Subgroups — #17960963

Order 4352.

414 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#63216	C₂	1	2	yes	maximal
2	C₂×#11763697	C₂	1	2	yes	maximal
3	#14543822	C₂	1	2	yes	maximal
4	C₄²×#3622	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63216	C₂²	1	4	yes	—
6	C₄⋊#63216	C₄	1	8	yes	complemented
7	#11763697	—	2	4	no	—
8	C₂×C₄×#3622	C₄	1	1	yes	characteristic
9	C₂×C₄×#3622	C₄	1	1	yes	characteristic
10	C₂×C₄×#3622	C₄	1	1	yes	characteristic
11	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
12	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
13	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
14	C₄³×C₁₇	C₄	1	1	yes	characteristic
15	C₂×C₄²×C₁₇	C₈	1	2	yes	—
16	C₂×C₄²×C₁₇	C₈	1	2	yes	—
17	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
21	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
22	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
23	C₂²×#3622	C₂³	1	1	yes	characteristic
24	C₂×#63216	—	2	4	no	—
25	C₂×#63216	—	2	4	no	—
26	C₂×#63216	C₂×C₄	1	8	yes	—
27	C₄×#3622	—	2	2	no	—
28	C₄×#3622	—	2	2	no	—
29	C₄×#3622	—	2	2	no	—
30	C₄×#3622	C₈	1	4	yes	—
31	C₄×#3622	C₂×C₄	1	4	yes	—
32	C₄×#3622	D₄	1	2	yes	—
33	C₄×#3622	Q₈	1	2	yes	—
34	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
36	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
37	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₁₇	—	2	4	no	—
40	C₄²×C₁₇	—	2	4	no	—
41	C₄²×C₁₇	—	2	1	no	—
42	C₄²×C₁₇	—	2	1	no	—
43	C₄²×C₁₇	C₄²	1	2	yes	—
44	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
45	C₄²×C₁₇	C₂×C₈	1	2	yes	—
46	C₄²×C₁₇	M₁₆	1	2	yes	—
47	C₂×#3622	—	2	1	no	—
48	C₂×#3622	—	2	1	no	—
49	C₂×#3622	—	2	1	no	—
50	C₂×#3622	—	2	1	no	—
51	C₂×#3622	C₄²	1	2	yes	—
52	C₂×#3622	C₄⋊C₄	1	2	yes	—
53	C₂×#3622	C₂×C₈	1	2	yes	—
54	C₂×#3622	C₂×C₈	1	2	yes	—
55	C₂×#3622	M₁₆	1	2	yes	—
56	C₂×#3622	C₂²×C₄	1	2	yes	—
57	C₂×#3622	C₂×D₄	1	1	yes	characteristic
58	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
59	C₂×#3622	#41	1	2	yes	—
60	#63216	—	2	8	no	—
61	#12653	—	17	1	no	maximal, Sylow
62	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
63	C₂×C₄×C₁₇	—	2	2	no	—
64	C₂×C₄×C₁₇	—	2	2	no	—
65	C₂×C₄×C₁₇	—	2	2	no	—
66	C₂×C₄×C₁₇	—	2	2	no	—
67	C₂×C₄×C₁₇	—	2	2	no	—
68	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
69	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
70	C₂×C₄×C₁₇	C₄⋊C₈	1	4	yes	—
71	C₂×C₄×C₁₇	M₃₂	1	4	yes	—
72	C₂×C₄×C₁₇	M₃₂	1	4	yes	—
73	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
79	#3622	—	2	2	no	—
80	#3622	—	2	2	no	—
81	#3622	C₄⋊C₈	1	4	yes	—
82	#3622	C₄×D₄	1	2	yes	—
83	#3622	C₄×Q₈	1	2	yes	—
84	C₂×C₄⋊C₁₆	—	17	2	no	—
85	C₂×#346	—	17	2	no	—
86	C₄²×C₈	—	17	1	no	—
87	#1374	—	17	2	no	—
88	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
89	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
90	C₂²×C₁₇	#431	1	1	yes	characteristic
91	C₂²×C₁₇	C₂×M₃₂	1	2	yes	—
92	C₂²×C₁₇	C₂×M₃₂	1	2	yes	—
93	C₄×C₁₇	—	2	2	no	—
94	C₄×C₁₇	—	2	4	no	—
95	C₄×C₁₇	—	2	4	no	—
96	C₄×C₁₇	#346	1	4	yes	complemented
97	C₄×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
98	C₄×C₁₇	C₄×M₁₆	1	1	yes	characteristic
99	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₄×C₁₇	#431	1	1	yes	characteristic
101	C₂²×C₁₆	—	17	4	no	—
102	C₂×C₄×C₈	—	17	1	no	—
103	C₂×C₄×C₈	—	17	1	no	—
104	C₂×C₄×C₈	—	17	1	no	—
105	C₂×C₄×C₈	—	17	1	no	—
106	C₂×C₄×C₈	—	17	1	no	—
107	C₂×C₄×C₈	—	17	1	no	—
108	C₄³	#595	1	1	yes	characteristic
109	C₄⋊C₁₆	—	17	8	no	—
110	#346	—	34	4	no	—
111	C₂×C₁₇	C₄×C₄⋊C₈	1	1	yes	characteristic
112	C₂×C₁₇	C₂×#346	1	1	yes	characteristic
113	C₂×C₁₇	C₄×M₃₂	1	1	yes	characteristic
114	C₂×C₁₇	#2150	1	2	yes	—
115	C₂×C₁₇	#2163	1	2	yes	—
116	C₂²×C₈	—	17	1	no	—
117	C₂²×C₈	—	17	1	no	—
118	C₂²×C₈	—	17	1	no	—
119	C₂²×C₈	—	17	1	no	—
120	C₂×C₄²	#3624	1	2	yes	—
121	C₂×C₄²	#3624	1	2	yes	—
122	C₂×C₄²	C₂×#595	1	1	yes	characteristic
123	C₂×C₄²	C₂×#595	1	1	yes	characteristic
124	C₂×C₄²	C₂×#595	1	1	yes	characteristic
125	C₂×C₁₆	—	17	8	no	—
126	C₂×C₁₆	—	34	4	no	—
127	C₂×C₁₆	—	34	4	no	—
128	C₄×C₈	—	17	4	no	—
129	C₄×C₈	—	17	4	no	—
130	C₄×C₈	—	17	2	no	—
131	C₄×C₈	—	17	2	no	—
132	C₄×C₈	—	34	2	no	—
133	C₄×C₈	—	34	2	no	—
134	C₄×C₈	—	34	2	no	—
135	C₁₇	#12653	1	1	yes	characteristic, complemented, Sylow
136	C₂²×C₄	C₂×#3624	1	2	yes	—
137	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
138	C₂²×C₄	C₂×#3624	1	2	yes	—
139	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
140	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
141	C₂×C₈	—	17	2	no	—
142	C₂×C₈	—	17	2	no	—
143	C₂×C₈	—	17	2	no	—
144	C₂×C₈	—	17	1	no	—
145	C₂×C₈	—	17	2	no	—
146	C₂×C₈	—	17	1	no	—
147	C₂×C₈	—	17	2	no	—
148	C₂×C₈	—	17	2	no	—
149	C₂×C₈	—	17	2	no	—
150	C₂×C₈	—	34	1	no	—
151	C₂×C₈	—	34	1	no	—
152	C₂×C₈	—	34	1	no	—
153	C₂×C₈	—	34	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	2	1	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	2	4	no	—
158	C₄²	#63242	1	2	yes	—
159	C₄²	#63243	1	2	yes	—
160	C₄²	C₄×#595	1	2	yes	—
161	C₄²	C₄⋊#595	1	2	yes	—
162	C₁₆	—	34	8	no	—
163	C₂³	C₂²×#3624	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	#10587538	1	4	yes	—
170	C₂×C₄	#10587538	1	4	yes	—
171	C₂×C₄	C₄×#3624	1	2	yes	—
172	C₂×C₄	C₄×#3624	1	2	yes	—
173	C₂×C₄	C₄⋊#3624	1	4	yes	—
174	C₂×C₄	C₂×#63242	1	1	yes	characteristic
175	C₂×C₄	C₂×#63243	1	1	yes	characteristic
176	C₂×C₄	D₁₇⋊M₁₆	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
179	C₂×C₄	#10587677	1	1	yes	characteristic
180	C₈	—	17	2	no	—
181	C₈	—	17	2	no	—
182	C₈	—	17	4	no	—
183	C₈	—	34	2	no	—
184	C₈	—	34	2	no	—
185	C₂²	C₂×C₄×#3624	1	1	yes	characteristic
186	C₂²	C₂×C₄⋊#3624	1	1	yes	characteristic
187	C₂²	#11763825	1	1	yes	characteristic
188	C₂²	C₂×#10587538	1	2	yes	—
189	C₂²	C₂×#10587538	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	4	no	—
193	C₄	#11763697	1	4	yes	complemented
194	C₄	C₄×C₄⋊#595	1	1	yes	characteristic
195	C₄	C₄×#63243	1	1	yes	characteristic
196	C₄	#11763810	1	1	yes	characteristic
197	C₄	#11763828	1	1	yes	characteristic
198	C₂	C₄×C₄⋊#3624	1	1	yes	characteristic
199	C₂	C₂×#11763697	1	1	yes	characteristic
200	C₂	C₄×#10587538	1	1	yes	characteristic
201	C₂	#14545116	1	2	yes	—
202	C₂	#14545145	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#595
commutator	C₂²×C₁₇	C₂×C₄×C₈
Fitting	C₄³×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂²×#3622	C₂³
hypercenter	C₄³	#595
hypocenter	C₁₇	#12653
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
6. C₄⋊#63216	C₄	semidirect
96. C₄×C₁₇	#346	semidirect
135. C₁₇	#12653	semidirect
193. C₄	#11763697	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#63216	2
C₂×#11763697	2
#14543822	2
C₄²×#3622	2
#12653	17