Subgroups — C₆₇×#21889 #30701140

Order 17152.

1118 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2283	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1988	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1981	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1892	C₂	1	4	yes	maximal
7	C₆₇×#1805	C₂	1	4	yes	maximal, complemented
8	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
9	#533×C₆₇	—	2	4	no	—
10	#533×C₆₇	C₄	1	8	yes	complemented
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#125×C₆₇	—	2	1	no	—
14	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
15	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
17	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
18	C₂×#117×C₆₇	—	2	1	no	—
19	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₆₇	—	2	4	no	—
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
23	C₂×#103×C₆₇	C₂²	1	8	yes	—
24	#383×C₆₇	—	2	1	no	—
25	#382×C₆₇	C₂²	1	2	yes	—
26	#381×C₆₇	—	2	1	no	—
27	#380×C₆₇	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
29	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
30	#376×C₆₇	—	2	1	no	—
31	#340×C₆₇	C₂²	1	8	yes	—
32	C₂²×Q₈×C₆₇	—	2	1	no	—
33	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
34	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	—	2	4	no	—
38	C₂²×C₈×C₆₇	—	4	4	no	—
39	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
40	C₄⋊Q₈×C₆₇	—	2	2	no	—
41	C₄⋊Q₈×C₆₇	—	2	2	no	—
42	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
43	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
44	#125×C₆₇	—	2	8	no	—
45	C₂²⋊Q₈×C₆₇	—	2	8	no	—
46	C₂²⋊Q₈×C₆₇	—	2	4	no	—
47	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
48	C₄×Q₈×C₆₇	—	2	8	no	—
49	C₄×Q₈×C₆₇	—	2	4	no	—
50	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	—
51	#117×C₆₇	—	2	8	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
58	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
59	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
61	C₂×C₄²×C₆₇	—	2	2	no	—
62	C₂×C₄²×C₆₇	—	2	2	no	—
63	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
64	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
65	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
66	C₂×C₄²×C₆₇	D₄	1	4	yes	—
67	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₆₇	—	2	4	no	—
70	C₄⋊C₈×C₆₇	—	2	4	no	—
71	#103×C₆₇	—	4	8	no	—
72	#95×C₆₇	—	2	2	no	—
73	#95×C₆₇	—	2	2	no	—
74	#95×C₆₇	—	2	2	no	—
75	#95×C₆₇	—	2	2	no	—
76	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂×Q₈×C₆₇	—	2	4	no	—
78	C₂×Q₈×C₆₇	—	2	2	no	—
79	C₂×Q₈×C₆₇	—	2	4	no	—
80	C₂×Q₈×C₆₇	—	2	4	no	—
81	C₂×Q₈×C₆₇	—	4	4	no	—
82	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
83	C₂²×C₄×C₆₇	—	2	4	no	—
84	C₂²×C₄×C₆₇	—	2	4	no	—
85	C₂²×C₄×C₆₇	—	2	2	no	—
86	C₂²×C₄×C₆₇	—	2	2	no	—
87	C₂²×C₄×C₆₇	—	2	2	no	—
88	C₂²×C₄×C₆₇	—	2	2	no	—
89	C₂²×C₄×C₆₇	—	2	4	no	—
90	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
91	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
92	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
100	C₂×C₈×C₆₇	—	2	4	no	—
101	C₂×C₈×C₆₇	—	2	8	no	—
102	C₂×C₈×C₆₇	—	4	8	no	—
103	C₄⋊C₄×C₆₇	—	2	4	no	—
104	C₄⋊C₄×C₆₇	—	2	4	no	—
105	C₄⋊C₄×C₆₇	—	2	4	no	—
106	C₄⋊C₄×C₆₇	—	2	8	no	—
107	C₄⋊C₄×C₆₇	—	2	8	no	—
108	C₄⋊C₄×C₆₇	—	2	2	no	—
109	C₄⋊C₄×C₆₇	—	2	4	no	—
110	C₄⋊C₄×C₆₇	—	2	4	no	—
111	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
112	C₂²⋊C₄×C₆₇	—	2	8	no	—
113	C₂²⋊C₄×C₆₇	—	4	4	no	—
114	C₄²×C₆₇	—	2	8	no	—
115	C₄²×C₆₇	—	4	4	no	—
116	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
117	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
118	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
119	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
120	C₄²×C₆₇	C₂×D₄	1	2	yes	—
121	C₄²×C₆₇	C₂×D₄	1	2	yes	—
122	C₂³×C₆₇	—	2	2	no	—
123	C₂³×C₆₇	—	2	2	no	—
124	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
125	C₂³×C₆₇	#137	1	4	yes	—
126	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
127	Q₈×C₆₇	—	2	8	no	—
128	Q₈×C₆₇	—	4	2	no	—
129	Q₈×C₆₇	—	4	2	no	—
130	Q₈×C₆₇	—	4	8	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	2	8	no	—
136	C₂×C₄×C₆₇	—	2	4	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	2	8	no	—
140	C₂×C₄×C₆₇	—	2	4	no	—
141	C₂×C₄×C₆₇	—	2	4	no	—
142	C₂×C₄×C₆₇	—	4	4	no	—
143	C₂×C₄×C₆₇	—	4	4	no	—
144	C₂×C₄×C₆₇	—	4	4	no	—
145	C₂×C₄×C₆₇	—	4	4	no	—
146	C₂×C₄×C₆₇	—	4	4	no	—
147	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
148	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
149	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
150	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
151	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
152	C₂×C₄×C₆₇	#135	1	4	yes	—
153	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
156	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
157	C₈×C₆₇	—	4	4	no	—
158	C₂²×C₆₇	—	2	2	no	—
159	C₂²×C₆₇	—	2	2	no	—
160	C₂²×C₆₇	—	2	8	no	—
161	C₂²×C₆₇	#419	1	4	yes	—
162	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
163	C₂²×C₆₇	#537	1	2	yes	—
164	C₂²×C₆₇	C₂×#135	1	2	yes	—
165	C₂²×C₆₇	C₂×#137	1	2	yes	—
166	C₄×C₆₇	—	2	2	no	—
167	C₄×C₆₇	—	2	2	no	—
168	C₄×C₆₇	—	2	2	no	—
169	C₄×C₆₇	—	2	2	no	—
170	C₄×C₆₇	—	4	4	no	—
171	C₄×C₆₇	—	4	4	no	—
172	C₄×C₆₇	—	8	4	no	—
173	C₄×C₆₇	#416	1	4	yes	—
174	C₄×C₆₇	#520	1	2	yes	—
175	C₄×C₆₇	#535	1	1	yes	characteristic
176	C₄×C₆₇	#536	1	1	yes	characteristic
177	#21889	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂×C₆₇	—	2	4	no	—
179	C₂×C₆₇	#2311	1	1	yes	characteristic
180	C₂×C₆₇	C₂×#416	1	1	yes	characteristic
181	C₂×C₆₇	C₂×#419	1	1	yes	characteristic
182	C₂×C₆₇	#3248	1	4	yes	—
183	C₂×#533	C₂×C₆₇	1	1	yes	characteristic
184	C₂²×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic
185	#2283	C₂×C₆₇	1	1	yes	characteristic
186	#1988	C₂×C₆₇	1	2	yes	complemented
187	#1981	C₂×C₆₇	1	2	yes	complemented
188	#1892	C₂×C₆₇	1	4	yes	—
189	#1805	C₂×C₆₇	1	4	yes	complemented
190	C₆₇	#21889	1	1	yes	characteristic, direct product factor, Sylow
191	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
192	#533	—	2	4	no	—
193	#533	C₄×C₆₇	1	8	yes	complemented
194	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
195	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
196	C₂×#125	—	2	1	no	—
197	C₂×C₂²⋊Q₈	—	2	1	no	—
198	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
199	C₂×C₄×Q₈	—	2	1	no	—
200	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
201	C₂×#117	—	2	1	no	—
202	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
203	C₂×C₄⋊C₈	—	2	4	no	—
204	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
205	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
206	C₂×#103	C₂²×C₆₇	1	8	yes	—
207	#383	—	2	1	no	—
208	#382	C₂²×C₆₇	1	2	yes	—
209	#381	—	2	1	no	—
210	#380	C₂²×C₆₇	1	2	yes	—
211	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
212	C₄×C₂²⋊C₄	—	2	1	no	—
213	#376	—	2	1	no	—
214	#340	C₂²×C₆₇	1	8	yes	—
215	C₂²×Q₈	—	2	1	no	—
216	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
217	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
218	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
219	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
220	C₂²×C₈	—	2	4	no	—
221	C₂²×C₈	—	4	4	no	—
222	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
223	C₄⋊Q₈	—	2	2	no	—
224	C₄⋊Q₈	—	2	2	no	—
225	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
226	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
227	#125	—	2	8	no	—
228	C₂²⋊Q₈	—	2	8	no	—
229	C₂²⋊Q₈	—	2	4	no	—
230	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
231	C₄×Q₈	—	2	8	no	—
232	C₄×Q₈	—	2	4	no	—
233	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	—
234	#117	—	2	8	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄⋊C₄	—	2	2	no	—
239	C₂×C₄⋊C₄	—	2	2	no	—
240	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
241	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
242	C₂×C₂²⋊C₄	—	2	2	no	—
243	C₂×C₂²⋊C₄	—	2	2	no	—
244	C₂×C₄²	—	2	2	no	—
245	C₂×C₄²	—	2	2	no	—
246	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
247	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
248	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
249	C₂×C₄²	D₄×C₆₇	1	4	yes	—
250	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
251	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
252	C₄⋊C₈	—	2	4	no	—
253	C₄⋊C₈	—	2	4	no	—
254	#103	—	4	8	no	—
255	#95	—	2	2	no	—
256	#95	—	2	2	no	—
257	#95	—	2	2	no	—
258	#95	—	2	2	no	—
259	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
260	C₂×Q₈	—	2	4	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	4	no	—
263	C₂×Q₈	—	2	4	no	—
264	C₂×Q₈	—	4	4	no	—
265	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
266	C₂²×C₄	—	2	4	no	—
267	C₂²×C₄	—	2	4	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	2	4	no	—
273	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
274	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
275	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
276	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
277	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
278	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
279	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
280	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
281	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
282	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	—	2	8	no	—
285	C₂×C₈	—	4	8	no	—
286	C₄⋊C₄	—	2	4	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	2	4	no	—
289	C₄⋊C₄	—	2	8	no	—
290	C₄⋊C₄	—	2	8	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	4	no	—
293	C₄⋊C₄	—	2	4	no	—
294	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
295	C₂²⋊C₄	—	2	8	no	—
296	C₂²⋊C₄	—	4	4	no	—
297	C₄²	—	2	8	no	—
298	C₄²	—	4	4	no	—
299	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
300	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
301	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
302	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
303	C₄²	C₂×D₄×C₆₇	1	2	yes	—
304	C₄²	C₂×D₄×C₆₇	1	2	yes	—
305	C₂³	—	2	2	no	—
306	C₂³	—	2	2	no	—
307	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
308	C₂³	#137×C₆₇	1	4	yes	—
309	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
310	Q₈	—	2	8	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	4	8	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	8	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	8	no	—
323	C₂×C₄	—	2	4	no	—
324	C₂×C₄	—	2	4	no	—
325	C₂×C₄	—	4	4	no	—
326	C₂×C₄	—	4	4	no	—
327	C₂×C₄	—	4	4	no	—
328	C₂×C₄	—	4	4	no	—
329	C₂×C₄	—	4	4	no	—
330	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
331	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
332	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
333	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
334	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
335	C₂×C₄	#135×C₆₇	1	4	yes	—
336	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
337	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
338	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
339	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
340	C₈	—	4	4	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	8	no	—
344	C₂²	#419×C₆₇	1	4	yes	—
345	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
346	C₂²	#537×C₆₇	1	2	yes	—
347	C₂²	C₂×#135×C₆₇	1	2	yes	—
348	C₂²	C₂×#137×C₆₇	1	2	yes	—
349	C₄	—	2	2	no	—
350	C₄	—	2	2	no	—
351	C₄	—	2	2	no	—
352	C₄	—	2	2	no	—
353	C₄	—	4	4	no	—
354	C₄	—	4	4	no	—
355	C₄	—	8	4	no	—
356	C₄	#416×C₆₇	1	4	yes	—
357	C₄	#520×C₆₇	1	2	yes	—
358	C₄	#535×C₆₇	1	1	yes	characteristic
359	C₄	#536×C₆₇	1	1	yes	characteristic
360	C₂	—	2	4	no	—
361	C₂	C₆₇×#2311	1	1	yes	characteristic
362	C₂	C₂×#416×C₆₇	1	1	yes	characteristic
363	C₂	C₂×#419×C₆₇	1	1	yes	characteristic
364	C₂	C₆₇×#3248	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1988	C₂	semidirect
5. C₆₇×#1981	C₂	semidirect
7. C₆₇×#1805	C₂	semidirect
10. #533×C₆₇	C₄	semidirect
177. #21889	C₆₇	direct (N × H)
186. #1988	C₂×C₆₇	semidirect
187. #1981	C₂×C₆₇	semidirect
189. #1805	C₂×C₆₇	semidirect
190. C₆₇	#21889	direct (N × H)
193. #533	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#533×C₆₇	2
C₂²×C₄⋊C₈×C₆₇	2
C₆₇×#2283	2
C₆₇×#1988	2
C₆₇×#1981	2
C₆₇×#1892	2
C₆₇×#1805	2
#21889	67

Subgroups — C67×#21889 #30701140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21889 #30701140