Subgroups — #30749783

Order 17152.

516 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863475	C₂	1	1	yes	maximal, characteristic
2	C₄×#14523963	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863146	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16863123	C₂	1	1	yes	maximal, characteristic
5	#22558050	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1939	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11761611	—	2	1	no	—
9	C₂×C₄×D₂₆₈	C₂²	1	1	yes	characteristic
10	#16863475	—	2	2	no	—
11	#16863475	—	2	2	no	—
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×#14523963	C₄	1	1	yes	characteristic
14	C₂×#14523963	C₄	1	1	yes	characteristic
15	C₂×(C₂×D₆₇)⋊C₈	—	2	1	no	—
16	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
18	#16863158	C₄	1	1	yes	characteristic
19	#16863158	C₄	1	1	yes	characteristic
20	#16863146	—	2	2	no	—
21	#16863123	—	2	2	no	—
22	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
26	#336×C₆₇	—	2	1	no	—
27	C₂²×D₂₆₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₆₇	—	2	1	no	—
29	C₂×#11761611	—	2	4	no	—
30	C₂×#11761611	—	4	2	no	—
31	C₂×#11761611	—	4	2	no	—
32	C₂²×C₆₇⋊C₈	—	2	1	no	—
33	C₂²×C₆₇⋊C₈	—	2	1	no	—
34	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₄×D₂₆₈	—	2	1	no	—
37	C₄×D₂₆₈	—	2	1	no	—
38	C₄×D₂₆₈	D₄	1	2	yes	—
39	C₄×D₂₆₈	Q₈	1	2	yes	—
40	C₂×C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂×(C₂×D₆₇)⋊C₄	—	2	1	no	—
42	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
43	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
44	#14523963	—	2	2	no	—
45	#14523963	—	2	2	no	—
46	(C₂×D₆₇)⋊C₈	—	2	4	no	—
47	C₄×C₆₇⋊C₈	—	2	1	no	—
48	C₄×C₆₇⋊C₈	—	2	1	no	—
49	C₄×C₆₇⋊C₈	—	2	1	no	—
50	C₄×C₆₇⋊C₈	—	2	1	no	—
51	C₂²×C₈×C₆₇	—	2	1	no	—
52	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	—	2	1	no	—
57	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₆₇	—	2	1	no	—
59	C₄⋊C₈×C₆₇	—	2	1	no	—
60	C₄⋊C₈×C₆₇	—	4	2	no	—
61	C₄×C₈×C₆₇	—	2	1	no	—
62	C₄×C₈×C₆₇	—	2	1	no	—
63	#95×C₆₇	—	2	1	no	—
64	C₂³×D₆₇	—	2	1	no	—
65	C₂×D₂₆₈	—	2	1	no	—
66	C₂×D₂₆₈	—	2	1	no	—
67	C₂×D₂₆₈	—	2	1	no	—
68	C₂×D₂₆₈	—	2	1	no	—
69	C₂×D₂₆₈	—	2	1	no	—
70	C₂×D₂₆₈	C₄⋊C₄	1	2	yes	—
71	C₂²×Dic₆₇	—	2	1	no	—
72	C₂×C₄×D₆₇	—	2	2	no	—
73	C₂×C₄×D₆₇	—	2	2	no	—
74	C₂×C₄×D₆₇	—	4	1	no	—
75	C₂×C₄×D₆₇	—	4	1	no	—
76	C₂×C₄×D₆₇	—	4	1	no	—
77	C₂×C₄×D₆₇	—	4	1	no	—
78	#11761611	—	4	8	no	—
79	C₂×C₆₇⋊C₈	—	2	1	no	—
80	C₂×C₆₇⋊C₈	—	2	1	no	—
81	C₂×C₆₇⋊C₈	—	2	2	no	—
82	C₂×C₆₇⋊C₈	—	2	1	no	—
83	C₂×C₆₇⋊C₈	—	2	2	no	—
84	C₂×C₆₇⋊C₈	—	2	1	no	—
85	C₂×C₆₇⋊C₈	—	2	1	no	—
86	C₂×C₆₇⋊C₈	—	4	1	no	—
87	C₂×C₆₇⋊C₈	—	4	1	no	—
88	C₂×C₆₇⋊C₈	—	4	1	no	—
89	C₂×C₆₇⋊C₈	—	4	1	no	—
90	C₂×C₆₇⋊C₈	M₁₆	1	2	yes	complemented
91	C₄⋊Dic₆₇	—	2	1	no	—
92	C₄⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
93	(C₂×D₆₇)⋊C₄	—	2	2	no	—
94	(C₂×D₆₇)⋊C₄	—	2	2	no	—
95	C₂²×C₄×C₆₇	—	2	1	no	—
96	C₂²×C₄×C₆₇	—	2	1	no	—
97	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
100	C₂×C₈×C₆₇	—	2	1	no	—
101	C₂×C₈×C₆₇	—	2	1	no	—
102	C₂×C₈×C₆₇	—	2	1	no	—
103	C₂×C₈×C₆₇	—	2	1	no	—
104	C₂×C₈×C₆₇	—	2	1	no	—
105	C₂×C₈×C₆₇	—	4	1	no	—
106	C₂×C₈×C₆₇	—	4	1	no	—
107	C₂×C₈×C₆₇	—	4	1	no	—
108	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
109	C₄⋊C₄×C₆₇	—	2	1	no	—
110	C₄⋊C₄×C₆₇	—	2	1	no	—
111	C₄⋊C₄×C₆₇	—	2	1	no	—
112	C₄⋊C₄×C₆₇	M₁₆	1	2	yes	complemented
113	C₄²×C₆₇	—	4	1	no	—
114	C₄²×C₆₇	—	4	1	no	—
115	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
118	C₄²×C₆₇	#41	1	1	yes	characteristic
119	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
120	C₂²×D₆₇	—	2	2	no	—
121	C₂²×D₆₇	—	4	1	no	—
122	C₂²×D₆₇	—	4	1	no	—
123	C₂²×D₆₇	—	4	1	no	—
124	C₂²×D₆₇	—	4	1	no	—
125	C₂²×D₆₇	—	4	1	no	—
126	C₂²×D₆₇	—	4	1	no	—
127	C₂×C₄×C₆₇	—	2	1	no	—
128	C₂×C₄×C₆₇	—	2	1	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	4	1	no	—
131	C₂×C₄×C₆₇	—	4	1	no	—
132	C₂×C₄×C₆₇	—	4	1	no	—
133	C₂×C₄×C₆₇	—	4	1	no	—
134	C₂×C₄×C₆₇	—	4	1	no	—
135	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
140	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
143	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
144	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
146	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
147	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×Dic₆₇	—	2	2	no	—
150	C₂×Dic₆₇	—	4	1	no	—
151	C₂×Dic₆₇	—	4	1	no	—
152	D₂₆₈	—	2	2	no	—
153	D₂₆₈	—	2	2	no	—
154	D₂₆₈	—	4	1	no	—
155	D₂₆₈	—	4	1	no	—
156	C₄×D₆₇	—	4	2	no	—
157	C₄×D₆₇	—	4	2	no	—
158	C₄×D₆₇	—	4	2	no	—
159	C₄×D₆₇	—	4	2	no	—
160	C₈×C₆₇	—	2	2	no	—
161	C₈×C₆₇	—	4	1	no	—
162	C₈×C₆₇	—	8	1	no	—
163	C₆₇⋊C₈	—	2	2	no	—
164	C₆₇⋊C₈	—	4	2	no	—
165	C₆₇⋊C₈	—	4	1	no	—
166	C₆₇⋊C₈	—	4	2	no	—
167	C₂²×C₆₇	#385	1	1	yes	characteristic
168	C₂²×C₆₇	#435	1	1	yes	characteristic
169	C₂²×C₆₇	#436	1	1	yes	characteristic
170	C₂²×C₆₇	C₄×SD₁₆	1	1	yes	characteristic
171	C₂²×C₆₇	#442	1	1	yes	characteristic
172	C₂²×C₆₇	#443	1	1	yes	characteristic
173	C₂²×C₆₇	#444	1	1	yes	characteristic
174	C₂×D₆₇	—	4	2	no	—
175	C₂×D₆₇	—	4	2	no	—
176	C₂×D₆₇	—	4	2	no	—
177	C₂×D₆₇	—	8	1	no	—
178	C₂×D₆₇	—	8	1	no	—
179	C₂×D₆₇	—	8	1	no	—
180	C₂×D₆₇	—	8	1	no	—
181	C₄×C₆₇	—	2	1	no	—
182	C₄×C₆₇	—	2	1	no	—
183	C₄×C₆₇	—	4	2	no	—
184	C₄×C₆₇	—	8	1	no	—
185	C₄×C₆₇	#423	1	1	yes	characteristic
186	C₄×C₆₇	#424	1	1	yes	characteristic
187	C₄×C₆₇	#435	1	1	yes	characteristic
188	C₄×C₆₇	#436	1	1	yes	characteristic
189	C₄×C₆₇	#449	1	1	yes	characteristic
190	C₄×C₆₇	#450	1	1	yes	characteristic
191	C₄×C₆₇	#476	1	1	yes	characteristic
192	C₄×C₆₇	#478	1	1	yes	characteristic
193	Dic₆₇	—	4	2	no	—
194	#8849	—	67	1	no	maximal, Sylow
195	C₂×C₆₇	#1583	1	1	yes	characteristic
196	C₂×C₆₇	#1586	1	1	yes	characteristic
197	C₂×C₆₇	#1590	1	1	yes	characteristic
198	C₂×C₆₇	#1594	1	1	yes	characteristic
199	C₂×C₆₇	#1851	1	1	yes	characteristic
200	C₂×C₆₇	#1864	1	1	yes	characteristic
201	C₂×C₆₇	#1866	1	1	yes	characteristic
202	D₆₇	—	8	2	no	—
203	C₂×#436	—	67	1	no	—
204	#1939	D₆₇	1	1	yes	characteristic, complemented
205	C₄×#102	—	67	1	no	—
206	C₂×#334	—	67	1	no	—
207	C₂×D₄⋊C₈	—	67	1	no	—
208	#1279	—	67	1	no	—
209	#1273	—	67	1	no	—
210	C₆₇	#8849	1	1	yes	characteristic, complemented, Sylow
211	C₂²×M₁₆	—	134	1	no	—
212	C₂×C₄×D₄	—	67	1	no	—
213	#436	—	134	2	no	—
214	#436	—	134	2	no	—
215	C₂×C₄⋊C₈	—	2	1	no	—
216	C₂×C₄⋊C₈	—	67	1	no	—
217	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
218	C₂×#102	—	67	1	no	—
219	C₂×#102	—	67	1	no	—
220	C₂×C₂²⋊C₈	—	134	1	no	—
221	C₂×C₄×C₈	—	67	1	no	—
222	C₂×C₄×C₈	—	67	1	no	—
223	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
224	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
225	#340	—	67	1	no	—
226	#340	—	67	1	no	—
227	#336	—	2	1	no	—
228	#334	—	134	2	no	—
229	D₄⋊C₈	—	134	2	no	—
230	C₂²×D₄	—	67	1	no	—
231	C₂³×C₄	—	134	1	no	—
232	C₂×M₁₆	—	268	2	no	—
233	C₂×M₁₆	—	268	2	no	—
234	C₂×M₁₆	—	134	4	no	—
235	C₂²×C₈	—	2	1	no	—
236	C₂²×C₈	—	67	1	no	—
237	C₂²×C₈	—	67	1	no	—
238	C₂²×C₈	—	134	1	no	—
239	C₂²×C₈	—	134	1	no	—
240	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
241	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
242	C₄×D₄	—	67	2	no	—
243	C₄×D₄	—	67	2	no	—
244	C₄×D₄	—	134	1	no	—
245	C₄×D₄	—	134	1	no	—
246	C₂×C₄⋊C₄	—	67	1	no	—
247	C₂×C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
248	C₂×C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
249	C₂×C₂²⋊C₄	—	134	1	no	—
250	C₂×C₄²	—	2	1	no	—
251	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
252	C₄⋊C₈	—	2	1	no	—
253	C₄⋊C₈	—	2	1	no	—
254	C₄⋊C₈	—	4	2	no	—
255	C₄⋊C₈	—	67	2	no	—
256	C₄⋊C₈	—	134	1	no	—
257	#102	—	134	2	no	—
258	#102	—	134	2	no	—
259	C₂²⋊C₈	—	134	4	no	—
260	C₄×C₈	—	2	1	no	—
261	C₄×C₈	—	2	1	no	—
262	C₄×C₈	—	134	1	no	—
263	C₄×C₈	—	134	1	no	—
264	C₄×C₈	—	134	1	no	—
265	C₄×C₈	—	134	1	no	—
266	#95	—	2	1	no	—
267	C₂⁴	—	134	1	no	—
268	C₂×D₄	—	67	2	no	—
269	C₂×D₄	—	134	1	no	—
270	C₂×D₄	—	134	1	no	—
271	C₂×D₄	—	134	1	no	—
272	C₂×D₄	—	134	1	no	—
273	C₂×D₄	—	134	1	no	—
274	C₂²×C₄	—	2	1	no	—
275	C₂²×C₄	—	2	1	no	—
276	C₂²×C₄	—	268	1	no	—
277	C₂²×C₄	—	268	1	no	—
278	C₂²×C₄	—	268	1	no	—
279	C₂²×C₄	—	268	1	no	—
280	C₂²×C₄	—	134	1	no	—
281	C₂²×C₄	—	134	2	no	—
282	C₂²×C₄	—	134	2	no	—
283	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
284	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
285	C₂²×C₄	#11761628	1	1	yes	characteristic
286	M₁₆	—	268	8	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	4	1	no	—
293	C₂×C₈	—	4	1	no	—
294	C₂×C₈	—	4	1	no	—
295	C₂×C₈	—	268	1	no	—
296	C₂×C₈	—	268	1	no	—
297	C₂×C₈	—	268	1	no	—
298	C₂×C₈	—	268	1	no	—
299	C₂×C₈	—	67	2	no	—
300	C₂×C₈	—	134	2	no	—
301	C₂×C₈	—	134	1	no	—
302	C₂×C₈	—	134	2	no	—
303	C₂×C₈	—	134	1	no	—
304	C₂×C₈	—	134	1	no	—
305	C₂×C₈	—	134	1	no	—
306	C₂×C₈	—	134	1	no	—
307	C₂×C₈	#11761615	1	2	yes	—
308	C₄⋊C₄	—	2	1	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	67	2	no	—
312	C₄⋊C₄	—	134	1	no	—
313	C₄⋊C₄	#11761611	1	2	yes	complemented
314	C₂²⋊C₄	—	134	2	no	—
315	C₂²⋊C₄	—	134	2	no	—
316	C₄²	—	4	1	no	—
317	C₄²	—	4	1	no	—
318	C₄²	D₄×D₆₇	1	1	yes	characteristic
319	C₄²	D₄×D₆₇	1	1	yes	characteristic
320	C₄²	Q₈×D₆₇	1	1	yes	characteristic
321	C₄²	#11761629	1	1	yes	characteristic
322	C₂³	—	268	1	no	—
323	C₂³	—	268	1	no	—
324	C₂³	—	268	1	no	—
325	C₂³	—	268	1	no	—
326	C₂³	—	268	1	no	—
327	C₂³	—	268	1	no	—
328	C₂³	—	134	2	no	—
329	C₂³	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
330	D₄	—	268	1	no	—
331	D₄	—	268	1	no	—
332	D₄	—	134	2	no	—
333	D₄	—	134	2	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	268	1	no	—
343	C₂×C₄	—	268	1	no	—
344	C₂×C₄	—	268	2	no	—
345	C₂×C₄	—	268	2	no	—
346	C₂×C₄	—	268	2	no	—
347	C₂×C₄	—	268	2	no	—
348	C₂×C₄	—	134	2	no	—
349	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
350	C₂×C₄	#14524005	1	1	yes	characteristic
351	C₂×C₄	#14524011	1	1	yes	characteristic
352	C₂×C₄	#14524019	1	1	yes	characteristic
353	C₂×C₄	#14524021	1	1	yes	characteristic
354	C₂×C₄	#14524028	1	1	yes	characteristic
355	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
356	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
357	C₂×C₄	#14524049	1	1	yes	characteristic
358	C₂×C₄	#14524051	1	1	yes	characteristic
359	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
360	C₂×C₄	#14524063	1	1	yes	characteristic
361	C₂×C₄	#14524068	1	1	yes	characteristic
362	C₂×C₄	#14524069	1	1	yes	characteristic
363	C₈	—	2	2	no	—
364	C₈	—	4	1	no	—
365	C₈	—	8	1	no	—
366	C₈	—	268	2	no	—
367	C₈	—	268	2	no	—
368	C₈	—	268	1	no	—
369	C₈	—	134	2	no	—
370	C₂²	—	268	2	no	—
371	C₂²	—	268	2	no	—
372	C₂²	—	268	2	no	—
373	C₂²	—	536	1	no	—
374	C₂²	—	536	1	no	—
375	C₂²	—	536	1	no	—
376	C₂²	—	536	1	no	—
377	C₂²	#16863267	1	1	yes	characteristic
378	C₂²	#16863467	1	1	yes	characteristic
379	C₂²	#16863472	1	1	yes	characteristic
380	C₂²	C₄×#11761615	1	1	yes	characteristic
381	C₂²	#16863506	1	1	yes	characteristic
382	C₂²	#16863511	1	1	yes	characteristic
383	C₂²	#16863516	1	1	yes	characteristic
384	C₄	—	2	1	no	—
385	C₄	—	2	1	no	—
386	C₄	—	4	2	no	—
387	C₄	—	8	1	no	—
388	C₄	—	268	2	no	—
389	C₄	#16863418	1	1	yes	characteristic
390	C₄	#16863421	1	1	yes	characteristic
391	C₄	#16863466	1	1	yes	characteristic
392	C₄	#16863473	1	1	yes	characteristic
393	C₄	#16863539	1	1	yes	characteristic
394	C₄	#16863546	1	1	yes	characteristic
395	C₄	#16863683	1	1	yes	characteristic
396	C₄	#16863693	1	1	yes	characteristic
397	C₂	—	536	2	no	—
398	C₂	#22559052	1	1	yes	characteristic
399	C₂	#22559073	1	1	yes	characteristic
400	C₂	#22559099	1	1	yes	characteristic
401	C₂	#22559121	1	1	yes	characteristic
402	C₂	#22560390	1	1	yes	characteristic
403	C₂	#22560469	1	1	yes	characteristic
404	C₂	#22560483	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1939	C₂
socle	C₂³×C₆₇	C₄×D₄
prime radical	C₂²×D₂₆₈	C₂×C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1939	D₆₇
hypocenter	C₆₇	#8849
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1939

Complements

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

N	H ≅ G/N	type
3. C₂×#16863146	C₂	semidirect
5. #22558050	C₂	semidirect
6. C₆₇⋊#1273	C₂	semidirect
7. C₆₇×#1939	C₂	semidirect
90. C₂×C₆₇⋊C₈	M₁₆	semidirect
112. C₄⋊C₄×C₆₇	M₁₆	semidirect
204. #1939	D₆₇	semidirect
210. C₆₇	#8849	semidirect
313. C₄⋊C₄	#11761611	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863475	2
C₄×#14523963	2
C₂×#16863146	2
C₂×#16863123	2
#22558050	2
C₆₇⋊#1273	2
C₆₇×#1939	2
#8849	67