Subgroups — C₆₇×#21932 #30701183

Order 17152.

1258 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#407×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1851	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1850	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1797	C₂	1	4	yes	maximal, complemented
7	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
9	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
10	C₂²×C₂²⋊C₄×C₆₇	C₄	1	2	yes	—
11	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₆₇	C₂²	1	8	yes	complemented
13	#407×C₆₇	—	2	4	no	—
14	C₂×C₂²⋊C₈×C₆₇	—	2	4	no	—
15	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
19	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
20	C₄×C₂²⋊C₄×C₆₇	C₄	1	2	yes	—
21	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
22	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
24	C₂×#95×C₆₇	C₄	1	2	yes	—
25	#336×C₆₇	C₂²	1	8	yes	complemented
26	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₆₇	—	4	4	no	—
28	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
29	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₆₇	D₄	1	4	yes	—
33	C₂³×C₄×C₆₇	Q₈	1	4	yes	—
34	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₆₇	—	2	4	no	—
38	C₂×M₁₆×C₆₇	—	4	4	no	—
39	C₂²×C₈×C₆₇	—	2	4	no	—
40	C₂²×C₈×C₆₇	—	4	4	no	—
41	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
42	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
43	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
44	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
46	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₆₇	—	4	4	no	—
48	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
51	C₄⋊C₈×C₆₇	—	4	8	no	—
52	C₂²⋊C₈×C₆₇	—	2	4	no	—
53	C₂²⋊C₈×C₆₇	—	2	4	no	—
54	C₂²⋊C₈×C₆₇	—	2	4	no	—
55	C₂²⋊C₈×C₆₇	—	2	4	no	—
56	#95×C₆₇	C₂×C₄	1	2	yes	—
57	#95×C₆₇	C₂×C₄	1	4	yes	—
58	#95×C₆₇	C₂×C₄	1	2	yes	—
59	C₂⁴×C₆₇	—	2	4	no	—
60	C₂⁴×C₆₇	—	2	4	no	—
61	C₂⁴×C₆₇	—	4	2	no	—
62	C₂⁴×C₆₇	C₄⋊C₄	1	4	yes	—
63	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	—	2	4	no	—
68	C₂²×C₄×C₆₇	—	2	4	no	—
69	C₂²×C₄×C₆₇	—	2	8	no	—
70	C₂²×C₄×C₆₇	—	2	8	no	—
71	C₂²×C₄×C₆₇	—	4	8	no	—
72	C₂²×C₄×C₆₇	—	4	8	no	—
73	C₂²×C₄×C₆₇	—	4	4	no	—
74	C₂²×C₄×C₆₇	—	4	2	no	—
75	C₂²×C₄×C₆₇	—	4	2	no	—
76	C₂²×C₄×C₆₇	—	4	2	no	—
77	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
78	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
79	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
80	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
85	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
89	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
90	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
91	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
92	M₁₆×C₆₇	—	4	8	no	—
93	C₂×C₈×C₆₇	—	2	4	no	—
94	C₂×C₈×C₆₇	—	2	4	no	—
95	C₂×C₈×C₆₇	—	2	8	no	—
96	C₂×C₈×C₆₇	—	2	8	no	—
97	C₂×C₈×C₆₇	—	4	8	no	—
98	C₂²⋊C₄×C₆₇	—	2	4	no	—
99	C₂²⋊C₄×C₆₇	—	2	8	no	—
100	C₂²⋊C₄×C₆₇	M₁₆	1	8	yes	complemented
101	C₄²×C₆₇	—	2	8	no	—
102	C₄²×C₆₇	—	4	2	no	—
103	C₄²×C₆₇	—	4	2	no	—
104	C₂³×C₆₇	—	2	2	no	—
105	C₂³×C₆₇	—	2	2	no	—
106	C₂³×C₆₇	—	2	2	no	—
107	C₂³×C₆₇	—	2	2	no	—
108	C₂³×C₆₇	—	2	2	no	—
109	C₂³×C₆₇	—	2	2	no	—
110	C₂³×C₆₇	—	2	2	no	—
111	C₂³×C₆₇	—	2	8	no	—
112	C₂³×C₆₇	—	2	8	no	—
113	C₂³×C₆₇	—	4	4	no	—
114	C₂³×C₆₇	—	4	16	no	—
115	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
116	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂³×C₆₇	C₂×M₁₆	1	4	yes	—
119	C₂³×C₆₇	#131	1	4	yes	—
120	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	—	2	4	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	4	no	—
126	C₂×C₄×C₆₇	—	2	4	no	—
127	C₂×C₄×C₆₇	—	2	4	no	—
128	C₂×C₄×C₆₇	—	2	4	no	—
129	C₂×C₄×C₆₇	—	2	4	no	—
130	C₂×C₄×C₆₇	—	2	4	no	—
131	C₂×C₄×C₆₇	—	2	4	no	—
132	C₂×C₄×C₆₇	—	2	4	no	—
133	C₂×C₄×C₆₇	—	2	4	no	—
134	C₂×C₄×C₆₇	—	2	4	no	—
135	C₂×C₄×C₆₇	—	4	4	no	—
136	C₂×C₄×C₆₇	—	4	4	no	—
137	C₂×C₄×C₆₇	—	4	4	no	—
138	C₂×C₄×C₆₇	—	4	16	no	—
139	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×C₄×C₆₇	C₂×M₁₆	1	8	yes	—
143	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
145	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
146	C₈×C₆₇	—	4	4	no	—
147	C₈×C₆₇	—	4	4	no	—
148	C₂²×C₆₇	—	2	2	no	—
149	C₂²×C₆₇	—	2	2	no	—
150	C₂²×C₆₇	—	2	2	no	—
151	C₂²×C₆₇	—	2	2	no	—
152	C₂²×C₆₇	—	2	8	no	—
153	C₂²×C₆₇	—	2	8	no	—
154	C₂²×C₆₇	—	2	2	no	—
155	C₂²×C₆₇	—	2	2	no	—
156	C₂²×C₆₇	—	2	2	no	—
157	C₂²×C₆₇	—	4	16	no	—
158	C₂²×C₆₇	—	4	4	no	—
159	C₂²×C₆₇	#423	1	4	yes	—
160	C₂²×C₆₇	#424	1	4	yes	—
161	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
162	C₂²×C₆₇	#518	1	2	yes	—
163	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
164	C₂²×C₆₇	C₂×#131	1	2	yes	—
165	C₄×C₆₇	—	2	2	no	—
166	C₄×C₆₇	—	2	2	no	—
167	C₄×C₆₇	—	2	2	no	—
168	C₄×C₆₇	—	2	8	no	—
169	C₄×C₆₇	—	4	4	no	—
170	C₄×C₆₇	#534	1	2	yes	—
171	C₄×C₆₇	#543	1	2	yes	—
172	#21932	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂×C₆₇	—	2	4	no	—
174	C₂×C₆₇	—	2	4	no	—
175	C₂×C₆₇	—	4	2	no	—
176	C₂×C₆₇	#2315	1	1	yes	characteristic
177	C₂×C₆₇	C₂×#423	1	1	yes	characteristic
178	C₂×C₆₇	C₂×#424	1	1	yes	characteristic
179	C₂×C₆₇	#2974	1	4	yes	—
180	C₂×#407	C₂×C₆₇	1	1	yes	characteristic
181	C₂²×C₂²⋊C₈	C₂×C₆₇	1	1	yes	characteristic
182	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic
183	#1851	C₂×C₆₇	1	4	yes	complemented
184	#1850	C₂×C₆₇	1	4	yes	complemented
185	#1797	C₂×C₆₇	1	4	yes	complemented
186	C₆₇	#21932	1	1	yes	characteristic, direct product factor, Sylow
187	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
188	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
189	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
190	C₂²×C₂²⋊C₄	C₄×C₆₇	1	2	yes	—
191	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
192	C₂×C₄⋊C₈	C₂²×C₆₇	1	8	yes	complemented
193	#407	—	2	4	no	—
194	C₂×C₂²⋊C₈	—	2	4	no	—
195	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
196	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
197	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
198	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
199	C₄×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
200	C₄×C₂²⋊C₄	C₄×C₆₇	1	2	yes	—
201	C₄×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
202	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
203	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
204	C₂×#95	C₄×C₆₇	1	2	yes	—
205	#336	C₂²×C₆₇	1	8	yes	complemented
206	C₂⁵	C₂×C₄×C₆₇	1	1	yes	characteristic
207	C₂³×C₄	—	4	4	no	—
208	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
209	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
210	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
211	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
212	C₂³×C₄	D₄×C₆₇	1	4	yes	—
213	C₂³×C₄	Q₈×C₆₇	1	4	yes	—
214	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
215	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
216	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
217	C₂×M₁₆	—	2	4	no	—
218	C₂×M₁₆	—	4	4	no	—
219	C₂²×C₈	—	2	4	no	—
220	C₂²×C₈	—	4	4	no	—
221	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
222	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
223	C₂×C₂²⋊C₄	—	2	8	no	—
224	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
225	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
226	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
227	C₂×C₄²	—	4	4	no	—
228	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
229	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
230	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
231	C₄⋊C₈	—	4	8	no	—
232	C₂²⋊C₈	—	2	4	no	—
233	C₂²⋊C₈	—	2	4	no	—
234	C₂²⋊C₈	—	2	4	no	—
235	C₂²⋊C₈	—	2	4	no	—
236	#95	C₂×C₄×C₆₇	1	2	yes	—
237	#95	C₂×C₄×C₆₇	1	4	yes	—
238	#95	C₂×C₄×C₆₇	1	2	yes	—
239	C₂⁴	—	2	4	no	—
240	C₂⁴	—	2	4	no	—
241	C₂⁴	—	4	2	no	—
242	C₂⁴	C₄⋊C₄×C₆₇	1	4	yes	—
243	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
244	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
245	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	2	8	no	—
250	C₂²×C₄	—	2	8	no	—
251	C₂²×C₄	—	4	8	no	—
252	C₂²×C₄	—	4	8	no	—
253	C₂²×C₄	—	4	4	no	—
254	C₂²×C₄	—	4	2	no	—
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
258	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
259	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
260	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
261	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
262	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
263	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
264	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
265	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
266	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
267	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
268	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
269	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
270	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
271	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
272	M₁₆	—	4	8	no	—
273	C₂×C₈	—	2	4	no	—
274	C₂×C₈	—	2	4	no	—
275	C₂×C₈	—	2	8	no	—
276	C₂×C₈	—	2	8	no	—
277	C₂×C₈	—	4	8	no	—
278	C₂²⋊C₄	—	2	4	no	—
279	C₂²⋊C₄	—	2	8	no	—
280	C₂²⋊C₄	M₁₆×C₆₇	1	8	yes	complemented
281	C₄²	—	2	8	no	—
282	C₄²	—	4	2	no	—
283	C₄²	—	4	2	no	—
284	C₂³	—	2	2	no	—
285	C₂³	—	2	2	no	—
286	C₂³	—	2	2	no	—
287	C₂³	—	2	2	no	—
288	C₂³	—	2	2	no	—
289	C₂³	—	2	2	no	—
290	C₂³	—	2	2	no	—
291	C₂³	—	2	8	no	—
292	C₂³	—	2	8	no	—
293	C₂³	—	4	4	no	—
294	C₂³	—	4	16	no	—
295	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
296	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
297	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
298	C₂³	C₂×M₁₆×C₆₇	1	4	yes	—
299	C₂³	#131×C₆₇	1	4	yes	—
300	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	4	4	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	—	4	4	no	—
318	C₂×C₄	—	4	16	no	—
319	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
320	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
321	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
322	C₂×C₄	C₂×M₁₆×C₆₇	1	8	yes	—
323	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
324	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
325	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
326	C₈	—	4	4	no	—
327	C₈	—	4	4	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	8	no	—
333	C₂²	—	2	8	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	4	16	no	—
338	C₂²	—	4	4	no	—
339	C₂²	#423×C₆₇	1	4	yes	—
340	C₂²	#424×C₆₇	1	4	yes	—
341	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
342	C₂²	#518×C₆₇	1	2	yes	—
343	C₂²	C₂²×M₁₆×C₆₇	1	2	yes	—
344	C₂²	C₂×#131×C₆₇	1	2	yes	—
345	C₄	—	2	2	no	—
346	C₄	—	2	2	no	—
347	C₄	—	2	2	no	—
348	C₄	—	2	8	no	—
349	C₄	—	4	4	no	—
350	C₄	#534×C₆₇	1	2	yes	—
351	C₄	#543×C₆₇	1	2	yes	—
352	C₂	—	2	4	no	—
353	C₂	—	2	4	no	—
354	C₂	—	4	2	no	—
355	C₂	C₆₇×#2315	1	1	yes	characteristic
356	C₂	C₂×#423×C₆₇	1	1	yes	characteristic
357	C₂	C₂×#424×C₆₇	1	1	yes	characteristic
358	C₂	C₆₇×#2974	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁵×C₆₇	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#1851	C₂	semidirect
5. C₆₇×#1850	C₂	semidirect
6. C₆₇×#1797	C₂	semidirect
12. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
25. #336×C₆₇	C₂²	semidirect
100. C₂²⋊C₄×C₆₇	M₁₆	semidirect
172. #21932	C₆₇	direct (N × H)
183. #1851	C₂×C₆₇	semidirect
184. #1850	C₂×C₆₇	semidirect
185. #1797	C₂×C₆₇	semidirect
186. C₆₇	#21932	direct (N × H)
192. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
205. #336	C₂²×C₆₇	semidirect
280. C₂²⋊C₄	M₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#407×C₆₇	2
C₂²×C₂²⋊C₈×C₆₇	2
C₂×C₄×C₂²⋊C₄×C₆₇	2
C₆₇×#1851	2
C₆₇×#1850	2
C₆₇×#1797	2
#21932	67

Subgroups — C67×#21932 #30701183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21932 #30701183