Subgroups — C₄×#16863141 #30746780

Order 17152.

540 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14524043	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863141	C₂	1	2	yes	maximal
4	#22558093	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524043	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₃₄	—	2	1	no	—
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
10	#16863302	—	2	1	no	—
11	#16863294	—	2	1	no	—
12	#16863294	C₄	1	2	yes	—
13	#16863266	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊Dic₆₇	—	2	1	no	—
15	C₄×C₄⋊Dic₆₇	C₄	1	2	yes	—
16	#16863151	—	2	1	no	—
17	#16863141	C₄	1	8	yes	direct product factor
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	—	2	1	no	—
20	C₄³×C₆₇	C₂²	1	1	yes	characteristic
21	#336×C₆₇	—	2	1	no	—
22	C₂²×Dic₁₃₄	—	2	1	no	—
23	C₂²×C₆₇⋊C₈	—	2	2	no	—
24	#14524043	—	2	2	no	—
25	#14524043	C₂×C₄	1	4	yes	—
26	C₄×Dic₁₃₄	—	4	2	no	—
27	C₄×Dic₁₃₄	—	4	2	no	—
28	C₂×C₄⋊Dic₆₇	—	2	1	no	—
29	C₂×C₄⋊Dic₆₇	—	2	1	no	—
30	C₂×C₄⋊Dic₆₇	—	2	1	no	—
31	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
33	C₂×Dic₆₇⋊C₄	—	2	2	no	—
34	C₂×C₄×Dic₆₇	—	2	2	no	—
35	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
36	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
37	C₄×C₆₇⋊C₈	—	4	2	no	—
38	C₆₇⋊#95	—	2	2	no	—
39	C₆₇⋊#95	—	2	2	no	—
40	C₂²×C₈×C₆₇	—	2	2	no	—
41	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₆₇	—	2	2	no	—
49	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₆₇	—	4	2	no	—
51	C₂×Dic₁₃₄	—	2	4	no	—
52	C₂×Dic₁₃₄	—	4	1	no	—
53	C₂×Dic₁₃₄	—	4	1	no	—
54	C₂×Dic₁₃₄	—	4	2	no	—
55	C₂²×Dic₆₇	—	2	2	no	—
56	C₂²×Dic₆₇	—	2	2	no	—
57	C₂×C₆₇⋊C₈	—	2	4	no	—
58	C₂×C₆₇⋊C₈	—	4	2	no	—
59	C₂×C₆₇⋊C₈	—	4	2	no	—
60	C₄⋊Dic₆₇	—	2	2	no	—
61	C₄⋊Dic₆₇	—	2	4	no	—
62	C₄⋊Dic₆₇	—	4	1	no	—
63	C₄⋊Dic₆₇	—	4	1	no	—
64	C₄⋊Dic₆₇	—	4	2	no	—
65	C₄⋊Dic₆₇	—	4	2	no	—
66	C₄⋊Dic₆₇	C₄²	1	4	yes	—
67	Dic₆₇⋊C₄	—	4	4	no	—
68	C₄×Dic₆₇	—	4	4	no	—
69	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	—	2	4	no	—
77	C₂×C₈×C₆₇	—	4	2	no	—
78	C₂×C₈×C₆₇	—	4	2	no	—
79	C₄²×C₆₇	—	2	2	no	—
80	C₄²×C₆₇	—	2	2	no	—
81	C₄²×C₆₇	—	2	2	no	—
82	C₄²×C₆₇	—	4	1	no	—
83	C₄²×C₆₇	D₈	1	2	yes	—
84	C₄²×C₆₇	SD₁₆	1	2	yes	—
85	C₄²×C₆₇	SD₁₆	1	2	yes	—
86	C₄²×C₆₇	Q₁₆	1	2	yes	—
87	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
88	C₄²×C₆₇	#41	1	2	yes	—
89	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₆₇	—	2	1	no	—
91	C₂×C₄×C₆₇	—	2	1	no	—
92	C₂×C₄×C₆₇	—	2	1	no	—
93	C₂×C₄×C₆₇	—	2	1	no	—
94	C₂×C₄×C₆₇	—	2	1	no	—
95	C₂×C₄×C₆₇	—	2	1	no	—
96	C₂×C₄×C₆₇	—	2	1	no	—
97	C₂×C₄×C₆₇	—	2	1	no	—
98	C₂×C₄×C₆₇	—	2	1	no	—
99	C₂×C₄×C₆₇	#101	1	2	yes	—
100	C₂×C₄×C₆₇	#102	1	2	yes	—
101	C₂×C₄×C₆₇	#102	1	2	yes	—
102	C₂×C₄×C₆₇	#103	1	2	yes	—
103	C₂×C₄×C₆₇	#103	1	2	yes	—
104	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
112	C₂×C₄×C₆₇	#135	1	2	yes	—
113	C₂×C₄×C₆₇	#135	1	2	yes	—
114	C₂×Dic₆₇	—	2	4	no	—
115	C₂×Dic₆₇	—	4	2	no	—
116	C₂×Dic₆₇	—	4	2	no	—
117	C₂×Dic₆₇	—	4	2	no	—
118	C₂×Dic₆₇	—	4	2	no	—
119	C₂×Dic₆₇	—	4	2	no	—
120	Dic₁₃₄	—	4	4	no	—
121	Dic₁₃₄	—	4	4	no	—
122	C₈×C₆₇	—	4	4	no	—
123	C₆₇⋊C₈	—	4	4	no	—
124	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
125	C₂²×C₆₇	C₂×#101	1	1	yes	characteristic
126	C₂²×C₆₇	#413	1	1	yes	characteristic
127	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
128	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
129	C₂²×C₆₇	#416	1	1	yes	characteristic
130	C₂²×C₆₇	#416	1	1	yes	characteristic
131	C₄×C₆₇	—	2	2	no	—
132	C₄×C₆₇	—	2	2	no	—
133	C₄×C₆₇	—	2	1	no	—
134	C₄×C₆₇	—	2	1	no	—
135	C₄×C₆₇	—	4	1	no	—
136	C₄×C₆₇	#332	1	4	yes	complemented
137	C₄×C₆₇	C₄×D₈	1	2	yes	—
138	C₄×C₆₇	C₄×SD₁₆	1	2	yes	—
139	C₄×C₆₇	C₄×SD₁₆	1	2	yes	—
140	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
141	Dic₆₇	—	4	4	no	—
142	Dic₆₇	—	8	2	no	—
143	C₄×#332	—	67	1	no	maximal, Sylow
144	C₂×C₆₇	C₂×#332	1	1	yes	characteristic
145	C₂×C₆₇	#1540	1	1	yes	characteristic
146	C₂×C₆₇	#1542	1	1	yes	characteristic
147	C₂×C₆₇	#1544	1	1	yes	characteristic
148	C₂×C₆₇	C₄×#101	1	1	yes	characteristic
149	C₂×C₆₇	C₄×#102	1	1	yes	characteristic
150	C₂×C₆₇	C₄×#103	1	1	yes	characteristic
151	C₄×C₄⋊Q₈	—	67	1	no	—
152	C₄×C₄⋊C₈	—	67	1	no	—
153	C₄×C₄⋊C₈	D₆₇	1	1	yes	characteristic
154	C₂×#332	—	67	2	no	—
155	#1296	—	67	2	no	—
156	C₆₇	C₄×#332	1	1	yes	characteristic, complemented, Sylow
157	C₂×C₄⋊Q₈	—	67	1	no	—
158	C₂×C₄×Q₈	—	134	1	no	—
159	C₂×C₄⋊C₈	—	67	2	no	—
160	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
161	C₂×C₄×C₈	—	2	1	no	—
162	C₂×C₄×C₈	—	134	1	no	—
163	#391	—	134	1	no	—
164	#389	—	67	2	no	—
165	#389	—	134	1	no	—
166	#384	—	67	1	no	—
167	C₄×C₄⋊C₄	—	67	2	no	—
168	C₄×C₄⋊C₄	—	134	1	no	—
169	C₄³	C₂×D₆₇	1	1	yes	characteristic
170	#336	—	2	1	no	—
171	#336	—	134	1	no	—
172	#332	—	67	8	no	—
173	C₂²×Q₈	—	134	1	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	134	2	no	—
176	C₄⋊Q₈	—	67	4	no	—
177	C₄⋊Q₈	—	134	2	no	—
178	C₄×Q₈	—	268	2	no	—
179	C₄×Q₈	—	268	2	no	—
180	C₂×C₄⋊C₄	—	67	2	no	—
181	C₂×C₄⋊C₄	—	67	2	no	—
182	C₂×C₄⋊C₄	—	134	2	no	—
183	C₂×C₄⋊C₄	—	134	1	no	—
184	C₂×C₄⋊C₄	—	134	1	no	—
185	C₂×C₄⋊C₄	—	134	1	no	—
186	C₂×C₄²	—	134	2	no	—
187	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
188	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
189	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
190	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
191	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
192	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
193	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	—	67	4	no	—
196	C₄⋊C₈	—	134	2	no	—
197	C₄⋊C₈	C₄×D₆₇	1	4	yes	—
198	C₄×C₈	—	4	2	no	—
199	C₄×C₈	—	268	2	no	—
200	#95	—	134	2	no	—
201	#95	—	134	2	no	—
202	C₂×Q₈	—	268	1	no	—
203	C₂×Q₈	—	268	1	no	—
204	C₂×Q₈	—	268	2	no	—
205	C₂×Q₈	—	134	4	no	—
206	C₂²×C₄	—	134	2	no	—
207	C₂²×C₄	—	134	2	no	—
208	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
209	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
211	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	268	2	no	—
219	C₂×C₈	—	268	2	no	—
220	C₂×C₈	—	134	4	no	—
221	C₄⋊C₄	—	268	2	no	—
222	C₄⋊C₄	—	268	4	no	—
223	C₄⋊C₄	—	268	1	no	—
224	C₄⋊C₄	—	268	1	no	—
225	C₄⋊C₄	—	268	2	no	—
226	C₄⋊C₄	—	67	4	no	—
227	C₄⋊C₄	—	134	2	no	—
228	C₄⋊C₄	—	134	4	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	4	1	no	—
233	C₄²	—	268	4	no	—
234	C₄²	D₅₃₆	1	2	yes	—
235	C₄²	#11761616	1	2	yes	—
236	C₄²	#11761617	1	2	yes	—
237	C₄²	#11761618	1	2	yes	—
238	C₄²	C₂×C₄×D₆₇	1	2	yes	—
239	C₄²	#11761628	1	2	yes	—
240	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
241	Q₈	—	268	4	no	—
242	Q₈	—	268	4	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	268	2	no	—
253	C₂×C₄	—	268	2	no	—
254	C₂×C₄	—	268	2	no	—
255	C₂×C₄	—	268	2	no	—
256	C₂×C₄	—	268	2	no	—
257	C₂×C₄	—	134	4	no	—
258	C₂×C₄	#14523961	1	2	yes	—
259	C₂×C₄	#14523965	1	2	yes	—
260	C₂×C₄	#14523965	1	2	yes	—
261	C₂×C₄	#14523966	1	2	yes	—
262	C₂×C₄	#14523966	1	2	yes	—
263	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
264	C₂×C₄	#14524000	1	1	yes	characteristic
265	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
266	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
267	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
268	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
269	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
270	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
271	C₂×C₄	#14524063	1	2	yes	—
272	C₂×C₄	#14524065	1	2	yes	—
273	C₈	—	4	4	no	—
274	C₈	—	268	4	no	—
275	C₂²	C₄×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
276	C₂²	C₂×#14523961	1	1	yes	characteristic
277	C₂²	#16863378	1	1	yes	characteristic
278	C₂²	C₂×#14523965	1	1	yes	characteristic
279	C₂²	C₂×#14523966	1	1	yes	characteristic
280	C₂²	#16863390	1	1	yes	characteristic
281	C₂²	#16863393	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	4	1	no	—
287	C₄	—	268	4	no	—
288	C₄	—	536	2	no	—
289	C₄	#16863141	1	4	yes	direct product factor
290	C₄	C₄×D₅₃₆	1	2	yes	—
291	C₄	C₄×#11761616	1	2	yes	—
292	C₄	C₄×#11761617	1	2	yes	—
293	C₄	C₄×#11761618	1	2	yes	—
294	C₂	C₂×#16863141	1	1	yes	characteristic
295	C₂	#22558856	1	1	yes	characteristic
296	C₂	#22558867	1	1	yes	characteristic
297	C₂	#22558873	1	1	yes	characteristic
298	C₂	C₄×#14523961	1	1	yes	characteristic
299	C₂	C₄×#14523965	1	1	yes	characteristic
300	C₂	C₄×#14523966	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₆₇)⋊C₄
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×C₄⋊C₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×C₄⋊C₈	D₆₇
hypocenter	C₆₇	C₄×#332
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
17. #16863141	C₄	direct (N × H)
136. C₄×C₆₇	#332	semidirect
156. C₆₇	C₄×#332	semidirect
289. C₄	#16863141	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#14524043	2
C₄×C₄⋊(C₆₇⋊C₈)	2
C₂×#16863141	2
#22558093	2
C₄×C₄⋊C₈×C₆₇	2
C₄×#332	67

Subgroups — C4×#16863141 #30746780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16863141 #30746780