Subgroups — #17960106

Order 4352.

262 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545141	C₂	1	2	yes	maximal
2	#14543813	C₂	1	2	yes	maximal, complemented
3	#14543806	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×#3622	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#63216	C₂²	1	4	yes	—
6	C₂²⋊#63216	—	2	2	no	—
7	C₄×#63216	—	2	2	no	—
8	C₂×C₄×#3622	C₄	1	1	yes	characteristic
9	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
10	C₈×#3622	C₄	1	1	yes	characteristic
11	C₈×#3622	C₄	1	1	yes	characteristic
12	C₈×#3622	C₂²	1	1	yes	characteristic
13	C₈×#3622	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₇	C₈	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	C₈	1	1	yes	characteristic
17	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×#3622	—	2	1	no	—
19	C₂²×#3622	D₄	1	1	yes	characteristic
20	C₂²×#3622	Q₈	1	1	yes	characteristic
21	C₂×#63216	—	2	4	no	—
22	C₄×C₈×C₁₇	C₈	1	1	yes	characteristic
23	C₄×C₈×C₁₇	C₈	1	1	yes	characteristic
24	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₄×#3622	—	2	1	no	—
27	C₄×#3622	C₂×C₄	1	1	yes	characteristic
28	C₄×#3622	C₂×C₄	1	1	yes	characteristic
29	C₄×#3622	C₂×C₄	1	1	yes	characteristic
30	C₄×#3622	D₄	1	1	yes	characteristic
31	C₄×#3622	Q₈	1	1	yes	characteristic
32	C₄×#3622	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
36	C₂×C₈×C₁₇	—	2	2	no	—
37	C₂×C₈×C₁₇	C₁₆	1	4	yes	complemented
38	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
44	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
45	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×#3622	—	2	4	no	—
47	C₂×#3622	—	2	1	no	—
48	C₂×#3622	—	2	1	no	—
49	C₂×#3622	—	2	2	no	—
50	C₂×#3622	—	2	2	no	—
51	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
52	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
53	C₂×#3622	C₂×D₄	1	1	yes	characteristic
54	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
55	#63216	—	4	4	no	—
56	#12012	—	17	1	no	maximal, Sylow
57	C₂³×C₁₇	C₄⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₁₇	—	2	2	no	—
59	C₂×C₄×C₁₇	—	2	1	no	—
60	C₂×C₄×C₁₇	—	2	1	no	—
61	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
65	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
66	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
69	C₈×C₁₇	—	2	1	no	—
70	C₈×C₁₇	—	2	1	no	—
71	C₈×C₁₇	C₂×C₁₆	1	2	yes	—
72	C₈×C₁₇	C₂×C₁₆	1	2	yes	—
73	#3622	—	2	2	no	—
74	#3622	—	2	2	no	—
75	#3622	—	2	1	no	—
76	#3622	—	2	1	no	—
77	#3622	#108	1	2	yes	—
78	#3622	#135	1	2	yes	—
79	#2162	—	17	2	no	—
80	C₂×C₈²	—	17	1	no	—
81	#1370	—	17	2	no	—
82	#1367	—	17	2	no	—
83	C₂²×C₁₇	—	2	1	no	—
84	C₂²×C₁₇	—	2	1	no	—
85	C₂²×C₁₇	C₄⋊C₁₆	1	2	yes	—
86	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₄×C₁₇	—	2	1	no	—
88	C₄×C₁₇	—	2	1	no	—
89	C₄×C₁₇	C₄⋊C₁₆	1	2	yes	—
90	C₄×C₁₇	#364	1	2	yes	—
91	C₄×C₁₇	#427	1	1	yes	characteristic
92	C₄×C₁₇	C₂×#108	1	1	yes	characteristic
93	C₄×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
94	C₄×C₁₇	C₂×M₃₂	1	1	yes	characteristic
95	C₂×C₄×C₈	—	17	1	no	—
96	C₂×C₄×C₈	—	17	1	no	—
97	C₂×C₄×C₈	#595	1	1	yes	characteristic
98	C₄⋊C₁₆	—	17	4	no	—
99	C₂²⋊C₁₆	—	34	2	no	—
100	C₄×C₁₆	—	34	2	no	—
101	C₈²	—	17	1	no	—
102	C₈²	—	17	1	no	—
103	C₈²	—	17	1	no	—
104	C₈²	—	17	1	no	—
105	C₂×C₁₇	—	2	2	no	—
106	C₂×C₁₇	#1564	1	1	yes	characteristic
107	C₂×C₁₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
108	C₂×C₁₇	C₂×#364	1	1	yes	characteristic
109	C₂²×C₈	—	17	1	no	—
110	C₂²×C₈	—	17	1	no	—
111	C₂²×C₈	—	34	1	no	—
112	C₂²×C₈	#3624	1	1	yes	characteristic
113	C₂²×C₈	#3624	1	1	yes	characteristic
114	C₂×C₄²	C₂×#595	1	1	yes	characteristic
115	C₂×C₁₆	—	34	4	no	—
116	C₄×C₈	—	17	1	no	—
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	—	17	1	no	—
122	C₄×C₈	—	34	1	no	—
123	C₄×C₈	#3624	1	1	yes	characteristic
124	C₄×C₈	#3624	1	1	yes	characteristic
125	C₄×C₈	C₂×#595	1	1	yes	characteristic
126	C₄×C₈	C₂×#595	1	1	yes	characteristic
127	C₁₇	#12012	1	1	yes	characteristic, complemented, Sylow
128	C₂²×C₄	#63243	1	1	yes	characteristic
129	C₂²×C₄	C₄⋊#595	1	1	yes	characteristic
130	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	17	1	no	—
133	C₂×C₈	—	17	1	no	—
134	C₂×C₈	—	17	1	no	—
135	C₂×C₈	—	17	1	no	—
136	C₂×C₈	—	34	2	no	—
137	C₂×C₈	—	34	2	no	—
138	C₂×C₈	—	34	4	no	—
139	C₂×C₈	—	34	1	no	—
140	C₂×C₈	—	34	1	no	—
141	C₂×C₈	#63216	1	4	yes	complemented
142	C₂×C₈	C₂×#3624	1	1	yes	characteristic
143	C₂×C₈	C₂×#3624	1	1	yes	characteristic
144	C₂×C₈	C₂×#3624	1	1	yes	characteristic
145	C₂×C₈	C₂×#3624	1	1	yes	characteristic
146	C₄²	#63243	1	1	yes	characteristic
147	C₄²	C₄⋊#595	1	1	yes	characteristic
148	C₄²	C₂×#3624	1	1	yes	characteristic
149	C₄²	C₂²×#595	1	1	yes	characteristic
150	C₁₆	—	68	4	no	—
151	C₂³	C₄⋊#3624	1	1	yes	characteristic
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	C₂×#63216	1	2	yes	—
156	C₂×C₄	#10587538	1	2	yes	—
157	C₂×C₄	C₄⋊#3624	1	1	yes	characteristic
158	C₂×C₄	C₄⋊#3624	1	1	yes	characteristic
159	C₂×C₄	C₄⋊#3624	1	1	yes	characteristic
160	C₂×C₄	C₂×#63243	1	1	yes	characteristic
161	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
162	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	17	2	no	—
166	C₈	—	17	2	no	—
167	C₈	—	34	2	no	—
168	C₈	—	34	2	no	—
169	C₈	—	34	1	no	—
170	C₈	—	34	1	no	—
171	C₈	C₂×#63216	1	2	yes	—
172	C₈	C₂×#63216	1	2	yes	—
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	C₄⋊#63216	1	2	yes	—
176	C₂²	C₂×C₄⋊#3624	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	C₄⋊#63216	1	2	yes	—
180	C₄	#11763731	1	2	yes	—
181	C₄	#11763819	1	1	yes	characteristic
182	C₄	D₁₇⋊#108	1	1	yes	characteristic
183	C₄	C₂²×#63216	1	1	yes	characteristic
184	C₄	C₂×#10587538	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#14544331	1	1	yes	characteristic
187	C₂	C₂×C₄⋊#63216	1	1	yes	characteristic
188	C₂	C₂×#11763731	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₄⋊#595
commutator	C₄×C₁₇	C₂²×C₁₆
Frattini	C₄²	C₂²×#595
Fitting	C₂×C₄×C₈×C₁₇	C₄
socle	C₂²×C₁₇	C₂×C₄⋊C₈
prime radical	C₂³×C₁₇	C₄⋊C₈
cosocle	C₄×#3622	C₂³
hypercenter	C₂×C₄×C₈	#595
hypocenter	C₁₇	#12012
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #14543813	C₂	semidirect
3. #14543806	C₂	semidirect
37. C₂×C₈×C₁₇	C₁₆	semidirect
127. C₁₇	#12012	semidirect
141. C₂×C₈	#63216	semidirect

Maximal subgroups

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

subgroup	index
#14545141	2
#14543813	2
#14543806	2
C₂×C₈×#3622	2
#12012	17