Subgroups — #30853830

Order 17152.

1138 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863354	C₂	1	2	yes	maximal
3	#22562616	C₂	1	1	yes	maximal, characteristic
4	#22560081	C₂	1	4	yes	maximal, complemented
5	#22560076	C₂	1	2	yes	maximal, complemented
6	C₈×C₄⋊Dic₆₇	C₂	1	2	yes	maximal, complemented
7	#22559907	C₂	1	2	yes	maximal
8	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₆₇⋊C₈	C₂²	1	2	yes	—
10	C₂×#14524002	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	8	yes	complemented
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
15	#16863354	—	2	8	no	—
16	C₂×#14523953	C₂²	1	4	yes	—
17	C₂×C₈×Dic₆₇	C₂²	1	4	yes	—
18	#16863266	C₄	1	2	yes	—
19	#16863262	C₄	1	2	yes	—
20	#16863256	C₄	1	2	yes	—
21	#16863256	C₄	1	2	yes	—
22	C₄×C₄⋊Dic₆₇	C₂²	1	2	yes	—
23	C₄×C₄⋊Dic₆₇	C₂²	1	2	yes	—
24	#16863150	C₂²	1	4	yes	—
25	#16863150	C₂²	1	4	yes	—
26	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₆₇	C₄	1	8	yes	complemented
30	C₂×#97×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
32	C₂²×C₆₇⋊C₈	—	2	4	no	—
33	C₂²×C₆₇⋊C₈	—	2	4	no	—
34	C₂²×C₆₇⋊C₈	C₂×C₄	1	8	yes	—
35	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
36	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
37	#14524002	—	2	4	no	—
38	#14524002	—	2	4	no	—
39	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
42	C₂×C₂²⋊Dic₆₇	C₂×C₄	1	4	yes	—
43	C₂×C₄×Dic₆₇	C₂³	1	4	yes	—
44	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
45	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
46	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	8	yes	—
47	#14523953	—	2	8	no	—
48	C₈×Dic₆₇	—	2	8	no	—
49	C₆₇⋊#95	C₂×C₄	1	4	yes	—
50	C₆₇⋊#95	C₂×C₄	1	4	yes	—
51	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×M₁₆×C₆₇	—	2	2	no	—
55	C₂×M₁₆×C₆₇	—	2	2	no	—
56	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
57	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
58	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
59	C₂×M₁₆×C₆₇	Q₈	1	4	yes	—
60	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
61	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
62	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
65	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
66	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
68	#97×C₆₇	C₂×C₄	1	8	yes	—
69	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	—
70	C₂²×Dic₆₇	C₂²×C₄	1	8	yes	—
71	C₂×C₆₇⋊C₈	—	2	16	no	—
72	C₂×C₆₇⋊C₈	—	2	2	no	—
73	C₂×C₆₇⋊C₈	—	2	2	no	—
74	C₂×C₆₇⋊C₈	—	2	2	no	—
75	C₂×C₆₇⋊C₈	—	2	2	no	—
76	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
77	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
78	C₄⋊Dic₆₇	—	2	8	no	—
79	C₄⋊Dic₆₇	—	2	8	no	—
80	C₄⋊Dic₆₇	—	4	4	no	—
81	C₂²⋊Dic₆₇	—	2	8	no	—
82	C₄×Dic₆₇	—	2	4	no	—
83	C₄×Dic₆₇	—	2	4	no	—
84	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	—	2	4	no	—
86	C₂²×C₄×C₆₇	—	4	2	no	—
87	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
88	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
89	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
98	C₂²×C₄×C₆₇	#41	1	2	yes	—
99	C₂²×C₄×C₆₇	#41	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
101	M₁₆×C₆₇	—	2	8	no	—
102	M₁₆×C₆₇	C₄⋊C₄	1	16	yes	complemented
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
106	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
107	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
108	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
109	C₄²×C₆₇	C₄²	1	4	yes	—
110	C₄²×C₆₇	C₄²	1	4	yes	—
111	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
112	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
113	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
114	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
115	C₂³×C₆₇	—	2	2	no	—
116	C₂³×C₆₇	—	2	2	no	—
117	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
118	C₂³×C₆₇	C₄×D₄	1	2	yes	—
119	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
120	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	8	no	—
125	C₂×C₄×C₆₇	—	4	4	no	—
126	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
127	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
128	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
130	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
132	C₂×C₄×C₆₇	#117	1	4	yes	—
133	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
134	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
136	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
137	C₂×C₄×C₆₇	#131	1	4	yes	—
138	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
144	C₂×Dic₆₇	—	2	8	no	—
145	C₂×Dic₆₇	—	2	8	no	—
146	C₂×Dic₆₇	—	2	8	no	—
147	C₈×C₆₇	—	2	4	no	—
148	C₈×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
149	C₆₇⋊C₈	—	2	4	no	—
150	C₆₇⋊C₈	—	2	4	no	—
151	C₂²×C₆₇	—	2	2	no	—
152	C₂²×C₆₇	—	2	2	no	—
153	C₂²×C₆₇	—	2	4	no	—
154	C₂²×C₆₇	—	2	4	no	—
155	C₂²×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
156	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
157	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
158	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
159	C₂²×C₆₇	C₂×#131	1	2	yes	—
160	C₂²×C₆₇	#568	1	2	yes	—
161	C₄×C₆₇	—	2	2	no	—
162	C₄×C₆₇	—	2	2	no	—
163	C₄×C₆₇	—	2	2	no	—
164	C₄×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
165	C₄×C₆₇	#405	1	4	yes	complemented
166	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
167	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
168	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
169	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
170	Dic₆₇	—	4	8	no	—
171	#20441	—	67	1	no	maximal, Sylow
172	C₂×C₆₇	—	2	4	no	—
173	C₂×C₆₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₆₇	C₂×#405	1	1	yes	characteristic
175	C₂×C₆₇	#2873	1	1	yes	characteristic
176	C₂×C₆₇	#2935	1	2	yes	—
177	C₂×C₆₇	#2962	1	2	yes	—
178	C₂²×C₄⋊C₈	—	67	1	no	—
179	C₂×#405	—	67	2	no	—
180	C₂×C₄×M₁₆	D₆₇	1	1	yes	characteristic
181	#2285	—	67	1	no	—
182	#1771	—	67	4	no	—
183	#1770	—	67	2	no	—
184	C₈×C₄⋊C₄	—	67	2	no	—
185	#1729	—	67	2	no	—
186	C₆₇	#20441	1	1	yes	characteristic, complemented, Sylow
187	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
188	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
189	C₂³×C₈	—	67	2	no	—
190	C₂×#117	—	67	2	no	—
191	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
192	C₂×C₄⋊C₈	—	67	8	no	—
193	C₂×C₄⋊C₈	—	67	1	no	—
194	C₂×C₄⋊C₈	—	67	1	no	—
195	C₂×C₄⋊C₈	—	67	2	no	—
196	#405	—	134	8	no	—
197	C₄×M₁₆	Dic₆₇	1	8	yes	complemented
198	C₂×#97	—	67	4	no	—
199	C₂×#97	C₂×D₆₇	1	2	yes	—
200	C₂×C₄×C₈	—	67	4	no	—
201	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
202	#384	—	67	2	no	—
203	#382	—	67	2	no	—
204	#380	—	67	2	no	—
205	#380	—	67	2	no	—
206	C₄×C₄⋊C₄	—	67	2	no	—
207	C₄×C₄⋊C₄	—	67	2	no	—
208	#336	—	67	4	no	—
209	#336	—	67	4	no	—
210	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
211	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
212	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
213	C₂×M₁₆	—	2	2	no	—
214	C₂×M₁₆	—	2	2	no	—
215	C₂×M₁₆	Dic₁₃₄	1	4	yes	—
216	C₂×M₁₆	D₂₆₈	1	4	yes	—
217	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
218	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
219	C₂²×C₈	—	67	8	no	—
220	C₂²×C₈	—	67	2	no	—
221	C₂²×C₈	—	67	2	no	—
222	C₂²×C₈	—	134	4	no	—
223	C₂²×C₈	—	134	4	no	—
224	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
225	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
226	#117	—	134	4	no	—
227	#117	—	134	4	no	—
228	C₂×C₄⋊C₄	—	67	4	no	—
229	C₂×C₄⋊C₄	—	67	4	no	—
230	C₂×C₄⋊C₄	—	67	4	no	—
231	C₂×C₂²⋊C₄	—	67	4	no	—
232	C₂×C₄²	—	67	4	no	—
233	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
234	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
235	C₂×C₄²	C₄×D₆₇	1	4	yes	—
236	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
237	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
239	C₄⋊C₈	—	67	4	no	—
240	C₄⋊C₈	—	67	4	no	—
241	C₄⋊C₈	—	67	8	no	—
242	#97	—	134	8	no	—
243	#97	C₂×Dic₆₇	1	8	yes	—
244	C₄×C₈	—	134	8	no	—
245	C₄×C₈	C₂×Dic₆₇	1	8	yes	—
246	#95	—	67	4	no	—
247	#95	—	67	4	no	—
248	C₂⁴	C₂×C₄×D₆₇	1	1	yes	characteristic
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	—	67	8	no	—
252	C₂²×C₄	C₄×Dic₆₇	1	4	yes	—
253	C₂²×C₄	C₄×Dic₆₇	1	4	yes	—
254	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
260	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
261	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
262	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
263	C₂²×C₄	#11761628	1	2	yes	—
264	C₂²×C₄	#11761628	1	2	yes	—
265	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
266	M₁₆	—	2	8	no	—
267	M₁₆	C₄⋊Dic₆₇	1	16	yes	complemented
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	67	4	no	—
271	C₂×C₈	—	67	4	no	—
272	C₂×C₈	—	134	16	no	—
273	C₂×C₈	—	134	2	no	—
274	C₂×C₈	—	134	2	no	—
275	C₂×C₈	—	134	2	no	—
276	C₂×C₈	—	134	2	no	—
277	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
278	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
279	C₂×C₈	C₂×D₂₆₈	1	4	yes	—
280	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
281	C₄⋊C₄	—	268	4	no	—
282	C₄⋊C₄	—	134	8	no	—
283	C₄⋊C₄	—	134	8	no	—
284	C₂²⋊C₄	—	134	8	no	—
285	C₄²	—	134	4	no	—
286	C₄²	—	134	4	no	—
287	C₄²	C₄×Dic₆₇	1	4	yes	—
288	C₄²	C₄×Dic₆₇	1	4	yes	—
289	C₄²	C₂×C₄×D₆₇	1	2	yes	—
290	C₄²	C₂×C₄×D₆₇	1	2	yes	—
291	C₄²	C₂²×Dic₆₇	1	2	yes	—
292	C₄²	C₂²×Dic₆₇	1	2	yes	—
293	C₂³	—	2	2	no	—
294	C₂³	—	2	2	no	—
295	C₂³	C₂×C₄×Dic₆₇	1	2	yes	—
296	C₂³	C₄×D₂₆₈	1	2	yes	—
297	C₂³	C₄×Dic₁₃₄	1	2	yes	—
298	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	8	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	134	8	no	—
305	C₂×C₄	—	134	8	no	—
306	C₂×C₄	—	134	8	no	—
307	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
308	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
309	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
310	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
311	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
312	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
313	C₂×C₄	#14524002	1	4	yes	—
314	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
315	C₂×C₄	C₄×D₂₆₈	1	4	yes	—
316	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
317	C₂×C₄	C₄×Dic₁₃₄	1	4	yes	—
318	C₂×C₄	#14524051	1	4	yes	—
319	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
320	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
321	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
322	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
323	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
324	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
325	C₈	—	2	4	no	—
326	C₈	—	134	4	no	—
327	C₈	—	134	4	no	—
328	C₈	C₂×C₄⋊Dic₆₇	1	8	yes	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	4	no	—
332	C₂²	—	2	4	no	—
333	C₂²	C₄×C₄⋊Dic₆₇	1	4	yes	—
334	C₂²	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
335	C₂²	C₂×C₄×D₂₆₈	1	1	yes	characteristic
336	C₂²	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
337	C₂²	C₂×#14524051	1	2	yes	—
338	C₂²	#16864162	1	2	yes	—
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	268	8	no	—
343	C₄	C₄×C₄⋊Dic₆₇	1	4	yes	—
344	C₄	#16863354	1	4	yes	complemented
345	C₄	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
346	C₄	C₂×#14524002	1	1	yes	characteristic
347	C₄	C₂×C₄×D₂₆₈	1	1	yes	characteristic
348	C₄	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
349	C₂	—	2	4	no	—
350	C₂	C₂×C₄×C₄⋊Dic₆₇	1	1	yes	characteristic
351	C₂	C₂×#16863354	1	1	yes	characteristic
352	C₂	#22568627	1	1	yes	characteristic
353	C₂	#22568955	1	2	yes	—
354	C₂	#22569156	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂²×C₄²
Fitting	C₂×C₄×M₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₆₇
hypocenter	C₆₇	#20441
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. #22560081	C₂	semidirect
5. #22560076	C₂	semidirect
6. C₈×C₄⋊Dic₆₇	C₂	semidirect
11. C₂×C₄⋊(C₆₇⋊C₈)	C₄	semidirect
29. C₄×M₁₆×C₆₇	C₄	semidirect
102. M₁₆×C₆₇	C₄⋊C₄	semidirect
165. C₄×C₆₇	#405	semidirect
186. C₆₇	#20441	semidirect
197. C₄×M₁₆	Dic₆₇	semidirect
267. M₁₆	C₄⋊Dic₆₇	semidirect
344. C₄	#16863354	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₆₇⋊C₈)	2
C₂×#16863354	2
#22562616	2
#22560081	2
#22560076	2
C₈×C₄⋊Dic₆₇	2
#22559907	2
C₂×C₄×M₁₆×C₆₇	2
#20441	67