Subgroups — #17982462

Order 4352.

414 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763912	C₂	1	2	yes	maximal
2	C₂×#11763903	C₂	1	2	yes	maximal
3	#14545854	C₂	1	2	yes	maximal
4	C₄²×#3624	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63241	C₂²	1	4	yes	—
6	#11763912	C₄	1	8	yes	complemented
7	#11763903	—	2	4	no	—
8	C₂×C₄×#3624	C₄	1	1	yes	characteristic
9	C₂×C₄×#3624	C₄	1	1	yes	characteristic
10	C₂×C₄×#3624	C₄	1	1	yes	characteristic
11	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
12	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
13	C₂×C₄×#3624	C₂²	1	1	yes	characteristic
14	C₄²×Dic₁₇	C₄	1	1	yes	characteristic
15	C₂²×#3624	C₂×C₄	1	1	yes	characteristic
16	C₂²×#3624	C₂×C₄	1	1	yes	characteristic
17	C₂²×#3624	C₂×C₄	1	1	yes	characteristic
18	C₂²×#3624	C₂³	1	1	yes	characteristic
19	C₂×#63241	—	2	4	no	—
20	C₂×#63241	—	2	4	no	—
21	C₂×#63241	C₂×C₄	1	8	yes	—
22	C₂×C₄²×C₁₇	C₈	1	1	yes	characteristic
23	C₂×C₄×Dic₁₇	C₈	1	1	yes	characteristic
24	C₂×C₄×Dic₁₇	C₈	1	1	yes	characteristic
25	C₂×C₄×Dic₁₇	C₈	1	1	yes	characteristic
26	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
29	C₄×#3624	—	2	2	no	—
30	C₄×#3624	—	2	2	no	—
31	C₄×#3624	—	2	2	no	—
32	C₄×#3624	C₈	1	4	yes	—
33	C₄×#3624	C₂×C₄	1	4	yes	—
34	C₄×#3624	D₄	1	2	yes	—
35	C₄×#3624	Q₈	1	2	yes	—
36	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
39	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
40	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×Dic₁₇	C₂×C₈	1	1	yes	characteristic
42	C₂²×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×#3624	—	2	1	no	—
44	C₂×#3624	—	2	1	no	—
45	C₂×#3624	—	2	1	no	—
46	C₂×#3624	—	2	1	no	—
47	C₂×#3624	C₄²	1	2	yes	—
48	C₂×#3624	C₄⋊C₄	1	2	yes	—
49	C₂×#3624	C₂×C₈	1	2	yes	—
50	C₂×#3624	C₂×C₈	1	2	yes	—
51	C₂×#3624	M₁₆	1	2	yes	—
52	C₂×#3624	C₂²×C₄	1	2	yes	—
53	C₂×#3624	C₂×D₄	1	1	yes	characteristic
54	C₂×#3624	C₂×Q₈	1	1	yes	characteristic
55	C₂×#3624	#41	1	2	yes	—
56	#63241	—	2	8	no	—
57	C₄²×C₁₇	—	2	2	no	—
58	C₄×Dic₁₇	—	2	2	no	—
59	C₄×Dic₁₇	—	2	2	no	—
60	C₄×Dic₁₇	—	2	2	no	—
61	C₄×Dic₁₇	—	2	1	no	—
62	C₄×Dic₁₇	—	2	1	no	—
63	C₄×Dic₁₇	C₄²	1	2	yes	—
64	C₄×Dic₁₇	C₄⋊C₄	1	2	yes	—
65	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
66	C₄×Dic₁₇	M₁₆	1	2	yes	—
67	#12653	—	17	1	no	maximal, Sylow
68	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₇	—	2	1	no	—
70	C₂×C₄×C₁₇	—	2	1	no	—
71	C₂×C₄×C₁₇	—	2	1	no	—
72	C₂×C₄×C₁₇	—	2	1	no	—
73	C₂×C₄×C₁₇	—	2	1	no	—
74	C₂×C₄×C₁₇	—	2	1	no	—
75	C₂×C₄×C₁₇	C₄×C₈	1	2	yes	—
76	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
77	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
78	C₂×Dic₁₇	—	2	1	no	—
79	C₂×Dic₁₇	—	2	1	no	—
80	C₂×Dic₁₇	—	2	1	no	—
81	C₂×Dic₁₇	—	2	1	no	—
82	C₂×Dic₁₇	C₄×C₈	1	2	yes	—
83	C₂×Dic₁₇	C₄⋊C₈	1	2	yes	—
84	C₂×Dic₁₇	M₃₂	1	2	yes	—
85	C₂×Dic₁₇	M₃₂	1	2	yes	—
86	C₂×Dic₁₇	M₃₂	1	2	yes	—
87	C₂×Dic₁₇	C₂×C₄²	1	1	yes	characteristic
88	C₂×Dic₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×Dic₁₇	#117	1	1	yes	characteristic
90	C₂×Dic₁₇	C₂²×C₈	1	1	yes	characteristic
91	C₂×Dic₁₇	C₂×M₁₆	1	1	yes	characteristic
92	C₂×Dic₁₇	C₂×M₁₆	1	1	yes	characteristic
93	#3624	—	2	2	no	—
94	#3624	—	2	2	no	—
95	#3624	C₄⋊C₈	1	4	yes	—
96	#3624	C₄×D₄	1	2	yes	—
97	#3624	C₄×Q₈	1	2	yes	—
98	C₂×C₄⋊C₁₆	—	17	2	no	—
99	C₂×#346	—	17	2	no	—
100	C₄²×C₈	—	17	1	no	—
101	#1374	—	17	2	no	—
102	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
103	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
104	C₂²×C₁₇	#431	1	1	yes	characteristic
105	C₂²×C₁₇	C₂×M₃₂	1	1	yes	characteristic
106	C₂²×C₁₇	C₂×M₃₂	1	1	yes	characteristic
107	C₂²×C₁₇	C₂×M₃₂	1	1	yes	characteristic
108	C₂²×C₁₇	C₂×M₃₂	1	1	yes	characteristic
109	C₄×C₁₇	—	2	2	no	—
110	C₄×C₁₇	—	2	2	no	—
111	C₄×C₁₇	—	2	2	no	—
112	Dic₁₇	—	2	2	no	—
113	Dic₁₇	—	2	2	no	—
114	Dic₁₇	#346	1	4	yes	complemented
115	Dic₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
116	Dic₁₇	C₄×M₁₆	1	1	yes	characteristic
117	Dic₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
118	Dic₁₇	#431	1	1	yes	characteristic
119	C₂²×C₁₆	—	17	4	no	—
120	C₂×C₄×C₈	—	17	1	no	—
121	C₂×C₄×C₈	—	17	1	no	—
122	C₂×C₄×C₈	—	17	1	no	—
123	C₂×C₄×C₈	—	17	1	no	—
124	C₂×C₄×C₈	—	17	1	no	—
125	C₂×C₄×C₈	—	17	1	no	—
126	C₄³	—	17	1	no	—
127	C₄⋊C₁₆	—	17	8	no	—
128	#346	—	34	4	no	—
129	C₂×C₁₇	C₄×C₄⋊C₈	1	1	yes	characteristic
130	C₂×C₁₇	C₂×#346	1	1	yes	characteristic
131	C₂×C₁₇	C₄×M₃₂	1	1	yes	characteristic
132	C₂×C₁₇	#2150	1	1	yes	characteristic
133	C₂×C₁₇	#2150	1	1	yes	characteristic
134	C₂×C₁₇	#2163	1	1	yes	characteristic
135	C₂×C₁₇	#2163	1	1	yes	characteristic
136	C₂²×C₈	—	17	1	no	—
137	C₂²×C₈	—	17	1	no	—
138	C₂²×C₈	—	17	1	no	—
139	C₂²×C₈	—	17	1	no	—
140	C₂×C₄²	—	17	1	no	—
141	C₂×C₄²	—	17	1	no	—
142	C₂×C₄²	—	17	1	no	—
143	C₂×C₄²	—	17	1	no	—
144	C₂×C₄²	—	17	1	no	—
145	C₂×C₄²	—	17	1	no	—
146	C₂×C₄²	#3633	1	1	yes	characteristic
147	C₂×C₁₆	—	17	8	no	—
148	C₂×C₁₆	—	34	4	no	—
149	C₂×C₁₆	—	34	4	no	—
150	C₄×C₈	—	17	2	no	—
151	C₄×C₈	—	17	4	no	—
152	C₄×C₈	—	17	4	no	—
153	C₄×C₈	—	17	2	no	—
154	C₄×C₈	—	34	2	no	—
155	C₄×C₈	—	34	2	no	—
156	C₄×C₈	—	34	2	no	—
157	C₁₇	#12653	1	1	yes	characteristic, complemented, Sylow
158	C₂²×C₄	—	17	1	no	—
159	C₂²×C₄	—	17	1	no	—
160	C₂²×C₄	—	17	1	no	—
161	C₂²×C₄	—	17	1	no	—
162	C₂²×C₄	C₂×#3633	1	1	yes	characteristic
163	C₂²×C₄	C₂×#3633	1	1	yes	characteristic
164	C₂²×C₄	C₂×#3633	1	1	yes	characteristic
165	C₂×C₈	—	17	2	no	—
166	C₂×C₈	—	17	2	no	—
167	C₂×C₈	—	17	2	no	—
168	C₂×C₈	—	17	2	no	—
169	C₂×C₈	—	17	1	no	—
170	C₂×C₈	—	17	1	no	—
171	C₂×C₈	—	17	2	no	—
172	C₂×C₈	—	17	2	no	—
173	C₂×C₈	—	17	2	no	—
174	C₂×C₈	—	34	1	no	—
175	C₂×C₈	—	34	1	no	—
176	C₂×C₈	—	34	1	no	—
177	C₂×C₈	—	34	1	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	17	2	no	—
180	C₄²	—	17	2	no	—
181	C₄²	—	17	2	no	—
182	C₄²	—	17	2	no	—
183	C₄²	—	34	1	no	—
184	C₄²	—	34	1	no	—
185	C₄²	—	34	2	no	—
186	C₄²	—	34	2	no	—
187	C₄²	—	34	2	no	—
188	C₁₆	—	34	8	no	—
189	C₂³	C₂²×#3633	1	1	yes	characteristic
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	17	1	no	—
197	C₂×C₄	—	17	1	no	—
198	C₂×C₄	—	17	2	no	—
199	C₂×C₄	—	17	1	no	—
200	C₂×C₄	—	17	1	no	—
201	C₂×C₄	—	17	1	no	—
202	C₂×C₄	—	17	1	no	—
203	C₂×C₄	—	17	2	no	—
204	C₂×C₄	—	17	2	no	—
205	C₂×C₄	—	17	2	no	—
206	C₂×C₄	—	17	2	no	—
207	C₂×C₄	—	34	1	no	—
208	C₂×C₄	—	34	1	no	—
209	C₂×C₄	—	34	1	no	—
210	C₂×C₄	—	34	1	no	—
211	C₂×C₄	C₄×#3633	1	2	yes	—
212	C₂×C₄	C₄⋊#3633	1	2	yes	—
213	C₂×C₄	#10587670	1	2	yes	—
214	C₈	—	17	2	no	—
215	C₈	—	17	4	no	—
216	C₈	—	17	2	no	—
217	C₈	—	34	2	no	—
218	C₈	—	34	2	no	—
219	C₂²	C₂×C₄×#3633	1	1	yes	characteristic
220	C₂²	C₂×C₄⋊#3633	1	1	yes	characteristic
221	C₂²	#11763922	1	1	yes	characteristic
222	C₂²	C₂×#10587670	1	1	yes	characteristic
223	C₂²	#11763928	1	1	yes	characteristic
224	C₂²	#11763928	1	1	yes	characteristic
225	C₂²	#11763928	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	17	1	no	—
230	C₄	—	17	1	no	—
231	C₄	—	17	1	no	—
232	C₄	—	17	1	no	—
233	C₄	—	17	4	no	—
234	C₄	—	34	2	no	—
235	C₄	—	34	2	no	—
236	C₂	C₄×C₄⋊#3633	1	1	yes	characteristic
237	C₂	#14545940	1	1	yes	characteristic
238	C₂	C₄×#10587670	1	1	yes	characteristic
239	C₂	#14545972	1	1	yes	characteristic
240	C₂	#14545974	1	1	yes	characteristic
241	C₂	#14545985	1	1	yes	characteristic
242	C₂	#14545988	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×#3633
commutator	C₂²×C₁₇	C₂×C₄×C₈
Fitting	C₂×C₄²×C₁₇	C₈
socle, prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂²×#3624	C₂³
hypercenter	C₂×C₄²	#3633
hypocenter	C₁₇	#12653
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
6. #11763912	C₄	semidirect
114. Dic₁₇	#346	semidirect
157. C₁₇	#12653	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763912	2
C₂×#11763903	2
#14545854	2
C₄²×#3624	2
#12653	17