Subgroups — C₆₇×#60182 #30739433

Order 17152.

2628 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#575×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#568×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2957	C₂	1	8	yes	maximal, complemented
4	C₆₇×#2956	C₂	1	8	yes	maximal, complemented
5	C₆₇×#2955	C₂	1	8	yes	maximal, complemented
6	C₂×#430×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#421×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2873	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₆₇	C₂²	1	2	yes	—
10	#575×C₆₇	C₄	1	8	yes	complemented
11	#575×C₆₇	C₂²	1	4	yes	—
12	#575×C₆₇	C₂²	1	4	yes	—
13	C₂×#137×C₆₇	C₄	1	4	yes	complemented
14	C₂×AGL(1,ℤ/8)×C₆₇	C₄	1	4	yes	complemented
15	C₂×#135×C₆₇	C₂²	1	4	yes	—
16	#568×C₆₇	—	2	4	no	—
17	#568×C₆₇	—	2	4	no	—
18	#568×C₆₇	C₂²	1	8	yes	complemented
19	#568×C₆₇	C₂²	1	8	yes	complemented
20	C₂×#131×C₆₇	—	2	4	no	—
21	C₂×#131×C₆₇	C₂²	1	8	yes	complemented
22	C₂²×M₁₆×C₆₇	—	2	4	no	—
23	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
24	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
27	#444×C₆₇	C₂²	1	32	yes	complemented
28	#443×C₆₇	C₂²	1	32	yes	complemented
29	#430×C₆₇	C₂²	1	4	yes	complemented
30	#430×C₆₇	C₂²	1	4	yes	complemented
31	C₂×#108×C₆₇	C₂²	1	4	yes	complemented
32	#421×C₆₇	C₂²	1	8	yes	complemented
33	#421×C₆₇	C₂²	1	8	yes	complemented
34	C₂×#104×C₆₇	C₂²	1	8	yes	complemented
35	#405×C₆₇	C₂²	1	4	yes	complemented
36	#405×C₆₇	C₂²	1	4	yes	complemented
37	C₄×M₁₆×C₆₇	C₂²	1	4	yes	complemented
38	C₂×#41×C₆₇	—	2	2	no	—
39	C₂×#41×C₆₇	—	2	2	no	—
40	C₂×#41×C₆₇	—	2	2	no	—
41	C₂×#41×C₆₇	—	2	2	no	—
42	C₂×#41×C₆₇	—	2	2	no	—
43	C₂×#41×C₆₇	—	2	2	no	—
44	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
45	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
46	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
47	C₂×#41×C₆₇	C₂³	1	4	yes	—
48	C₂×#41×C₆₇	C₂³	1	4	yes	—
49	C₂×#41×C₆₇	C₂³	1	4	yes	—
50	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
51	C₂²×D₄×C₆₇	—	2	2	no	—
52	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
53	C₂³×C₄×C₆₇	—	2	2	no	—
54	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
55	#137×C₆₇	C₂×C₄	1	16	yes	—
56	#137×C₆₇	C₂×C₄	1	16	yes	complemented
57	AGL(1,ℤ/8)×C₆₇	C₂×C₄	1	16	yes	—
58	AGL(1,ℤ/8)×C₆₇	C₂×C₄	1	16	yes	complemented
59	#135×C₆₇	C₂×C₄	1	16	yes	complemented
60	#135×C₆₇	C₂³	1	16	yes	—
61	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
62	C₂×SD₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
63	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	complemented
64	#131×C₆₇	—	2	16	no	—
65	#131×C₆₇	—	2	16	no	—
66	#131×C₆₇	—	2	16	no	—
67	#131×C₆₇	C₂³	1	32	yes	complemented
68	C₂×M₁₆×C₆₇	—	2	2	no	—
69	C₂×M₁₆×C₆₇	—	2	2	no	—
70	C₂×M₁₆×C₆₇	—	2	2	no	—
71	C₂×M₁₆×C₆₇	—	2	16	no	—
72	C₂×M₁₆×C₆₇	—	2	16	no	—
73	C₂×M₁₆×C₆₇	—	2	8	no	—
74	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
75	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
76	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
77	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
78	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
79	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
80	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
81	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
82	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
83	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
84	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
85	C₂²×C₈×C₆₇	—	2	2	no	—
86	C₂²×C₈×C₆₇	—	2	8	no	—
87	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
88	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
89	#117×C₆₇	—	2	1	no	—
90	#117×C₆₇	—	2	1	no	—
91	#117×C₆₇	C₂³	1	2	yes	—
92	#117×C₆₇	C₂³	1	2	yes	—
93	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
94	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
95	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
96	#108×C₆₇	C₂³	1	16	yes	complemented
97	#104×C₆₇	C₂³	1	32	yes	complemented
98	#97×C₆₇	C₂³	1	8	yes	complemented
99	C₄×C₈×C₆₇	C₂³	1	8	yes	complemented
100	C₂⁴×C₆₇	—	2	2	no	—
101	C₂⁴×C₆₇	#41	1	1	yes	characteristic
102	#41×C₆₇	—	2	4	no	—
103	#41×C₆₇	—	2	4	no	—
104	#41×C₆₇	—	2	4	no	—
105	#41×C₆₇	—	2	4	no	—
106	#41×C₆₇	—	2	4	no	—
107	#41×C₆₇	—	2	4	no	—
108	#41×C₆₇	—	2	4	no	—
109	#41×C₆₇	—	2	4	no	—
110	#41×C₆₇	—	2	4	no	—
111	#41×C₆₇	—	2	4	no	—
112	#41×C₆₇	—	2	4	no	—
113	#41×C₆₇	—	2	4	no	—
114	#41×C₆₇	C₂²×C₄	1	8	yes	—
115	#41×C₆₇	C₂²×C₄	1	8	yes	—
116	#41×C₆₇	C₂²×C₄	1	8	yes	—
117	#41×C₆₇	C₂⁴	1	8	yes	—
118	C₂×Q₈×C₆₇	—	2	2	no	—
119	C₂×Q₈×C₆₇	—	2	2	no	—
120	C₂×Q₈×C₆₇	—	2	2	no	—
121	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
122	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
123	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
124	C₂×D₄×C₆₇	—	2	4	no	—
125	C₂×D₄×C₆₇	—	2	4	no	—
126	C₂×D₄×C₆₇	—	2	4	no	—
127	C₂×D₄×C₆₇	—	2	4	no	—
128	C₂×D₄×C₆₇	—	2	4	no	—
129	C₂×D₄×C₆₇	—	2	4	no	—
130	C₂×D₄×C₆₇	—	2	2	no	—
131	C₂×D₄×C₆₇	—	2	2	no	—
132	C₂×D₄×C₆₇	—	2	2	no	—
133	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
134	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
135	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
136	C₂²×C₄×C₆₇	—	2	2	no	—
137	C₂²×C₄×C₆₇	—	2	4	no	—
138	C₂²×C₄×C₆₇	—	2	4	no	—
139	C₂²×C₄×C₆₇	—	2	4	no	—
140	C₂²×C₄×C₆₇	—	2	4	no	—
141	C₂²×C₄×C₆₇	—	2	4	no	—
142	C₂²×C₄×C₆₇	—	2	4	no	—
143	C₂²×C₄×C₆₇	—	2	2	no	—
144	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
145	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
146	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
147	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
148	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
150	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
154	Q₁₆×C₆₇	C₂²×C₄	1	16	yes	—
155	SD₁₆×C₆₇	C₂²×C₄	1	32	yes	—
156	D₈×C₆₇	C₂²×C₄	1	16	yes	—
157	M₁₆×C₆₇	—	2	4	no	—
158	M₁₆×C₆₇	—	2	4	no	—
159	M₁₆×C₆₇	—	2	4	no	—
160	M₁₆×C₆₇	—	2	32	no	—
161	M₁₆×C₆₇	C₂²×C₄	1	8	yes	—
162	M₁₆×C₆₇	C₂²×C₄	1	8	yes	—
163	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
164	M₁₆×C₆₇	C₂×D₄	1	8	yes	—
165	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
166	M₁₆×C₆₇	C₂⁴	1	8	yes	—
167	C₂×C₈×C₆₇	—	2	4	no	—
168	C₂×C₈×C₆₇	—	2	4	no	—
169	C₂×C₈×C₆₇	—	2	4	no	—
170	C₂×C₈×C₆₇	—	2	32	no	—
171	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
172	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
173	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
174	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
175	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
176	C₄⋊C₄×C₆₇	—	2	2	no	—
177	C₄⋊C₄×C₆₇	—	2	2	no	—
178	C₂²⋊C₄×C₆₇	—	2	2	no	—
179	C₂²⋊C₄×C₆₇	—	2	2	no	—
180	C₄²×C₆₇	—	2	2	no	—
181	C₄²×C₆₇	C₂⁴	1	4	yes	—
182	C₂³×C₆₇	—	2	4	no	—
183	C₂³×C₆₇	—	2	4	no	—
184	C₂³×C₆₇	—	2	4	no	—
185	C₂³×C₆₇	—	8	1	no	—
186	C₂³×C₆₇	—	8	4	no	—
187	C₂³×C₆₇	C₄×D₄	1	2	yes	—
188	C₂³×C₆₇	C₄×D₄	1	2	yes	—
189	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
190	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
191	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
192	Q₈×C₆₇	—	2	4	no	—
193	Q₈×C₆₇	—	2	4	no	—
194	Q₈×C₆₇	—	2	4	no	—
195	Q₈×C₆₇	C₂³×C₄	1	8	yes	—
196	D₄×C₆₇	—	2	8	no	—
197	D₄×C₆₇	—	2	8	no	—
198	D₄×C₆₇	—	2	8	no	—
199	D₄×C₆₇	—	2	8	no	—
200	D₄×C₆₇	—	2	4	no	—
201	D₄×C₆₇	—	2	4	no	—
202	D₄×C₆₇	—	2	4	no	—
203	D₄×C₆₇	C₂³×C₄	1	8	yes	—
204	C₂×C₄×C₆₇	—	2	4	no	—
205	C₂×C₄×C₆₇	—	2	4	no	—
206	C₂×C₄×C₆₇	—	2	4	no	—
207	C₂×C₄×C₆₇	—	2	8	no	—
208	C₂×C₄×C₆₇	—	2	8	no	—
209	C₂×C₄×C₆₇	—	2	8	no	—
210	C₂×C₄×C₆₇	—	2	8	no	—
211	C₂×C₄×C₆₇	—	2	4	no	—
212	C₂×C₄×C₆₇	—	8	2	no	—
213	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
214	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
215	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
216	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
217	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
218	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
219	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
220	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
221	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
222	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
223	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
225	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
226	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
228	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
229	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
230	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
231	C₈×C₆₇	—	2	8	no	—
232	C₈×C₆₇	C₂³×C₄	1	8	yes	—
233	C₈×C₆₇	C₂²×D₄	1	8	yes	—
234	C₂²×C₆₇	—	2	8	no	—
235	C₂²×C₆₇	—	4	1	no	—
236	C₂²×C₆₇	—	4	2	no	—
237	C₂²×C₆₇	—	4	2	no	—
238	C₂²×C₆₇	—	4	1	no	—
239	C₂²×C₆₇	—	4	1	no	—
240	C₂²×C₆₇	—	4	8	no	—
241	C₂²×C₆₇	—	8	4	no	—
242	C₂²×C₆₇	—	8	4	no	—
243	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
244	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
245	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
246	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
247	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
248	C₄×C₆₇	—	2	8	no	—
249	C₄×C₆₇	—	4	4	no	—
250	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
251	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
252	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
253	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
254	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
255	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
256	#60182	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
257	C₂×C₆₇	—	2	1	no	—
258	C₂×C₆₇	—	2	2	no	—
259	C₂×C₆₇	—	2	1	no	—
260	C₂×C₆₇	—	2	2	no	—
261	C₂×C₆₇	—	2	1	no	—
262	C₂×C₆₇	—	4	8	no	—
263	C₂×C₆₇	C₂²×C₄×D₄	1	1	yes	characteristic
264	C₂×#575	C₂×C₆₇	1	1	yes	characteristic
265	C₂×#568	C₂×C₆₇	1	2	yes	complemented
266	#2957	C₂×C₆₇	1	8	yes	complemented
267	#2956	C₂×C₆₇	1	8	yes	complemented
268	#2955	C₂×C₆₇	1	8	yes	complemented
269	C₂×#430	C₂×C₆₇	1	1	yes	characteristic, complemented
270	C₂×#421	C₂×C₆₇	1	2	yes	complemented
271	#2873	C₂×C₆₇	1	1	yes	characteristic, complemented
272	C₆₇	#60182	1	1	yes	characteristic, direct product factor, Sylow
273	C₂²×#41	C₂²×C₆₇	1	2	yes	—
274	#575	C₄×C₆₇	1	8	yes	complemented
275	#575	C₂²×C₆₇	1	4	yes	—
276	#575	C₂²×C₆₇	1	4	yes	—
277	C₂×#137	C₄×C₆₇	1	4	yes	complemented
278	C₂×AGL(1,ℤ/8)	C₄×C₆₇	1	4	yes	complemented
279	C₂×#135	C₂²×C₆₇	1	4	yes	—
280	#568	—	2	4	no	—
281	#568	—	2	4	no	—
282	#568	C₂²×C₆₇	1	8	yes	complemented
283	#568	C₂²×C₆₇	1	8	yes	complemented
284	C₂×#131	—	2	4	no	—
285	C₂×#131	C₂²×C₆₇	1	8	yes	complemented
286	C₂²×M₁₆	—	2	4	no	—
287	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
288	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
289	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
290	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
291	#444	C₂²×C₆₇	1	32	yes	complemented
292	#443	C₂²×C₆₇	1	32	yes	complemented
293	#430	C₂²×C₆₇	1	4	yes	complemented
294	#430	C₂²×C₆₇	1	4	yes	complemented
295	C₂×#108	C₂²×C₆₇	1	4	yes	complemented
296	#421	C₂²×C₆₇	1	8	yes	complemented
297	#421	C₂²×C₆₇	1	8	yes	complemented
298	C₂×#104	C₂²×C₆₇	1	8	yes	complemented
299	#405	C₂²×C₆₇	1	4	yes	complemented
300	#405	C₂²×C₆₇	1	4	yes	complemented
301	C₄×M₁₆	C₂²×C₆₇	1	4	yes	complemented
302	C₂×#41	—	2	2	no	—
303	C₂×#41	—	2	2	no	—
304	C₂×#41	—	2	2	no	—
305	C₂×#41	—	2	2	no	—
306	C₂×#41	—	2	2	no	—
307	C₂×#41	—	2	2	no	—
308	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
309	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
310	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
311	C₂×#41	C₂³×C₆₇	1	4	yes	—
312	C₂×#41	C₂³×C₆₇	1	4	yes	—
313	C₂×#41	C₂³×C₆₇	1	4	yes	—
314	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
315	C₂²×D₄	—	2	2	no	—
316	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
317	C₂³×C₄	—	2	2	no	—
318	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
319	#137	C₂×C₄×C₆₇	1	16	yes	—
320	#137	C₂×C₄×C₆₇	1	16	yes	complemented
321	AGL(1,ℤ/8)	C₂×C₄×C₆₇	1	16	yes	—
322	AGL(1,ℤ/8)	C₂×C₄×C₆₇	1	16	yes	complemented
323	#135	C₂×C₄×C₆₇	1	16	yes	complemented
324	#135	C₂³×C₆₇	1	16	yes	—
325	C₂×Q₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
326	C₂×SD₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
327	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	complemented
328	#131	—	2	16	no	—
329	#131	—	2	16	no	—
330	#131	—	2	16	no	—
331	#131	C₂³×C₆₇	1	32	yes	complemented
332	C₂×M₁₆	—	2	2	no	—
333	C₂×M₁₆	—	2	2	no	—
334	C₂×M₁₆	—	2	2	no	—
335	C₂×M₁₆	—	2	16	no	—
336	C₂×M₁₆	—	2	16	no	—
337	C₂×M₁₆	—	2	8	no	—
338	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
339	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
340	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
341	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
342	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
343	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
344	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
345	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
346	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
347	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
348	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
349	C₂²×C₈	—	2	2	no	—
350	C₂²×C₈	—	2	8	no	—
351	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
352	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
353	#117	—	2	1	no	—
354	#117	—	2	1	no	—
355	#117	C₂³×C₆₇	1	2	yes	—
356	#117	C₂³×C₆₇	1	2	yes	—
357	C₂×C₄⋊C₄	—	2	1	no	—
358	C₂×C₂²⋊C₄	—	2	1	no	—
359	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
360	#108	C₂³×C₆₇	1	16	yes	complemented
361	#104	C₂³×C₆₇	1	32	yes	complemented
362	#97	C₂³×C₆₇	1	8	yes	complemented
363	C₄×C₈	C₂³×C₆₇	1	8	yes	complemented
364	C₂⁴	—	2	2	no	—
365	C₂⁴	#41×C₆₇	1	1	yes	characteristic
366	#41	—	2	4	no	—
367	#41	—	2	4	no	—
368	#41	—	2	4	no	—
369	#41	—	2	4	no	—
370	#41	—	2	4	no	—
371	#41	—	2	4	no	—
372	#41	—	2	4	no	—
373	#41	—	2	4	no	—
374	#41	—	2	4	no	—
375	#41	—	2	4	no	—
376	#41	—	2	4	no	—
377	#41	—	2	4	no	—
378	#41	C₂²×C₄×C₆₇	1	8	yes	—
379	#41	C₂²×C₄×C₆₇	1	8	yes	—
380	#41	C₂²×C₄×C₆₇	1	8	yes	—
381	#41	C₂⁴×C₆₇	1	8	yes	—
382	C₂×Q₈	—	2	2	no	—
383	C₂×Q₈	—	2	2	no	—
384	C₂×Q₈	—	2	2	no	—
385	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
386	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
387	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
388	C₂×D₄	—	2	4	no	—
389	C₂×D₄	—	2	4	no	—
390	C₂×D₄	—	2	4	no	—
391	C₂×D₄	—	2	4	no	—
392	C₂×D₄	—	2	4	no	—
393	C₂×D₄	—	2	4	no	—
394	C₂×D₄	—	2	2	no	—
395	C₂×D₄	—	2	2	no	—
396	C₂×D₄	—	2	2	no	—
397	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
398	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
399	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	2	4	no	—
402	C₂²×C₄	—	2	4	no	—
403	C₂²×C₄	—	2	4	no	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	—	2	4	no	—
406	C₂²×C₄	—	2	4	no	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
409	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
410	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
411	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
412	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
413	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
414	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
415	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
416	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
417	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
418	Q₁₆	C₂²×C₄×C₆₇	1	16	yes	—
419	SD₁₆	C₂²×C₄×C₆₇	1	32	yes	—
420	D₈	C₂²×C₄×C₆₇	1	16	yes	—
421	M₁₆	—	2	4	no	—
422	M₁₆	—	2	4	no	—
423	M₁₆	—	2	4	no	—
424	M₁₆	—	2	32	no	—
425	M₁₆	C₂²×C₄×C₆₇	1	8	yes	—
426	M₁₆	C₂²×C₄×C₆₇	1	8	yes	—
427	M₁₆	C₂×D₄×C₆₇	1	4	yes	—
428	M₁₆	C₂×D₄×C₆₇	1	8	yes	—
429	M₁₆	C₂×D₄×C₆₇	1	4	yes	—
430	M₁₆	C₂⁴×C₆₇	1	8	yes	—
431	C₂×C₈	—	2	4	no	—
432	C₂×C₈	—	2	4	no	—
433	C₂×C₈	—	2	4	no	—
434	C₂×C₈	—	2	32	no	—
435	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
436	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
437	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
438	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
439	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
440	C₄⋊C₄	—	2	2	no	—
441	C₄⋊C₄	—	2	2	no	—
442	C₂²⋊C₄	—	2	2	no	—
443	C₂²⋊C₄	—	2	2	no	—
444	C₄²	—	2	2	no	—
445	C₄²	C₂⁴×C₆₇	1	4	yes	—
446	C₂³	—	2	4	no	—
447	C₂³	—	2	4	no	—
448	C₂³	—	2	4	no	—
449	C₂³	—	8	1	no	—
450	C₂³	—	8	4	no	—
451	C₂³	C₄×D₄×C₆₇	1	2	yes	—
452	C₂³	C₄×D₄×C₆₇	1	2	yes	—
453	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
454	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
455	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
456	Q₈	—	2	4	no	—
457	Q₈	—	2	4	no	—
458	Q₈	—	2	4	no	—
459	Q₈	C₂³×C₄×C₆₇	1	8	yes	—
460	D₄	—	2	8	no	—
461	D₄	—	2	8	no	—
462	D₄	—	2	8	no	—
463	D₄	—	2	8	no	—
464	D₄	—	2	4	no	—
465	D₄	—	2	4	no	—
466	D₄	—	2	4	no	—
467	D₄	C₂³×C₄×C₆₇	1	8	yes	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	8	no	—
472	C₂×C₄	—	2	8	no	—
473	C₂×C₄	—	2	8	no	—
474	C₂×C₄	—	2	8	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	8	2	no	—
477	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
478	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
479	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
480	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
481	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
482	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
483	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
484	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
485	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
486	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
487	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
488	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
489	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
490	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
491	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
492	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
493	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
494	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
495	C₈	—	2	8	no	—
496	C₈	C₂³×C₄×C₆₇	1	8	yes	—
497	C₈	C₂²×D₄×C₆₇	1	8	yes	—
498	C₂²	—	2	8	no	—
499	C₂²	—	4	1	no	—
500	C₂²	—	4	2	no	—
501	C₂²	—	4	2	no	—
502	C₂²	—	4	1	no	—
503	C₂²	—	4	1	no	—
504	C₂²	—	4	8	no	—
505	C₂²	—	8	4	no	—
506	C₂²	—	8	4	no	—
507	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
508	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
509	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
510	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
511	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
512	C₄	—	2	8	no	—
513	C₄	—	4	4	no	—
514	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
515	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
516	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
517	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
518	C₄	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
519	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
520	C₂	—	2	1	no	—
521	C₂	—	2	2	no	—
522	C₂	—	2	1	no	—
523	C₂	—	2	2	no	—
524	C₂	—	2	1	no	—
525	C₂	—	4	8	no	—
526	C₂	C₂²×C₄×D₄×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₆₇	C₂³×D₄
commutator	C₄	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂×C₆₇	C₂²×C₄×D₄
prime radical	C₂×#575×C₆₇	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₇ ◁ C₂²×M₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#568×C₆₇	C₂	semidirect
3. C₆₇×#2957	C₂	semidirect
4. C₆₇×#2956	C₂	semidirect
5. C₆₇×#2955	C₂	semidirect
6. C₂×#430×C₆₇	C₂	semidirect
7. C₂×#421×C₆₇	C₂	semidirect
8. C₆₇×#2873	C₂	semidirect
10. #575×C₆₇	C₄	semidirect
13. C₂×#137×C₆₇	C₄	semidirect
14. C₂×AGL(1,ℤ/8)×C₆₇	C₄	semidirect
18. #568×C₆₇	C₂²	semidirect
19. #568×C₆₇	C₂²	semidirect
21. C₂×#131×C₆₇	C₂²	semidirect
27. #444×C₆₇	C₂²	semidirect
28. #443×C₆₇	C₂²	semidirect
29. #430×C₆₇	C₂²	semidirect
30. #430×C₆₇	C₂²	semidirect
31. C₂×#108×C₆₇	C₂²	semidirect
32. #421×C₆₇	C₂²	semidirect
33. #421×C₆₇	C₂²	semidirect
34. C₂×#104×C₆₇	C₂²	semidirect
35. #405×C₆₇	C₂²	semidirect
36. #405×C₆₇	C₂²	semidirect
37. C₄×M₁₆×C₆₇	C₂²	semidirect
56. #137×C₆₇	C₂×C₄	semidirect
58. AGL(1,ℤ/8)×C₆₇	C₂×C₄	semidirect
59. #135×C₆₇	C₂×C₄	semidirect
61. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
62. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
63. C₂×D₈×C₆₇	C₂×C₄	semidirect
67. #131×C₆₇	C₂³	semidirect
96. #108×C₆₇	C₂³	semidirect
97. #104×C₆₇	C₂³	semidirect
98. #97×C₆₇	C₂³	semidirect
99. C₄×C₈×C₆₇	C₂³	semidirect
256. #60182	C₆₇	direct (N × H)
265. C₂×#568	C₂×C₆₇	semidirect
266. #2957	C₂×C₆₇	semidirect
267. #2956	C₂×C₆₇	semidirect
268. #2955	C₂×C₆₇	semidirect
269. C₂×#430	C₂×C₆₇	semidirect
270. C₂×#421	C₂×C₆₇	semidirect
271. #2873	C₂×C₆₇	semidirect
272. C₆₇	#60182	direct (N × H)
274. #575	C₄×C₆₇	semidirect
277. C₂×#137	C₄×C₆₇	semidirect
278. C₂×AGL(1,ℤ/8)	C₄×C₆₇	semidirect
282. #568	C₂²×C₆₇	semidirect
283. #568	C₂²×C₆₇	semidirect
285. C₂×#131	C₂²×C₆₇	semidirect
291. #444	C₂²×C₆₇	semidirect
292. #443	C₂²×C₆₇	semidirect
293. #430	C₂²×C₆₇	semidirect
294. #430	C₂²×C₆₇	semidirect
295. C₂×#108	C₂²×C₆₇	semidirect
296. #421	C₂²×C₆₇	semidirect
297. #421	C₂²×C₆₇	semidirect
298. C₂×#104	C₂²×C₆₇	semidirect
299. #405	C₂²×C₆₇	semidirect
300. #405	C₂²×C₆₇	semidirect
301. C₄×M₁₆	C₂²×C₆₇	semidirect
320. #137	C₂×C₄×C₆₇	semidirect
322. AGL(1,ℤ/8)	C₂×C₄×C₆₇	semidirect
323. #135	C₂×C₄×C₆₇	semidirect
325. C₂×Q₁₆	C₂×C₄×C₆₇	semidirect
326. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
327. C₂×D₈	C₂×C₄×C₆₇	semidirect
331. #131	C₂³×C₆₇	semidirect
360. #108	C₂³×C₆₇	semidirect
361. #104	C₂³×C₆₇	semidirect
362. #97	C₂³×C₆₇	semidirect
363. C₄×C₈	C₂³×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#575×C₆₇	2
C₂×#568×C₆₇	2
C₆₇×#2957	2
C₆₇×#2956	2
C₆₇×#2955	2
C₂×#430×C₆₇	2
C₂×#421×C₆₇	2
C₆₇×#2873	2
#60182	67

Subgroups — C67×#60182 #30739433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60182 #30739433