Subgroups — #17948583

Order 4352.

286 conjugacy classes of subgroups.

The subgroup lattice has 216 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×C₄×#63216	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×#3622	C₂	1	1	yes	maximal, characteristic
4	C₂²×#63216	—	2	1	no	—
5	C₂²×#63216	—	2	1	no	—
6	C₄⋊#63216	—	2	1	no	—
7	C₄⋊#63216	C₄	1	2	yes	complemented
8	C₄×#63216	—	2	1	no	—
9	C₄×#63216	C₄	1	2	yes	complemented
10	C₂×C₄⋊#3622	C₄	1	1	yes	characteristic
11	C₂×C₄×#3622	—	2	1	no	—
12	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
13	#11762564	—	2	1	no	—
14	C₄×C₄⋊C₄×C₁₇	C₄	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₁₇	C₈	1	2	yes	—
16	C₂×C₄²×C₁₇	—	2	1	no	—
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	D₄	1	1	yes	characteristic
21	C₂×#63216	—	2	2	no	—
22	C₂×#63216	—	2	2	no	—
23	C₂×#63216	—	2	2	no	—
24	C₂×#63216	—	2	2	no	—
25	C₂×#63216	—	4	1	no	—
26	C₂×#63216	—	4	1	no	—
27	C₁₇×#95	—	2	1	no	—
28	C₄⋊#3622	—	2	1	no	—
29	C₄⋊#3622	C₈	1	2	yes	—
30	C₄×#3622	—	4	1	no	—
31	C₄×#3622	—	4	1	no	—
32	C₄×#3622	C₂×C₄	1	1	yes	characteristic
33	C₄×#3622	C₂×C₄	1	1	yes	characteristic
34	C₄×#3622	D₄	1	1	yes	characteristic
35	C₄×#3622	Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₇	—	2	2	no	—
37	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
40	C₄⋊C₄×C₁₇	—	2	2	no	—
41	C₄⋊C₄×C₁₇	C₁₆	1	4	yes	complemented
42	C₄²×C₁₇	—	4	1	no	—
43	C₄²×C₁₇	—	4	1	no	—
44	C₄²×C₁₇	C₄²	1	1	yes	characteristic
45	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
48	C₂×#3622	—	2	1	no	—
49	C₂×#3622	—	2	1	no	—
50	C₂×#3622	—	2	2	no	—
51	C₂×#3622	—	4	1	no	—
52	C₂×#3622	—	4	1	no	—
53	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
56	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
57	C₂×#3622	D₈	1	1	yes	characteristic
58	C₂×#3622	SD₁₆	1	1	yes	characteristic
59	C₂×#3622	SD₁₆	1	1	yes	characteristic
60	C₂×#3622	Q₁₆	1	1	yes	characteristic
61	#63216	—	2	4	no	—
62	#63216	—	4	2	no	—
63	#7167	—	17	1	no	maximal, Sylow
64	C₂³×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₁₇	—	2	2	no	—
66	C₂×C₄×C₁₇	—	2	4	no	—
67	C₂×C₄×C₁₇	—	4	2	no	—
68	C₂×C₄×C₁₇	—	4	2	no	—
69	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
71	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
73	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
74	C₂×C₄×C₁₇	#103	1	1	yes	characteristic
75	C₂×C₄×C₁₇	#104	1	1	yes	characteristic
76	C₂×C₄×C₁₇	#104	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₄⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
80	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
81	#3622	—	2	1	no	—
82	#3622	—	2	1	no	—
83	#3622	—	4	2	no	—
84	#3622	#102	1	1	yes	characteristic
85	#3622	#103	1	1	yes	characteristic
86	#3622	#106	1	1	yes	characteristic
87	#3622	#107	1	1	yes	characteristic
88	C₂×C₄⋊C₁₆	—	17	1	no	—
89	C₂×C₄×C₁₆	—	17	1	no	—
90	C₈×C₄⋊C₄	—	17	1	no	—
91	C₂²×C₁₇	D₄⋊C₈	1	1	yes	characteristic
92	C₂²×C₁₇	Q₈⋊C₈	1	1	yes	characteristic
93	C₂²×C₁₇	#336	1	1	yes	characteristic
94	C₂²×C₁₇	C₂²⋊C₁₆	1	2	yes	—
95	C₂²×C₁₇	#350	1	2	yes	—
96	C₄×C₁₇	—	2	1	no	—
97	C₄×C₁₇	—	2	1	no	—
98	C₄×C₁₇	—	4	4	no	—
99	C₄×C₁₇	D₄⋊C₈	1	1	yes	characteristic
100	C₄×C₁₇	Q₈⋊C₈	1	1	yes	characteristic
101	C₄×C₁₇	#339	1	1	yes	characteristic
102	C₄×C₁₇	#340	1	1	yes	characteristic
103	C₄×C₁₇	C₄×C₁₆	1	1	yes	characteristic
104	C₄×C₁₇	#346	1	1	yes	characteristic
105	C₄×C₁₇	C₄⋊C₁₆	1	2	yes	—
106	C₂²×C₁₆	—	34	1	no	—
107	C₂²×C₁₆	—	34	1	no	—
108	C₂×C₄⋊C₈	—	17	1	no	—
109	C₂×C₄×C₈	—	17	1	no	—
110	C₂×C₄×C₈	—	34	1	no	—
111	C₄×C₄⋊C₄	#595	1	1	yes	characteristic
112	C₄⋊C₁₆	—	17	2	no	—
113	C₄⋊C₁₆	—	34	1	no	—
114	C₄×C₁₆	—	17	2	no	—
115	C₄×C₁₆	—	34	1	no	—
116	#336	—	34	1	no	—
117	C₂×C₁₇	#1279	1	1	yes	characteristic
118	C₂×C₁₇	D₄⋊C₁₆	1	2	yes	—
119	C₂×C₁₇	Q₈⋊C₁₆	1	2	yes	—
120	C₂×C₁₇	#1374	1	1	yes	characteristic
121	C₂×C₁₇	#1379	1	1	yes	characteristic
122	C₂²×C₈	—	17	1	no	—
123	C₂²×C₈	—	17	1	no	—
124	C₂²×C₈	—	34	1	no	—
125	C₂×C₄⋊C₄	#3624	1	2	yes	—
126	C₂×C₄²	—	2	1	no	—
127	C₂×C₄²	C₂×#595	1	1	yes	characteristic
128	C₂×C₁₆	—	34	2	no	—
129	C₂×C₁₆	—	34	2	no	—
130	C₂×C₁₆	—	34	2	no	—
131	C₂×C₁₆	—	34	2	no	—
132	C₂×C₁₆	—	68	1	no	—
133	C₂×C₁₆	—	68	1	no	—
134	C₄⋊C₈	—	17	2	no	—
135	C₄⋊C₈	—	34	1	no	—
136	C₄×C₈	—	17	1	no	—
137	C₄×C₈	—	17	1	no	—
138	C₄×C₈	—	17	1	no	—
139	C₄×C₈	—	17	1	no	—
140	C₄×C₈	—	68	1	no	—
141	C₄×C₈	—	68	1	no	—
142	#95	—	2	1	no	—
143	C₁₇	#7167	1	1	yes	characteristic, complemented, Sylow
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
146	C₂²×C₄	#63247	1	1	yes	characteristic
147	C₂²×C₄	C₂²⋊#595	1	1	yes	characteristic
148	C₂×C₈	—	17	1	no	—
149	C₂×C₈	—	17	1	no	—
150	C₂×C₈	—	17	1	no	—
151	C₂×C₈	—	17	1	no	—
152	C₂×C₈	—	17	1	no	—
153	C₂×C₈	—	17	1	no	—
154	C₂×C₈	—	17	1	no	—
155	C₂×C₈	—	17	1	no	—
156	C₂×C₈	—	34	1	no	—
157	C₂×C₈	—	34	1	no	—
158	C₂×C₈	—	34	2	no	—
159	C₂×C₈	—	68	1	no	—
160	C₂×C₈	—	68	1	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	#63216	1	4	yes	complemented
163	C₄²	—	4	1	no	—
164	C₄²	—	4	1	no	—
165	C₄²	#63242	1	1	yes	characteristic
166	C₄²	#63243	1	1	yes	characteristic
167	C₄²	C₄×#595	1	1	yes	characteristic
168	C₄²	C₄⋊#595	1	1	yes	characteristic
169	C₁₆	—	34	4	no	—
170	C₁₆	—	68	2	no	—
171	C₂³	C₂²⋊#3624	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	C₂×#63216	1	2	yes	—
177	C₂×C₄	#10587538	1	2	yes	—
178	C₂×C₄	C₄×#3624	1	1	yes	characteristic
179	C₂×C₄	C₄⋊#3624	1	1	yes	characteristic
180	C₂×C₄	#10587542	1	1	yes	characteristic
181	C₂×C₄	#10587543	1	1	yes	characteristic
182	C₂×C₄	#10587544	1	1	yes	characteristic
183	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
184	C₂×C₄	#10587547	1	1	yes	characteristic
185	C₂×C₄	#10587548	1	1	yes	characteristic
186	C₂×C₄	#10587549	1	1	yes	characteristic
187	C₂×C₄	#10587550	1	1	yes	characteristic
188	C₈	—	17	1	no	—
189	C₈	—	17	1	no	—
190	C₈	—	17	1	no	—
191	C₈	—	17	1	no	—
192	C₈	—	34	1	no	—
193	C₈	—	34	1	no	—
194	C₈	—	68	2	no	—
195	C₂²	D₄⋊#3624	1	1	yes	characteristic
196	C₂²	Q₈⋊#3624	1	1	yes	characteristic
197	C₂²	#11763675	1	1	yes	characteristic
198	C₂²	C₂²⋊#63216	1	2	yes	—
199	C₂²	#11763707	1	2	yes	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	4	4	no	—
203	C₄	#11763657	1	1	yes	characteristic
204	C₄	#11763659	1	1	yes	characteristic
205	C₄	#11763679	1	1	yes	characteristic
206	C₄	#11763682	1	1	yes	characteristic
207	C₄	C₄×#63216	1	1	yes	characteristic
208	C₄	#11763697	1	1	yes	characteristic
209	C₄	C₄⋊#63216	1	2	yes	—
210	C₂	#14543620	1	1	yes	characteristic
211	C₂	D₄⋊#63216	1	2	yes	—
212	C₂	Q₈⋊#63216	1	2	yes	—
213	C₂	#14543822	1	1	yes	characteristic
214	C₂	#14543838	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊#595
commutator	C₄×C₁₇	C₄×C₁₆
Frattini	C₂×C₄²	C₂×#595
Fitting	C₄×C₄⋊C₄×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂²⋊C₈
cosocle	C₂×C₄×#3622	C₂²
hypercenter	C₄×C₄⋊C₄	#595
hypocenter	C₁₇	#7167
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
7. C₄⋊#63216	C₄	semidirect
9. C₄×#63216	C₄	semidirect
41. C₄⋊C₄×C₁₇	C₁₆	semidirect
143. C₁₇	#7167	semidirect
162. C₄⋊C₄	#63216	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×C₄×#63216	2
C₄⋊C₄×#3622	2
#7167	17