Subgroups — #17978714

Order 4352.

548 conjugacy classes of subgroups.

The subgroup lattice has 184 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	#14545168	C₂	1	6	yes	maximal, complemented
3	#14545145	C₂	1	3	yes	maximal, complemented
4	C₄×#10587538	C₂	1	3	yes	maximal, complemented
5	#41×#3622	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587538	C₂²	1	6	yes	—
7	C₄⋊#63216	C₂²	1	6	yes	—
8	C₂²⋊#63216	C₂²	1	6	yes	—
9	#11763697	C₂²	1	1	yes	characteristic
10	#11763697	C₂²	1	3	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	#10587538	—	2	12	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	3	yes	—
37	C₄×#3622	C₂³	1	1	yes	characteristic
38	#41×C₁₇	—	2	4	no	—
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	6	no	—
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	C₂⁴	1	1	yes	characteristic
57	#63216	—	2	6	no	—
58	#63216	#41	1	4	yes	complemented
59	#33485	—	17	1	no	maximal, Sylow
60	C₂³×C₁₇	C₂²×C₈	1	3	yes	—
61	Q₈×C₁₇	M₃₂	1	4	yes	complemented
62	D₄×C₁₇	—	2	6	no	—
63	C₂×C₄×C₁₇	—	2	3	no	—
64	C₂×C₄×C₁₇	—	2	3	no	—
65	C₂×C₄×C₁₇	—	2	6	no	—
66	C₂×C₄×C₁₇	—	4	3	no	—
67	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
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	3	yes	—
72	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
73	#3622	—	2	3	no	—
74	#3622	—	2	3	no	—
75	#3622	#131	1	2	yes	—
76	#3622	C₂×#41	1	2	yes	—
77	C₈×#41	—	17	1	no	—
78	Q₈×C₁₆	—	17	2	no	—
79	#2169	—	17	6	no	—
80	#2163	—	17	3	no	—
81	C₄×M₃₂	—	17	3	no	—
82	C₂²×C₁₇	—	2	3	no	—
83	C₂²×C₁₇	—	2	3	no	—
84	C₂²×C₁₇	—	2	3	no	—
85	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
86	C₄×C₁₇	—	2	1	no	—
87	C₄×C₁₇	—	2	1	no	—
88	C₄×C₁₇	—	2	3	no	—
89	C₄×C₁₇	—	2	3	no	—
90	C₄×C₁₇	C₂×M₃₂	1	6	yes	—
91	C₄×C₁₇	C₄×#41	1	1	yes	characteristic
92	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
93	C₄×#41	#595	1	1	yes	characteristic
94	C₂×M₃₂	—	17	6	no	—
95	C₈×Q₈	—	17	1	no	—
96	C₈×Q₈	—	17	1	no	—
97	C₈×D₄	—	17	3	no	—
98	C₈×D₄	—	17	3	no	—
99	#431	—	17	3	no	—
100	C₂×C₄×C₈	—	17	3	no	—
101	C₄⋊C₁₆	—	17	6	no	—
102	C₂²⋊C₁₆	—	17	6	no	—
103	#346	—	17	1	no	—
104	#346	—	17	3	no	—
105	C₄×C₁₆	—	17	6	no	—
106	C₂×C₁₇	—	4	3	no	—
107	C₂×C₁₇	C₈×#41	1	1	yes	characteristic
108	C₂×C₁₇	C₂²×M₃₂	1	1	yes	characteristic
109	C₂×C₁₇	#3407	1	1	yes	characteristic
110	C₂×#41	#3624	1	1	yes	characteristic
111	C₂²×C₈	—	17	3	no	—
112	C₂²×C₈	—	17	3	no	—
113	C₄×Q₈	#3624	1	1	yes	characteristic
114	C₄×Q₈	C₂×#595	1	1	yes	characteristic
115	C₄×D₄	#3624	1	3	yes	—
116	C₄×D₄	C₂×#595	1	3	yes	—
117	#117	#3624	1	3	yes	—
118	C₂×C₄²	C₂×#595	1	3	yes	—
119	M₃₂	—	34	12	no	—
120	C₂×C₁₆	—	17	2	no	—
121	C₂×C₁₆	—	17	6	no	—
122	C₄⋊C₈	—	17	3	no	—
123	C₄⋊C₈	—	17	3	no	—
124	C₂²⋊C₈	—	17	6	no	—
125	C₄×C₈	—	17	3	no	—
126	C₄×C₈	—	17	3	no	—
127	C₄×C₈	—	17	3	no	—
128	C₄×C₈	—	17	1	no	—
129	C₁₇	#33485	1	1	yes	characteristic, complemented, Sylow
130	#41	—	2	4	no	—
131	C₂×Q₈	C₂×#3624	1	1	yes	characteristic
132	C₂×D₄	C₂×#3624	1	3	yes	—
133	C₂²×C₄	C₂×#3624	1	3	yes	—
134	C₂²×C₄	C₂×#3624	1	3	yes	—
135	C₂²×C₄	C₂²×#595	1	3	yes	—
136	C₂×C₈	—	17	1	no	—
137	C₂×C₈	—	17	1	no	—
138	C₂×C₈	—	17	3	no	—
139	C₂×C₈	—	17	3	no	—
140	C₂×C₈	—	34	6	no	—
141	C₂×C₈	—	34	6	no	—
142	C₄⋊C₄	C₂×#3624	1	3	yes	—
143	C₄⋊C₄	C₂×#3624	1	3	yes	—
144	C₂²⋊C₄	C₂×#3624	1	6	yes	—
145	C₄²	C₂×#3624	1	3	yes	—
146	C₄²	C₂×#3624	1	3	yes	—
147	C₄²	C₂²×#595	1	1	yes	characteristic
148	C₄²	C₂²×#595	1	3	yes	—
149	C₁₆	—	17	4	no	—
150	C₁₆	—	34	6	no	—
151	C₂³	C₂²×#3624	1	3	yes	—
152	Q₈	#10587538	1	4	yes	complemented
153	D₄	—	2	6	no	—
154	C₂×C₄	—	2	6	no	—
155	C₂×C₄	—	2	3	no	—
156	C₂×C₄	—	2	3	no	—
157	C₂×C₄	—	4	3	no	—
158	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
159	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
160	C₂×C₄	C₂²×#3624	1	3	yes	—
161	C₂×C₄	C₂²×#3624	1	3	yes	—
162	C₂×C₄	C₂²×#3624	1	3	yes	—
163	C₂×C₄	C₂³×#595	1	1	yes	characteristic
164	C₈	—	17	2	no	—
165	C₈	—	17	2	no	—
166	C₈	—	34	3	no	—
167	C₈	—	34	3	no	—
168	C₂²	—	2	3	no	—
169	C₂²	—	2	3	no	—
170	C₂²	—	2	3	no	—
171	C₂²	C₂³×#3624	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	3	no	—
175	C₄	—	2	3	no	—
176	C₄	C₂×#10587538	1	6	yes	—
177	C₄	#41×#595	1	1	yes	characteristic
178	C₄	D₁₇⋊#131	1	1	yes	characteristic
179	C₂	—	4	3	no	—
180	C₂	#41×#3624	1	1	yes	characteristic
181	C₂	C₂²×#10587538	1	1	yes	characteristic
182	C₂	#14545751	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₈
Fitting	C₄×#41×C₁₇	C₄
prime radical	C₂×#41×C₁₇	C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	C₄×#41	#595
hypocenter	C₁₇	#33485
LCS	G ▷ C₂²×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. #14545168	C₂	semidirect
3. #14545145	C₂	semidirect
4. C₄×#10587538	C₂	semidirect
58. #63216	#41	semidirect
61. Q₈×C₁₇	M₃₂	semidirect
129. C₁₇	#33485	semidirect
152. Q₈	#10587538	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
#14545168	2
#14545145	2
C₄×#10587538	2
#41×#3622	2
#33485	17