Subgroups — #16950367

Order 4352.

446 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₆₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#63214	C₂	1	1	yes	maximal, characteristic
3	C₂×#11762612	C₂	1	2	yes	maximal
4	#14526875	C₂	1	2	yes	maximal
5	C₁₇×#1845	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₆₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₃₄	—	2	1	no	—
8	C₂²×#63214	C₂²	1	2	yes	—
9	C₄×Dic₆₈	—	2	2	no	—
10	C₄×Dic₆₈	C₄	1	4	yes	complemented
11	C₂×#10587489	C₂²	1	1	yes	characteristic
12	C₂×#10587487	—	2	1	no	—
13	C₄⋊#63214	—	2	2	no	—
14	#11762612	—	2	4	no	—
15	C₂×C₁₇×#107	—	2	1	no	—
16	C₂×C₁₇×#107	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×#384	—	2	1	no	—
19	C₂²×Dic₃₄	—	2	1	no	—
20	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₇	—	4	1	no	—
24	C₂×C₄⋊C₄×C₁₇	—	4	1	no	—
25	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₇	—	2	1	no	—
27	C₂×#63225	—	4	1	no	—
28	C₂×C₄×Dic₁₇	—	4	1	no	—
29	C₂×Dic₆₈	—	2	1	no	—
30	C₂×Dic₆₈	—	2	1	no	—
31	C₂×Dic₆₈	—	2	1	no	—
32	C₂×Dic₆₈	C₂×C₄	1	2	yes	—
33	C₂×Dic₆₈	D₄	1	2	yes	—
34	C₂×Dic₆₈	Q₈	1	2	yes	—
35	C₄×Dic₃₄	—	2	4	no	—
36	C₄×Dic₃₄	—	4	2	no	—
37	C₁₇×#107	—	2	2	no	—
38	C₁₇×#107	—	2	2	no	—
39	C₁₇×#107	—	4	2	no	—
40	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₁₇	D₄	1	2	yes	—
42	#10587489	—	2	1	no	—
43	#10587489	C₂×C₄	1	2	yes	—
44	#10587487	—	2	4	no	—
45	C₂×#63214	—	2	2	no	—
46	C₂×#63214	—	2	2	no	—
47	C₂×#63214	—	2	2	no	—
48	C₂²×C₄×C₁₇	—	4	2	no	—
49	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₁₇	—	4	1	no	—
53	C₂²×Dic₁₇	—	4	1	no	—
54	C₂×Dic₃₄	—	2	2	no	—
55	C₂×Dic₃₄	—	2	2	no	—
56	C₂×Dic₃₄	—	2	2	no	—
57	C₂×Dic₃₄	—	4	1	no	—
58	C₂×Dic₃₄	—	4	1	no	—
59	C₂×Dic₃₄	—	4	1	no	—
60	C₂×C₈×C₁₇	—	2	1	no	—
61	C₂×C₈×C₁₇	—	2	1	no	—
62	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
68	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
69	C₄⋊C₄×C₁₇	—	4	2	no	—
70	C₄⋊C₄×C₁₇	—	4	2	no	—
71	C₄⋊C₄×C₁₇	—	8	2	no	—
72	C₄⋊C₄×C₁₇	—	8	2	no	—
73	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₁₇	SD₁₆	1	2	yes	—
75	C₄⋊Dic₁₇	—	2	2	no	—
76	C₄⋊Dic₁₇	—	4	1	no	—
77	#63225	—	4	4	no	—
78	C₄×Dic₁₇	—	4	4	no	—
79	Dic₆₈	—	2	2	no	—
80	Dic₆₈	—	2	2	no	—
81	Dic₆₈	—	2	2	no	—
82	Dic₆₈	C₄⋊C₄	1	4	yes	complemented
83	#63214	—	4	2	no	—
84	#12812	—	17	1	no	maximal, Sylow
85	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₁₇	—	2	1	no	—
87	C₂×C₄×C₁₇	—	2	1	no	—
88	C₂×C₄×C₁₇	—	8	2	no	—
89	C₂×C₄×C₁₇	—	8	2	no	—
90	C₂×C₄×C₁₇	—	8	2	no	—
91	C₂×C₄×C₁₇	#102	1	2	yes	—
92	C₂×C₄×C₁₇	#102	1	2	yes	—
93	C₂×C₄×C₁₇	SD₃₂	1	2	yes	—
94	C₂×C₄×C₁₇	Q₃₂	1	2	yes	—
95	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
97	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
99	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
100	C₂×C₄×C₁₇	#137	1	1	yes	characteristic
101	C₂×Dic₁₇	—	4	2	no	—
102	C₂×Dic₁₇	—	4	2	no	—
103	C₂×Dic₁₇	—	4	2	no	—
104	C₂×Dic₁₇	—	4	2	no	—
105	C₂×Dic₁₇	—	8	1	no	—
106	C₂×Dic₁₇	—	8	1	no	—
107	Dic₃₄	—	2	4	no	—
108	Dic₃₄	—	4	2	no	—
109	Dic₃₄	—	4	2	no	—
110	Dic₃₄	—	4	2	no	—
111	C₈×C₁₇	—	2	2	no	—
112	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₈×C₁₇	#117	1	1	yes	characteristic
114	C₈×C₁₇	#121	1	1	yes	characteristic
115	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×Q₁₆	—	17	1	no	—
117	C₂×C₄⋊C₁₆	—	17	1	no	—
118	C₂×#358	—	17	2	no	—
119	#1845	D₁₇	1	1	yes	characteristic
120	#1380	—	17	2	no	—
121	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
122	C₂²×C₁₇	#459	1	1	yes	characteristic
123	C₂²×C₁₇	#474	1	1	yes	characteristic
124	C₂²×C₁₇	C₂×SD₃₂	1	1	yes	characteristic
125	C₂²×C₁₇	C₂×Q₃₂	1	1	yes	characteristic
126	C₂²×C₁₇	#509	1	1	yes	characteristic
127	C₂²×C₁₇	#510	1	1	yes	characteristic
128	C₄×C₁₇	—	2	2	no	—
129	C₄×C₁₇	—	16	2	no	—
130	C₄×C₁₇	#358	1	4	yes	complemented
131	C₄×C₁₇	#380	1	1	yes	characteristic
132	C₄×C₁₇	#417	1	1	yes	characteristic
133	C₄×C₁₇	#461	1	1	yes	characteristic
134	C₄×C₁₇	#476	1	1	yes	characteristic
135	Dic₁₇	—	4	4	no	—
136	Dic₁₇	—	8	2	no	—
137	C₂²×Q₁₆	—	17	1	no	—
138	C₂×C₄×Q₈	—	34	1	no	—
139	C₂²×C₁₆	—	17	2	no	—
140	C₄×Q₁₆	—	17	4	no	—
141	C₄×Q₁₆	—	34	2	no	—
142	C₂×#107	—	2	1	no	—
143	C₂×#107	—	17	1	no	—
144	C₂×#107	C₂×D₁₇	1	2	yes	—
145	C₂×#103	—	34	1	no	—
146	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
147	#384	—	2	1	no	—
148	C₄⋊C₁₆	—	34	2	no	—
149	#358	—	34	4	no	—
150	C₂×C₁₇	#1802	1	1	yes	characteristic
151	C₂×C₁₇	C₂×#358	1	1	yes	characteristic
152	C₂×C₁₇	#2139	1	1	yes	characteristic
153	C₂×C₁₇	#2207	1	1	yes	characteristic
154	C₂×C₁₇	#2209	1	1	yes	characteristic
155	C₂×C₁₇	#2225	1	1	yes	characteristic
156	C₂×C₁₇	#2227	1	1	yes	characteristic
157	C₂²×Q₈	—	34	1	no	—
158	C₂×Q₁₆	—	17	2	no	—
159	C₂×Q₁₆	—	17	2	no	—
160	C₂×Q₁₆	—	17	2	no	—
161	C₂×Q₁₆	—	34	1	no	—
162	C₂×Q₁₆	—	34	1	no	—
163	C₂×Q₁₆	—	34	1	no	—
164	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
165	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
166	C₄×Q₈	—	34	4	no	—
167	C₄×Q₈	—	68	2	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	4	1	no	—
170	C₂×C₄⋊C₄	—	4	1	no	—
171	C₂×C₄⋊C₄	—	34	1	no	—
172	C₂×C₄⋊C₄	—	68	1	no	—
173	C₂×C₄²	—	68	1	no	—
174	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
175	C₂×C₁₆	—	34	2	no	—
176	C₂×C₁₆	—	34	2	no	—
177	C₂×C₁₆	—	34	2	no	—
178	#107	—	2	2	no	—
179	#107	—	2	2	no	—
180	#107	—	4	2	no	—
181	#107	—	17	2	no	—
182	#107	—	34	1	no	—
183	#103	—	34	4	no	—
184	C₄×C₈	C₄×D₁₇	1	2	yes	—
185	C₄×C₈	D₆₈	1	2	yes	—
186	C₁₇	#12812	1	1	yes	characteristic, complemented, Sylow
187	C₂×Q₈	—	34	2	no	—
188	C₂×Q₈	—	34	2	no	—
189	C₂×Q₈	—	34	2	no	—
190	C₂×Q₈	—	68	1	no	—
191	C₂×Q₈	—	68	1	no	—
192	C₂×Q₈	—	68	1	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	68	1	no	—
195	C₂²×C₄	—	68	1	no	—
196	C₂²×C₄	#63228	1	1	yes	characteristic
197	C₂²×C₄	#63228	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
199	Q₁₆	—	17	4	no	—
200	Q₁₆	—	34	2	no	—
201	Q₁₆	—	34	2	no	—
202	Q₁₆	—	34	2	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	#63227	1	2	yes	—
206	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
207	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
208	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
209	C₂×C₈	#63254	1	1	yes	characteristic
210	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
211	C₂×C₈	#63256	1	1	yes	characteristic
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	8	2	no	—
215	C₄⋊C₄	—	8	2	no	—
216	C₄⋊C₄	—	34	2	no	—
217	C₄⋊C₄	—	68	4	no	—
218	C₄⋊C₄	—	68	1	no	—
219	C₄²	—	68	4	no	—
220	C₄²	#63227	1	2	yes	—
221	C₄²	#63230	1	2	yes	—
222	C₁₆	—	68	2	no	—
223	C₂³	C₂×#63228	1	1	yes	characteristic
224	Q₈	—	34	4	no	—
225	Q₈	—	68	2	no	—
226	Q₈	—	68	2	no	—
227	Q₈	—	68	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	—	68	2	no	—
234	C₂×C₄	—	68	2	no	—
235	C₂×C₄	—	68	2	no	—
236	C₂×C₄	—	68	2	no	—
237	C₂×C₄	—	136	1	no	—
238	C₂×C₄	—	136	1	no	—
239	C₂×C₄	#10587480	1	2	yes	—
240	C₂×C₄	#10587480	1	2	yes	—
241	C₂×C₄	#10587498	1	2	yes	—
242	C₂×C₄	#10587500	1	2	yes	—
243	C₂×C₄	C₂×#63227	1	1	yes	characteristic
244	C₂×C₄	#10587618	1	1	yes	characteristic
245	C₂×C₄	#10587624	1	1	yes	characteristic
246	C₂×C₄	C₂×#63230	1	1	yes	characteristic
247	C₂×C₄	#10587628	1	1	yes	characteristic
248	C₂×C₄	#10587632	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	C₄⋊C₄×D₁₇	1	1	yes	characteristic
251	C₈	#10587578	1	1	yes	characteristic
252	C₈	#10587581	1	1	yes	characteristic
253	C₈	#10587583	1	1	yes	characteristic
254	C₂²	C₂×#10587480	1	1	yes	characteristic
255	C₂²	#11763001	1	1	yes	characteristic
256	C₂²	#11763086	1	1	yes	characteristic
257	C₂²	C₂×#10587498	1	1	yes	characteristic
258	C₂²	C₂×#10587500	1	1	yes	characteristic
259	C₂²	#11763245	1	1	yes	characteristic
260	C₂²	#11763250	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	16	2	no	—
263	C₄	—	68	4	no	—
264	C₄	—	136	2	no	—
265	C₄	#11762612	1	4	yes	complemented
266	C₄	#11762666	1	1	yes	characteristic
267	C₄	#11762808	1	1	yes	characteristic
268	C₄	#11763011	1	1	yes	characteristic
269	C₄	#11763096	1	1	yes	characteristic
270	C₂	#14528781	1	1	yes	characteristic
271	C₂	C₂×#11762612	1	1	yes	characteristic
272	C₂	#14530494	1	1	yes	characteristic
273	C₂	#14530821	1	1	yes	characteristic
274	C₂	#14530831	1	1	yes	characteristic
275	C₂	#14530923	1	1	yes	characteristic
276	C₂	#14530933	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#63228
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1845	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1845	D₁₇
hypocenter	C₁₇	#12812
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
10. C₄×Dic₆₈	C₄	semidirect
82. Dic₆₈	C₄⋊C₄	semidirect
130. C₄×C₁₇	#358	semidirect
186. C₁₇	#12812	semidirect
265. C₄	#11762612	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₆₈	2
C₂×C₄⋊#63214	2
C₂×#11762612	2
#14526875	2
C₁₇×#1845	2
#12812	17