Subgroups — C₂×#11763681 #14544462

Order 2176.

488 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10587677	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#595	C₂	1	1	yes	maximal, characteristic
3	#11763681	C₂	1	4	yes	maximal, direct product factor
4	C₂×M₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×#595	—	2	1	no	—
6	C₂²×C₄×D₁₇	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊#595	—	2	1	no	—
8	#10587677	—	2	1	no	—
9	#10587677	—	2	1	no	—
10	#10587677	C₄	1	2	yes	complemented
11	#10587677	C₂²	1	2	yes	—
12	C₂×C₄⋊#595	C₄	1	2	yes	complemented
13	C₂×C₄×#595	—	2	1	no	—
14	C₂×C₄×#595	—	2	1	no	—
15	C₂×C₄×#595	—	2	1	no	—
16	C₂×C₄×#595	—	2	1	no	—
17	C₂×C₄×#595	C₄	1	2	yes	complemented
18	C₂×C₄×#595	C₄	1	2	yes	complemented
19	C₂×C₄×#595	C₂²	1	2	yes	—
20	C₂×M₁₆×C₁₇	C₄	1	1	yes	characteristic, complemented
21	C₂×#63223	C₄	1	1	yes	characteristic, complemented
22	M₁₆×D₁₇	—	2	1	no	—
23	M₁₆×D₁₇	—	2	1	no	—
24	M₁₆×D₁₇	C₄	1	2	yes	complemented
25	M₁₆×D₁₇	C₂²	1	2	yes	—
26	C₂×#63218	—	2	1	no	—
27	C₂×C₈×D₁₇	—	2	1	no	—
28	C₂³×D₁₇	D₄	1	1	yes	characteristic
29	C₂²×#595	—	2	8	no	—
30	C₂²×#595	—	2	2	no	—
31	C₂²×#595	—	2	2	no	—
32	C₂²×#595	—	2	2	no	—
33	C₂²×#595	—	2	2	no	—
34	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₁₇	Q₈	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	D₄	1	1	yes	characteristic
42	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
43	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
44	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
45	C₂×C₄×D₁₇	D₄	1	1	yes	characteristic
46	C₂×C₄×D₁₇	Q₈	1	1	yes	characteristic
47	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
48	C₂²⋊#595	—	2	2	no	—
49	C₂²⋊#595	—	2	2	no	—
50	C₄⋊#595	—	2	2	no	—
51	C₄⋊#595	C₂×C₄	1	4	yes	complemented
52	C₄×#595	—	2	2	no	—
53	C₄×#595	—	2	2	no	—
54	C₄×#595	—	2	2	no	—
55	C₄×#595	—	2	2	no	—
56	C₄×#595	—	2	2	no	—
57	C₄×#595	C₂×C₄	1	4	yes	complemented
58	M₁₆×C₁₇	—	2	1	no	—
59	M₁₆×C₁₇	C₂×C₄	1	2	yes	complemented
60	C₂×C₈×C₁₇	—	2	1	no	—
61	#63223	—	2	1	no	—
62	#63223	C₂×C₄	1	2	yes	complemented
63	C₂×#3622	—	2	1	no	—
64	#63218	—	2	2	no	—
65	#63218	—	2	2	no	—
66	C₈×D₁₇	—	2	2	no	—
67	C₈×D₁₇	—	2	2	no	—
68	C₂³×C₁₇	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×D₁₇	—	2	4	no	—
70	C₂²×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×D₁₇	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
76	C₂×#595	—	2	8	no	—
77	C₂×#595	—	2	8	no	—
78	C₂×#595	—	2	8	no	—
79	C₂×#595	—	2	4	no	—
80	C₂×#595	—	2	4	no	—
81	C₂×#595	—	4	4	no	—
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	C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
89	C₂×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
90	C₂×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
91	C₂×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
92	C₂×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
93	C₂×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
94	C₄×D₁₇	C₄²	1	1	yes	characteristic
95	C₄×D₁₇	C₄²	1	1	yes	characteristic
96	C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
97	C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
98	C₄×D₁₇	C₂²⋊C₄	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	C₄⋊C₄	1	1	yes	characteristic
103	C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
104	C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
105	C₄×D₁₇	C₄⋊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	C₂×D₄	1	1	yes	characteristic
109	C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
110	C₈×C₁₇	—	2	2	no	—
111	#3622	—	2	2	no	—
112	C₂×#339	—	17	1	no	maximal, Sylow
113	C₂²×C₁₇	—	2	2	no	—
114	C₂²×C₁₇	#95	1	1	yes	characteristic
115	C₂²×C₁₇	#95	1	1	yes	characteristic
116	C₂²×C₁₇	C₂×C₄⋊C₄	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	—	2	2	no	—
121	C₂×D₁₇	—	2	2	no	—
122	C₂×D₁₇	#95	1	1	yes	characteristic
123	C₂×D₁₇	#95	1	1	yes	characteristic
124	C₂×D₁₇	#104	1	4	yes	—
125	C₂×D₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×D₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	#595	—	2	8	no	—
128	#595	—	4	4	no	—
129	C₄×C₁₇	#95	1	1	yes	characteristic
130	C₄×C₁₇	#95	1	1	yes	characteristic
131	C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
132	C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	Dic₁₇	#95	1	1	yes	characteristic
134	Dic₁₇	#95	1	1	yes	characteristic
135	Dic₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
136	Dic₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×M₁₆	—	17	1	no	—
138	C₂×#117	—	17	1	no	—
139	C₂²×C₄²	—	17	1	no	—
140	#339	—	17	4	no	—
141	C₂×C₁₇	—	2	1	no	—
142	C₂×C₁₇	—	2	1	no	—
143	C₂×C₁₇	#339	1	2	yes	complemented
144	C₂×C₁₇	C₂×#95	1	1	yes	characteristic
145	D₁₇	—	2	1	no	—
146	D₁₇	—	2	1	no	—
147	D₁₇	#339	1	2	yes	complemented
148	D₁₇	C₂×#104	1	2	yes	—
149	C₂³×C₄	—	17	1	no	—
150	C₂³×C₄	—	34	1	no	—
151	C₂×M₁₆	—	17	1	no	—
152	C₂×M₁₆	—	17	2	no	—
153	C₂×M₁₆	—	17	2	no	—
154	C₂×M₁₆	—	34	1	no	—
155	C₂×M₁₆	—	34	1	no	—
156	C₂×M₁₆	—	34	1	no	—
157	C₂×M₁₆	#595	1	1	yes	characteristic, complemented
158	C₂²×C₈	—	34	1	no	—
159	#117	—	17	2	no	—
160	#117	—	17	2	no	—
161	#117	—	34	1	no	—
162	#117	—	34	1	no	—
163	C₂×C₄⋊C₄	—	17	2	no	—
164	C₂×C₂²⋊C₄	—	34	1	no	—
165	C₂×C₄²	—	17	2	no	—
166	C₂×C₄²	—	17	2	no	—
167	C₂×C₄²	—	17	2	no	—
168	C₂×C₄²	—	34	1	no	—
169	C₂×C₄²	—	34	1	no	—
170	C₂×C₄²	—	34	1	no	—
171	C₂×C₄²	—	34	1	no	—
172	C₁₇	C₂×#339	1	1	yes	characteristic, complemented, Sylow
173	C₂⁴	—	17	1	no	—
174	C₂²×C₄	—	17	1	no	—
175	C₂²×C₄	—	17	1	no	—
176	C₂²×C₄	—	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	—	34	2	no	—
188	C₂²×C₄	—	34	2	no	—
189	C₂²×C₄	—	34	2	no	—
190	C₂²×C₄	—	34	8	no	—
191	C₂²×C₄	—	34	2	no	—
192	C₂²×C₄	C₂×#595	1	1	yes	characteristic
193	M₁₆	—	2	1	no	—
194	M₁₆	—	17	2	no	—
195	M₁₆	—	34	2	no	—
196	M₁₆	—	34	2	no	—
197	M₁₆	—	34	1	no	—
198	M₁₆	C₂×#595	1	2	yes	complemented
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	34	2	no	—
201	C₂×C₈	—	34	2	no	—
202	C₂×C₈	—	34	1	no	—
203	C₄⋊C₄	—	17	4	no	—
204	C₄⋊C₄	—	34	2	no	—
205	C₂²⋊C₄	—	34	2	no	—
206	C₂²⋊C₄	—	34	2	no	—
207	C₄²	—	17	4	no	—
208	C₄²	—	34	2	no	—
209	C₄²	—	34	2	no	—
210	C₄²	—	34	2	no	—
211	C₄²	—	34	2	no	—
212	C₄²	—	34	2	no	—
213	C₂³	—	17	1	no	—
214	C₂³	—	17	1	no	—
215	C₂³	—	17	1	no	—
216	C₂³	—	17	1	no	—
217	C₂³	—	17	1	no	—
218	C₂³	—	17	1	no	—
219	C₂³	—	34	4	no	—
220	C₂³	C₄⋊#595	1	1	yes	characteristic
221	C₂×C₄	—	17	1	no	—
222	C₂×C₄	—	17	1	no	—
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	—	34	4	no	—
244	C₂×C₄	—	34	8	no	—
245	C₂×C₄	—	34	8	no	—
246	C₂×C₄	—	34	8	no	—
247	C₂×C₄	—	34	4	no	—
248	C₂×C₄	—	68	4	no	—
249	C₂×C₄	C₄×#595	1	1	yes	characteristic
250	C₂×C₄	C₄×#595	1	1	yes	characteristic
251	C₂×C₄	C₄⋊#595	1	1	yes	characteristic
252	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
253	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
254	C₂×C₄	C₂²×#595	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	34	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	17	1	no	—
259	C₂²	—	17	1	no	—
260	C₂²	—	17	4	no	—
261	C₂²	—	17	1	no	—
262	C₂²	—	17	1	no	—
263	C₂²	—	34	2	no	—
264	C₂²	—	34	2	no	—
265	C₂²	—	34	2	no	—
266	C₂²	—	34	2	no	—
267	C₂²	—	34	2	no	—
268	C₂²	(C₂×#595)⋊C₄	1	1	yes	characteristic
269	C₂²	(C₂×#595)⋊C₄	1	1	yes	characteristic
270	C₂²	C₂×C₄⋊#595	1	1	yes	characteristic
271	C₄	—	17	1	no	—
272	C₄	—	17	1	no	—
273	C₄	—	17	1	no	—
274	C₄	—	17	1	no	—
275	C₄	—	34	8	no	—
276	C₄	—	68	4	no	—
277	C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
278	C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
279	C₄	C₂×C₄×#595	1	1	yes	characteristic
280	C₄	C₂×C₂²⋊#595	1	1	yes	characteristic
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	17	2	no	—
284	C₂	—	17	2	no	—
285	C₂	—	34	1	no	—
286	C₂	—	34	1	no	—
287	C₂	#11763681	1	2	yes	direct product factor
288	C₂	C₂×(C₂×#595)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#595
commutator	C₄×C₁₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×#595
Fitting	C₂×M₁₆×C₁₇	C₄
socle	C₂²×C₁₇	#95
prime radical	C₂³×D₁₇	D₄
cosocle	C₂×C₄×D₁₇	C₂³
hypercenter	C₂×M₁₆	#595
hypocenter	C₁₇	C₂×#339
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #11763681	C₂	direct (N × H)
10. #10587677	C₄	semidirect
12. C₂×C₄⋊#595	C₄	semidirect
17. C₂×C₄×#595	C₄	semidirect
18. C₂×C₄×#595	C₄	semidirect
20. C₂×M₁₆×C₁₇	C₄	semidirect
21. C₂×#63223	C₄	semidirect
24. M₁₆×D₁₇	C₄	semidirect
51. C₄⋊#595	C₂×C₄	semidirect
57. C₄×#595	C₂×C₄	semidirect
59. M₁₆×C₁₇	C₂×C₄	semidirect
62. #63223	C₂×C₄	semidirect
143. C₂×C₁₇	#339	semidirect
147. D₁₇	#339	semidirect
157. C₂×M₁₆	#595	semidirect
172. C₁₇	C₂×#339	semidirect
198. M₁₆	C₂×#595	semidirect
287. C₂	#11763681	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#10587677	2
C₂²×C₄×#595	2
#11763681	2
C₂×M₁₆×D₁₇	2
C₂×#339	17

Subgroups — C2×#11763681 #14544462

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11763681 #14544462