Subgroups — C₂×#14528278 #17016930

Order 4352.

682 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11763189	C₂	1	1	yes	maximal, characteristic
2	C₂×#11763094	C₂	1	2	yes	maximal
3	#14528278	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11762560	C₂	1	1	yes	maximal, characteristic
5	C₂×#11762556	C₂	1	2	yes	maximal
6	C₂×C₁₇×#446	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₆₈	—	2	1	no	—
8	C₂²×Dic₆₈	—	2	1	no	—
9	C₂×#10587555	C₂²	1	2	yes	—
10	#11763189	—	2	2	no	—
11	#11763189	C₂²	1	4	yes	—
12	#11763094	—	2	4	no	—
13	#11763094	C₂²	1	8	yes	—
14	C₂×#10587479	—	2	2	no	—
15	C₂×C₄⋊#3622	C₂²	1	2	yes	—
16	C₂×#10587506	—	2	2	no	—
17	#11762560	C₂²	1	4	yes	—
18	#11762556	C₂²	1	8	yes	—
19	C₂×C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₁₇×#446	—	2	2	no	—
21	C₁₇×#446	C₂²	1	4	yes	—
22	C₂×C₁₇×C₄⋊C₈	—	2	1	no	—
23	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₁₇×#97	—	2	1	no	—
25	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
26	C₂²×Q₈×C₁₇	D₄	1	1	yes	characteristic
27	C₂²×Dic₃₄	—	2	2	no	—
28	C₂²×C₈×C₁₇	—	2	1	no	—
29	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
31	C₄×Q₈×C₁₇	—	2	1	no	—
32	C₄×Q₈×C₁₇	—	2	1	no	—
33	C₄×Q₈×C₁₇	D₄	1	2	yes	—
34	C₄×Q₈×C₁₇	C₂³	1	2	yes	—
35	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₇	—	2	1	no	—
38	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊Dic₁₇	—	2	2	no	—
40	C₂×C₄⋊Dic₁₇	—	2	2	no	—
41	C₂²×#3622	—	2	2	no	—
42	C₂×Dic₆₈	—	2	4	no	—
43	C₂×Dic₆₈	—	2	4	no	—
44	C₂×Dic₆₈	—	4	4	no	—
45	C₂×Dic₆₈	—	4	4	no	—
46	#10587555	—	2	2	no	—
47	#10587555	—	2	2	no	—
48	#10587555	—	2	2	no	—
49	#10587555	C₂³	1	4	yes	—
50	C₁₇×C₄⋊C₈	—	2	4	no	—
51	C₁₇×C₄⋊C₈	—	2	1	no	—
52	C₁₇×C₄⋊C₈	C₂³	1	2	yes	—
53	C₁₇×#97	—	2	4	no	—
54	C₄×C₈×C₁₇	D₄	1	2	yes	—
55	C₄×C₈×C₁₇	C₂³	1	2	yes	—
56	#10587506	—	2	8	no	—
57	#10587479	—	2	8	no	—
58	C₄⋊#3622	—	2	2	no	—
59	C₄⋊#3622	C₂³	1	4	yes	—
60	C₂×Q₈×C₁₇	—	2	1	no	—
61	C₂×Q₈×C₁₇	—	2	4	no	—
62	C₂×Q₈×C₁₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₇	—	2	1	no	—
64	C₂²×C₄×C₁₇	—	2	1	no	—
65	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
68	C₂²×Dic₁₇	—	4	2	no	—
69	C₂×Dic₃₄	—	2	4	no	—
70	C₂×Dic₃₄	—	4	4	no	—
71	C₂×Dic₃₄	—	4	4	no	—
72	C₂×Dic₃₄	—	4	2	no	—
73	C₂×C₈×C₁₇	—	2	2	no	—
74	C₂×C₈×C₁₇	—	2	2	no	—
75	C₂×C₈×C₁₇	—	4	2	no	—
76	C₂×C₈×C₁₇	SD₁₆	1	4	yes	—
77	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
79	C₄⋊C₄×C₁₇	—	2	2	no	—
80	C₄⋊C₄×C₁₇	—	2	1	no	—
81	C₄⋊C₄×C₁₇	—	2	2	no	—
82	C₄⋊C₄×C₁₇	C₂×D₄	1	2	yes	—
83	C₄²×C₁₇	—	2	2	no	—
84	C₄²×C₁₇	—	2	2	no	—
85	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₇	#41	1	1	yes	characteristic
88	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
89	C₄⋊Dic₁₇	—	2	4	no	—
90	C₄⋊Dic₁₇	—	2	4	no	—
91	C₄⋊Dic₁₇	—	4	2	no	—
92	C₄⋊Dic₁₇	—	4	2	no	—
93	C₂×#3622	—	2	4	no	—
94	C₂×#3622	—	4	4	no	—
95	Dic₆₈	—	4	8	no	—
96	Dic₆₈	—	4	8	no	—
97	C₂×#1690	—	17	1	no	maximal, Sylow
98	C₂³×C₁₇	#121	1	1	yes	characteristic
99	Q₈×C₁₇	—	2	4	no	—
100	Q₈×C₁₇	—	4	2	no	—
101	C₂×C₄×C₁₇	—	2	2	no	—
102	C₂×C₄×C₁₇	—	2	2	no	—
103	C₂×C₄×C₁₇	—	2	2	no	—
104	C₂×C₄×C₁₇	—	4	2	no	—
105	C₂×C₄×C₁₇	—	4	2	no	—
106	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
107	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
108	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
109	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
110	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
111	C₂×C₄×C₁₇	#137	1	4	yes	—
112	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
115	C₂×Dic₁₇	—	4	4	no	—
116	C₂×Dic₁₇	—	8	4	no	—
117	Dic₃₄	—	4	4	no	—
118	Dic₃₄	—	4	4	no	—
119	Dic₃₄	—	8	4	no	—
120	C₈×C₁₇	—	2	2	no	—
121	C₈×C₁₇	—	4	2	no	—
122	C₈×C₁₇	C₂×SD₁₆	1	4	yes	—
123	#3622	—	4	4	no	—
124	C₂×#494	—	17	1	no	—
125	C₂×#475	—	17	2	no	—
126	C₂×#446	D₁₇	1	1	yes	characteristic
127	#1690	—	17	8	no	—
128	C₂×#334	—	17	1	no	—
129	C₂×#332	—	17	2	no	—
130	C₂²×C₁₇	#460	1	2	yes	—
131	C₂²×C₁₇	#470	1	2	yes	—
132	C₂²×C₁₇	#473	1	2	yes	—
133	C₂²×C₁₇	C₂×#121	1	1	yes	characteristic
134	C₄×C₁₇	—	2	1	no	—
135	C₄×C₁₇	—	2	1	no	—
136	C₄×C₁₇	—	4	2	no	—
137	C₄×C₁₇	—	4	2	no	—
138	C₄×C₁₇	#460	1	2	yes	—
139	C₄×C₁₇	#470	1	2	yes	—
140	C₄×C₁₇	C₂²×SD₁₆	1	1	yes	characteristic
141	C₄×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₄×C₁₇	C₂×#137	1	2	yes	—
143	Dic₁₇	—	8	4	no	—
144	C₂²×Q₁₆	—	34	1	no	—
145	C₂²×Q₁₆	—	34	1	no	—
146	C₂×C₄⋊Q₈	—	17	2	no	—
147	C₂×C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
148	#494	—	17	4	no	—
149	#494	—	34	2	no	—
150	#475	—	17	8	no	—
151	#475	—	34	4	no	—
152	#446	—	2	2	no	—
153	#446	C₂×D₁₇	1	4	yes	—
154	C₂×#106	—	34	2	no	—
155	C₂×C₄⋊C₈	—	2	1	no	—
156	C₂×C₄⋊C₈	—	17	2	no	—
157	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
158	C₂×#103	—	34	2	no	—
159	C₂×#97	—	2	1	no	—
160	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
161	#334	—	17	4	no	—
162	#332	—	17	8	no	—
163	C₂×C₁₇	#1690	1	4	yes	complemented
164	C₂×C₁₇	C₂×#460	1	1	yes	characteristic
165	C₂×C₁₇	C₂×#470	1	1	yes	characteristic
166	C₂×C₁₇	C₂×#473	1	1	yes	characteristic
167	C₂²×Q₈	—	34	2	no	—
168	C₂²×Q₈	D₆₈	1	1	yes	characteristic
169	C₂×Q₁₆	—	34	4	no	—
170	C₂×Q₁₆	—	34	4	no	—
171	C₂×Q₁₆	—	68	4	no	—
172	C₂×Q₁₆	—	68	4	no	—
173	C₂²×C₈	—	2	1	no	—
174	C₂²×C₈	—	34	2	no	—
175	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
176	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
177	C₄⋊Q₈	—	17	4	no	—
178	C₄⋊Q₈	—	34	2	no	—
179	C₄⋊Q₈	—	34	2	no	—
180	C₄⋊Q₈	—	34	2	no	—
181	C₄×Q₈	—	2	1	no	—
182	C₄×Q₈	—	2	1	no	—
183	C₄×Q₈	D₆₈	1	2	yes	—
184	C₄×Q₈	C₂²×D₁₇	1	2	yes	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	—	34	2	no	—
187	C₂×C₄⋊C₄	—	34	2	no	—
188	C₂×C₄⋊C₄	D₆₈	1	1	yes	characteristic
189	C₂×C₄²	—	2	1	no	—
190	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
191	#106	—	34	8	no	—
192	C₄⋊C₈	—	2	4	no	—
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	—	17	4	no	—
195	C₄⋊C₈	—	34	2	no	—
196	C₄⋊C₈	C₂²×D₁₇	1	2	yes	—
197	#103	—	34	8	no	—
198	#97	—	2	4	no	—
199	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
200	C₄×C₈	C₂²×D₁₇	1	2	yes	—
201	C₁₇	C₂×#1690	1	1	yes	characteristic, complemented, Sylow
202	C₂×Q₈	—	2	1	no	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	—	34	4	no	—
205	C₂×Q₈	—	68	2	no	—
206	C₂×Q₈	—	68	4	no	—
207	C₂×Q₈	—	68	4	no	—
208	C₂×Q₈	C₂×D₆₈	1	2	yes	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	68	2	no	—
212	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
213	C₂²×C₄	#63252	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
215	Q₁₆	—	68	8	no	—
216	Q₁₆	—	68	8	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	34	4	no	—
221	C₂×C₈	—	68	4	no	—
222	C₂×C₈	#63230	1	4	yes	—
223	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
224	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	34	4	no	—
229	C₄⋊C₄	—	34	4	no	—
230	C₄⋊C₄	—	68	2	no	—
231	C₄⋊C₄	—	68	2	no	—
232	C₄⋊C₄	C₂×D₆₈	1	2	yes	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	C₂×D₆₈	1	1	yes	characteristic
236	C₄²	#63252	1	1	yes	characteristic
237	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
238	C₄²	C₂³×D₁₇	1	1	yes	characteristic
239	C₂³	#10587616	1	1	yes	characteristic
240	Q₈	—	2	4	no	—
241	Q₈	—	4	2	no	—
242	Q₈	—	68	4	no	—
243	Q₈	—	68	4	no	—
244	Q₈	—	136	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	68	4	no	—
251	C₂×C₄	—	136	4	no	—
252	C₂×C₄	#10587595	1	4	yes	—
253	C₂×C₄	#10587616	1	1	yes	characteristic
254	C₂×C₄	#10587616	1	1	yes	characteristic
255	C₂×C₄	#10587616	1	1	yes	characteristic
256	C₂×C₄	C₂×#63230	1	2	yes	—
257	C₂×C₄	#10587635	1	2	yes	—
258	C₂×C₄	C₂²×D₆₈	1	1	yes	characteristic
259	C₂×C₄	C₂×#63252	1	1	yes	characteristic
260	C₂×C₄	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	4	2	no	—
263	C₈	—	68	4	no	—
264	C₈	C₂×#63230	1	4	yes	—
265	C₂²	#11763009	1	2	yes	—
266	C₂²	#11763067	1	2	yes	—
267	C₂²	#11763084	1	2	yes	—
268	C₂²	C₂×#10587616	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	2	no	—
273	C₄	—	136	4	no	—
274	C₄	#11763009	1	2	yes	—
275	C₄	#11763067	1	2	yes	—
276	C₄	C₂²×#63230	1	1	yes	characteristic
277	C₄	C₂×#10587635	1	1	yes	characteristic
278	C₄	C₂×#10587595	1	2	yes	—
279	C₂	#14528278	1	4	yes	direct product factor
280	C₂	C₂×#11763009	1	1	yes	characteristic
281	C₂	C₂×#11763067	1	1	yes	characteristic
282	C₂	C₂×#11763084	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10587616
commutator, cosocle	C₄²×C₁₇	C₂⁴
Frattini	C₄²	C₂³×D₁₇
Fitting	C₂×C₁₇×#446	C₂
socle, prime radical	C₂³×C₁₇	#121
hypercenter	C₂×#446	D₁₇
hypocenter	C₁₇	C₂×#1690
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
3. #14528278	C₂	direct (N × H)
163. C₂×C₁₇	#1690	semidirect
201. C₁₇	C₂×#1690	semidirect
279. C₂	#14528278	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×#11763189	2
C₂×#11763094	2
#14528278	2
C₂×#11762560	2
C₂×#11762556	2
C₂×C₁₇×#446	2
C₂×#1690	17

Subgroups — C2×#14528278 #17016930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14528278 #17016930