Subgroups — #17948487

Order 4352.

306 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763695	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#3624	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#63216	C₄	1	2	yes	—
5	C₂×C₄×#3624	C₄	1	2	yes	—
6	#11763695	—	2	1	no	—
7	#11763695	—	2	1	no	—
8	C₈×#3624	—	2	1	no	—
9	C₈×#3624	—	2	1	no	—
10	C₂²×#63214	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
12	C₁₆×Dic₁₇	—	2	1	no	—
13	C₁₆×Dic₁₇	—	2	1	no	—
14	C₂²×C₁₆×C₁₇	C₄	1	1	yes	characteristic
15	C₂²×#3624	C₈	1	4	yes	—
16	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
19	C₄×#3624	—	2	2	no	—
20	C₄×#3624	—	2	2	no	—
21	C₂×#63216	—	2	2	no	—
22	C₂×#63216	—	2	2	no	—
23	C₂×#63216	C₈	1	4	yes	complemented
24	C₂×C₁₆×C₁₇	—	2	1	no	—
25	C₂×C₁₆×C₁₇	—	2	1	no	—
26	C₂×C₁₆×C₁₇	C₈	1	2	yes	complemented
27	C₈×Dic₁₇	D₄	1	1	yes	characteristic
28	C₈×Dic₁₇	D₄	1	1	yes	characteristic
29	C₈×Dic₁₇	D₄	1	1	yes	characteristic
30	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
31	C₂×#63214	—	2	1	no	—
32	C₂×#63214	—	2	1	no	—
33	C₂×#63214	C₈	1	2	yes	complemented
34	C₂²×C₄×C₁₇	C₄²	1	1	yes	characteristic
35	C₂²×Dic₁₇	C₂×C₈	1	2	yes	—
36	C₂×#3624	—	2	4	no	—
37	C₂×#3624	—	2	4	no	—
38	C₂×#3624	C₁₆	1	8	yes	complemented
39	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₁₇	M₁₆	1	1	yes	characteristic
44	C₂×C₈×C₁₇	M₁₆	1	1	yes	characteristic
45	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
47	C₄×Dic₁₇	M₁₆	1	2	yes	—
48	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
50	C₂×#3622	C₂×C₈	1	2	yes	—
51	C₂×#3622	M₁₆	1	2	yes	—
52	#63216	—	2	4	no	—
53	C₁₆×C₁₇	—	2	2	no	—
54	#63214	—	2	2	no	—
55	#7136	—	17	1	no	maximal, Sylow
56	C₂³×C₁₇	C₄×C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
58	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
59	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
60	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
61	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
62	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
63	C₂×Dic₁₇	C₂²⋊C₈	1	2	yes	—
64	C₂×Dic₁₇	C₄⋊C₈	1	2	yes	—
65	C₂×Dic₁₇	C₂×C₁₆	1	4	yes	—
66	C₂×Dic₁₇	M₃₂	1	4	yes	—
67	#3624	—	2	8	no	—
68	C₈×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
69	C₈×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
70	C₈×C₁₇	C₄⋊C₈	1	1	yes	characteristic
71	C₈×C₁₇	C₄⋊C₈	1	1	yes	characteristic
72	#3622	C₂²⋊C₈	1	2	yes	—
73	#3622	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₁₆	—	17	1	no	—
75	C₂×C₄×C₁₆	—	17	1	no	—
76	C₂×C₈²	—	17	1	no	—
77	C₂²×C₁₇	C₈²	1	1	yes	characteristic
78	C₂²×C₁₇	#322	1	1	yes	characteristic
79	C₂²×C₁₇	#336	1	1	yes	characteristic
80	C₂²×C₁₇	C₄×C₁₆	1	2	yes	—
81	C₂²×C₁₇	#346	1	2	yes	—
82	C₄×C₁₇	#322	1	1	yes	characteristic
83	C₄×C₁₇	#322	1	1	yes	characteristic
84	C₄×C₁₇	#336	1	1	yes	characteristic
85	C₄×C₁₇	#336	1	1	yes	characteristic
86	Dic₁₇	C₂²⋊C₁₆	1	4	yes	—
87	Dic₁₇	C₄⋊C₁₆	1	4	yes	—
88	C₂²×C₁₆	—	17	2	no	—
89	C₂²×C₁₆	—	17	1	no	—
90	C₂²×C₁₆	#595	1	1	yes	characteristic
91	C₂×C₄×C₈	—	17	1	no	—
92	C₂×C₄×C₈	—	17	2	no	—
93	C₄×C₁₆	—	34	1	no	—
94	C₄×C₁₆	—	34	1	no	—
95	C₄×C₁₆	—	34	1	no	—
96	C₄×C₁₆	—	34	1	no	—
97	C₈²	—	34	1	no	—
98	C₈²	—	34	1	no	—
99	C₂×C₁₇	#1273	1	1	yes	characteristic
100	C₂×C₁₇	C₈×C₁₆	1	1	yes	characteristic
101	C₂×C₁₇	#1311	1	1	yes	characteristic
102	C₂×C₁₇	#1312	1	2	yes	—
103	C₂×C₁₇	#1374	1	2	yes	—
104	C₂²×C₈	—	17	1	no	—
105	C₂²×C₈	—	17	4	no	—
106	C₂²×C₈	C₂×#595	1	1	yes	characteristic
107	C₂×C₄²	—	17	1	no	—
108	C₂×C₁₆	—	2	1	no	—
109	C₂×C₁₆	—	2	1	no	—
110	C₂×C₁₆	—	17	4	no	—
111	C₂×C₁₆	—	17	2	no	—
112	C₂×C₁₆	—	34	2	no	—
113	C₂×C₁₆	—	34	2	no	—
114	C₂×C₁₆	—	34	1	no	—
115	C₂×C₁₆	—	34	1	no	—
116	C₂×C₁₆	#3624	1	2	yes	complemented
117	C₄×C₈	—	17	1	no	—
118	C₄×C₈	—	17	1	no	—
119	C₄×C₈	—	17	1	no	—
120	C₄×C₈	—	17	1	no	—
121	C₄×C₈	—	34	2	no	—
122	C₄×C₈	—	34	2	no	—
123	C₁₇	#7136	1	1	yes	characteristic, complemented, Sylow
124	C₂²×C₄	—	17	2	no	—
125	C₂²×C₄	C₄×#595	1	1	yes	characteristic
126	C₂×C₈	—	17	2	no	—
127	C₂×C₈	—	17	2	no	—
128	C₂×C₈	—	17	1	no	—
129	C₂×C₈	—	17	1	no	—
130	C₂×C₈	—	17	8	no	—
131	C₂×C₈	—	34	4	no	—
132	C₂×C₈	—	34	4	no	—
133	C₂×C₈	#63242	1	1	yes	characteristic
134	C₂×C₈	#63243	1	1	yes	characteristic
135	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
136	C₂×C₈	C₂×#3624	1	1	yes	characteristic
137	C₂×C₈	#63247	1	1	yes	characteristic
138	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
139	C₄²	—	17	2	no	—
140	C₄²	—	17	1	no	—
141	C₄²	—	17	1	no	—
142	C₁₆	—	2	2	no	—
143	C₁₆	—	34	4	no	—
144	C₁₆	—	34	2	no	—
145	C₂³	C₈×#595	1	1	yes	characteristic
146	C₂×C₄	—	17	2	no	—
147	C₂×C₄	—	17	2	no	—
148	C₂×C₄	—	17	4	no	—
149	C₂×C₄	—	17	4	no	—
150	C₂×C₄	C₈×#595	1	1	yes	characteristic
151	C₂×C₄	#10587532	1	1	yes	characteristic
152	C₂×C₄	#10587532	1	1	yes	characteristic
153	C₂×C₄	C₄×#3624	1	1	yes	characteristic
154	C₂×C₄	#10587542	1	1	yes	characteristic
155	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
156	C₈	—	17	2	no	—
157	C₈	—	17	2	no	—
158	C₈	—	34	8	no	—
159	C₈	C₄⋊#3624	1	1	yes	characteristic
160	C₈	#10587543	1	1	yes	characteristic
161	C₈	#10587544	1	1	yes	characteristic
162	C₈	C₂²⋊#3624	1	1	yes	characteristic
163	C₂²	C₈×#3624	1	1	yes	characteristic
164	C₂²	#11763651	1	1	yes	characteristic
165	C₂²	#11763674	1	1	yes	characteristic
166	C₂²	C₁₆×#595	1	2	yes	—
167	C₂²	#11763696	1	2	yes	—
168	C₄	—	17	4	no	—
169	C₄	—	17	4	no	—
170	C₄	#11763650	1	1	yes	characteristic
171	C₄	#11763651	1	1	yes	characteristic
172	C₄	#11763674	1	1	yes	characteristic
173	C₄	#11763675	1	1	yes	characteristic
174	C₂	#14543606	1	1	yes	characteristic
175	C₂	C₁₆×#3624	1	1	yes	characteristic
176	C₂	#14543695	1	1	yes	characteristic
177	C₂	#14543698	1	2	yes	—
178	C₂	#14543821	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂×#595
commutator	C₂×C₁₇	C₈×C₁₆
Fitting	C₂²×C₁₆×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₄×C₈
cosocle	C₂×C₈×Dic₁₇	C₂²
hypercenter	C₂²×C₁₆	#595
hypocenter	C₁₇	#7136
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
23. C₂×#63216	C₈	semidirect
26. C₂×C₁₆×C₁₇	C₈	semidirect
33. C₂×#63214	C₈	semidirect
38. C₂×#3624	C₁₆	semidirect
116. C₂×C₁₆	#3624	semidirect
123. C₁₇	#7136	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763695	2
C₂×C₈×#3624	2
C₂×C₁₆×Dic₁₇	2
#7136	17