Subgroups — #30748760

Order 17152.

398 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×C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	#22560202	C₂	1	2	yes	maximal
3	#22558968	C₂	1	1	yes	maximal, characteristic
4	#22558083	C₂	1	2	yes	maximal
5	C₆₇×#1567	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₃₄	—	2	1	no	—
8	C₈×Dic₁₃₄	—	2	1	no	—
9	C₈×Dic₁₃₄	—	2	1	no	—
10	C₈×Dic₁₃₄	—	2	1	no	—
11	#16863420	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₆₇⋊C₈	C₄	1	2	yes	—
14	C₂×#14523961	—	2	2	no	—
15	C₂×C₈×Dic₆₇	C₄	1	2	yes	—
16	#16863164	—	2	2	no	—
17	#16863149	—	2	1	no	—
18	#16863149	—	2	1	no	—
19	#423×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	#335×C₆₇	—	2	1	no	—
22	#335×C₆₇	—	2	1	no	—
23	C₂²×Dic₁₃₄	D₄	1	1	yes	characteristic
24	C₂×#11761612	—	2	2	no	—
25	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×Dic₁₃₄	—	2	1	no	—
27	C₄×Dic₁₃₄	—	2	1	no	—
28	C₄×Dic₁₃₄	—	2	1	no	—
29	C₄×Dic₁₃₄	—	2	1	no	—
30	C₂×C₄⋊Dic₆₇	D₄	1	1	yes	characteristic
31	C₂×Dic₆₇⋊C₄	Q₈	1	2	yes	—
32	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
33	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
34	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
35	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
36	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
37	Dic₆₇⋊C₈	—	2	2	no	—
38	Dic₆₇⋊C₈	—	2	2	no	—
39	#14523961	—	4	4	no	—
40	C₈×Dic₆₇	—	2	2	no	—
41	C₈×Dic₆₇	—	2	2	no	—
42	C₂×M₁₆×C₆₇	—	2	2	no	—
43	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₆₇	—	2	2	no	—
47	C₄⋊C₈×C₆₇	—	2	2	no	—
48	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
49	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
52	C₂×Dic₁₃₄	—	2	1	no	—
53	C₂×Dic₁₃₄	—	2	1	no	—
54	C₂×Dic₁₃₄	—	2	1	no	—
55	C₂×Dic₁₃₄	—	2	1	no	—
56	C₂×Dic₁₃₄	D₈	1	2	yes	—
57	C₂×Dic₁₃₄	Q₁₆	1	2	yes	—
58	C₂²×Dic₆₇	C₄⋊C₄	1	2	yes	—
59	C₂²×Dic₆₇	C₄⋊C₄	1	2	yes	—
60	#11761612	—	4	2	no	—
61	#11761612	—	4	2	no	—
62	C₂×C₆₇⋊C₈	—	2	2	no	—
63	C₂×C₆₇⋊C₈	—	4	2	no	—
64	C₂×C₆₇⋊C₈	—	4	2	no	—
65	C₄⋊Dic₆₇	—	2	1	no	—
66	C₄⋊Dic₆₇	—	2	1	no	—
67	Dic₆₇⋊C₄	—	2	2	no	—
68	Dic₆₇⋊C₄	—	2	2	no	—
69	C₄×Dic₆₇	—	2	2	no	—
70	C₄×Dic₆₇	—	2	2	no	—
71	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
74	M₁₆×C₆₇	—	4	2	no	—
75	M₁₆×C₆₇	—	4	2	no	—
76	C₂×C₈×C₆₇	—	2	2	no	—
77	C₂×C₈×C₆₇	—	4	1	no	—
78	C₂×C₈×C₆₇	—	4	2	no	—
79	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₆₇	#41	1	1	yes	characteristic
86	C₄²×C₆₇	#41	1	1	yes	characteristic
87	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₆₇	—	2	1	no	—
89	C₂×C₄×C₆₇	—	2	1	no	—
90	C₂×C₄×C₆₇	—	4	1	no	—
91	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
93	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
94	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
100	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
101	C₂×Dic₆₇	—	2	2	no	—
102	C₂×Dic₆₇	—	2	2	no	—
103	C₂×Dic₆₇	—	2	2	no	—
104	C₂×Dic₆₇	—	4	2	no	—
105	C₂×Dic₆₇	#107	1	4	yes	—
106	Dic₁₃₄	—	2	2	no	—
107	Dic₁₃₄	—	2	2	no	—
108	Dic₁₃₄	—	2	2	no	—
109	Dic₁₃₄	—	2	2	no	—
110	C₈×C₆₇	—	2	1	no	—
111	C₈×C₆₇	—	2	1	no	—
112	C₈×C₆₇	—	2	1	no	—
113	C₈×C₆₇	—	8	2	no	—
114	C₈×C₆₇	#101	1	2	yes	complemented
115	C₆₇⋊C₈	—	4	2	no	—
116	C₆₇⋊C₈	—	8	2	no	—
117	C₂²×C₆₇	—	2	1	no	—
118	C₂²×C₆₇	—	2	1	no	—
119	C₂²×C₆₇	#380	1	1	yes	characteristic
120	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
121	C₂²×C₆₇	#427	1	1	yes	characteristic
122	C₄×C₆₇	—	2	1	no	—
123	C₄×C₆₇	—	2	1	no	—
124	C₄×C₆₇	—	2	1	no	—
125	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
126	C₄×C₆₇	#413	1	1	yes	characteristic
127	C₄×C₆₇	#466	1	1	yes	characteristic
128	C₄×C₆₇	#466	1	1	yes	characteristic
129	C₄×C₆₇	#467	1	1	yes	characteristic
130	C₄×C₆₇	#467	1	1	yes	characteristic
131	Dic₆₇	—	2	4	no	—
132	Dic₆₇	—	4	2	no	—
133	#8617	—	67	1	no	maximal, Sylow
134	C₂×C₆₇	—	2	1	no	—
135	C₂×C₆₇	—	2	1	no	—
136	C₂×C₆₇	#1801	1	1	yes	characteristic
137	C₂×C₆₇	#1808	1	1	yes	characteristic
138	C₂×C₆₇	#1813	1	1	yes	characteristic
139	C₂×C₈×Q₈	—	67	1	no	—
140	#1801	—	67	2	no	—
141	#1567	—	67	1	no	—
142	#1567	D₆₇	1	1	yes	characteristic
143	#1292	—	67	2	no	—
144	C₆₇	#8617	1	1	yes	characteristic, complemented, Sylow
145	C₂×C₄×Q₈	—	67	1	no	—
146	C₈×Q₈	—	134	1	no	—
147	C₈×Q₈	—	134	1	no	—
148	C₈×Q₈	—	134	1	no	—
149	C₈×Q₈	—	134	1	no	—
150	#423	—	67	2	no	—
151	#423	C₂×D₆₇	1	2	yes	—
152	C₂×C₄⋊C₈	—	67	2	no	—
153	C₂×C₄⋊C₈	—	67	1	no	—
154	C₂×#101	—	134	2	no	—
155	C₂×C₄×C₈	—	67	2	no	—
156	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
157	#343	—	134	2	no	—
158	#335	—	2	1	no	—
159	#335	—	2	1	no	—
160	#335	—	134	1	no	—
161	#335	—	134	1	no	—
162	C₂²×Q₈	—	67	1	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	134	2	no	—
165	C₂²×C₈	—	67	2	no	—
166	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
167	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
168	C₄×Q₈	—	134	1	no	—
169	C₄×Q₈	—	134	1	no	—
170	C₄×Q₈	—	134	1	no	—
171	C₄×Q₈	—	134	1	no	—
172	C₂×C₄⋊C₄	—	67	1	no	—
173	C₂×C₄⋊C₄	—	67	2	no	—
174	C₂×C₄²	—	67	2	no	—
175	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	—	134	2	no	—
179	C₄⋊C₈	—	134	2	no	—
180	C₄⋊C₈	—	134	1	no	—
181	C₄⋊C₈	—	134	2	no	—
182	C₄⋊C₈	—	134	1	no	—
183	C₄⋊C₈	—	134	2	no	—
184	#101	—	268	4	no	—
185	C₄×C₈	—	134	2	no	—
186	C₄×C₈	—	134	2	no	—
187	C₄×C₈	D₂₆₈	1	1	yes	characteristic
188	C₄×C₈	D₂₆₈	1	1	yes	characteristic
189	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
190	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
191	C₂×Q₈	—	67	2	no	—
192	C₂×Q₈	—	67	2	no	—
193	C₂×Q₈	—	134	1	no	—
194	C₂×Q₈	—	134	1	no	—
195	C₂×Q₈	—	134	1	no	—
196	C₂×Q₈	—	134	1	no	—
197	C₂²×C₄	—	67	2	no	—
198	C₂²×C₄	—	67	2	no	—
199	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
200	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
201	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
202	M₁₆	—	4	2	no	—
203	M₁₆	—	4	2	no	—
204	M₁₆	—	268	2	no	—
205	M₁₆	—	268	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	268	2	no	—
210	C₂×C₈	—	268	2	no	—
211	C₂×C₈	—	134	2	no	—
212	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
213	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
214	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
215	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
216	C₄⋊C₄	—	134	1	no	—
217	C₄⋊C₄	—	134	1	no	—
218	C₄⋊C₄	—	134	2	no	—
219	C₄⋊C₄	—	134	2	no	—
220	C₄²	—	134	2	no	—
221	C₄²	—	134	2	no	—
222	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
223	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
224	C₄²	#11761627	1	1	yes	characteristic
225	C₄²	#11761629	1	1	yes	characteristic
226	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
227	Q₈	—	134	2	no	—
228	Q₈	—	134	2	no	—
229	Q₈	—	134	2	no	—
230	Q₈	—	134	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	268	2	no	—
235	C₂×C₄	—	67	4	no	—
236	C₂×C₄	—	134	2	no	—
237	C₂×C₄	—	134	2	no	—
238	C₂×C₄	—	134	2	no	—
239	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
240	C₂×C₄	#14524001	1	1	yes	characteristic
241	C₂×C₄	#14524014	1	1	yes	characteristic
242	C₂×C₄	#14524018	1	1	yes	characteristic
243	C₂×C₄	#14524019	1	1	yes	characteristic
244	C₂×C₄	#14524023	1	1	yes	characteristic
245	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
246	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
247	C₂×C₄	#14524062	1	1	yes	characteristic
248	C₂×C₄	#14524066	1	1	yes	characteristic
249	C₈	—	2	1	no	—
250	C₈	—	2	1	no	—
251	C₈	—	2	1	no	—
252	C₈	—	8	2	no	—
253	C₈	—	268	2	no	—
254	C₈	—	536	2	no	—
255	C₈	#14523961	1	2	yes	complemented
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	#16863253	1	1	yes	characteristic
259	C₂²	#107×D₆₇	1	1	yes	characteristic
260	C₂²	#16863433	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	268	2	no	—
265	C₄	—	134	4	no	—
266	C₄	C₂×#14523961	1	1	yes	characteristic
267	C₄	#16863378	1	1	yes	characteristic
268	C₄	#16863629	1	1	yes	characteristic
269	C₄	#16863633	1	1	yes	characteristic
270	C₄	#16863634	1	1	yes	characteristic
271	C₄	#16863638	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	#22560202	1	1	yes	characteristic
275	C₂	#22560235	1	1	yes	characteristic
276	C₂	#22560256	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863253
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1567	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1567	D₆₇
hypocenter	C₆₇	#8617
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
114. C₈×C₆₇	#101	semidirect
144. C₆₇	#8617	semidirect
255. C₈	#14523961	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₃₄	2
#22560202	2
#22558968	2
#22558083	2
C₆₇×#1567	2
#8617	67