Subgroups — C₆₇×#9666 #30688917

Order 17152.

534 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1984	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#1954	C₂	1	1	yes	maximal, characteristic
4	C₂×#335×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₈⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₆₇	C₄	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
11	C₄×SD₁₆×C₆₇	—	2	2	no	—
12	C₄×SD₁₆×C₆₇	—	2	2	no	—
13	C₂×#106×C₆₇	C₄	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#103×C₆₇	C₄	1	1	yes	characteristic
17	C₂×#102×C₆₇	C₄	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	#336×C₆₇	—	2	2	no	—
23	#335×C₆₇	—	2	2	no	—
24	Q₈⋊C₈×C₆₇	—	2	2	no	—
25	D₄⋊C₈×C₆₇	—	2	2	no	—
26	#322×C₆₇	—	2	2	no	—
27	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₆₇	—	2	1	no	—
30	C₂×SD₁₆×C₆₇	—	2	2	no	—
31	C₂×SD₁₆×C₆₇	—	2	2	no	—
32	C₂×SD₁₆×C₆₇	—	2	2	no	—
33	C₂²×C₈×C₆₇	—	2	1	no	—
34	C₂²×C₈×C₆₇	—	2	1	no	—
35	C₂²×C₈×C₆₇	—	2	1	no	—
36	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	D₄	1	2	yes	—
39	C₄×Q₈×C₆₇	—	2	1	no	—
40	C₄×Q₈×C₆₇	—	2	1	no	—
41	C₄×Q₈×C₆₇	—	2	1	no	—
42	C₄×Q₈×C₆₇	—	2	1	no	—
43	C₄×D₄×C₆₇	—	2	1	no	—
44	C₄×D₄×C₆₇	—	2	1	no	—
45	C₄×D₄×C₆₇	—	2	1	no	—
46	C₄×D₄×C₆₇	—	2	1	no	—
47	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
48	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
51	C₂×C₄²×C₆₇	—	2	1	no	—
52	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
53	#106×C₆₇	—	2	2	no	—
54	C₄⋊C₈×C₆₇	—	2	1	no	—
55	C₄⋊C₈×C₆₇	—	2	1	no	—
56	C₄⋊C₈×C₆₇	—	2	1	no	—
57	C₄⋊C₈×C₆₇	—	2	1	no	—
58	#103×C₆₇	—	2	2	no	—
59	#102×C₆₇	—	2	2	no	—
60	C₂²⋊C₈×C₆₇	—	4	4	no	—
61	C₄×C₈×C₆₇	—	2	1	no	—
62	C₄×C₈×C₆₇	—	2	1	no	—
63	C₄×C₈×C₆₇	—	2	1	no	—
64	C₄×C₈×C₆₇	D₄	1	2	yes	—
65	C₄×C₈×C₆₇	D₄	1	2	yes	—
66	C₄×C₈×C₆₇	D₄	1	2	yes	—
67	C₂⁴×C₆₇	—	2	1	no	—
68	C₂×Q₈×C₆₇	—	2	1	no	—
69	C₂×Q₈×C₆₇	—	2	1	no	—
70	C₂×Q₈×C₆₇	—	2	1	no	—
71	C₂×Q₈×C₆₇	—	2	1	no	—
72	C₂×Q₈×C₆₇	—	2	1	no	—
73	C₂×Q₈×C₆₇	M₁₆	1	2	yes	—
74	C₂×D₄×C₆₇	—	2	1	no	—
75	C₂×D₄×C₆₇	—	2	1	no	—
76	C₂×D₄×C₆₇	—	2	1	no	—
77	C₂×D₄×C₆₇	—	2	1	no	—
78	C₂×D₄×C₆₇	—	2	1	no	—
79	C₂×D₄×C₆₇	M₁₆	1	2	yes	—
80	C₂²×C₄×C₆₇	—	2	1	no	—
81	C₂²×C₄×C₆₇	—	2	1	no	—
82	C₂²×C₄×C₆₇	—	2	1	no	—
83	C₂²×C₄×C₆₇	—	4	1	no	—
84	C₂²×C₄×C₆₇	—	4	1	no	—
85	C₂²×C₄×C₆₇	—	4	2	no	—
86	C₂²×C₄×C₆₇	—	4	2	no	—
87	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
90	SD₁₆×C₆₇	—	2	8	no	—
91	C₂×C₈×C₆₇	—	2	4	no	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	—	2	1	no	—
94	C₂×C₈×C₆₇	—	2	1	no	—
95	C₂×C₈×C₆₇	—	2	1	no	—
96	C₂×C₈×C₆₇	—	2	1	no	—
97	C₂×C₈×C₆₇	—	2	2	no	—
98	C₂×C₈×C₆₇	—	2	2	no	—
99	C₂×C₈×C₆₇	—	2	2	no	—
100	C₂×C₈×C₆₇	—	4	1	no	—
101	C₂×C₈×C₆₇	—	4	1	no	—
102	C₂×C₈×C₆₇	—	4	2	no	—
103	C₂×C₈×C₆₇	—	4	2	no	—
104	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
105	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
106	C₄⋊C₄×C₆₇	—	2	1	no	—
107	C₄⋊C₄×C₆₇	—	2	1	no	—
108	C₄⋊C₄×C₆₇	—	2	1	no	—
109	C₄⋊C₄×C₆₇	—	2	1	no	—
110	C₄⋊C₄×C₆₇	—	4	1	no	—
111	C₄⋊C₄×C₆₇	—	4	1	no	—
112	C₂²⋊C₄×C₆₇	—	4	1	no	—
113	C₂²⋊C₄×C₆₇	—	4	1	no	—
114	C₄²×C₆₇	—	4	1	no	—
115	C₄²×C₆₇	—	4	1	no	—
116	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₆₇	#41	1	1	yes	characteristic
120	C₂³×C₆₇	—	2	2	no	—
121	C₂³×C₆₇	—	4	1	no	—
122	C₂³×C₆₇	—	4	1	no	—
123	C₂³×C₆₇	—	4	2	no	—
124	C₂³×C₆₇	—	4	2	no	—
125	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
126	Q₈×C₆₇	—	2	2	no	—
127	Q₈×C₆₇	—	2	2	no	—
128	Q₈×C₆₇	—	4	1	no	—
129	Q₈×C₆₇	—	4	1	no	—
130	D₄×C₆₇	—	2	2	no	—
131	D₄×C₆₇	—	2	2	no	—
132	D₄×C₆₇	—	4	1	no	—
133	D₄×C₆₇	—	4	1	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	4	1	no	—
137	C₂×C₄×C₆₇	—	4	1	no	—
138	C₂×C₄×C₆₇	—	4	1	no	—
139	C₂×C₄×C₆₇	—	4	1	no	—
140	C₂×C₄×C₆₇	—	4	2	no	—
141	C₂×C₄×C₆₇	—	4	2	no	—
142	C₂×C₄×C₆₇	—	8	2	no	—
143	C₂×C₄×C₆₇	—	8	1	no	—
144	C₂×C₄×C₆₇	—	8	1	no	—
145	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
148	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
149	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
150	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
153	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
154	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
155	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
156	C₂×C₄×C₆₇	#137	1	2	yes	—
157	C₈×C₆₇	—	2	4	no	—
158	C₈×C₆₇	—	2	2	no	—
159	C₈×C₆₇	—	2	2	no	—
160	C₈×C₆₇	—	4	2	no	—
161	C₈×C₆₇	—	8	1	no	—
162	C₈×C₆₇	—	8	1	no	—
163	C₂²×C₆₇	—	4	4	no	—
164	C₂²×C₆₇	—	4	2	no	—
165	C₂²×C₆₇	—	8	2	no	—
166	C₂²×C₆₇	—	8	1	no	—
167	C₂²×C₆₇	—	8	1	no	—
168	C₂²×C₆₇	#390	1	1	yes	characteristic
169	C₂²×C₆₇	#435	1	1	yes	characteristic
170	C₂²×C₆₇	#435	1	1	yes	characteristic
171	C₂²×C₆₇	#440	1	2	yes	—
172	C₂²×C₆₇	#443	1	2	yes	—
173	C₄×C₆₇	—	2	1	no	—
174	C₄×C₆₇	—	2	1	no	—
175	C₄×C₆₇	—	4	2	no	—
176	C₄×C₆₇	—	8	1	no	—
177	C₄×C₆₇	—	8	1	no	—
178	C₄×C₆₇	#407	1	1	yes	characteristic
179	C₄×C₆₇	#408	1	1	yes	characteristic
180	C₄×C₆₇	#435	1	1	yes	characteristic
181	C₄×C₆₇	#435	1	1	yes	characteristic
182	C₄×C₆₇	#468	1	2	yes	—
183	C₄×C₆₇	#496	1	1	yes	characteristic
184	C₄×C₆₇	#497	1	1	yes	characteristic
185	#9666	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
186	C₂×C₆₇	—	8	2	no	—
187	C₂×C₆₇	#1585	1	2	yes	—
188	C₂×C₆₇	#1588	1	2	yes	—
189	C₂×C₆₇	#1955	1	1	yes	characteristic
190	C₂×C₆₇	#1973	1	1	yes	characteristic
191	C₂×C₆₇	#1974	1	1	yes	characteristic
192	C₂×C₄×SD₁₆	C₂×C₆₇	1	1	yes	characteristic
193	#1984	C₂×C₆₇	1	1	yes	characteristic, complemented
194	#1954	C₂×C₆₇	1	1	yes	characteristic
195	C₂×#335	C₂×C₆₇	1	1	yes	characteristic, complemented
196	C₂×Q₈⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
197	C₂×D₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic
198	C₂×#322	C₂×C₆₇	1	1	yes	characteristic, complemented
199	C₆₇	#9666	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²×SD₁₆	C₄×C₆₇	1	1	yes	characteristic
201	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
202	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
203	C₄×SD₁₆	—	2	2	no	—
204	C₄×SD₁₆	—	2	2	no	—
205	C₂×#106	C₄×C₆₇	1	1	yes	characteristic
206	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
207	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
208	C₂×#103	C₄×C₆₇	1	1	yes	characteristic
209	C₂×#102	C₄×C₆₇	1	1	yes	characteristic
210	C₂×C₂²⋊C₈	—	2	2	no	—
211	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
212	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
213	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
214	#336	—	2	2	no	—
215	#335	—	2	2	no	—
216	Q₈⋊C₈	—	2	2	no	—
217	D₄⋊C₈	—	2	2	no	—
218	#322	—	2	2	no	—
219	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
220	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
221	C₂³×C₄	—	2	1	no	—
222	C₂×SD₁₆	—	2	2	no	—
223	C₂×SD₁₆	—	2	2	no	—
224	C₂×SD₁₆	—	2	2	no	—
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	2	1	no	—
227	C₂²×C₈	—	2	1	no	—
228	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
229	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
230	C₂²×C₈	D₄×C₆₇	1	2	yes	—
231	C₄×Q₈	—	2	1	no	—
232	C₄×Q₈	—	2	1	no	—
233	C₄×Q₈	—	2	1	no	—
234	C₄×Q₈	—	2	1	no	—
235	C₄×D₄	—	2	1	no	—
236	C₄×D₄	—	2	1	no	—
237	C₄×D₄	—	2	1	no	—
238	C₄×D₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	2	1	no	—
240	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
241	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
242	C₂×C₂²⋊C₄	—	2	1	no	—
243	C₂×C₄²	—	2	1	no	—
244	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
245	#106	—	2	2	no	—
246	C₄⋊C₈	—	2	1	no	—
247	C₄⋊C₈	—	2	1	no	—
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	2	1	no	—
250	#103	—	2	2	no	—
251	#102	—	2	2	no	—
252	C₂²⋊C₈	—	4	4	no	—
253	C₄×C₈	—	2	1	no	—
254	C₄×C₈	—	2	1	no	—
255	C₄×C₈	—	2	1	no	—
256	C₄×C₈	D₄×C₆₇	1	2	yes	—
257	C₄×C₈	D₄×C₆₇	1	2	yes	—
258	C₄×C₈	D₄×C₆₇	1	2	yes	—
259	C₂⁴	—	2	1	no	—
260	C₂×Q₈	—	2	1	no	—
261	C₂×Q₈	—	2	1	no	—
262	C₂×Q₈	—	2	1	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	M₁₆×C₆₇	1	2	yes	—
266	C₂×D₄	—	2	1	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	2	1	no	—
269	C₂×D₄	—	2	1	no	—
270	C₂×D₄	—	2	1	no	—
271	C₂×D₄	M₁₆×C₆₇	1	2	yes	—
272	C₂²×C₄	—	2	1	no	—
273	C₂²×C₄	—	2	1	no	—
274	C₂²×C₄	—	2	1	no	—
275	C₂²×C₄	—	4	1	no	—
276	C₂²×C₄	—	4	1	no	—
277	C₂²×C₄	—	4	2	no	—
278	C₂²×C₄	—	4	2	no	—
279	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
280	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
281	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
282	SD₁₆	—	2	8	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	4	1	no	—
293	C₂×C₈	—	4	1	no	—
294	C₂×C₈	—	4	2	no	—
295	C₂×C₈	—	4	2	no	—
296	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
297	C₂×C₈	C₂²⋊C₄×C₆₇	1	2	yes	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	1	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₂²⋊C₄	—	4	1	no	—
305	C₂²⋊C₄	—	4	1	no	—
306	C₄²	—	4	1	no	—
307	C₄²	—	4	1	no	—
308	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
309	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
310	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
311	C₄²	#41×C₆₇	1	1	yes	characteristic
312	C₂³	—	2	2	no	—
313	C₂³	—	4	1	no	—
314	C₂³	—	4	1	no	—
315	C₂³	—	4	2	no	—
316	C₂³	—	4	2	no	—
317	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
318	Q₈	—	2	2	no	—
319	Q₈	—	2	2	no	—
320	Q₈	—	4	1	no	—
321	Q₈	—	4	1	no	—
322	D₄	—	2	2	no	—
323	D₄	—	2	2	no	—
324	D₄	—	4	1	no	—
325	D₄	—	4	1	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	8	2	no	—
335	C₂×C₄	—	8	1	no	—
336	C₂×C₄	—	8	1	no	—
337	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
338	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
339	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
340	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
341	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
342	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
343	C₂×C₄	C₂×M₁₆×C₆₇	1	1	yes	characteristic
344	C₂×C₄	C₂×M₁₆×C₆₇	1	1	yes	characteristic
345	C₂×C₄	#131×C₆₇	1	1	yes	characteristic
346	C₂×C₄	#131×C₆₇	1	1	yes	characteristic
347	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
348	C₂×C₄	#137×C₆₇	1	2	yes	—
349	C₈	—	2	4	no	—
350	C₈	—	2	2	no	—
351	C₈	—	2	2	no	—
352	C₈	—	4	2	no	—
353	C₈	—	8	1	no	—
354	C₈	—	8	1	no	—
355	C₂²	—	4	4	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	8	2	no	—
358	C₂²	—	8	1	no	—
359	C₂²	—	8	1	no	—
360	C₂²	#390×C₆₇	1	1	yes	characteristic
361	C₂²	#435×C₆₇	1	1	yes	characteristic
362	C₂²	#435×C₆₇	1	1	yes	characteristic
363	C₂²	#440×C₆₇	1	2	yes	—
364	C₂²	#443×C₆₇	1	2	yes	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	4	2	no	—
368	C₄	—	8	1	no	—
369	C₄	—	8	1	no	—
370	C₄	#407×C₆₇	1	1	yes	characteristic
371	C₄	#408×C₆₇	1	1	yes	characteristic
372	C₄	#435×C₆₇	1	1	yes	characteristic
373	C₄	#435×C₆₇	1	1	yes	characteristic
374	C₄	#468×C₆₇	1	2	yes	—
375	C₄	#496×C₆₇	1	1	yes	characteristic
376	C₄	#497×C₆₇	1	1	yes	characteristic
377	C₂	—	8	2	no	—
378	C₂	C₆₇×#1585	1	2	yes	—
379	C₂	C₆₇×#1588	1	2	yes	—
380	C₂	C₆₇×#1955	1	1	yes	characteristic
381	C₂	C₆₇×#1973	1	1	yes	characteristic
382	C₂	C₆₇×#1974	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
socle	C₂³×C₆₇	C₄×D₄
prime radical	C₂²×D₄×C₆₇	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#1984	C₂	semidirect
4. C₂×#335×C₆₇	C₂	semidirect
5. C₂×Q₈⋊C₈×C₆₇	C₂	semidirect
7. C₂×#322×C₆₇	C₂	semidirect
185. #9666	C₆₇	direct (N × H)
193. #1984	C₂×C₆₇	semidirect
195. C₂×#335	C₂×C₆₇	semidirect
196. C₂×Q₈⋊C₈	C₂×C₆₇	semidirect
198. C₂×#322	C₂×C₆₇	semidirect
199. C₆₇	#9666	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×SD₁₆×C₆₇	2
C₆₇×#1984	2
C₆₇×#1954	2
C₂×#335×C₆₇	2
C₂×Q₈⋊C₈×C₆₇	2
C₂×D₄⋊C₈×C₆₇	2
C₂×#322×C₆₇	2
#9666	67

Subgroups — C67×#9666 #30688917

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#9666 #30688917