Subgroups — #17962207

Order 4352.

338 conjugacy classes of subgroups.

The subgroup lattice has 212 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×#11763697	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	#11763697	—	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	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	2	yes	—
21	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
22	C₂²×#3622	C₂³	1	1	yes	characteristic
23	C₂×#63216	—	2	1	no	—
24	C₂×#63216	—	2	1	no	—
25	C₂×#63216	—	2	1	no	—
26	C₂×#63216	—	2	1	no	—
27	C₂×#63216	—	2	2	no	—
28	C₂×#63216	—	2	2	no	—
29	C₂×#63216	—	2	2	no	—
30	C₂×#63216	D₄	1	2	yes	—
31	C₂×#63216	Q₈	1	2	yes	—
32	C₁₇×#95	C₈	1	1	yes	characteristic
33	C₁₇×#95	C₈	1	1	yes	characteristic
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	1	yes	characteristic
42	C₂²×C₄×C₁₇	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₄⋊C₄×C₁₇	—	2	4	no	—
45	C₄²×C₁₇	—	2	2	no	—
46	C₄²×C₁₇	—	2	2	no	—
47	C₄²×C₁₇	C₄⋊C₄	1	2	yes	—
48	C₄²×C₁₇	M₁₆	1	2	yes	—
49	C₂×#3622	—	2	1	no	—
50	C₂×#3622	—	2	1	no	—
51	C₂×#3622	—	2	2	no	—
52	C₂×#3622	—	2	2	no	—
53	C₂×#3622	—	2	2	no	—
54	C₂×#3622	C₄⋊C₄	1	2	yes	—
55	C₂×#3622	C₂×D₄	1	1	yes	characteristic
56	C₂×#3622	C₂×D₄	1	1	yes	characteristic
57	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
58	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
59	C₂×#3622	#41	1	1	yes	characteristic
60	C₂×#3622	#41	1	1	yes	characteristic
61	#63216	—	2	2	no	—
62	#63216	—	2	2	no	—
63	#63216	—	4	2	no	—
64	#13160	—	17	1	no	maximal, Sylow
65	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₇	—	2	2	no	—
67	C₂×C₄×C₁₇	—	2	2	no	—
68	C₂×C₄×C₁₇	—	2	2	no	—
69	C₂×C₄×C₁₇	—	2	2	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	C₄⋊C₈	1	2	yes	—
74	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
75	C₂×C₄×C₁₇	M₃₂	1	4	yes	—
76	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
81	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
82	#3622	—	2	2	no	—
83	#3622	—	4	2	no	—
84	#3622	#121	1	1	yes	characteristic
85	#3622	C₂²⋊Q₈	1	1	yes	characteristic
86	#3622	#125	1	1	yes	characteristic
87	#3622	C₄⋊Q₈	1	1	yes	characteristic
88	C₂×C₄⋊C₁₆	—	17	1	no	—
89	C₂×C₄⋊C₁₆	—	17	2	no	—
90	C₂×#346	—	17	1	no	—
91	C₈×C₄⋊C₄	—	17	1	no	—
92	#1374	—	17	2	no	—
93	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
94	C₂²×C₁₇	C₈×D₄	1	1	yes	characteristic
95	C₂²×C₁₇	C₈×Q₈	1	1	yes	characteristic
96	C₂²×C₁₇	C₂×M₃₂	1	2	yes	—
97	C₂²×C₁₇	#504	1	2	yes	—
98	C₄×C₁₇	—	2	2	no	—
99	C₄×C₁₇	—	2	2	no	—
100	C₄×C₁₇	—	2	4	no	—
101	C₄×C₁₇	—	4	2	no	—
102	C₄×C₁₇	#389	1	1	yes	characteristic
103	C₄×C₁₇	#424	1	1	yes	characteristic
104	C₄×C₁₇	#436	1	1	yes	characteristic
105	C₄×C₁₇	#446	1	1	yes	characteristic
106	C₂²×C₁₆	—	17	1	no	—
107	C₂²×C₁₆	—	17	1	no	—
108	C₂²×C₁₆	—	17	2	no	—
109	C₂×C₄⋊C₈	—	17	1	no	—
110	C₂×C₄×C₈	—	17	1	no	—
111	C₂×C₄×C₈	—	17	2	no	—
112	C₄×C₄⋊C₄	#595	1	1	yes	characteristic
113	C₄⋊C₁₆	—	34	2	no	—
114	C₄⋊C₁₆	—	34	4	no	—
115	#346	—	34	2	no	—
116	#336	—	17	2	no	—
117	C₂×C₁₇	#1939	1	1	yes	characteristic
118	C₂×C₁₇	#2150	1	1	yes	characteristic
119	C₂×C₁₇	#2151	1	1	yes	characteristic
120	C₂×C₁₇	#2168	1	2	yes	—
121	C₂×C₁₇	#2183	1	2	yes	—
122	C₂²×C₈	—	17	1	no	—
123	C₂²×C₈	—	17	1	no	—
124	C₂²×C₈	—	17	2	no	—
125	C₂×C₄⋊C₄	#3624	1	2	yes	—
126	C₂×C₄²	C₂×#595	1	2	yes	—
127	C₂×C₄²	C₂×#595	1	1	yes	characteristic
128	C₂×C₁₆	—	17	2	no	—
129	C₂×C₁₆	—	17	2	no	—
130	C₂×C₁₆	—	34	1	no	—
131	C₂×C₁₆	—	34	1	no	—
132	C₂×C₁₆	—	34	1	no	—
133	C₂×C₁₆	—	34	1	no	—
134	C₂×C₁₆	—	34	2	no	—
135	C₂×C₁₆	—	34	2	no	—
136	C₂×C₁₆	—	34	2	no	—
137	C₄⋊C₈	—	34	2	no	—
138	C₄×C₈	—	17	2	no	—
139	C₄×C₈	—	17	2	no	—
140	C₄×C₈	—	34	2	no	—
141	C₄×C₈	—	34	2	no	—
142	#95	#3624	1	1	yes	characteristic
143	#95	#3624	1	1	yes	characteristic
144	C₁₇	#13160	1	1	yes	characteristic, complemented, Sylow
145	C₂²×C₄	C₂×#3624	1	2	yes	—
146	C₂²×C₄	C₂×#3624	1	2	yes	—
147	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
148	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
149	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
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	—	17	2	no	—
157	C₂×C₈	—	34	1	no	—
158	C₂×C₈	—	34	1	no	—
159	C₂×C₈	—	34	2	no	—
160	C₂×C₈	—	34	2	no	—
161	C₂×C₈	—	34	2	no	—
162	C₄⋊C₄	—	2	4	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	#63243	1	2	yes	—
166	C₄²	C₄⋊#595	1	2	yes	—
167	C₁₆	—	34	2	no	—
168	C₁₆	—	34	2	no	—
169	C₁₆	—	68	2	no	—
170	C₂³	C₂²×#3624	1	1	yes	characteristic
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	#10587538	1	4	yes	—
179	C₂×C₄	C₄⋊#3624	1	2	yes	—
180	C₂×C₄	C₄⋊#3624	1	2	yes	—
181	C₂×C₄	C₂×#63243	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
183	C₂×C₄	#10587678	1	1	yes	characteristic
184	C₂×C₄	D₄×#595	1	1	yes	characteristic
185	C₂×C₄	#10587680	1	1	yes	characteristic
186	C₂×C₄	Q₈×#595	1	1	yes	characteristic
187	C₈	—	17	1	no	—
188	C₈	—	17	1	no	—
189	C₈	—	17	1	no	—
190	C₈	—	17	1	no	—
191	C₈	—	34	2	no	—
192	C₈	—	68	2	no	—
193	C₂²	C₂×C₄⋊#3624	1	1	yes	characteristic
194	C₂²	D₄×#3624	1	1	yes	characteristic
195	C₂²	Q₈×#3624	1	1	yes	characteristic
196	C₂²	C₂×#10587538	1	2	yes	—
197	C₂²	#11763867	1	2	yes	—
198	C₄	—	2	4	no	—
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	4	2	no	—
202	C₄	#11763760	1	1	yes	characteristic
203	C₄	#11763816	1	1	yes	characteristic
204	C₄	#11763843	1	1	yes	characteristic
205	C₄	#11763859	1	1	yes	characteristic
206	C₂	#14544881	1	1	yes	characteristic
207	C₂	#14545116	1	1	yes	characteristic
208	C₂	#14545119	1	1	yes	characteristic
209	C₂	#14545164	1	2	yes	—
210	C₂	#14545199	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#595
commutator, socle, prime radical	C₂³×C₁₇	C₂²×C₈
Fitting	C₄×C₄⋊C₄×C₁₇	C₄
cosocle	C₂²×#3622	C₂³
hypercenter	C₄×C₄⋊C₄	#595
hypocenter	C₁₇	#13160
LCS	G ▷ C₂³×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
144. C₁₇	#13160	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
C₄⋊C₄×#3622	2
#13160	17