Subgroups — #17978661

Order 4352.

490 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545199	C₂	1	8	yes	maximal, complemented
2	#14545119	C₂	1	2	yes	maximal
3	#14545116	C₂	1	1	yes	maximal, characteristic
4	#14544993	C₂	1	2	yes	maximal
5	C₂×#11763697	C₂	1	1	yes	maximal, characteristic
6	#14537719	C₂	1	1	yes	maximal, characteristic
7	C₂×#10587538	C₂²	1	2	yes	—
8	C₂²×#63216	C₂²	1	1	yes	characteristic
9	C₂²×#63216	C₂²	1	1	yes	characteristic
10	C₄⋊#63216	C₂²	1	4	yes	—
11	C₄⋊#63216	C₂²	1	8	yes	—
12	#11763697	C₂²	1	4	yes	—
13	#11763697	C₂²	1	4	yes	—
14	C₄×#63216	C₂²	1	4	yes	—
15	#11762940	C₄	1	4	yes	—
16	Q₈×#3622	C₂²	1	4	yes	—
17	#11762836	C₄	1	2	yes	—
18	#11762833	C₄	1	1	yes	characteristic
19	#11762765	C₂²	1	2	yes	—
20	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
21	C₁₇×#533	C₄	1	1	yes	characteristic
22	C₁₇×C₄⋊Q₈	C₈	1	2	yes	—
23	C₁₇×#125	C₈	1	2	yes	—
24	C₁₇×C₂²⋊Q₈	C₈	1	4	yes	—
25	C₄×Q₈×C₁₇	C₂×C₄	1	4	yes	—
26	C₁₇×#117	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂×#63223	C₂×C₄	1	2	yes	—
29	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
30	C₂²×#3622	C₂³	1	1	yes	characteristic
31	#10587538	—	2	4	no	—
32	C₂×#63216	—	2	2	no	—
33	C₂×#63216	Q₈	1	4	yes	complemented
34	C₂×#63216	C₂³	1	2	yes	—
35	C₂×#63216	C₂³	1	4	yes	—
36	C₂×#63216	C₂³	1	2	yes	—
37	C₄⋊#3622	C₂×C₄	1	4	yes	—
38	C₄⋊#3622	C₂×C₄	1	8	yes	—
39	#10587474	C₂×C₄	1	4	yes	—
40	C₄×#3622	C₂×C₄	1	2	yes	—
41	C₄×#3622	C₂³	1	2	yes	—
42	C₄×#3622	C₂³	1	4	yes	—
43	C₂×Q₈×C₁₇	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
46	C₄⋊C₄×C₁₇	C₂×C₈	1	8	yes	—
47	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
48	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
49	C₂²⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
50	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
51	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
52	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
53	C₄²×C₁₇	C₂²×C₄	1	4	yes	—
54	#63223	—	2	4	no	—
55	C₂×#3622	—	2	2	no	—
56	C₂×#3622	C₂²×C₄	1	2	yes	—
57	C₂×#3622	C₂²×C₄	1	4	yes	—
58	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
59	C₂×#3622	C₂×Q₈	1	2	yes	—
60	C₂×#3622	#41	1	2	yes	—
61	C₂×#3622	C₂⁴	1	1	yes	characteristic
62	#63216	—	2	2	no	—
63	#63216	—	2	4	no	—
64	#63216	C₂×Q₈	1	4	yes	—
65	#33475	—	17	1	no	maximal, Sylow
66	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
67	Q₈×C₁₇	—	4	4	no	—
68	C₂×C₄×C₁₇	—	4	2	no	—
69	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
70	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
71	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
72	C₂×C₄×C₁₇	C₂²×C₈	1	8	yes	—
73	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₇	#131	1	2	yes	—
75	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
76	#3622	—	2	2	no	—
77	#3622	—	2	4	no	—
78	#3622	C₄×Q₈	1	2	yes	—
79	#3622	C₂²×Q₈	1	1	yes	characteristic
80	#3622	C₂×#41	1	1	yes	characteristic
81	#2960	—	17	1	no	—
82	#2183	—	17	8	no	—
83	#2151	—	17	2	no	—
84	#2150	—	17	1	no	—
85	#2108	—	17	2	no	—
86	C₂×#346	—	17	1	no	—
87	C₂²×C₁₇	—	2	2	no	—
88	C₂²×C₁₇	C₈×Q₈	1	2	yes	—
89	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
90	C₄×C₁₇	—	2	2	no	—
91	C₄×C₁₇	—	2	2	no	—
92	C₄×C₁₇	—	2	8	no	—
93	C₄×C₁₇	C₈×Q₈	1	2	yes	—
94	C₄×C₁₇	C₂×C₄×Q₈	1	1	yes	characteristic
95	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
96	#533	#595	1	1	yes	characteristic
97	C₂×M₃₂	—	17	2	no	—
98	C₂²×C₁₆	—	17	1	no	—
99	C₂²×C₁₆	—	17	1	no	—
100	#446	—	17	4	no	—
101	C₈×Q₈	—	17	4	no	—
102	#424	—	17	2	no	—
103	#423	—	17	1	no	—
104	#405	—	17	2	no	—
105	C₂×C₄×C₈	—	17	1	no	—
106	C₄⋊C₁₆	—	17	4	no	—
107	C₄⋊C₁₆	—	17	8	no	—
108	#346	—	17	4	no	—
109	#346	—	17	4	no	—
110	C₄×C₁₆	—	17	4	no	—
111	C₂×C₁₇	—	2	2	no	—
112	C₂×C₁₇	C₂×C₈×Q₈	1	1	yes	characteristic
113	C₂×C₁₇	#3407	1	2	yes	—
114	C₂×M₁₆	—	17	2	no	—
115	C₂²×C₈	—	17	1	no	—
116	C₂²×C₈	—	17	1	no	—
117	C₄⋊Q₈	#3624	1	2	yes	—
118	#125	#3624	1	2	yes	—
119	C₂²⋊Q₈	#3624	1	4	yes	—
120	C₄×Q₈	C₂×#595	1	4	yes	—
121	#117	C₂×#595	1	2	yes	—
122	C₂×C₄²	C₂×#595	1	1	yes	characteristic
123	M₃₂	—	34	4	no	—
124	C₂×C₁₆	—	17	2	no	—
125	C₂×C₁₆	—	17	4	no	—
126	C₂×C₁₆	—	17	4	no	—
127	C₂×C₁₆	—	17	2	no	—
128	C₂×C₁₆	—	34	2	no	—
129	C₄⋊C₈	—	17	4	no	—
130	C₄⋊C₈	—	17	8	no	—
131	#97	—	17	4	no	—
132	C₄×C₈	—	17	2	no	—
133	C₄×C₈	—	17	4	no	—
134	C₄×C₈	—	17	2	no	—
135	C₁₇	#33475	1	1	yes	characteristic, complemented, Sylow
136	C₂×Q₈	C₂×#3624	1	4	yes	—
137	C₂²×C₄	C₂×#3624	1	2	yes	—
138	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
139	M₁₆	—	34	4	no	—
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	2	no	—
145	C₂×C₈	—	17	4	no	—
146	C₂×C₈	—	34	2	no	—
147	C₄⋊C₄	C₂×#3624	1	4	yes	—
148	C₄⋊C₄	C₂×#3624	1	8	yes	—
149	C₄⋊C₄	C₂×#3624	1	4	yes	—
150	C₂²⋊C₄	C₂×#3624	1	4	yes	—
151	C₄²	C₂×#3624	1	1	yes	characteristic
152	C₄²	C₂×#3624	1	1	yes	characteristic
153	C₄²	C₂²×#595	1	2	yes	—
154	C₄²	C₂²×#595	1	4	yes	—
155	C₁₆	—	17	4	no	—
156	C₁₆	—	34	2	no	—
157	C₁₆	—	34	4	no	—
158	C₂³	C₂²×#3624	1	1	yes	characteristic
159	Q₈	—	4	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	#10587680	1	2	yes	—
162	C₂×C₄	Q₈×#595	1	2	yes	—
163	C₂×C₄	C₂²×#3624	1	2	yes	—
164	C₂×C₄	C₂²×#3624	1	2	yes	—
165	C₂×C₄	C₂²×#3624	1	8	yes	—
166	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
167	C₂×C₄	C₂³×#595	1	1	yes	characteristic
168	C₈	—	17	1	no	—
169	C₈	—	17	1	no	—
170	C₈	—	17	2	no	—
171	C₈	—	34	2	no	—
172	C₈	—	34	4	no	—
173	C₂²	—	2	2	no	—
174	C₂²	Q₈×#3624	1	2	yes	—
175	C₂²	C₂³×#3624	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	2	8	no	—
179	C₄	Q₈×#3624	1	2	yes	—
180	C₄	C₂×Q₈×#595	1	1	yes	characteristic
181	C₄	C₂×#10587680	1	1	yes	characteristic
182	C₂	—	2	2	no	—
183	C₂	C₂×Q₈×#3624	1	1	yes	characteristic
184	C₂	#14545751	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	#33475
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
33. C₂×#63216	Q₈	semidirect
135. C₁₇	#33475	semidirect

Maximal subgroups

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

subgroup	index
#14545199	2
#14545119	2
#14545116	2
#14544993	2
C₂×#11763697	2
#14537719	2
#33475	17