Subgroups — #17962204

Order 4352.

350 conjugacy classes of subgroups.

The subgroup lattice has 226 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×C₄⋊#63216	C₂	1	1	yes	maximal, characteristic
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	2	yes	—
7	C₂²×#63216	C₂²	1	1	yes	characteristic
8	C₂²×#63216	C₂²	1	1	yes	characteristic
9	C₄⋊#63216	—	2	4	no	—
10	C₄⋊#63216	—	2	2	no	—
11	C₄×#63216	—	2	2	no	—
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	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₇	C₈	1	1	yes	characteristic
19	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×#3622	C₂×C₄	1	2	yes	—
22	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
23	C₂²×#3622	C₂³	1	1	yes	characteristic
24	C₂×#63216	—	2	1	no	—
25	C₂×#63216	—	2	1	no	—
26	C₂×#63216	—	2	1	no	—
27	C₂×#63216	—	2	1	no	—
28	C₂×#63216	—	2	2	no	—
29	C₂×#63216	—	2	2	no	—
30	C₂×#63216	—	2	2	no	—
31	C₂×#63216	D₄	1	2	yes	—
32	C₂×#63216	Q₈	1	2	yes	—
33	C₁₇×#95	C₈	1	2	yes	—
34	C₄⋊#3622	—	2	2	no	—
35	C₄×#3622	—	2	2	no	—
36	C₄×#3622	—	2	2	no	—
37	C₄×#3622	D₄	1	2	yes	—
38	C₄×#3622	Q₈	1	2	yes	—
39	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
40	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₄⋊C₄×C₁₇	—	2	2	no	—
45	C₄⋊C₄×C₁₇	C₁₆	1	4	yes	complemented
46	C₄²×C₁₇	—	2	2	no	—
47	C₄²×C₁₇	—	2	2	no	—
48	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
49	C₄²×C₁₇	M₁₆	1	2	yes	—
50	C₂×#3622	—	2	1	no	—
51	C₂×#3622	—	2	1	no	—
52	C₂×#3622	—	2	2	no	—
53	C₂×#3622	—	2	2	no	—
54	C₂×#3622	—	2	2	no	—
55	C₂×#3622	C₄⋊C₄	1	2	yes	—
56	C₂×#3622	C₂×D₄	1	1	yes	characteristic
57	C₂×#3622	C₂×D₄	1	1	yes	characteristic
58	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
59	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
60	C₂×#3622	#41	1	1	yes	characteristic
61	C₂×#3622	#41	1	1	yes	characteristic
62	#63216	—	2	2	no	—
63	#63216	—	2	2	no	—
64	#63216	—	4	2	no	—
65	#13159	—	17	1	no	maximal, Sylow
66	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₁₇	—	2	2	no	—
68	C₂×C₄×C₁₇	—	2	1	no	—
69	C₂×C₄×C₁₇	—	2	1	no	—
70	C₂×C₄×C₁₇	—	2	2	no	—
71	C₂×C₄×C₁₇	—	2	2	no	—
72	C₂×C₄×C₁₇	—	2	2	no	—
73	C₂×C₄×C₁₇	—	2	2	no	—
74	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
75	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
76	C₂×C₄×C₁₇	C₂×C₁₆	1	4	yes	—
77	C₂×C₄×C₁₇	C₂×C₁₆	1	2	yes	—
78	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
79	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
84	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
85	#3622	—	2	2	no	—
86	#3622	—	4	2	no	—
87	#3622	#121	1	1	yes	characteristic
88	#3622	C₂²⋊Q₈	1	1	yes	characteristic
89	#3622	#125	1	1	yes	characteristic
90	#3622	C₄⋊Q₈	1	1	yes	characteristic
91	C₂×C₄⋊C₁₆	—	17	1	no	—
92	C₂×C₄⋊C₁₆	—	17	2	no	—
93	C₂×C₄×C₁₆	—	17	1	no	—
94	C₈×C₄⋊C₄	—	17	1	no	—
95	#1374	—	17	2	no	—
96	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
97	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
98	C₂²×C₁₇	C₈×Q₈	1	1	yes	characteristic
99	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
100	C₂²×C₁₇	C₂×M₃₂	1	1	yes	characteristic
101	C₂²×C₁₇	#504	1	1	yes	characteristic
102	C₂²×C₁₇	#504	1	1	yes	characteristic
103	C₄×C₁₇	—	2	2	no	—
104	C₄×C₁₇	—	2	4	no	—
105	C₄×C₁₇	—	4	2	no	—
106	C₄×C₁₇	C₄⋊C₁₆	1	4	yes	complemented
107	C₄×C₁₇	#389	1	1	yes	characteristic
108	C₄×C₁₇	#424	1	1	yes	characteristic
109	C₄×C₁₇	#436	1	1	yes	characteristic
110	C₄×C₁₇	#446	1	1	yes	characteristic
111	C₂²×C₁₆	—	17	1	no	—
112	C₂²×C₁₆	—	17	1	no	—
113	C₂²×C₁₆	—	17	2	no	—
114	C₂×C₄⋊C₈	—	17	1	no	—
115	C₂×C₄×C₈	—	17	1	no	—
116	C₂×C₄×C₈	—	17	2	no	—
117	C₄×C₄⋊C₄	#595	1	1	yes	characteristic
118	C₄⋊C₁₆	—	34	2	no	—
119	C₄⋊C₁₆	—	34	4	no	—
120	C₄×C₁₆	—	34	2	no	—
121	#336	—	17	2	no	—
122	C₂×C₁₇	#1939	1	1	yes	characteristic
123	C₂×C₁₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
124	C₂×C₁₇	#2151	1	1	yes	characteristic
125	C₂×C₁₇	D₄×C₁₆	1	1	yes	characteristic
126	C₂×C₁₇	#2169	1	1	yes	characteristic
127	C₂×C₁₇	Q₈×C₁₆	1	1	yes	characteristic
128	C₂×C₁₇	#2183	1	1	yes	characteristic
129	C₂²×C₈	—	17	1	no	—
130	C₂²×C₈	—	17	1	no	—
131	C₂²×C₈	—	17	2	no	—
132	C₂×C₄⋊C₄	#3624	1	1	yes	characteristic
133	C₂×C₄⋊C₄	#3624	1	1	yes	characteristic
134	C₂×C₄²	C₂×#595	1	2	yes	—
135	C₂×C₄²	C₂×#595	1	1	yes	characteristic
136	C₂×C₁₆	—	17	2	no	—
137	C₂×C₁₆	—	17	2	no	—
138	C₂×C₁₆	—	34	1	no	—
139	C₂×C₁₆	—	34	1	no	—
140	C₂×C₁₆	—	34	1	no	—
141	C₂×C₁₆	—	34	1	no	—
142	C₂×C₁₆	—	34	2	no	—
143	C₂×C₁₆	—	34	2	no	—
144	C₂×C₁₆	—	34	2	no	—
145	C₄⋊C₈	—	34	2	no	—
146	C₄×C₈	—	17	2	no	—
147	C₄×C₈	—	17	2	no	—
148	C₄×C₈	—	34	2	no	—
149	C₄×C₈	—	34	2	no	—
150	#95	#3624	1	2	yes	—
151	C₁₇	#13159	1	1	yes	characteristic, complemented, Sylow
152	C₂²×C₄	C₂×#3624	1	2	yes	—
153	C₂²×C₄	C₂×#3624	1	2	yes	—
154	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
155	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
156	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
157	C₂×C₈	—	17	1	no	—
158	C₂×C₈	—	17	1	no	—
159	C₂×C₈	—	17	1	no	—
160	C₂×C₈	—	17	1	no	—
161	C₂×C₈	—	17	1	no	—
162	C₂×C₈	—	17	1	no	—
163	C₂×C₈	—	17	2	no	—
164	C₂×C₈	—	34	1	no	—
165	C₂×C₈	—	34	1	no	—
166	C₂×C₈	—	34	2	no	—
167	C₂×C₈	—	34	2	no	—
168	C₂×C₈	—	34	2	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	#63216	1	4	yes	complemented
171	C₄²	—	2	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	#63243	1	2	yes	—
174	C₄²	C₄⋊#595	1	2	yes	—
175	C₁₆	—	34	2	no	—
176	C₁₆	—	34	2	no	—
177	C₁₆	—	68	2	no	—
178	C₂³	C₂²×#3624	1	1	yes	characteristic
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	C₂×#63216	1	4	yes	—
187	C₂×C₄	C₂×#63216	1	2	yes	—
188	C₂×C₄	#10587538	1	2	yes	—
189	C₂×C₄	C₄⋊#3624	1	2	yes	—
190	C₂×C₄	C₄⋊#3624	1	2	yes	—
191	C₂×C₄	C₂×#63243	1	1	yes	characteristic
192	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
193	C₂×C₄	#10587678	1	1	yes	characteristic
194	C₂×C₄	D₄×#595	1	1	yes	characteristic
195	C₂×C₄	#10587680	1	1	yes	characteristic
196	C₂×C₄	Q₈×#595	1	1	yes	characteristic
197	C₈	—	17	1	no	—
198	C₈	—	17	1	no	—
199	C₈	—	17	1	no	—
200	C₈	—	17	1	no	—
201	C₈	—	34	2	no	—
202	C₈	—	68	2	no	—
203	C₂²	C₂×C₄⋊#3624	1	1	yes	characteristic
204	C₂²	D₄×#3624	1	1	yes	characteristic
205	C₂²	Q₈×#3624	1	1	yes	characteristic
206	C₂²	C₂²×#63216	1	1	yes	characteristic
207	C₂²	C₂×#10587538	1	1	yes	characteristic
208	C₂²	#11763867	1	1	yes	characteristic
209	C₂²	#11763867	1	1	yes	characteristic
210	C₄	—	2	4	no	—
211	C₄	—	2	2	no	—
212	C₄	—	4	2	no	—
213	C₄	C₄⋊#63216	1	4	yes	complemented
214	C₄	#11763760	1	1	yes	characteristic
215	C₄	#11763816	1	1	yes	characteristic
216	C₄	#11763843	1	1	yes	characteristic
217	C₄	#11763859	1	1	yes	characteristic
218	C₂	#14544881	1	1	yes	characteristic
219	C₂	C₂×C₄⋊#63216	1	1	yes	characteristic
220	C₂	#14545119	1	1	yes	characteristic
221	C₂	D₄×#63216	1	1	yes	characteristic
222	C₂	#14545168	1	1	yes	characteristic
223	C₂	Q₈×#63216	1	1	yes	characteristic
224	C₂	#14545199	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#595
commutator	C₂²×C₁₇	C₂²×C₁₆
Fitting	C₄×C₄⋊C₄×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂²×#3622	C₂³
hypercenter	C₄×C₄⋊C₄	#595
hypocenter	C₁₇	#13159
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
45. C₄⋊C₄×C₁₇	C₁₆	semidirect
106. C₄×C₁₇	C₄⋊C₁₆	semidirect
151. C₁₇	#13159	semidirect
170. C₄⋊C₄	#63216	semidirect
213. 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
#14543822	2
C₄⋊C₄×#3622	2
#13159	17