Subgroups — #131×#3633 #17982830

Order 4352.

922 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#3633	C₂	1	2	yes	maximal
2	#14545893	C₂	1	3	yes	maximal, complemented
3	M₁₆×#3633	C₂	1	6	yes	maximal, complemented
4	C₂×C₈×#3633	C₂	1	3	yes	maximal, complemented
5	#131×#595	C₂	1	1	yes	maximal, characteristic
6	Q₈×#3633	C₄	1	4	yes	—
7	D₄×#3633	C₄	1	12	yes	—
8	#11763922	C₂²	1	6	yes	—
9	C₂×C₄×#3633	C₂²	1	6	yes	—
10	#11763893	C₂²	1	6	yes	—
11	C₈×#3633	C₂²	1	1	yes	characteristic
12	C₈×#3633	C₂²	1	6	yes	—
13	C₈×#3633	C₂²	1	3	yes	—
14	D₁₇⋊#131	C₄	1	1	yes	characteristic
15	#41×#595	C₂²	1	1	yes	characteristic
16	#11763775	C₄	1	3	yes	—
17	#11763772	C₄	1	3	yes	—
18	M₁₆×#595	C₂²	1	3	yes	—
19	C₂×C₈×#595	C₂²	1	3	yes	—
20	#131×D₁₇	C₄	1	1	yes	characteristic
21	C₂²×#3633	C₂×C₄	1	12	yes	—
22	#41×D₁₇	C₂×C₄	1	1	yes	characteristic
23	Q₈×#595	C₂×C₄	1	2	yes	—
24	#10587680	C₈	1	2	yes	complemented
25	D₄×#595	C₂×C₄	1	6	yes	—
26	#10587678	C₈	1	6	yes	complemented
27	#10587677	C₂×C₄	1	3	yes	—
28	C₂×C₄×#595	C₂³	1	3	yes	—
29	D₁₇⋊M₁₆	C₂×C₄	1	3	yes	—
30	C₂×#63242	C₂×C₄	1	3	yes	—
31	C₂²⋊#3633	C₂×C₄	1	12	yes	—
32	C₄⋊#3633	C₂×C₄	1	12	yes	—
33	C₄×#3633	C₂×C₄	1	12	yes	—
34	C₄×#3633	C₂³	1	2	yes	—
35	C₄×#3633	C₂³	1	6	yes	—
36	C₁₇×#131	C₈	1	1	yes	characteristic, complemented
37	#10587634	C₈	1	1	yes	characteristic, complemented
38	#10587593	C₈	1	3	yes	complemented
39	M₁₆×D₁₇	C₂×C₄	1	3	yes	—
40	#10587587	C₈	1	3	yes	complemented
41	C₂×C₈×D₁₇	C₂×C₄	1	3	yes	—
42	#10587532	C₂×C₄	1	6	yes	—
43	C₈×#595	C₂×C₄	1	3	yes	—
44	C₈×#595	C₂×C₄	1	3	yes	—
45	C₈×#595	C₂³	1	1	yes	characteristic
46	C₈×#595	C₂³	1	3	yes	—
47	C₂²×#595	C₂²×C₄	1	6	yes	—
48	C₂×#3633	—	2	24	no	—
49	C₂×#3633	C₂²×C₄	1	12	yes	—
50	C₂×#3633	C₂²×C₄	1	4	yes	—
51	#41×C₁₇	C₂×C₈	1	1	yes	characteristic
52	#63256	C₂×C₈	1	1	yes	characteristic
53	Q₈×D₁₇	C₄²	1	1	yes	characteristic
54	#63254	C₂×C₈	1	3	yes	—
55	D₄×D₁₇	C₄²	1	3	yes	—
56	#63252	C₂×C₈	1	3	yes	—
57	C₂×C₄×D₁₇	C₂²×C₄	1	3	yes	—
58	C₂²⋊#595	C₄²	1	6	yes	—
59	#63247	C₂×C₈	1	6	yes	complemented
60	C₂×#3624	C₂×C₈	1	6	yes	complemented
61	C₄⋊#595	C₄²	1	6	yes	—
62	C₄×#595	C₂²×C₄	1	6	yes	—
63	C₄×#595	C₂²×C₄	1	3	yes	—
64	C₄×#595	C₂⁴	1	1	yes	characteristic
65	#63243	C₂×C₈	1	6	yes	complemented
66	#63242	C₂×C₈	1	6	yes	complemented
67	#63242	C₂²×C₄	1	1	yes	characteristic
68	#63242	C₂²×C₄	1	3	yes	—
69	M₁₆×C₁₇	C₂×C₈	1	3	yes	complemented
70	C₂×C₈×C₁₇	C₂×C₈	1	3	yes	complemented
71	#63223	C₂×C₈	1	3	yes	complemented
72	C₂×#3622	C₂×C₈	1	3	yes	complemented
73	#63218	C₂×C₈	1	6	yes	complemented
74	C₈×D₁₇	C₂×C₈	1	3	yes	complemented
75	C₈×D₁₇	C₂×C₈	1	3	yes	complemented
76	C₈×D₁₇	C₂²×C₄	1	1	yes	characteristic
77	C₈×D₁₇	C₂²×C₄	1	3	yes	—
78	C₈×#131	—	17	1	no	maximal, Sylow
79	C₂²×D₁₇	C₂×C₄²	1	3	yes	—
80	C₂×#595	—	2	12	no	—
81	C₂×#595	C₂×C₄²	1	6	yes	—
82	C₂×#595	C₂³×C₄	1	2	yes	—
83	#3633	—	2	12	no	—
84	#3633	#131	1	8	yes	direct product factor
85	Q₈×C₁₇	C₄×C₈	1	1	yes	characteristic
86	D₄×C₁₇	C₄×C₈	1	3	yes	—
87	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
88	C₂²⋊D₁₇	C₄×C₈	1	6	yes	—
89	C₂×Dic₁₇	C₂²×C₈	1	3	yes	—
90	D₆₈	C₄×C₈	1	3	yes	—
91	C₄×D₁₇	C₂×C₄²	1	3	yes	—
92	C₄×D₁₇	C₂²×C₈	1	3	yes	—
93	C₄×D₁₇	C₂²×C₈	1	3	yes	—
94	C₄×D₁₇	C₂³×C₄	1	1	yes	characteristic
95	Dic₃₄	C₄×C₈	1	3	yes	—
96	#3624	C₂²×C₈	1	2	yes	—
97	#3624	C₂²×C₈	1	6	yes	—
98	C₈×C₁₇	C₂²×C₈	1	3	yes	—
99	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
100	#3622	C₂²×C₈	1	1	yes	characteristic
101	#3622	C₂²×C₈	1	3	yes	—
102	C₈×#41	—	17	2	no	—
103	C₄×#131	—	17	1	no	—
104	#1450	—	17	3	no	—
105	C₈×M₁₆	—	17	6	no	—
106	C₂×C₈²	—	17	3	no	—
107	C₂²×C₁₇	C₂×C₄×C₈	1	3	yes	—
108	C₂×D₁₇	—	2	6	no	—
109	C₂×D₁₇	C₂×C₄×C₈	1	3	yes	—
110	C₂×D₁₇	C₂²×C₄²	1	1	yes	characteristic
111	#595	—	2	6	no	—
112	#595	C₂×#131	1	4	yes	—
113	C₄×C₁₇	C₂×C₄×C₈	1	3	yes	—
114	C₄×C₁₇	C₂³×C₈	1	1	yes	characteristic
115	Dic₁₇	C₂×C₄×C₈	1	3	yes	—
116	Dic₁₇	C₂³×C₈	1	1	yes	characteristic
117	C₂×#131	—	17	1	no	—
118	C₂×#131	—	17	1	no	—
119	C₄×#41	—	17	1	no	—
120	C₈×Q₈	—	17	4	no	—
121	C₈×D₄	—	17	12	no	—
122	#431	—	17	6	no	—
123	#405	—	17	3	no	—
124	#405	—	17	3	no	—
125	C₄×M₁₆	—	17	3	no	—
126	C₂×C₄×C₈	—	17	3	no	—
127	C₂×C₄×C₈	—	17	6	no	—
128	#322	—	17	6	no	—
129	C₈²	—	17	1	no	—
130	C₈²	—	17	6	no	—
131	C₈²	—	17	3	no	—
132	C₂×C₁₇	—	2	3	no	—
133	C₂×C₁₇	C₂²×C₄×C₈	1	1	yes	characteristic
134	D₁₇	—	2	3	no	—
135	D₁₇	C₄×#131	1	2	yes	—
136	C₂×#41	—	17	1	no	—
137	#131	—	17	6	no	—
138	#131	—	17	2	no	—
139	#131	—	17	1	no	—
140	#131	—	17	3	no	—
141	#131	—	17	3	no	—
142	#131	#3633	1	1	yes	characteristic, direct product factor
143	C₂×M₁₆	—	17	3	no	—
144	C₂×M₁₆	—	17	3	no	—
145	C₂²×C₈	—	17	3	no	—
146	C₂²×C₈	—	17	3	no	—
147	C₂²×C₈	—	17	12	no	—
148	C₄×Q₈	—	17	2	no	—
149	C₄×D₄	—	17	6	no	—
150	#117	—	17	3	no	—
151	C₂×C₄²	—	17	3	no	—
152	C₄⋊C₈	—	17	12	no	—
153	C₂²⋊C₈	—	17	12	no	—
154	#97	—	17	6	no	—
155	C₄×C₈	—	17	12	no	—
156	C₄×C₈	—	17	2	no	—
157	C₄×C₈	—	17	6	no	—
158	C₄×C₈	—	17	1	no	—
159	C₄×C₈	—	17	3	no	—
160	C₄×C₈	—	17	3	no	—
161	C₄×C₈	—	17	3	no	—
162	C₁₇	C₈×#131	1	1	yes	characteristic, complemented, Sylow
163	#41	—	17	1	no	—
164	#41	—	17	3	no	—
165	#41	—	17	3	no	—
166	#41	C₂×#3633	1	1	yes	characteristic
167	C₂×Q₈	—	17	1	no	—
168	C₂×D₄	—	17	3	no	—
169	C₂²×C₄	—	17	3	no	—
170	C₂²×C₄	—	17	6	no	—
171	M₁₆	—	17	6	no	—
172	M₁₆	—	17	6	no	—
173	M₁₆	—	17	3	no	—
174	M₁₆	—	17	6	no	—
175	M₁₆	C₂×#3633	1	3	yes	complemented
176	C₂×C₈	—	17	6	no	—
177	C₂×C₈	—	17	1	no	—
178	C₂×C₈	—	17	3	no	—
179	C₂×C₈	—	17	6	no	—
180	C₂×C₈	—	17	4	no	—
181	C₂×C₈	—	17	3	no	—
182	C₂×C₈	—	17	3	no	—
183	C₂×C₈	—	17	3	no	—
184	C₂×C₈	—	17	1	no	—
185	C₂×C₈	—	17	3	no	—
186	C₂×C₈	—	17	12	no	—
187	C₂×C₈	—	34	24	no	—
188	C₂×C₈	C₂×#3633	1	3	yes	complemented
189	C₄⋊C₄	—	17	6	no	—
190	C₂²⋊C₄	—	17	6	no	—
191	C₄²	—	17	3	no	—
192	C₄²	—	17	6	no	—
193	C₄²	—	17	1	no	—
194	C₂³	—	17	3	no	—
195	Q₈	—	17	3	no	—
196	Q₈	C₄×#3633	1	1	yes	characteristic
197	D₄	—	17	6	no	—
198	D₄	—	17	3	no	—
199	D₄	C₄×#3633	1	3	yes	—
200	C₂×C₄	—	17	2	no	—
201	C₂×C₄	—	17	3	no	—
202	C₂×C₄	—	17	3	no	—
203	C₂×C₄	—	17	3	no	—
204	C₂×C₄	—	17	1	no	—
205	C₂×C₄	—	17	3	no	—
206	C₂×C₄	—	17	6	no	—
207	C₂×C₄	—	34	12	no	—
208	C₂×C₄	C₂²×#3633	1	3	yes	—
209	C₈	—	17	2	no	—
210	C₈	—	17	6	no	—
211	C₈	—	17	8	no	—
212	C₈	—	17	1	no	—
213	C₈	—	17	3	no	—
214	C₈	—	34	12	no	—
215	C₈	C₂²×#3633	1	3	yes	—
216	C₈	C₂²×#3633	1	1	yes	characteristic
217	C₂²	—	17	1	no	—
218	C₂²	—	17	3	no	—
219	C₂²	—	34	6	no	—
220	C₂²	C₂×C₄×#3633	1	3	yes	—
221	C₄	—	17	4	no	—
222	C₄	—	17	1	no	—
223	C₄	—	17	3	no	—
224	C₄	—	34	6	no	—
225	C₄	C₂×C₄×#3633	1	3	yes	—
226	C₄	C₂³×#3633	1	1	yes	characteristic
227	C₂	—	2	3	no	—
228	C₂	—	17	2	no	—
229	C₂	—	34	3	no	—
230	C₂	C₂²×C₄×#3633	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×#3633
commutator, socle	C₂×C₁₇	C₂²×C₄×C₈
Frattini	C₄	C₂³×#3633
Fitting	C₁₇×#131	C₈
prime radical	#41×D₁₇	C₂×C₄
cosocle	C₄×#595	C₂⁴
hypercenter	#131	#3633
hypocenter	C₁₇	C₈×#131
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
2. #14545893	C₂	semidirect
3. M₁₆×#3633	C₂	semidirect
4. C₂×C₈×#3633	C₂	semidirect
24. #10587680	C₈	semidirect
26. #10587678	C₈	semidirect
36. C₁₇×#131	C₈	semidirect
37. #10587634	C₈	semidirect
38. #10587593	C₈	semidirect
40. #10587587	C₈	semidirect
59. #63247	C₂×C₈	semidirect
60. C₂×#3624	C₂×C₈	semidirect
65. #63243	C₂×C₈	semidirect
66. #63242	C₂×C₈	semidirect
69. M₁₆×C₁₇	C₂×C₈	semidirect
70. C₂×C₈×C₁₇	C₂×C₈	semidirect
71. #63223	C₂×C₈	semidirect
72. C₂×#3622	C₂×C₈	semidirect
73. #63218	C₂×C₈	semidirect
74. C₈×D₁₇	C₂×C₈	semidirect
75. C₈×D₁₇	C₂×C₈	semidirect
84. #3633	#131	direct (N × H)
142. #131	#3633	direct (N × H)
162. C₁₇	C₈×#131	semidirect
175. M₁₆	C₂×#3633	semidirect
188. C₂×C₈	C₂×#3633	semidirect

Maximal subgroups

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

subgroup	index
#41×#3633	2
#14545893	2
M₁₆×#3633	2
C₂×C₈×#3633	2
#131×#595	2
C₈×#131	17