Subgroups — #17961035

Order 4352.

318 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	#14545116	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763701	C₂	1	2	yes	maximal
3	C₄×#10587538	C₂	1	1	yes	maximal, characteristic
4	#14543825	C₂	1	2	yes	maximal
5	#14528653	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587538	C₂²	1	1	yes	characteristic
7	C₂×#10587538	C₂²	1	2	yes	—
8	C₂×#10587538	C₂²	1	1	yes	characteristic
9	C₄⋊#63216	C₄	1	2	yes	complemented
10	C₄⋊#63216	C₄	1	2	yes	complemented
11	#11763701	—	2	4	no	—
12	#11763697	C₄	1	2	yes	complemented
13	C₄×#63216	C₄	1	2	yes	complemented
14	C₂×C₄⋊#3622	C₄	1	1	yes	characteristic
15	C₂×#10587474	C₂²	1	2	yes	—
16	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
17	#11762564	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₁₇	C₄	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
20	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×#3622	C₂×C₄	1	2	yes	—
23	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
24	C₂²×#3622	C₂³	1	1	yes	characteristic
25	#10587538	—	2	2	no	—
26	#10587538	—	2	1	no	—
27	#10587538	—	2	1	no	—
28	#10587538	—	2	2	no	—
29	#10587538	D₄	1	2	yes	—
30	#10587538	Q₈	1	2	yes	—
31	C₂×#63216	C₂×C₄	1	4	yes	—
32	C₂×#63216	C₂×C₄	1	2	yes	—
33	C₂×#63216	C₂×C₄	1	2	yes	—
34	C₁₇×#95	—	2	1	no	—
35	C₄⋊#3622	—	2	1	no	—
36	C₄⋊#3622	—	2	1	no	—
37	#10587474	—	2	2	no	—
38	#10587474	—	2	2	no	—
39	C₄×#3622	C₂×C₄	1	1	yes	characteristic
40	C₄×#3622	C₂×C₄	1	1	yes	characteristic
41	C₄×#3622	C₂×C₄	1	1	yes	characteristic
42	C₄×#3622	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₇	—	2	2	no	—
44	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₁₇	M₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
47	C₄⋊C₄×C₁₇	—	4	2	no	—
48	C₄²×C₁₇	—	2	2	no	—
49	C₄²×C₁₇	—	2	2	no	—
50	C₄²×C₁₇	C₄²	1	1	yes	characteristic
51	C₄²×C₁₇	C₄²	1	1	yes	characteristic
52	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
53	C₄²×C₁₇	M₁₆	1	1	yes	characteristic
54	C₂×#3622	—	2	1	no	—
55	C₂×#3622	—	2	1	no	—
56	C₂×#3622	—	2	2	no	—
57	C₂×#3622	—	2	2	no	—
58	C₂×#3622	—	2	2	no	—
59	C₂×#3622	C₄²	1	1	yes	characteristic
60	C₂×#3622	C₄²	1	1	yes	characteristic
61	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
62	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
63	C₂×#3622	C₂×D₄	1	1	yes	characteristic
64	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
65	C₂×#3622	#41	1	1	yes	characteristic
66	C₂×#3622	#41	1	1	yes	characteristic
67	#63216	—	2	2	no	—
68	#63216	—	2	4	no	—
69	#63216	C₄⋊C₄	1	4	yes	complemented
70	#12666	—	17	1	no	maximal, Sylow
71	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₁₇	—	2	2	no	—
73	C₂×C₄×C₁₇	—	4	1	no	—
74	C₂×C₄×C₁₇	—	4	2	no	—
75	C₂×C₄×C₁₇	—	4	2	no	—
76	C₂×C₄×C₁₇	—	4	2	no	—
77	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
78	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
79	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
80	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
86	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
87	#3622	—	2	1	no	—
88	#3622	—	2	1	no	—
89	#3622	—	4	2	no	—
90	#3622	C₂×C₄⋊C₄	1	1	yes	characteristic
91	#3622	#117	1	1	yes	characteristic
92	#3622	C₄×D₄	1	1	yes	characteristic
93	#3622	C₄×Q₈	1	1	yes	characteristic
94	#2150	—	17	1	no	—
95	C₂×#347	—	17	2	no	—
96	C₄×M₃₂	—	17	1	no	—
97	#1770	—	17	1	no	—
98	#1375	—	17	2	no	—
99	C₂²×C₁₇	—	2	1	no	—
100	C₂²×C₁₇	—	2	1	no	—
101	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
102	C₂²×C₁₇	#436	1	1	yes	characteristic
103	C₂²×C₁₇	#446	1	1	yes	characteristic
104	C₄×C₁₇	—	2	1	no	—
105	C₄×C₁₇	—	2	1	no	—
106	C₄×C₁₇	—	2	1	no	—
107	C₄×C₁₇	—	8	2	no	—
108	C₄×C₁₇	#347	1	2	yes	complemented
109	C₄×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
110	C₄×C₁₇	#405	1	1	yes	characteristic
111	C₄×C₁₇	#436	1	1	yes	characteristic
112	C₄×C₁₇	#446	1	1	yes	characteristic
113	C₂×M₃₂	—	17	1	no	—
114	C₂×M₃₂	—	17	1	no	—
115	C₂×M₃₂	—	17	2	no	—
116	C₂×C₄⋊C₈	—	17	1	no	—
117	C₂×#97	—	17	2	no	—
118	C₂×C₄×C₈	—	17	1	no	—
119	C₄×C₄⋊C₄	#595	1	1	yes	characteristic
120	C₄⋊C₁₆	—	17	2	no	—
121	C₄⋊C₁₆	—	17	2	no	—
122	#347	—	34	4	no	—
123	#346	—	17	2	no	—
124	C₄×C₁₆	—	17	2	no	—
125	#336	—	17	2	no	—
126	C₂×C₁₇	—	2	1	no	—
127	C₂×C₁₇	—	2	1	no	—
128	C₂×C₁₇	#1770	1	1	yes	characteristic
129	C₂×C₁₇	C₂×#347	1	1	yes	characteristic
130	C₂×C₁₇	#2110	1	1	yes	characteristic
131	C₂²×C₈	—	17	1	no	—
132	C₂²×C₈	—	17	1	no	—
133	C₂²×C₈	—	17	2	no	—
134	C₂×C₄⋊C₄	—	2	1	no	—
135	C₂×C₄²	C₂×#595	1	2	yes	—
136	C₂×C₄²	C₂×#595	1	1	yes	characteristic
137	M₃₂	—	17	2	no	—
138	M₃₂	—	17	2	no	—
139	M₃₂	—	34	2	no	—
140	M₃₂	—	34	1	no	—
141	M₃₂	—	34	1	no	—
142	M₃₂	—	34	2	no	—
143	C₂×C₁₆	—	17	2	no	—
144	C₂×C₁₆	—	17	2	no	—
145	C₂×C₁₆	—	17	4	no	—
146	C₄⋊C₈	—	34	1	no	—
147	C₄⋊C₈	—	34	1	no	—
148	#97	—	34	2	no	—
149	#97	—	34	2	no	—
150	C₄×C₈	—	17	1	no	—
151	C₄×C₈	—	17	1	no	—
152	C₄×C₈	—	17	1	no	—
153	C₄×C₈	—	17	1	no	—
154	#95	—	2	1	no	—
155	C₁₇	#12666	1	1	yes	characteristic, complemented, Sylow
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	#63247	1	1	yes	characteristic
158	C₂²×C₄	#63247	1	1	yes	characteristic
159	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
160	C₂×C₈	—	17	1	no	—
161	C₂×C₈	—	17	1	no	—
162	C₂×C₈	—	17	1	no	—
163	C₂×C₈	—	17	1	no	—
164	C₂×C₈	—	17	1	no	—
165	C₂×C₈	—	17	1	no	—
166	C₂×C₈	—	17	1	no	—
167	C₂×C₈	—	17	1	no	—
168	C₂×C₈	—	34	1	no	—
169	C₂×C₈	—	34	1	no	—
170	C₂×C₈	—	34	2	no	—
171	C₂×C₈	—	34	2	no	—
172	C₂×C₈	—	34	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	#63243	1	1	yes	characteristic
177	C₄²	#63243	1	1	yes	characteristic
178	C₄²	C₄×#595	1	1	yes	characteristic
179	C₄²	C₄×#595	1	1	yes	characteristic
180	C₁₆	—	17	4	no	—
181	C₁₆	—	34	4	no	—
182	C₁₆	—	34	2	no	—
183	C₂³	C₂×#63247	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	#10587542	1	1	yes	characteristic
190	C₂×C₄	#10587542	1	1	yes	characteristic
191	C₂×C₄	#10587542	1	1	yes	characteristic
192	C₂×C₄	#10587542	1	1	yes	characteristic
193	C₂×C₄	C₂×#63243	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
195	C₂×C₄	#10587678	1	1	yes	characteristic
196	C₂×C₄	D₄×#595	1	1	yes	characteristic
197	C₂×C₄	#10587680	1	1	yes	characteristic
198	C₂×C₄	Q₈×#595	1	1	yes	characteristic
199	C₈	—	17	1	no	—
200	C₈	—	17	1	no	—
201	C₈	—	17	1	no	—
202	C₈	—	17	1	no	—
203	C₈	—	34	1	no	—
204	C₈	—	34	1	no	—
205	C₈	—	68	2	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	C₂×#10587542	1	1	yes	characteristic
209	C₂²	#11763844	1	1	yes	characteristic
210	C₂²	#11763860	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	8	2	no	—
215	C₄	#11763701	1	2	yes	complemented
216	C₄	C₄⋊C₄×#595	1	1	yes	characteristic
217	C₄	#11763774	1	1	yes	characteristic
218	C₄	#11763843	1	1	yes	characteristic
219	C₄	#11763859	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#14544580	1	1	yes	characteristic
223	C₂	C₂×#11763701	1	1	yes	characteristic
224	C₂	#14545005	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#595
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₄×C₄⋊C₄×C₁₇	C₄
socle	C₂²×C₁₇	C₂×#97
prime radical	C₂³×C₁₇	C₂×M₁₆
cosocle	C₂²×#3622	C₂³
hypercenter	C₄×C₄⋊C₄	#595
hypocenter	C₁₇	#12666
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×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
10. C₄⋊#63216	C₄	semidirect
12. #11763697	C₄	semidirect
13. C₄×#63216	C₄	semidirect
69. #63216	C₄⋊C₄	semidirect
108. C₄×C₁₇	#347	semidirect
155. C₁₇	#12666	semidirect
215. C₄	#11763701	semidirect

Maximal subgroups

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

subgroup	index
#14545116	2
C₂×#11763701	2
C₄×#10587538	2
#14543825	2
#14528653	2
#12666	17