Subgroups — #17958752

Order 4352.

380 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763731	C₂	1	1	yes	maximal, characteristic, complemented
2	#14543987	C₂	1	1	yes	maximal, characteristic, complemented
3	#14543748	C₂	1	4	yes	maximal, complemented
4	C₂×#11762926	C₂	1	1	yes	maximal, characteristic
5	C₂×#10587538	—	2	1	no	—
6	#11763825	—	2	1	no	—
7	#11763731	—	2	1	no	—
8	#11763731	C₂²	1	2	yes	complemented
9	#11763650	C₂²	1	2	yes	complemented
10	C₈×#3624	—	2	1	no	—
11	C₂×#10587634	—	2	1	no	—
12	#11762926	—	2	1	no	—
13	#11762926	—	2	1	no	—
14	#11762926	C₄	1	2	yes	—
15	#11762926	C₂²	1	2	yes	—
16	C₂×#10587490	C₄	1	1	yes	characteristic
17	C₂×#10587508	—	2	1	no	—
18	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
19	C₂×C₁₇×#135	C₄	1	1	yes	characteristic
20	C₂×#41×C₁₇	—	2	1	no	—
21	C₁₇×#135	—	2	1	no	—
22	C₁₇×#135	—	2	1	no	—
23	C₁₇×#135	C₈	1	2	yes	complemented
24	C₁₇×#135	C₂×C₄	1	2	yes	—
25	C₂×Q₁₆×C₁₇	C₈	1	1	yes	characteristic, complemented
26	C₂×SD₁₆×C₁₇	—	2	1	no	—
27	C₂×D₈×C₁₇	C₈	1	1	yes	characteristic, complemented
28	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
29	#10587634	—	2	2	no	—
30	#10587634	—	2	2	no	—
31	#10587634	—	4	1	no	—
32	#10587634	—	4	1	no	—
33	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
34	C₂×#63223	—	2	1	no	—
35	C₂×#63223	—	4	1	no	—
36	C₂²×#3622	—	4	1	no	—
37	C₂²×#3622	D₄	1	1	yes	characteristic
38	C₂²⋊#3624	—	2	2	no	—
39	C₄⋊#3624	—	2	2	no	—
40	C₄×#3624	—	2	2	no	—
41	#10587538	—	2	2	no	—
42	#10587538	—	4	1	no	—
43	C₂×#63216	—	4	1	no	—
44	#10587508	—	2	2	no	—
45	#10587508	—	2	2	no	—
46	#10587490	—	2	1	no	—
47	#10587490	C₂×C₄	1	2	yes	—
48	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
49	C₈×Dic₁₇	D₄	1	1	yes	characteristic
50	C₈×Dic₁₇	D₄	1	1	yes	characteristic
51	C₈×Dic₁₇	C₂³	1	1	yes	characteristic
52	#41×C₁₇	—	2	2	no	—
53	#41×C₁₇	—	2	2	no	—
54	#41×C₁₇	—	4	1	no	—
55	#41×C₁₇	—	4	1	no	—
56	C₂×Q₈×C₁₇	—	2	1	no	—
57	C₂×D₄×C₁₇	—	2	1	no	—
58	C₂×D₄×C₁₇	—	4	1	no	—
59	C₂²×C₄×C₁₇	—	4	1	no	—
60	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×Dic₁₇	—	2	1	no	—
62	C₂×#3624	—	2	4	no	—
63	Q₁₆×C₁₇	—	2	1	no	—
64	Q₁₆×C₁₇	C₂×C₈	1	2	yes	—
65	SD₁₆×C₁₇	—	2	2	no	—
66	SD₁₆×C₁₇	—	2	2	no	—
67	D₈×C₁₇	—	2	1	no	—
68	D₈×C₁₇	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
72	C₂×C₈×C₁₇	M₁₆	1	1	yes	characteristic
73	C₂×C₈×C₁₇	M₁₆	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
75	C₄×Dic₁₇	—	2	1	no	—
76	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
78	#63223	—	2	2	no	—
79	#63223	—	4	1	no	—
80	#63223	—	4	2	no	—
81	#63223	—	4	2	no	—
82	C₂×#3622	—	4	1	no	—
83	C₂×#3622	—	4	2	no	—
84	C₂×#3622	—	4	2	no	—
85	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×#3622	D₈	1	1	yes	characteristic
87	C₂×#3622	D₈	1	1	yes	characteristic
88	C₂×#3622	SD₁₆	1	1	yes	characteristic
89	C₂×#3622	SD₁₆	1	1	yes	characteristic
90	C₂×#3622	C₂×D₄	1	1	yes	characteristic
91	#63216	—	4	2	no	—
92	#11604	—	17	1	no	maximal, Sylow
93	C₂³×C₁₇	—	4	1	no	—
94	C₂³×C₁₇	#104	1	1	yes	characteristic
95	Q₈×C₁₇	—	2	2	no	—
96	Q₈×C₁₇	—	4	1	no	—
97	D₄×C₁₇	—	2	2	no	—
98	D₄×C₁₇	—	4	2	no	—
99	D₄×C₁₇	—	4	2	no	—
100	D₄×C₁₇	—	4	1	no	—
101	C₂×C₄×C₁₇	—	4	1	no	—
102	C₂×C₄×C₁₇	—	4	2	no	—
103	C₂×C₄×C₁₇	—	4	2	no	—
104	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
106	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
107	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
108	C₂×C₄×C₁₇	#104	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×Dic₁₇	—	2	1	no	—
111	C₂×Dic₁₇	—	2	1	no	—
112	C₂×Dic₁₇	—	4	2	no	—
113	#3624	—	4	4	no	—
114	C₈×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
115	C₈×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
116	C₈×C₁₇	C₂²×C₈	1	1	yes	characteristic
117	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
118	#3622	—	4	2	no	—
119	#3622	#102	1	1	yes	characteristic
120	#3622	#102	1	1	yes	characteristic
121	#3622	C₂×D₈	1	1	yes	characteristic
122	#3622	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×#443	—	17	1	no	—
124	C₂×#364	—	17	1	no	—
125	#1450	—	17	1	no	—
126	#1336	—	17	4	no	—
127	C₂²×C₁₇	—	4	1	no	—
128	C₂²×C₁₇	—	4	2	no	—
129	C₂²×C₁₇	—	8	2	no	—
130	C₂²×C₁₇	D₄⋊C₈	1	1	yes	characteristic
131	C₂²×C₁₇	D₄⋊C₈	1	1	yes	characteristic
132	C₂²×C₁₇	C₂×#104	1	1	yes	characteristic
133	C₄×C₁₇	—	4	2	no	—
134	C₄×C₁₇	D₄⋊C₈	1	1	yes	characteristic
135	C₄×C₁₇	D₄⋊C₈	1	1	yes	characteristic
136	C₄×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
137	C₄×C₁₇	C₂×#102	1	1	yes	characteristic
138	Dic₁₇	—	2	2	no	—
139	Dic₁₇	—	4	1	no	—
140	C₂×#135	#595	1	1	yes	characteristic
141	C₂×#131	—	34	1	no	—
142	C₂×M₃₂	—	34	1	no	—
143	#443	—	17	2	no	—
144	#443	—	17	2	no	—
145	#443	—	34	1	no	—
146	#443	—	34	1	no	—
147	#431	—	34	1	no	—
148	C₂×#108	—	17	1	no	—
149	C₂×#104	—	34	1	no	—
150	C₂×C₄×C₈	—	17	1	no	—
151	#364	—	17	2	no	—
152	#364	—	34	1	no	—
153	#322	—	17	2	no	—
154	C₈²	—	34	1	no	—
155	C₂×C₁₇	—	2	1	no	—
156	C₂×C₁₇	—	2	1	no	—
157	C₂×C₁₇	—	2	1	no	—
158	C₂×C₁₇	—	8	2	no	—
159	C₂×C₁₇	C₂×D₄⋊C₈	1	1	yes	characteristic
160	C₂×#41	—	2	1	no	—
161	#135	—	2	1	no	—
162	#135	—	2	1	no	—
163	#135	#3624	1	2	yes	complemented
164	#135	C₂×#595	1	2	yes	—
165	C₂×Q₁₆	#3624	1	1	yes	characteristic, complemented
166	C₂×SD₁₆	—	2	1	no	—
167	C₂×D₈	#3624	1	1	yes	characteristic, complemented
168	#131	—	34	2	no	—
169	#131	—	34	2	no	—
170	#131	—	68	1	no	—
171	#131	—	68	1	no	—
172	C₂×M₁₆	—	34	1	no	—
173	C₂×M₁₆	—	68	1	no	—
174	C₂²×C₈	—	17	1	no	—
175	C₂²×C₈	—	68	1	no	—
176	C₂²×C₈	C₂×#595	1	1	yes	characteristic
177	C₂×C₄²	—	17	1	no	—
178	M₃₂	—	34	2	no	—
179	M₃₂	—	68	1	no	—
180	C₂×C₁₆	—	68	1	no	—
181	#108	—	17	2	no	—
182	#108	—	34	1	no	—
183	C₄⋊C₈	—	34	2	no	—
184	#104	—	34	2	no	—
185	#104	—	34	2	no	—
186	C₂²⋊C₈	—	34	2	no	—
187	C₄×C₈	—	17	1	no	—
188	C₄×C₈	—	17	1	no	—
189	C₄×C₈	—	17	1	no	—
190	C₄×C₈	—	17	1	no	—
191	C₄×C₈	—	34	2	no	—
192	C₁₇	#11604	1	1	yes	characteristic, complemented, Sylow
193	#41	—	2	2	no	—
194	#41	—	2	2	no	—
195	#41	—	4	1	no	—
196	#41	—	4	1	no	—
197	C₂×Q₈	—	2	1	no	—
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂²×C₄	—	4	1	no	—
201	C₂²×C₄	—	34	1	no	—
202	C₂²×C₄	C₂²⋊#595	1	1	yes	characteristic
203	Q₁₆	—	2	1	no	—
204	Q₁₆	C₂×#3624	1	2	yes	—
205	SD₁₆	—	2	2	no	—
206	SD₁₆	—	2	2	no	—
207	D₈	—	2	1	no	—
208	D₈	C₂×#3624	1	2	yes	—
209	M₁₆	—	34	2	no	—
210	M₁₆	—	68	2	no	—
211	M₁₆	—	68	2	no	—
212	M₁₆	—	68	1	no	—
213	C₂×C₈	—	17	1	no	—
214	C₂×C₈	—	17	1	no	—
215	C₂×C₈	—	17	1	no	—
216	C₂×C₈	—	17	1	no	—
217	C₂×C₈	—	17	1	no	—
218	C₂×C₈	—	17	1	no	—
219	C₂×C₈	—	34	4	no	—
220	C₂×C₈	—	68	2	no	—
221	C₂×C₈	—	68	2	no	—
222	C₂×C₈	—	68	1	no	—
223	C₂×C₈	C₂×#3624	1	1	yes	characteristic
224	C₂×C₈	C₂×#3624	1	1	yes	characteristic
225	C₂×C₈	#63247	1	1	yes	characteristic
226	C₂×C₈	#63247	1	1	yes	characteristic
227	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
228	C₂×C₈	C₂²×#595	1	1	yes	characteristic
229	C₄²	—	17	1	no	—
230	C₄²	—	17	1	no	—
231	C₄²	—	34	1	no	—
232	C₁₆	—	68	2	no	—
233	C₂³	—	4	1	no	—
234	C₂³	#10587548	1	1	yes	characteristic
235	Q₈	—	2	2	no	—
236	Q₈	—	4	1	no	—
237	D₄	—	2	2	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	D₄	—	4	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	34	1	no	—
245	C₂×C₄	—	34	1	no	—
246	C₂×C₄	—	68	2	no	—
247	C₂×C₄	#10587547	1	1	yes	characteristic
248	C₂×C₄	#10587547	1	1	yes	characteristic
249	C₂×C₄	#10587548	1	1	yes	characteristic
250	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
251	C₂×C₄	C₂²⋊#3624	1	1	yes	characteristic
252	C₂×C₄	C₂×C₂²⋊#595	1	1	yes	characteristic
253	C₈	—	17	1	no	—
254	C₈	—	17	1	no	—
255	C₈	—	17	1	no	—
256	C₈	—	17	1	no	—
257	C₈	—	68	4	no	—
258	C₈	—	68	2	no	—
259	C₈	C₂²⋊#3624	1	1	yes	characteristic
260	C₈	C₂²⋊#3624	1	1	yes	characteristic
261	C₈	C₂²×#3624	1	1	yes	characteristic
262	C₈	C₂×#63247	1	1	yes	characteristic
263	C₂²	—	4	1	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	D₄⋊#3624	1	1	yes	characteristic
267	C₂²	D₄⋊#3624	1	1	yes	characteristic
268	C₂²	C₂×#10587548	1	1	yes	characteristic
269	C₄	—	4	2	no	—
270	C₄	—	34	2	no	—
271	C₄	—	68	1	no	—
272	C₄	D₄⋊#3624	1	1	yes	characteristic
273	C₄	D₄⋊#3624	1	1	yes	characteristic
274	C₄	C₂×C₂²⋊#3624	1	1	yes	characteristic
275	C₄	C₂×#10587547	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	2	1	no	—
279	C₂	—	8	2	no	—
280	C₂	C₂×D₄⋊#3624	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#10587547
commutator	C₈×C₁₇	C₂²×C₈
Frattini	C₂×C₈	C₂²×#595
Fitting, prime radical	C₂×C₁₇×#135	C₄
socle	C₂×C₁₇	C₂×D₄⋊C₈
cosocle	C₈×Dic₁₇	C₂³
hypercenter	C₂×#135	#595
hypocenter	C₁₇	#11604
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#11763731	C₂	semidirect
2. #14543987	C₂	semidirect
3. #14543748	C₂	semidirect
8. #11763731	C₂²	semidirect
9. #11763650	C₂²	semidirect
23. C₁₇×#135	C₈	semidirect
25. C₂×Q₁₆×C₁₇	C₈	semidirect
27. C₂×D₈×C₁₇	C₈	semidirect
163. #135	#3624	semidirect
165. C₂×Q₁₆	#3624	semidirect
167. C₂×D₈	#3624	semidirect
192. C₁₇	#11604	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11763731	2
#14543987	2
#14543748	2
C₂×#11762926	2
#11604	17