Subgroups — #17961655

Order 4352.

414 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763730	C₂	1	4	yes	maximal
2	C₂×C₄⋊#63216	C₂	1	2	yes	maximal
3	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#63216	C₂²	1	4	yes	—
5	#11763730	—	2	8	no	—
6	C₄⋊#63216	—	2	4	no	—
7	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₁₇	C₄	1	2	yes	—
9	C₂×C₈×Dic₁₇	C₂²	1	2	yes	—
10	C₄²×Dic₁₇	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
12	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₁₇	C₈	1	1	yes	characteristic
15	C₂×C₄×Dic₁₇	C₈	1	2	yes	—
16	C₂×C₄×Dic₁₇	C₈	1	1	yes	characteristic
17	C₂×C₄×Dic₁₇	C₂×C₄	1	2	yes	—
18	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
19	C₂²×#3622	C₂³	1	1	yes	characteristic
20	C₂×#63216	—	2	4	no	—
21	C₂×#63216	—	2	8	no	—
22	C₄×C₈×C₁₇	—	2	2	no	—
23	C₈×Dic₁₇	—	2	4	no	—
24	C₈×Dic₁₇	D₄	1	4	yes	—
25	C₈×Dic₁₇	Q₈	1	4	yes	—
26	C₄×#3622	D₄	1	2	yes	—
27	C₄×#3622	Q₈	1	2	yes	—
28	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
29	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
31	C₂²×Dic₁₇	C₂×C₈	1	2	yes	—
32	C₂²×Dic₁₇	C₂×C₈	1	2	yes	—
33	C₂×C₈×C₁₇	—	2	2	no	—
34	C₂×C₈×C₁₇	—	2	2	no	—
35	C₂×C₈×C₁₇	C₄⋊C₄	1	4	yes	—
36	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
37	C₄²×C₁₇	M₁₆	1	2	yes	—
38	C₄×Dic₁₇	—	2	2	no	—
39	C₄×Dic₁₇	—	2	4	no	—
40	C₄×Dic₁₇	C₁₆	1	4	yes	complemented
41	C₄×Dic₁₇	C₄⋊C₄	1	4	yes	—
42	C₄×Dic₁₇	M₁₆	1	4	yes	—
43	C₂×#3622	—	2	2	no	—
44	C₂×#3622	C₄⋊C₄	1	2	yes	—
45	C₂×#3622	C₂×D₄	1	1	yes	characteristic
46	C₂×#3622	C₂×D₄	1	2	yes	—
47	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
48	C₂×#3622	C₂×Q₈	1	2	yes	—
49	#63216	—	4	4	no	—
50	#12898	—	17	1	no	maximal, Sylow
51	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₇	—	2	2	no	—
53	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
54	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
56	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
57	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
59	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
60	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
61	C₂×Dic₁₇	—	2	2	no	—
62	C₂×Dic₁₇	—	2	2	no	—
63	C₂×Dic₁₇	C₄⋊C₈	1	4	yes	—
64	C₂×Dic₁₇	C₄⋊C₈	1	4	yes	—
65	C₂×Dic₁₇	C₂×C₁₆	1	4	yes	—
66	C₂×Dic₁₇	M₃₂	1	4	yes	—
67	C₈×C₁₇	—	2	4	no	—
68	#3622	—	2	2	no	—
69	#3622	#127	1	1	yes	characteristic
70	#3622	C₄⋊Q₈	1	1	yes	characteristic
71	#3622	C₄⋊Q₈	1	2	yes	—
72	C₂×C₄⋊C₁₆	—	17	2	no	—
73	C₂×C₄⋊C₁₆	—	17	4	no	—
74	C₄²×C₈	—	17	1	no	—
75	C₂²×C₁₇	C₂×C₄⋊C₈	1	2	yes	—
76	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
77	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
78	C₂²×C₁₇	C₂×M₃₂	1	2	yes	—
79	C₂²×C₁₇	C₂×M₃₂	1	1	yes	characteristic
80	C₄×C₁₇	—	2	2	no	—
81	C₄×C₁₇	C₄⋊C₁₆	1	4	yes	complemented
82	C₄×C₁₇	#384	1	1	yes	characteristic
83	C₄×C₁₇	#423	1	2	yes	—
84	C₄×C₁₇	#423	1	1	yes	characteristic
85	Dic₁₇	—	2	4	no	—
86	Dic₁₇	C₄⋊C₁₆	1	8	yes	complemented
87	C₂²×C₁₆	—	17	4	no	—
88	C₂×C₄×C₈	—	17	2	no	—
89	C₂×C₄×C₈	—	17	1	no	—
90	C₂×C₄×C₈	—	17	2	no	—
91	C₂×C₄×C₈	#595	1	1	yes	characteristic
92	C₄³	—	17	1	no	—
93	C₄⋊C₁₆	—	34	4	no	—
94	C₄⋊C₁₆	—	34	8	no	—
95	C₂×C₁₇	#1842	1	1	yes	characteristic
96	C₂×C₁₇	C₂×C₄⋊C₁₆	1	2	yes	—
97	C₂×C₁₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
98	C₂×C₁₇	#2150	1	2	yes	—
99	C₂×C₁₇	#2150	1	1	yes	characteristic
100	C₂²×C₈	—	17	1	no	—
101	C₂²×C₈	—	17	1	no	—
102	C₂²×C₈	C₂×#595	1	2	yes	—
103	C₂×C₄²	—	17	1	no	—
104	C₂×C₄²	—	17	2	no	—
105	C₂×C₄²	—	17	2	no	—
106	C₂×C₄²	—	17	1	no	—
107	C₂×C₄²	C₂×#595	1	1	yes	characteristic
108	C₂×C₁₆	—	34	4	no	—
109	C₂×C₁₆	—	34	8	no	—
110	C₄×C₈	—	2	2	no	—
111	C₄×C₈	—	17	4	no	—
112	C₄×C₈	—	17	4	no	—
113	C₄×C₈	—	17	2	no	—
114	C₄×C₈	—	17	2	no	—
115	C₄×C₈	—	34	4	no	—
116	C₁₇	#12898	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₄	—	17	2	no	—
118	C₂²×C₄	—	17	2	no	—
119	C₂²×C₄	#63242	1	1	yes	characteristic
120	C₂²×C₄	#63242	1	1	yes	characteristic
121	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	17	2	no	—
125	C₂×C₈	—	17	1	no	—
126	C₂×C₈	—	17	2	no	—
127	C₂×C₈	—	17	1	no	—
128	C₂×C₈	—	17	2	no	—
129	C₂×C₈	—	34	2	no	—
130	C₂×C₈	C₄⋊#595	1	4	yes	—
131	C₄²	—	17	4	no	—
132	C₄²	—	17	4	no	—
133	C₄²	—	17	4	no	—
134	C₄²	—	34	4	no	—
135	C₄²	—	34	2	no	—
136	C₄²	C₄⋊#595	1	2	yes	—
137	C₄²	#63247	1	2	yes	—
138	C₁₆	—	68	4	no	—
139	C₂³	C₂×#63242	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	17	4	no	—
142	C₂×C₄	—	17	4	no	—
143	C₂×C₄	—	17	4	no	—
144	C₂×C₄	—	17	4	no	—
145	C₂×C₄	—	34	2	no	—
146	C₂×C₄	—	34	2	no	—
147	C₂×C₄	#10587529	1	2	yes	—
148	C₂×C₄	#10587530	1	2	yes	—
149	C₂×C₄	#10587544	1	2	yes	—
150	C₂×C₄	#10587544	1	2	yes	—
151	C₂×C₄	D₁₇⋊M₁₆	1	2	yes	—
152	C₂×C₄	C₂×C₄⋊#595	1	2	yes	—
153	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
154	C₂×C₄	C₂×#63247	1	1	yes	characteristic
155	C₈	—	2	4	no	—
156	C₈	—	17	1	no	—
157	C₈	—	17	2	no	—
158	C₈	—	17	1	no	—
159	C₈	—	34	2	no	—
160	C₂²	#11763810	1	2	yes	—
161	C₂²	C₂×#10587544	1	1	yes	characteristic
162	C₂²	C₂×#10587529	1	1	yes	characteristic
163	C₂²	#11763865	1	2	yes	—
164	C₂²	C₂×#10587530	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	17	8	no	—
167	C₄	—	34	4	no	—
168	C₄	#11763730	1	4	yes	complemented
169	C₄	#11763755	1	1	yes	characteristic
170	C₄	#11763812	1	2	yes	—
171	C₄	#11763814	1	1	yes	characteristic
172	C₂	#14544717	1	1	yes	characteristic
173	C₂	#14545114	1	2	yes	—
174	C₂	C₂×#11763730	1	1	yes	characteristic
175	C₂	#14545117	1	2	yes	—
176	C₂	#14545118	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#595
commutator	C₂²×C₁₇	C₂²×C₁₆
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂²×#3622	C₂³
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#12898
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
40. C₄×Dic₁₇	C₁₆	semidirect
81. C₄×C₁₇	C₄⋊C₁₆	semidirect
86. Dic₁₇	C₄⋊C₁₆	semidirect
116. C₁₇	#12898	semidirect
168. C₄	#11763730	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763730	2
C₂×C₄⋊#63216	2
C₄×C₈×Dic₁₇	2
#12898	17