Subgroups — #17961010

Order 4352.

374 conjugacy classes of subgroups.

The subgroup lattice has 200 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	1	yes	maximal, characteristic
2	C₂×#11763697	C₂	1	2	yes	maximal
3	C₂×C₄×#63216	C₂	1	1	yes	maximal, characteristic
4	#14543822	C₂	1	2	yes	maximal
5	C₄⋊C₄×#3622	C₂	1	1	yes	maximal, characteristic
6	C₂²×#63216	C₂²	1	1	yes	characteristic
7	C₂²×#63216	C₂²	1	2	yes	—
8	C₂²×#63216	C₂²	1	1	yes	characteristic
9	C₄⋊#63216	C₄	1	4	yes	complemented
10	#11763697	—	2	4	no	—
11	C₄×#63216	C₄	1	4	yes	complemented
12	C₂×C₄⋊#3622	C₄	1	1	yes	characteristic
13	C₂×C₄×#3622	C₂²	1	2	yes	—
14	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
15	#11762564	C₄	1	2	yes	—
16	C₄×C₄⋊C₄×C₁₇	C₄	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₇	C₈	1	2	yes	—
18	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
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	2	yes	—
22	C₂²×#3622	C₂³	1	1	yes	characteristic
23	C₂×#63216	—	2	2	no	—
24	C₂×#63216	—	2	1	no	—
25	C₂×#63216	—	2	1	no	—
26	C₂×#63216	—	2	2	no	—
27	C₂×#63216	C₂×C₄	1	4	yes	—
28	C₂×#63216	C₂×C₄	1	2	yes	—
29	C₂×#63216	C₂×C₄	1	2	yes	—
30	C₂×#63216	D₄	1	2	yes	—
31	C₂×#63216	Q₈	1	2	yes	—
32	C₁₇×#95	C₈	1	2	yes	—
33	C₄⋊#3622	C₈	1	4	yes	—
34	C₄×#3622	—	2	2	no	—
35	C₄×#3622	—	2	2	no	—
36	C₄×#3622	C₂×C₄	1	2	yes	—
37	C₄×#3622	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₁₇	C₂×C₈	1	4	yes	—
39	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
42	C₄⋊C₄×C₁₇	—	2	4	no	—
43	C₄²×C₁₇	—	2	2	no	—
44	C₄²×C₁₇	—	2	2	no	—
45	C₄²×C₁₇	C₄²	1	2	yes	—
46	C₄²×C₁₇	C₂×C₈	1	2	yes	—
47	C₂×#3622	—	2	2	no	—
48	C₂×#3622	—	2	1	no	—
49	C₂×#3622	—	2	1	no	—
50	C₂×#3622	—	2	2	no	—
51	C₂×#3622	C₄²	1	2	yes	—
52	C₂×#3622	C₂×C₈	1	4	yes	—
53	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
54	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
55	C₂×#3622	C₂×D₄	1	1	yes	characteristic
56	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
57	C₂×#3622	#41	1	1	yes	characteristic
58	C₂×#3622	#41	1	1	yes	characteristic
59	#63216	—	2	2	no	—
60	#63216	—	2	4	no	—
61	#63216	C₄⋊C₄	1	4	yes	complemented
62	#12662	—	17	1	no	maximal, Sylow
63	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₇	—	2	2	no	—
65	C₂×C₄×C₁₇	—	2	4	no	—
66	C₂×C₄×C₁₇	—	2	4	no	—
67	C₂×C₄×C₁₇	—	2	4	no	—
68	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
69	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
70	C₂×C₄×C₁₇	M₃₂	1	4	yes	—
71	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
76	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
77	#3622	—	2	2	no	—
78	#3622	—	2	4	no	—
79	#3622	C₂×C₄⋊C₄	1	1	yes	characteristic
80	#3622	#117	1	1	yes	characteristic
81	#3622	C₄×D₄	1	1	yes	characteristic
82	#3622	C₄×Q₈	1	1	yes	characteristic
83	C₂×C₄⋊C₁₆	—	17	1	no	—
84	C₂×#346	—	17	2	no	—
85	C₂×C₄×C₁₆	—	17	1	no	—
86	C₈×C₄⋊C₄	—	17	1	no	—
87	#1374	—	17	2	no	—
88	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
89	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
90	C₂²×C₁₇	C₈×Q₈	1	1	yes	characteristic
91	C₂²×C₁₇	C₂×M₃₂	1	2	yes	—
92	C₂²×C₁₇	#504	1	2	yes	—
93	C₄×C₁₇	—	2	2	no	—
94	C₄×C₁₇	—	4	4	no	—
95	C₄×C₁₇	#346	1	4	yes	complemented
96	C₄×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
97	C₄×C₁₇	#405	1	1	yes	characteristic
98	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
99	C₄×C₁₇	C₈×Q₈	1	1	yes	characteristic
100	C₂²×C₁₆	—	17	1	no	—
101	C₂²×C₁₆	—	17	1	no	—
102	C₂²×C₁₆	—	17	2	no	—
103	C₂×C₄⋊C₈	—	17	1	no	—
104	C₂×C₄×C₈	—	17	1	no	—
105	C₂×C₄×C₈	—	17	2	no	—
106	C₄×C₄⋊C₄	#595	1	1	yes	characteristic
107	C₄⋊C₁₆	—	17	4	no	—
108	#346	—	34	4	no	—
109	C₄×C₁₆	—	17	4	no	—
110	#336	—	17	2	no	—
111	C₂×C₁₇	C₈×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₁₇	C₂×#346	1	1	yes	characteristic
113	C₂×C₁₇	#2108	1	1	yes	characteristic
114	C₂×C₁₇	#2169	1	2	yes	—
115	C₂×C₁₇	#2183	1	2	yes	—
116	C₂²×C₈	—	17	1	no	—
117	C₂²×C₈	—	17	1	no	—
118	C₂²×C₈	—	17	2	no	—
119	C₂×C₄⋊C₄	#3624	1	2	yes	—
120	C₂×C₄²	C₂×#595	1	2	yes	—
121	C₂×C₄²	C₂×#595	1	1	yes	characteristic
122	C₂×C₁₆	—	17	2	no	—
123	C₂×C₁₆	—	17	2	no	—
124	C₂×C₁₆	—	17	2	no	—
125	C₂×C₁₆	—	17	2	no	—
126	C₂×C₁₆	—	17	4	no	—
127	C₂×C₁₆	—	34	2	no	—
128	C₂×C₁₆	—	34	1	no	—
129	C₂×C₁₆	—	34	1	no	—
130	C₂×C₁₆	—	34	2	no	—
131	C₄⋊C₈	—	17	4	no	—
132	C₄×C₈	—	17	2	no	—
133	C₄×C₈	—	17	2	no	—
134	C₄×C₈	—	34	2	no	—
135	C₄×C₈	—	34	2	no	—
136	#95	#3624	1	2	yes	—
137	C₁₇	#12662	1	1	yes	characteristic, complemented, Sylow
138	C₂²×C₄	C₂×#3624	1	4	yes	—
139	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
140	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
141	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
142	C₂×C₈	—	17	1	no	—
143	C₂×C₈	—	17	1	no	—
144	C₂×C₈	—	17	1	no	—
145	C₂×C₈	—	17	1	no	—
146	C₂×C₈	—	17	1	no	—
147	C₂×C₈	—	17	1	no	—
148	C₂×C₈	—	17	4	no	—
149	C₂×C₈	—	17	2	no	—
150	C₂×C₈	—	34	1	no	—
151	C₂×C₈	—	34	2	no	—
152	C₂×C₈	—	34	1	no	—
153	C₂×C₈	—	34	2	no	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	#63242	1	2	yes	—
158	C₄²	C₄×#595	1	2	yes	—
159	C₁₆	—	17	4	no	—
160	C₁₆	—	34	4	no	—
161	C₁₆	—	34	2	no	—
162	C₂³	C₂²×#3624	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	#10587538	1	4	yes	—
168	C₂×C₄	C₄×#3624	1	2	yes	—
169	C₂×C₄	C₄×#3624	1	2	yes	—
170	C₂×C₄	C₂×#63242	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
172	C₂×C₄	#10587678	1	1	yes	characteristic
173	C₂×C₄	D₄×#595	1	1	yes	characteristic
174	C₂×C₄	#10587680	1	1	yes	characteristic
175	C₂×C₄	Q₈×#595	1	1	yes	characteristic
176	C₈	—	17	1	no	—
177	C₈	—	17	1	no	—
178	C₈	—	17	1	no	—
179	C₈	—	17	1	no	—
180	C₈	—	34	2	no	—
181	C₈	—	34	4	no	—
182	C₂²	C₂×C₄×#3624	1	1	yes	characteristic
183	C₂²	D₄×#3624	1	1	yes	characteristic
184	C₂²	Q₈×#3624	1	1	yes	characteristic
185	C₂²	C₂×#10587538	1	2	yes	—
186	C₂²	#11763867	1	2	yes	—
187	C₄	—	2	2	no	—
188	C₄	—	4	4	no	—
189	C₄	#11763697	1	4	yes	complemented
190	C₄	C₄⋊C₄×#595	1	1	yes	characteristic
191	C₄	#11763774	1	1	yes	characteristic
192	C₄	#11763837	1	1	yes	characteristic
193	C₄	#11763858	1	1	yes	characteristic
194	C₂	C₄⋊C₄×#3624	1	1	yes	characteristic
195	C₂	C₂×#11763697	1	1	yes	characteristic
196	C₂	#14544993	1	1	yes	characteristic
197	C₂	#14545168	1	2	yes	—
198	C₂	#14545199	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#595
commutator	C₂²×C₁₇	C₂×C₄×C₈
Fitting	C₄×C₄⋊C₄×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂²×#3622	C₂³
hypercenter	C₄×C₄⋊C₄	#595
hypocenter	C₁₇	#12662
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. C₄⋊#63216	C₄	semidirect
11. C₄×#63216	C₄	semidirect
61. #63216	C₄⋊C₄	semidirect
95. C₄×C₁₇	#346	semidirect
137. C₁₇	#12662	semidirect
189. 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
C₂×C₄×#63216	2
#14543822	2
C₄⋊C₄×#3622	2
#12662	17