156	C₂³	C₂²×#3624	1	1	yes	characteristic
157	Q₈	—	2	6	no	—
158	Q₈	#10587538	1	4	yes	direct product factor
159	C₂×C₄	—	2	6	no	—
160	C₂×C₄	—	4	3	no	—
161	C₂×C₄	#10587680	1	2	yes	—
162	C₂×C₄	Q₈×#595	1	2	yes	—
163	C₂×C₄	C₂²×#3624	1	3	yes	—
164	C₂×C₄	C₂²×#3624	1	3	yes	—
165	C₂×C₄	C₂²×#3624	1	3	yes	—
166	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
167	C₂×C₄	C₂²×#3624	1	3	yes	—
168	C₂×C₄	C₂³×#595	1	1	yes	characteristic
169	C₈	—	17	1	no	—
170	C₈	—	17	1	no	—
171	C₈	—	17	2	no	—
172	C₈	—	34	3	no	—
173	C₈	—	34	3	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	Q₈×#3624	1	2	yes	—
177	C₂²	C₂³×#3624	1	1	yes	characteristic
178	C₄	—	2	3	no	—
179	C₄	—	2	3	no	—
180	C₄	—	2	3	no	—
181	C₄	Q₈×#3624	1	2	yes	—
182	C₄	C₂×#10587538	1	6	yes	—
183	C₄	C₂×Q₈×#595	1	1	yes	characteristic
184	C₄	C₂×#10587680	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	C₂×Q₈×#3624	1	1	yes	characteristic
187	C₂	C₂²×#10587538	1	1	yes	characteristic
188	C₂	#14545751	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₈
Fitting	C₂×C₄×Q₈×C₁₇	C₄
prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	C₂×C₄×Q₈	#595
hypocenter	C₁₇	Q₈×M₃₂
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #14545199	C₂	semidirect
2. Q₈×#63216	C₂	semidirect
29. #10587538	Q₈	direct (N × H)
60. #63216	C₂×Q₈	semidirect
64. Q₈×C₁₇	M₃₂	semidirect
133. C₁₇	Q₈×M₃₂	semidirect
158. Q₈	#10587538	direct (N × H)

Maximal subgroups

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

subgroup	index
#14545199	2
Q₈×#63216	2
#14545116	2
C₄×#10587538	2
C₂×Q₈×#3622	2
Q₈×M₃₂	17

Subgroups — Q8×#10587538 #17978670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#10587538 #17978670