Subgroups — #17978777

Order 4352.

452 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14545199	C₂	1	2	yes	maximal, complemented
2	#14545199	C₂	1	4	yes	maximal, complemented
3	Q₈×#63216	C₂	1	2	yes	maximal, complemented
4	#14545149	C₂	1	4	yes	maximal
5	#14545145	C₂	1	1	yes	maximal, characteristic
6	#14545141	C₂	1	1	yes	maximal, characteristic
7	#14537719	C₂	1	1	yes	maximal, characteristic
8	C₄⋊#63216	C₂²	1	2	yes	—
9	C₄⋊#63216	C₂²	1	4	yes	—
10	C₄⋊#63216	C₂²	1	2	yes	—
11	C₄⋊#63216	C₂²	1	4	yes	—
12	C₂²⋊#63216	C₂²	1	2	yes	—
13	C₂²⋊#63216	C₂²	1	2	yes	—
14	#11763697	C₂²	1	2	yes	—
15	#11763697	C₂²	1	4	yes	—
16	C₄×#63216	C₂²	1	1	yes	characteristic
17	C₄×#63216	C₂²	1	4	yes	—
18	C₄×#63216	C₂²	1	1	yes	characteristic
19	#11762940	C₄	1	2	yes	—
20	#11762940	C₄	1	2	yes	—
21	Q₈×#3622	C₂²	1	2	yes	—
22	Q₈×#3622	C₂²	1	2	yes	—
23	#11762836	C₄	1	2	yes	—
24	#11762833	C₄	1	1	yes	characteristic
25	#11762765	C₂²	1	2	yes	—
26	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
27	C₁₇×#533	C₄	1	1	yes	characteristic
28	C₁₇×C₄⋊Q₈	C₈	1	1	yes	characteristic
29	C₁₇×C₄⋊Q₈	C₈	1	1	yes	characteristic
30	C₁₇×#125	C₈	1	2	yes	—
31	C₁₇×C₂²⋊Q₈	C₈	1	2	yes	—
32	C₁₇×C₂²⋊Q₈	C₈	1	2	yes	—
33	C₄×Q₈×C₁₇	C₂×C₄	1	2	yes	—
34	C₄×Q₈×C₁₇	C₂×C₄	1	2	yes	—
35	C₁₇×#117	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
37	C₂×#63223	C₂×C₄	1	2	yes	—
38	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
39	C₂²×#3622	C₂³	1	1	yes	characteristic
40	C₂×#63216	C₂³	1	2	yes	—
41	C₂×#63216	C₂³	1	4	yes	—
42	C₂×#63216	C₂³	1	2	yes	—
43	C₄⋊#3622	C₂×C₄	1	2	yes	—
44	C₄⋊#3622	C₂×C₄	1	2	yes	—
45	C₄⋊#3622	C₂×C₄	1	4	yes	—
46	C₄⋊#3622	C₂×C₄	1	4	yes	—
47	#10587474	C₂×C₄	1	4	yes	—
48	C₄×#3622	C₂×C₄	1	1	yes	characteristic
49	C₄×#3622	C₂×C₄	1	1	yes	characteristic
50	C₄×#3622	C₂³	1	1	yes	characteristic
51	C₄×#3622	C₂³	1	1	yes	characteristic
52	C₄×#3622	C₂³	1	4	yes	—
53	C₂×Q₈×C₁₇	C₂×C₈	1	2	yes	—
54	C₂×Q₈×C₁₇	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
58	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
59	C₄⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
60	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
61	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
62	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
63	C₄⋊C₄×C₁₇	C₂×C₈	1	2	yes	—
64	C₂²⋊C₄×C₁₇	C₂×C₈	1	4	yes	—
65	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
66	C₄²×C₁₇	C₂×C₈	1	1	yes	characteristic
67	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₁₇	C₂²×C₄	1	4	yes	—
70	#63223	—	4	2	no	—
71	C₂×#3622	—	2	2	no	—
72	C₂×#3622	—	4	1	no	—
73	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
74	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
75	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
76	C₂×#3622	C₂²×C₄	1	4	yes	—
77	C₂×#3622	C₂⁴	1	1	yes	characteristic
78	#63216	—	2	2	no	—
79	#63216	—	2	4	no	—
80	#63216	—	2	2	no	—
81	#33499	—	17	1	no	maximal, Sylow
82	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
83	Q₈×C₁₇	—	2	4	no	—
84	Q₈×C₁₇	—	4	2	no	—
85	C₂×C₄×C₁₇	—	2	1	no	—
86	C₂×C₄×C₁₇	—	2	1	no	—
87	C₂×C₄×C₁₇	—	2	2	no	—
88	C₂×C₄×C₁₇	—	4	1	no	—
89	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
94	C₂×C₄×C₁₇	C₂²×C₈	1	4	yes	—
95	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
97	#3622	—	2	1	no	—
98	#3622	—	2	1	no	—
99	#3622	—	2	1	no	—
100	#3622	—	2	4	no	—
101	#3622	2₋¹⁺⁴	1	2	yes	—
102	#2960	—	17	1	no	—
103	#2183	—	17	2	no	—
104	#2183	—	17	4	no	—
105	Q₈×C₁₆	—	17	2	no	—
106	#2164	—	17	4	no	—
107	#2163	—	17	1	no	—
108	#2162	—	17	1	no	—
109	C₂²×C₁₇	—	2	1	no	—
110	C₂²×C₁₇	—	2	1	no	—
111	C₂²×C₁₇	—	2	1	no	—
112	C₂²×C₁₇	C₂³×C₈	1	1	yes	characteristic
113	C₄×C₁₇	—	2	1	no	—
114	C₄×C₁₇	—	2	4	no	—
115	C₄×C₁₇	—	2	1	no	—
116	C₄×C₁₇	—	2	1	no	—
117	C₄×C₁₇	—	2	4	no	—
118	C₄×C₁₇	—	2	1	no	—
119	C₄×C₁₇	#504	1	2	yes	—
120	C₄×C₁₇	#519	1	1	yes	characteristic
121	C₄×C₁₇	#568	1	1	yes	characteristic
122	#533	#595	1	1	yes	characteristic
123	#446	—	17	2	no	—
124	#446	—	17	2	no	—
125	C₈×Q₈	—	17	2	no	—
126	C₈×Q₈	—	17	2	no	—
127	#424	—	17	2	no	—
128	#423	—	17	1	no	—
129	#405	—	17	2	no	—
130	C₂×C₄×C₈	—	17	1	no	—
131	C₄⋊C₁₆	—	17	2	no	—
132	C₄⋊C₁₆	—	17	2	no	—
133	C₄⋊C₁₆	—	17	4	no	—
134	C₄⋊C₁₆	—	17	4	no	—
135	C₂²⋊C₁₆	—	17	2	no	—
136	C₂²⋊C₁₆	—	17	2	no	—
137	#346	—	17	2	no	—
138	#346	—	17	4	no	—
139	C₄×C₁₆	—	17	1	no	—
140	C₄×C₁₆	—	17	1	no	—
141	C₄×C₁₆	—	17	4	no	—
142	C₂×C₁₇	—	4	1	no	—
143	C₂×C₁₇	#2982	1	1	yes	characteristic
144	C₂×C₁₇	C₂×#504	1	1	yes	characteristic
145	C₂×C₁₇	#3407	1	1	yes	characteristic
146	C₂×M₁₆	—	17	2	no	—
147	C₂²×C₈	—	17	1	no	—
148	C₂²×C₈	—	17	1	no	—
149	C₄⋊Q₈	#3624	1	1	yes	characteristic
150	C₄⋊Q₈	#3624	1	1	yes	characteristic
151	#125	#3624	1	2	yes	—
152	C₂²⋊Q₈	#3624	1	2	yes	—
153	C₂²⋊Q₈	#3624	1	2	yes	—
154	C₄×Q₈	C₂×#595	1	2	yes	—
155	C₄×Q₈	C₂×#595	1	2	yes	—
156	#117	C₂×#595	1	2	yes	—
157	C₂×C₄²	C₂×#595	1	1	yes	characteristic
158	C₂×C₁₆	—	17	2	no	—
159	C₂×C₁₆	—	17	2	no	—
160	C₂×C₁₆	—	17	4	no	—
161	C₄⋊C₈	—	17	2	no	—
162	C₄⋊C₈	—	17	2	no	—
163	C₄⋊C₈	—	17	4	no	—
164	C₄⋊C₈	—	17	4	no	—
165	#97	—	17	4	no	—
166	C₄×C₈	—	17	1	no	—
167	C₄×C₈	—	17	1	no	—
168	C₄×C₈	—	17	1	no	—
169	C₄×C₈	—	17	4	no	—
170	C₄×C₈	—	17	1	no	—
171	C₁₇	#33499	1	1	yes	characteristic, complemented, Sylow
172	C₂×Q₈	C₂×#3624	1	2	yes	—
173	C₂×Q₈	C₂×#3624	1	2	yes	—
174	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
175	C₂²×C₄	C₂×#3624	1	1	yes	characteristic
176	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
177	M₁₆	—	68	2	no	—
178	C₂×C₈	—	17	1	no	—
179	C₂×C₈	—	17	1	no	—
180	C₂×C₈	—	17	1	no	—
181	C₂×C₈	—	17	1	no	—
182	C₂×C₈	—	17	4	no	—
183	C₂×C₈	—	34	2	no	—
184	C₂×C₈	—	68	1	no	—
185	C₄⋊C₄	C₂×#3624	1	2	yes	—
186	C₄⋊C₄	C₂×#3624	1	2	yes	—
187	C₄⋊C₄	C₂×#3624	1	4	yes	—
188	C₄⋊C₄	C₂×#3624	1	4	yes	—
189	C₄⋊C₄	C₂×#3624	1	2	yes	—
190	C₄⋊C₄	C₂×#3624	1	2	yes	—
191	C₂²⋊C₄	C₂×#3624	1	4	yes	—
192	C₄²	C₂×#3624	1	1	yes	characteristic
193	C₄²	C₂×#3624	1	1	yes	characteristic
194	C₄²	C₂²×#595	1	1	yes	characteristic
195	C₄²	C₂²×#595	1	1	yes	characteristic
196	C₄²	C₂²×#595	1	4	yes	—
197	C₁₆	—	34	2	no	—
198	C₁₆	—	34	2	no	—
199	C₁₆	—	34	4	no	—
200	C₂³	C₂²×#3624	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	Q₈	—	4	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
208	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
209	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
210	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
211	C₂×C₄	C₂²×#3624	1	4	yes	—
212	C₂×C₄	C₂²×#3624	1	4	yes	—
213	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
214	C₂×C₄	C₂³×#595	1	1	yes	characteristic
215	C₈	—	17	2	no	—
216	C₈	—	34	1	no	—
217	C₈	—	34	1	no	—
218	C₈	—	34	1	no	—
219	C₈	—	34	4	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	C₂³×#3624	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	2	4	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	4	no	—
230	C₄	#11763867	1	2	yes	—
231	C₄	#11763877	1	1	yes	characteristic
232	C₄	#11763889	1	1	yes	characteristic
233	C₂	—	4	1	no	—
234	C₂	#14545677	1	1	yes	characteristic
235	C₂	C₂×#11763867	1	1	yes	characteristic
236	C₂	#14545751	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#595
commutator, socle	C₂²×C₁₇	C₂³×C₈
Fitting	C₁₇×#533	C₄
prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂×#3622	C₂⁴
hypercenter	#533	#595
hypocenter	C₁₇	#33499
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #14545199	C₂	semidirect
2. #14545199	C₂	semidirect
3. Q₈×#63216	C₂	semidirect
171. C₁₇	#33499	semidirect

Maximal subgroups

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

subgroup	index
#14545199	2
Q₈×#63216	2
#14545149	2
#14545145	2
#14545141	2
#14537719	2
#33499	17