Subgroups — #14545347

Order 2176.

564 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₇⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#10587677	C₂	1	1	yes	maximal, characteristic
3	#11763775	C₂	1	4	yes	maximal, complemented
4	#11763772	C₂	1	4	yes	maximal, complemented
5	M₁₆×#595	C₂	1	4	yes	maximal, complemented
6	C₂×M₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄×D₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊#595	C₄	1	2	yes	—
9	C₂×#63247	C₄	1	2	yes	complemented
10	#10587677	C₄	1	2	yes	—
11	#10587677	C₄	1	2	yes	—
12	#10587677	C₂²	1	2	yes	—
13	#10587677	C₂²	1	2	yes	—
14	C₂×C₄⋊#595	C₄	1	2	yes	—
15	C₂×C₄×#595	C₂²	1	2	yes	—
16	D₁₇⋊M₁₆	C₄	1	2	yes	complemented
17	D₁₇⋊M₁₆	C₄	1	2	yes	complemented
18	D₁₇⋊M₁₆	C₂²	1	2	yes	—
19	D₁₇⋊M₁₆	C₂²	1	2	yes	—
20	C₂×#63243	C₄	1	2	yes	complemented
21	C₂×#63242	C₂²	1	2	yes	—
22	C₂×M₁₆×C₁₇	C₄	1	1	yes	characteristic, complemented
23	C₂×#63223	C₄	1	1	yes	characteristic, complemented
24	M₁₆×D₁₇	C₄	1	2	yes	complemented
25	M₁₆×D₁₇	C₄	1	2	yes	complemented
26	M₁₆×D₁₇	C₂²	1	2	yes	—
27	M₁₆×D₁₇	C₂²	1	2	yes	—
28	C₂×#63218	C₄	1	2	yes	complemented
29	C₂×C₈×D₁₇	C₂²	1	2	yes	—
30	#10587532	C₂²	1	8	yes	complemented
31	C₈×#595	C₂²	1	8	yes	complemented
32	C₂³×D₁₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×#595	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₁₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
46	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
48	C₂²⋊#595	C₂×C₄	1	4	yes	—
49	C₂²⋊#595	C₂×C₄	1	4	yes	—
50	#63247	C₂×C₄	1	4	yes	—
51	#63247	C₂×C₄	1	4	yes	complemented
52	C₂×#3624	C₂×C₄	1	4	yes	complemented
53	C₄⋊#595	C₂×C₄	1	4	yes	—
54	C₄⋊#595	C₂×C₄	1	4	yes	—
55	C₄×#595	C₂×C₄	1	4	yes	—
56	C₄×#595	C₂³	1	4	yes	—
57	#63243	C₂×C₄	1	4	yes	—
58	#63243	C₂×C₄	1	4	yes	complemented
59	#63242	C₂×C₄	1	4	yes	complemented
60	#63242	C₂³	1	4	yes	—
61	M₁₆×C₁₇	C₂×C₄	1	2	yes	—
62	M₁₆×C₁₇	C₂×C₄	1	2	yes	complemented
63	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	complemented
64	#63223	C₂×C₄	1	2	yes	—
65	#63223	C₂×C₄	1	2	yes	complemented
66	C₂×#3622	C₂×C₄	1	2	yes	complemented
67	#63218	C₂×C₄	1	4	yes	—
68	#63218	C₂×C₄	1	4	yes	complemented
69	C₈×D₁₇	C₂×C₄	1	4	yes	complemented
70	C₈×D₁₇	C₂³	1	4	yes	—
71	C₂³×C₁₇	C₄²	1	1	yes	characteristic
72	C₂²×D₁₇	—	4	2	no	—
73	C₂²×D₁₇	C₄²	1	1	yes	characteristic
74	C₂²×D₁₇	C₄²	1	1	yes	characteristic
75	C₂²×D₁₇	C₄²	1	1	yes	characteristic
76	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×#595	—	4	4	no	—
80	C₂×#595	C₂²×C₄	1	8	yes	—
81	C₂×C₄×C₁₇	C₄²	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₄²	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₄²	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₁₇	C₄²	1	1	yes	characteristic
88	C₂×Dic₁₇	C₄²	1	1	yes	characteristic
89	C₂×Dic₁₇	C₄²	1	1	yes	characteristic
90	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
93	C₄×D₁₇	C₄²	1	1	yes	characteristic
94	C₄×D₁₇	C₄²	1	1	yes	characteristic
95	C₄×D₁₇	C₄²	1	1	yes	characteristic
96	C₄×D₁₇	C₄²	1	1	yes	characteristic
97	C₄×D₁₇	C₄²	1	1	yes	characteristic
98	C₄×D₁₇	C₄²	1	1	yes	characteristic
99	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
100	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
101	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
102	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
103	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
104	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
105	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
106	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
107	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
108	C₄×D₁₇	C₂⁴	1	1	yes	characteristic
109	#3624	C₂²×C₄	1	8	yes	—
110	C₈×C₁₇	C₂²×C₄	1	4	yes	—
111	#3622	C₂²×C₄	1	4	yes	—
112	#2873	—	17	1	no	maximal, Sylow
113	C₂²×C₁₇	—	4	1	no	—
114	C₂²×C₁₇	C₂×C₄²	1	1	yes	characteristic
115	C₂²×C₁₇	C₂×C₄²	1	1	yes	characteristic
116	C₂²×C₁₇	C₂×C₄²	1	1	yes	characteristic
117	C₂×D₁₇	—	2	2	no	—
118	C₂×D₁₇	—	2	2	no	—
119	C₂×D₁₇	—	2	2	no	—
120	C₂×D₁₇	—	4	1	no	—
121	C₂×D₁₇	—	4	1	no	—
122	C₂×D₁₇	—	4	1	no	—
123	C₂×D₁₇	C₂×C₄²	1	1	yes	characteristic
124	C₂×D₁₇	C₂×C₄²	1	1	yes	characteristic
125	C₂×D₁₇	C₂×C₄²	1	1	yes	characteristic
126	C₂×D₁₇	C₂³×C₄	1	1	yes	characteristic
127	#595	—	2	8	no	—
128	C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
129	C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
130	C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
131	C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
132	Dic₁₇	C₂×C₄²	1	1	yes	characteristic
133	Dic₁₇	C₂×C₄²	1	1	yes	characteristic
134	Dic₁₇	C₂×C₄²	1	1	yes	characteristic
135	Dic₁₇	C₂³×C₄	1	1	yes	characteristic
136	C₂²×M₁₆	—	17	1	no	—
137	C₂²×M₁₆	—	17	1	no	—
138	C₂×#117	—	17	1	no	—
139	#405	—	17	4	no	—
140	#405	—	17	4	no	—
141	C₄×M₁₆	—	17	4	no	—
142	C₂×C₁₇	—	2	1	no	—
143	C₂×C₁₇	—	2	1	no	—
144	C₂×C₁₇	—	2	1	no	—
145	C₂×C₁₇	C₂²×C₄²	1	1	yes	characteristic
146	D₁₇	—	2	1	no	—
147	D₁₇	—	2	1	no	—
148	D₁₇	—	2	1	no	—
149	D₁₇	#568	1	2	yes	—
150	C₂³×C₄	—	17	1	no	—
151	C₂×M₁₆	—	17	2	no	—
152	C₂×M₁₆	—	17	2	no	—
153	C₂×M₁₆	—	17	2	no	—
154	C₂×M₁₆	—	17	2	no	—
155	C₂×M₁₆	—	17	2	no	—
156	C₂×M₁₆	—	17	2	no	—
157	C₂×M₁₆	—	17	1	no	—
158	C₂×M₁₆	—	17	2	no	—
159	C₂×M₁₆	—	17	2	no	—
160	C₂×M₁₆	—	17	2	no	—
161	C₂×M₁₆	—	17	2	no	—
162	C₂×M₁₆	—	17	2	no	—
163	C₂×M₁₆	#595	1	1	yes	characteristic, complemented
164	C₂²×C₈	—	17	2	no	—
165	C₂²×C₈	—	17	2	no	—
166	#117	—	17	2	no	—
167	#117	—	17	2	no	—
168	#117	—	17	2	no	—
169	#117	—	17	2	no	—
170	C₂×C₄⋊C₄	—	17	2	no	—
171	C₂×C₂²⋊C₄	—	17	2	no	—
172	C₂×C₄²	—	17	2	no	—
173	#97	—	17	8	no	—
174	C₄×C₈	—	17	8	no	—
175	C₁₇	#2873	1	1	yes	characteristic, complemented, Sylow
176	C₂⁴	—	17	1	no	—
177	C₂²×C₄	—	17	1	no	—
178	C₂²×C₄	—	17	1	no	—
179	C₂²×C₄	—	17	1	no	—
180	C₂²×C₄	—	17	1	no	—
181	C₂²×C₄	—	17	1	no	—
182	C₂²×C₄	—	17	1	no	—
183	C₂²×C₄	—	17	1	no	—
184	C₂²×C₄	—	17	1	no	—
185	C₂²×C₄	—	17	1	no	—
186	C₂²×C₄	—	17	1	no	—
187	C₂²×C₄	—	17	1	no	—
188	C₂²×C₄	—	17	1	no	—
189	C₂²×C₄	—	17	1	no	—
190	C₂²×C₄	—	17	4	no	—
191	C₂²×C₄	C₂×#595	1	1	yes	characteristic
192	M₁₆	—	17	4	no	—
193	M₁₆	—	17	4	no	—
194	M₁₆	—	17	4	no	—
195	M₁₆	—	17	4	no	—
196	M₁₆	—	17	2	no	—
197	M₁₆	—	17	2	no	—
198	M₁₆	—	17	4	no	—
199	M₁₆	—	17	4	no	—
200	M₁₆	C₂×#595	1	2	yes	—
201	M₁₆	C₂×#595	1	2	yes	complemented
202	C₂×C₈	—	17	4	no	—
203	C₂×C₈	—	17	4	no	—
204	C₂×C₈	—	17	4	no	—
205	C₂×C₈	—	17	2	no	—
206	C₂×C₈	—	17	4	no	—
207	C₂×C₈	—	17	4	no	—
208	C₂×C₈	C₂×#595	1	2	yes	complemented
209	C₄⋊C₄	—	17	4	no	—
210	C₄⋊C₄	—	17	4	no	—
211	C₂²⋊C₄	—	17	4	no	—
212	C₂²⋊C₄	—	17	4	no	—
213	C₄²	—	17	4	no	—
214	C₄²	—	17	4	no	—
215	C₂³	—	17	1	no	—
216	C₂³	—	17	1	no	—
217	C₂³	—	17	1	no	—
218	C₂³	—	17	1	no	—
219	C₂³	—	17	1	no	—
220	C₂³	—	17	1	no	—
221	C₂³	—	68	2	no	—
222	C₂³	C₄×#595	1	1	yes	characteristic
223	C₂×C₄	—	17	1	no	—
224	C₂×C₄	—	17	1	no	—
225	C₂×C₄	—	17	1	no	—
226	C₂×C₄	—	17	1	no	—
227	C₂×C₄	—	17	1	no	—
228	C₂×C₄	—	17	1	no	—
229	C₂×C₄	—	17	1	no	—
230	C₂×C₄	—	17	1	no	—
231	C₂×C₄	—	17	1	no	—
232	C₂×C₄	—	17	1	no	—
233	C₂×C₄	—	17	1	no	—
234	C₂×C₄	—	17	1	no	—
235	C₂×C₄	—	17	1	no	—
236	C₂×C₄	—	17	1	no	—
237	C₂×C₄	—	17	1	no	—
238	C₂×C₄	—	17	1	no	—
239	C₂×C₄	—	17	1	no	—
240	C₂×C₄	—	17	1	no	—
241	C₂×C₄	—	17	1	no	—
242	C₂×C₄	—	17	1	no	—
243	C₂×C₄	—	17	1	no	—
244	C₂×C₄	—	17	1	no	—
245	C₂×C₄	—	17	8	no	—
246	C₂×C₄	—	68	4	no	—
247	C₂×C₄	C₄×#595	1	1	yes	characteristic
248	C₂×C₄	C₄×#595	1	1	yes	characteristic
249	C₂×C₄	C₄×#595	1	1	yes	characteristic
250	C₂×C₄	C₂²×#595	1	1	yes	characteristic
251	C₂×C₄	C₂²×#595	1	1	yes	characteristic
252	C₂×C₄	C₂²×#595	1	1	yes	characteristic
253	C₈	—	17	8	no	—
254	C₈	—	17	4	no	—
255	C₈	C₂²×#595	1	4	yes	—
256	C₂²	—	4	1	no	—
257	C₂²	—	17	1	no	—
258	C₂²	—	17	1	no	—
259	C₂²	—	17	1	no	—
260	C₂²	—	17	1	no	—
261	C₂²	—	34	2	no	—
262	C₂²	—	34	2	no	—
263	C₂²	—	34	2	no	—
264	C₂²	—	68	1	no	—
265	C₂²	—	68	1	no	—
266	C₂²	—	68	1	no	—
267	C₂²	C₂×C₄×#595	1	1	yes	characteristic
268	C₂²	C₂×C₄×#595	1	1	yes	characteristic
269	C₂²	C₂×C₄×#595	1	1	yes	characteristic
270	C₄	—	17	1	no	—
271	C₄	—	17	1	no	—
272	C₄	—	17	1	no	—
273	C₄	—	17	1	no	—
274	C₄	—	34	8	no	—
275	C₄	C₂×C₄×#595	1	1	yes	characteristic
276	C₄	C₂×C₄×#595	1	1	yes	characteristic
277	C₄	C₂×C₄×#595	1	1	yes	characteristic
278	C₄	C₂³×#595	1	1	yes	characteristic
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	17	2	no	—
283	C₂	—	34	1	no	—
284	C₂	—	34	1	no	—
285	C₂	—	34	1	no	—
286	C₂	C₂²×C₄×#595	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×#595
commutator, socle	C₂×C₁₇	C₂²×C₄²
Fitting	C₂×M₁₆×C₁₇	C₄
prime radical	C₂³×D₁₇	C₂×C₄
cosocle	C₄×D₁₇	C₂⁴
hypercenter	C₂×M₁₆	#595
hypocenter	C₁₇	#2873
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #11763775	C₂	semidirect
4. #11763772	C₂	semidirect
5. M₁₆×#595	C₂	semidirect
9. C₂×#63247	C₄	semidirect
16. D₁₇⋊M₁₆	C₄	semidirect
17. D₁₇⋊M₁₆	C₄	semidirect
20. C₂×#63243	C₄	semidirect
22. C₂×M₁₆×C₁₇	C₄	semidirect
23. C₂×#63223	C₄	semidirect
24. M₁₆×D₁₇	C₄	semidirect
25. M₁₆×D₁₇	C₄	semidirect
28. C₂×#63218	C₄	semidirect
30. #10587532	C₂²	semidirect
31. C₈×#595	C₂²	semidirect
51. #63247	C₂×C₄	semidirect
52. C₂×#3624	C₂×C₄	semidirect
58. #63243	C₂×C₄	semidirect
59. #63242	C₂×C₄	semidirect
62. M₁₆×C₁₇	C₂×C₄	semidirect
63. C₂×C₈×C₁₇	C₂×C₄	semidirect
65. #63223	C₂×C₄	semidirect
66. C₂×#3622	C₂×C₄	semidirect
68. #63218	C₂×C₄	semidirect
69. C₈×D₁₇	C₂×C₄	semidirect
163. C₂×M₁₆	#595	semidirect
175. C₁₇	#2873	semidirect
201. M₁₆	C₂×#595	semidirect
208. C₂×C₈	C₂×#595	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₇⋊M₁₆	2
C₂×#10587677	2
#11763775	2
#11763772	2
M₁₆×#595	2
C₂×M₁₆×D₁₇	2
#2873	17