Subgroups — #17978657

Order 4352.

506 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	#14545199	C₂	1	4	yes	maximal, complemented
2	Q₈×#63216	C₂	1	4	yes	maximal, complemented
3	#14545119	C₂	1	2	yes	maximal
4	#14545116	C₂	1	1	yes	maximal, characteristic
5	#14544993	C₂	1	2	yes	maximal
6	C₂×C₄×#63216	C₂	1	1	yes	maximal, characteristic
7	#14537719	C₂	1	1	yes	maximal, characteristic
8	C₂×#10587538	C₂²	1	2	yes	—
9	C₂²×#63216	C₂²	1	1	yes	characteristic
10	C₂²×#63216	C₂²	1	1	yes	characteristic
11	C₄⋊#63216	C₂²	1	4	yes	—
12	C₄⋊#63216	C₂²	1	8	yes	—
13	#11763697	C₂²	1	4	yes	—
14	C₄×#63216	C₂²	1	2	yes	—
15	C₄×#63216	C₂²	1	4	yes	—
16	C₄×#63216	C₂²	1	2	yes	—
17	#11762940	C₄	1	4	yes	—
18	Q₈×#3622	C₂²	1	4	yes	—
19	#11762836	C₄	1	2	yes	—
20	#11762833	C₄	1	1	yes	characteristic
21	#11762765	C₂²	1	2	yes	—
22	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
23	C₁₇×#533	C₄	1	1	yes	characteristic
24	C₁₇×C₄⋊Q₈	C₈	1	2	yes	—
25	C₁₇×#125	C₈	1	2	yes	—
26	C₁₇×C₂²⋊Q₈	C₈	1	4	yes	—
27	C₄×Q₈×C₁₇	C₂×C₄	1	4	yes	—
28	C₁₇×#117	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
30	C₂×#63223	C₂×C₄	1	2	yes	—
31	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
32	C₂²×#3622	C₂³	1	1	yes	characteristic
33	#10587538	—	2	4	no	—
34	C₂×#63216	—	2	2	no	—
35	C₂×#63216	Q₈	1	4	yes	complemented
36	C₂×#63216	C₂³	1	2	yes	—
37	C₂×#63216	C₂³	1	4	yes	—
38	C₂×#63216	C₂³	1	1	yes	characteristic
39	C₂×#63216	C₂³	1	1	yes	characteristic
40	C₄⋊#3622	C₂×C₄	1	4	yes	—
41	C₄⋊#3622	C₂×C₄	1	8	yes	—
42	#10587474	C₂×C₄	1	4	yes	—
43	C₄×#3622	C₂×C₄	1	2	yes	—
44	C₄×#3622	C₂³	1	2	yes	—
45	C₄×#3622	C₂³	1	4	yes	—
46	C₂×Q₈×C₁₇	C₂×C₈	1	4	yes	—
47	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
49	C₄⋊C₄×C₁₇	C₂×C₈	1	8	yes	—
50	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
51	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
52	C₂²⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
53	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
54	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
55	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
56	C₄²×C₁₇	C₂²×C₄	1	4	yes	—
57	#63223	—	2	4	no	—
58	C₂×#3622	—	2	2	no	—
59	C₂×#3622	C₂²×C₄	1	2	yes	—
60	C₂×#3622	C₂²×C₄	1	4	yes	—
61	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
62	C₂×#3622	C₂×Q₈	1	2	yes	—
63	C₂×#3622	#41	1	2	yes	—
64	C₂×#3622	C₂⁴	1	1	yes	characteristic
65	#63216	—	2	2	no	—
66	#63216	—	2	4	no	—
67	#63216	C₂×Q₈	1	2	yes	—
68	#63216	C₂×Q₈	1	2	yes	—
69	#33474	—	17	1	no	maximal, Sylow
70	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
71	Q₈×C₁₇	—	2	8	no	—
72	C₂×C₄×C₁₇	—	2	4	no	—
73	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
74	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
75	C₂×C₄×C₁₇	C₂²×C₈	1	8	yes	—
76	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
78	C₂×C₄×C₁₇	#131	1	2	yes	—
79	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
80	#3622	—	2	2	no	—
81	#3622	—	2	4	no	—
82	#3622	C₄×Q₈	1	2	yes	—
83	#3622	C₂²×Q₈	1	1	yes	characteristic
84	#3622	C₂×#41	1	1	yes	characteristic
85	#2960	—	17	1	no	—
86	#2183	—	17	4	no	—
87	Q₈×C₁₆	—	17	4	no	—
88	#2151	—	17	2	no	—
89	#2150	—	17	1	no	—
90	#2108	—	17	2	no	—
91	C₂×C₄×C₁₆	—	17	1	no	—
92	C₂²×C₁₇	—	2	2	no	—
93	C₂²×C₁₇	C₈×Q₈	1	2	yes	—
94	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
95	C₄×C₁₇	—	2	2	no	—
96	C₄×C₁₇	—	2	8	no	—
97	C₄×C₁₇	C₈×Q₈	1	2	yes	—
98	C₄×C₁₇	#504	1	4	yes	—
99	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
100	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
101	#533	#595	1	1	yes	characteristic
102	C₂×M₃₂	—	17	2	no	—
103	C₂²×C₁₆	—	17	1	no	—
104	C₂²×C₁₆	—	17	1	no	—
105	#446	—	17	4	no	—
106	C₈×Q₈	—	17	4	no	—
107	#424	—	17	2	no	—
108	#423	—	17	1	no	—
109	#405	—	17	2	no	—
110	C₂×C₄×C₈	—	17	1	no	—
111	C₄⋊C₁₆	—	17	4	no	—
112	C₄⋊C₁₆	—	17	8	no	—
113	#346	—	17	4	no	—
114	C₄×C₁₆	—	17	2	no	—
115	C₄×C₁₆	—	17	2	no	—
116	C₄×C₁₆	—	17	4	no	—
117	C₂×C₁₇	—	2	2	no	—
118	C₂×C₁₇	C₂×C₈×Q₈	1	1	yes	characteristic
119	C₂×C₁₇	C₂×#504	1	2	yes	—
120	C₂×M₁₆	—	17	2	no	—
121	C₂²×C₈	—	17	1	no	—
122	C₂²×C₈	—	17	1	no	—
123	C₄⋊Q₈	#3624	1	2	yes	—
124	#125	#3624	1	2	yes	—
125	C₂²⋊Q₈	#3624	1	4	yes	—
126	C₄×Q₈	C₂×#595	1	4	yes	—
127	#117	C₂×#595	1	2	yes	—
128	C₂×C₄²	C₂×#595	1	1	yes	characteristic
129	M₃₂	—	34	4	no	—
130	C₂×C₁₆	—	17	1	no	—
131	C₂×C₁₆	—	17	1	no	—
132	C₂×C₁₆	—	17	4	no	—
133	C₂×C₁₆	—	17	2	no	—
134	C₂×C₁₆	—	17	4	no	—
135	C₂×C₁₆	—	34	2	no	—
136	C₄⋊C₈	—	17	4	no	—
137	C₄⋊C₈	—	17	8	no	—
138	#97	—	17	4	no	—
139	C₄×C₈	—	17	2	no	—
140	C₄×C₈	—	17	2	no	—
141	C₄×C₈	—	17	4	no	—
142	C₁₇	#33474	1	1	yes	characteristic, complemented, Sylow
143	C₂×Q₈	C₂×#3624	1	4	yes	—
144	C₂²×C₄	C₂×#3624	1	2	yes	—
145	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
146	M₁₆	—	34	4	no	—
147	C₂×C₈	—	17	2	no	—
148	C₂×C₈	—	17	2	no	—
149	C₂×C₈	—	17	1	no	—
150	C₂×C₈	—	17	1	no	—
151	C₂×C₈	—	17	2	no	—
152	C₂×C₈	—	17	4	no	—
153	C₂×C₈	—	34	2	no	—
154	C₄⋊C₄	C₂×#3624	1	4	yes	—
155	C₄⋊C₄	C₂×#3624	1	8	yes	—
156	C₄⋊C₄	C₂×#3624	1	4	yes	—
157	C₂²⋊C₄	C₂×#3624	1	4	yes	—
158	C₄²	C₂×#3624	1	1	yes	characteristic
159	C₄²	C₂×#3624	1	1	yes	characteristic
160	C₄²	C₂²×#595	1	2	yes	—
161	C₄²	C₂²×#595	1	4	yes	—
162	C₁₆	—	17	2	no	—
163	C₁₆	—	17	2	no	—
164	C₁₆	—	34	2	no	—
165	C₁₆	—	34	4	no	—
166	C₂³	C₂²×#3624	1	1	yes	characteristic
167	Q₈	—	2	8	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	#10587680	1	2	yes	—
170	C₂×C₄	Q₈×#595	1	2	yes	—
171	C₂×C₄	C₂²×#3624	1	2	yes	—
172	C₂×C₄	C₂²×#3624	1	8	yes	—
173	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
174	C₂×C₄	C₂²×#3624	1	2	yes	—
175	C₂×C₄	C₂³×#595	1	1	yes	characteristic
176	C₈	—	17	1	no	—
177	C₈	—	17	1	no	—
178	C₈	—	17	2	no	—
179	C₈	—	34	2	no	—
180	C₈	—	34	4	no	—
181	C₂²	—	2	2	no	—
182	C₂²	Q₈×#3624	1	2	yes	—
183	C₂²	C₂³×#3624	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	2	8	no	—
186	C₄	Q₈×#3624	1	2	yes	—
187	C₄	#11763867	1	4	yes	—
188	C₄	C₂×Q₈×#595	1	1	yes	characteristic
189	C₄	C₂×#10587680	1	1	yes	characteristic
190	C₂	—	2	2	no	—
191	C₂	C₂×Q₈×#3624	1	1	yes	characteristic
192	C₂	C₂×#11763867	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₈
Fitting	C₁₇×#533	C₄
prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	#533	#595
hypocenter	C₁₇	#33474
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #533

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
35. C₂×#63216	Q₈	semidirect
142. C₁₇	#33474	semidirect

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
#14545119	2
#14545116	2
#14544993	2
C₂×C₄×#63216	2
#14537719	2
#33474	17