Subgroups — C₆₇×#21454 #30700705

Order 17152.

1520 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#448×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#440×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₄×#121×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1880	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1874	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1863	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1852	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1787	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1763	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1749	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₆₇	—	2	8	no	—
16	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
17	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
18	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₆₇	—	2	1	no	—
21	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
23	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄²×C₆₇	—	2	1	no	—
25	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
26	#448×C₆₇	C₄	1	8	yes	complemented
27	#440×C₆₇	—	2	8	no	—
28	C₂×#107×C₆₇	C₂²	1	2	yes	—
29	C₂×#103×C₆₇	C₂²	1	2	yes	—
30	C₂×#103×C₆₇	C₂²	1	2	yes	—
31	C₂×#103×C₆₇	C₂²	1	2	yes	—
32	C₂×#102×C₆₇	C₂²	1	2	yes	—
33	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#97×C₆₇	C₂²	1	2	yes	—
36	#390×C₆₇	C₂²	1	1	yes	characteristic
37	#389×C₆₇	C₂²	1	1	yes	characteristic
38	#388×C₆₇	—	2	1	no	—
39	#386×C₆₇	—	2	1	no	—
40	#380×C₆₇	C₂²	1	1	yes	characteristic
41	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
42	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
43	#340×C₆₇	C₂²	1	2	yes	—
44	#340×C₆₇	C₂²	1	2	yes	—
45	#336×C₆₇	C₂²	1	2	yes	—
46	C₂²×Q₈×C₆₇	—	2	4	no	—
47	C₂²×Q₈×C₆₇	—	2	4	no	—
48	C₂²×Q₈×C₆₇	—	2	1	no	—
49	C₂²×Q₈×C₆₇	—	2	2	no	—
50	C₂²×Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
51	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
52	C₂²×D₄×C₆₇	—	2	1	no	—
53	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₂³×C₄×C₆₇	—	2	1	no	—
55	C₂³×C₄×C₆₇	—	2	1	no	—
56	C₂³×C₄×C₆₇	—	2	1	no	—
57	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
58	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
59	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×SD₁₆×C₆₇	—	2	4	no	—
61	C₂×SD₁₆×C₆₇	—	2	4	no	—
62	C₂×SD₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
63	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
64	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
65	#121×C₆₇	—	2	2	no	—
66	#121×C₆₇	C₂×C₄	1	4	yes	—
67	C₄×Q₈×C₆₇	—	2	8	no	—
68	C₄×Q₈×C₆₇	—	2	8	no	—
69	C₄×Q₈×C₆₇	—	2	4	no	—
70	C₄×Q₈×C₆₇	—	2	4	no	—
71	C₄×Q₈×C₆₇	—	2	4	no	—
72	C₄×Q₈×C₆₇	D₄	1	8	yes	complemented
73	C₄×D₄×C₆₇	—	2	4	no	—
74	C₄×D₄×C₆₇	—	4	4	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
76	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
77	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
78	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
79	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
80	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
85	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
86	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
87	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
88	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
89	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
90	C₂×C₄²×C₆₇	—	2	8	no	—
91	C₂×C₄²×C₆₇	—	2	2	no	—
92	C₂×C₄²×C₆₇	—	2	2	no	—
93	C₂×C₄²×C₆₇	—	2	4	no	—
94	C₂×C₄²×C₆₇	—	2	1	no	—
95	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
96	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
99	#107×C₆₇	—	2	4	no	—
100	#103×C₆₇	—	2	4	no	—
101	#103×C₆₇	—	2	4	no	—
102	#103×C₆₇	C₂×C₄	1	8	yes	complemented
103	#102×C₆₇	—	2	4	no	—
104	C₂²⋊C₈×C₆₇	—	2	2	no	—
105	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	complemented
106	#97×C₆₇	—	2	4	no	—
107	#95×C₆₇	—	2	1	no	—
108	#95×C₆₇	—	2	1	no	—
109	#95×C₆₇	—	2	1	no	—
110	C₂⁴×C₆₇	—	2	1	no	—
111	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂×Q₈×C₆₇	—	2	4	no	—
113	C₂×Q₈×C₆₇	—	2	16	no	—
114	C₂×Q₈×C₆₇	—	2	4	no	—
115	C₂×Q₈×C₆₇	—	2	16	no	—
116	C₂×Q₈×C₆₇	—	2	4	no	—
117	C₂×Q₈×C₆₇	—	2	2	no	—
118	C₂×Q₈×C₆₇	—	2	2	no	—
119	C₂×Q₈×C₆₇	—	2	2	no	—
120	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
121	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
122	C₂×Q₈×C₆₇	#41	1	4	yes	—
123	C₂×D₄×C₆₇	—	2	4	no	—
124	C₂×D₄×C₆₇	—	2	2	no	—
125	C₂×D₄×C₆₇	—	2	2	no	—
126	C₂×D₄×C₆₇	—	2	1	no	—
127	C₂×D₄×C₆₇	—	4	2	no	—
128	C₂×D₄×C₆₇	—	4	2	no	—
129	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
130	C₂²×C₄×C₆₇	—	2	4	no	—
131	C₂²×C₄×C₆₇	—	2	4	no	—
132	C₂²×C₄×C₆₇	—	2	4	no	—
133	C₂²×C₄×C₆₇	—	2	2	no	—
134	C₂²×C₄×C₆₇	—	2	2	no	—
135	C₂²×C₄×C₆₇	—	2	2	no	—
136	C₂²×C₄×C₆₇	—	2	2	no	—
137	C₂²×C₄×C₆₇	—	2	2	no	—
138	C₂²×C₄×C₆₇	—	2	2	no	—
139	C₂²×C₄×C₆₇	—	2	4	no	—
140	C₂²×C₄×C₆₇	—	2	2	no	—
141	C₂²×C₄×C₆₇	—	2	2	no	—
142	C₂²×C₄×C₆₇	—	2	1	no	—
143	C₂²×C₄×C₆₇	—	2	2	no	—
144	C₂²×C₄×C₆₇	—	2	1	no	—
145	C₂²×C₄×C₆₇	—	2	1	no	—
146	C₂²×C₄×C₆₇	—	4	1	no	—
147	C₂²×C₄×C₆₇	—	4	2	no	—
148	C₂²×C₄×C₆₇	—	4	2	no	—
149	C₂²×C₄×C₆₇	—	4	2	no	—
150	C₂²×C₄×C₆₇	—	4	2	no	—
151	C₂²×C₄×C₆₇	—	4	2	no	—
152	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
153	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₇	#41	1	2	yes	—
159	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
160	SD₁₆×C₆₇	—	2	16	no	—
161	C₂×C₈×C₆₇	—	2	2	no	—
162	C₂×C₈×C₆₇	—	2	2	no	—
163	C₂×C₈×C₆₇	—	2	2	no	—
164	C₂×C₈×C₆₇	—	2	2	no	—
165	C₂×C₈×C₆₇	—	2	2	no	—
166	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
167	C₄⋊C₄×C₆₇	—	2	8	no	—
168	C₄⋊C₄×C₆₇	—	2	4	no	—
169	C₄⋊C₄×C₆₇	—	2	8	no	—
170	C₄⋊C₄×C₆₇	—	2	2	no	—
171	C₄⋊C₄×C₆₇	—	2	1	no	—
172	C₄⋊C₄×C₆₇	—	2	2	no	—
173	C₄⋊C₄×C₆₇	—	2	2	no	—
174	C₄⋊C₄×C₆₇	—	4	2	no	—
175	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
176	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
177	C₂²⋊C₄×C₆₇	—	2	4	no	—
178	C₂²⋊C₄×C₆₇	—	4	2	no	—
179	C₂²⋊C₄×C₆₇	—	4	2	no	—
180	C₂²⋊C₄×C₆₇	—	4	2	no	—
181	C₂²⋊C₄×C₆₇	—	4	2	no	—
182	C₄²×C₆₇	—	2	8	no	—
183	C₄²×C₆₇	—	2	8	no	—
184	C₄²×C₆₇	—	2	2	no	—
185	C₄²×C₆₇	—	2	2	no	—
186	C₄²×C₆₇	—	2	2	no	—
187	C₄²×C₆₇	—	4	2	no	—
188	C₄²×C₆₇	C₂×D₄	1	4	yes	—
189	C₂³×C₆₇	—	2	1	no	—
190	C₂³×C₆₇	—	2	1	no	—
191	C₂³×C₆₇	—	2	1	no	—
192	C₂³×C₆₇	—	2	1	no	—
193	C₂³×C₆₇	—	2	2	no	—
194	C₂³×C₆₇	—	4	1	no	—
195	C₂³×C₆₇	—	4	1	no	—
196	C₂³×C₆₇	—	4	1	no	—
197	C₂³×C₆₇	—	4	1	no	—
198	C₂³×C₆₇	—	4	1	no	—
199	C₂³×C₆₇	—	4	1	no	—
200	C₂³×C₆₇	C₄×D₄	1	2	yes	—
201	C₂³×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
202	C₂³×C₆₇	#137	1	2	yes	—
203	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
204	Q₈×C₆₇	—	2	16	no	—
205	Q₈×C₆₇	—	2	4	no	—
206	Q₈×C₆₇	—	2	4	no	—
207	Q₈×C₆₇	—	2	4	no	—
208	Q₈×C₆₇	C₄×D₄	1	8	yes	complemented
209	D₄×C₆₇	—	2	4	no	—
210	D₄×C₆₇	—	4	8	no	—
211	D₄×C₆₇	—	4	2	no	—
212	C₂×C₄×C₆₇	—	2	8	no	—
213	C₂×C₄×C₆₇	—	2	16	no	—
214	C₂×C₄×C₆₇	—	2	4	no	—
215	C₂×C₄×C₆₇	—	2	1	no	—
216	C₂×C₄×C₆₇	—	2	4	no	—
217	C₂×C₄×C₆₇	—	2	1	no	—
218	C₂×C₄×C₆₇	—	2	4	no	—
219	C₂×C₄×C₆₇	—	2	1	no	—
220	C₂×C₄×C₆₇	—	2	1	no	—
221	C₂×C₄×C₆₇	—	2	1	no	—
222	C₂×C₄×C₆₇	—	2	1	no	—
223	C₂×C₄×C₆₇	—	2	1	no	—
224	C₂×C₄×C₆₇	—	2	1	no	—
225	C₂×C₄×C₆₇	—	2	4	no	—
226	C₂×C₄×C₆₇	—	2	4	no	—
227	C₂×C₄×C₆₇	—	2	2	no	—
228	C₂×C₄×C₆₇	—	2	1	no	—
229	C₂×C₄×C₆₇	—	2	4	no	—
230	C₂×C₄×C₆₇	—	2	1	no	—
231	C₂×C₄×C₆₇	—	2	1	no	—
232	C₂×C₄×C₆₇	—	4	4	no	—
233	C₂×C₄×C₆₇	—	4	8	no	—
234	C₂×C₄×C₆₇	—	4	1	no	—
235	C₂×C₄×C₆₇	—	4	2	no	—
236	C₂×C₄×C₆₇	—	4	1	no	—
237	C₂×C₄×C₆₇	—	4	2	no	—
238	C₂×C₄×C₆₇	—	4	1	no	—
239	C₂×C₄×C₆₇	—	4	1	no	—
240	C₂×C₄×C₆₇	—	4	1	no	—
241	C₂×C₄×C₆₇	—	4	1	no	—
242	C₂×C₄×C₆₇	—	4	1	no	—
243	C₂×C₄×C₆₇	—	4	1	no	—
244	C₂×C₄×C₆₇	—	8	4	no	—
245	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
246	C₂×C₄×C₆₇	#120	1	2	yes	—
247	C₂×C₄×C₆₇	#120	1	2	yes	—
248	C₂×C₄×C₆₇	#137	1	2	yes	—
249	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
250	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
251	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
252	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
253	C₈×C₆₇	—	2	4	no	—
254	C₈×C₆₇	—	4	2	no	—
255	C₂²×C₆₇	—	2	2	no	—
256	C₂²×C₆₇	—	2	2	no	—
257	C₂²×C₆₇	—	2	2	no	—
258	C₂²×C₆₇	—	2	2	no	—
259	C₂²×C₆₇	—	2	2	no	—
260	C₂²×C₆₇	—	2	2	no	—
261	C₂²×C₆₇	—	4	2	no	—
262	C₂²×C₆₇	—	4	2	no	—
263	C₂²×C₆₇	—	4	2	no	—
264	C₂²×C₆₇	—	8	1	no	—
265	C₂²×C₆₇	—	8	1	no	—
266	C₂²×C₆₇	—	8	1	no	—
267	C₂²×C₆₇	—	8	1	no	—
268	C₂²×C₆₇	#440	1	4	yes	complemented
269	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
270	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
271	C₂²×C₆₇	#525	1	1	yes	characteristic
272	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
273	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
274	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
275	C₂²×C₆₇	#575	1	1	yes	characteristic
276	C₄×C₆₇	—	2	2	no	—
277	C₄×C₆₇	—	2	2	no	—
278	C₄×C₆₇	—	2	2	no	—
279	C₄×C₆₇	—	2	8	no	—
280	C₄×C₆₇	—	2	2	no	—
281	C₄×C₆₇	—	4	2	no	—
282	C₄×C₆₇	—	4	1	no	—
283	C₄×C₆₇	—	4	4	no	—
284	C₄×C₆₇	—	8	1	no	—
285	C₄×C₆₇	—	8	1	no	—
286	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
287	C₄×C₆₇	#525	1	2	yes	—
288	#21454	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
289	C₂×C₆₇	—	2	4	no	—
290	C₂×C₆₇	—	8	2	no	—
291	C₂×C₆₇	C₄×#120	1	1	yes	characteristic
292	C₂×C₆₇	C₂×#440	1	1	yes	characteristic
293	C₂×C₆₇	C₄×#137	1	1	yes	characteristic
294	C₂×C₆₇	#3004	1	1	yes	characteristic
295	C₂×C₆₇	#3006	1	1	yes	characteristic
296	C₂×C₆₇	#3112	1	1	yes	characteristic
297	C₂×C₆₇	#3117	1	1	yes	characteristic
298	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
299	C₂×#448	C₂×C₆₇	1	1	yes	characteristic
300	C₂×#440	C₂×C₆₇	1	2	yes	complemented
301	C₄×#121	C₂×C₆₇	1	1	yes	characteristic
302	#1880	C₂×C₆₇	1	2	yes	complemented
303	#1874	C₂×C₆₇	1	1	yes	characteristic
304	#1863	C₂×C₆₇	1	2	yes	complemented
305	#1852	C₂×C₆₇	1	1	yes	characteristic, complemented
306	#1787	C₂×C₆₇	1	1	yes	characteristic, complemented
307	#1763	C₂×C₆₇	1	2	yes	complemented
308	#1749	C₂×C₆₇	1	1	yes	characteristic, complemented
309	C₆₇	#21454	1	1	yes	characteristic, direct product factor, Sylow
310	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
311	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
312	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
313	C₂×C₄×Q₈	—	2	8	no	—
314	C₂×C₄×Q₈	—	2	1	no	—
315	C₂×C₄×Q₈	—	2	1	no	—
316	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
317	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
318	C₂×C₄×D₄	—	2	1	no	—
319	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
320	C₂²×C₄⋊C₄	—	2	1	no	—
321	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
322	C₂²×C₄²	—	2	1	no	—
323	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
324	#448	C₄×C₆₇	1	8	yes	complemented
325	#440	—	2	8	no	—
326	C₂×#107	C₂²×C₆₇	1	2	yes	—
327	C₂×#103	C₂²×C₆₇	1	2	yes	—
328	C₂×#103	C₂²×C₆₇	1	2	yes	—
329	C₂×#103	C₂²×C₆₇	1	2	yes	—
330	C₂×#102	C₂²×C₆₇	1	2	yes	—
331	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
332	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
333	C₂×#97	C₂²×C₆₇	1	2	yes	—
334	#390	C₂²×C₆₇	1	1	yes	characteristic
335	#389	C₂²×C₆₇	1	1	yes	characteristic
336	#388	—	2	1	no	—
337	#386	—	2	1	no	—
338	#380	C₂²×C₆₇	1	1	yes	characteristic
339	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
340	C₄×C₂²⋊C₄	—	2	1	no	—
341	#340	C₂²×C₆₇	1	2	yes	—
342	#340	C₂²×C₆₇	1	2	yes	—
343	#336	C₂²×C₆₇	1	2	yes	—
344	C₂²×Q₈	—	2	4	no	—
345	C₂²×Q₈	—	2	4	no	—
346	C₂²×Q₈	—	2	1	no	—
347	C₂²×Q₈	—	2	2	no	—
348	C₂²×Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
349	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
350	C₂²×D₄	—	2	1	no	—
351	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
352	C₂³×C₄	—	2	1	no	—
353	C₂³×C₄	—	2	1	no	—
354	C₂³×C₄	—	2	1	no	—
355	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
356	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
357	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
358	C₂×SD₁₆	—	2	4	no	—
359	C₂×SD₁₆	—	2	4	no	—
360	C₂×SD₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
361	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
362	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
363	#121	—	2	2	no	—
364	#121	C₂×C₄×C₆₇	1	4	yes	—
365	C₄×Q₈	—	2	8	no	—
366	C₄×Q₈	—	2	8	no	—
367	C₄×Q₈	—	2	4	no	—
368	C₄×Q₈	—	2	4	no	—
369	C₄×Q₈	—	2	4	no	—
370	C₄×Q₈	D₄×C₆₇	1	8	yes	complemented
371	C₄×D₄	—	2	4	no	—
372	C₄×D₄	—	4	4	no	—
373	C₂×C₄⋊C₄	—	2	8	no	—
374	C₂×C₄⋊C₄	—	2	2	no	—
375	C₂×C₄⋊C₄	—	2	2	no	—
376	C₂×C₄⋊C₄	—	2	1	no	—
377	C₂×C₄⋊C₄	—	2	4	no	—
378	C₂×C₄⋊C₄	—	2	1	no	—
379	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
380	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
381	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
382	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
383	C₂×C₂²⋊C₄	—	2	1	no	—
384	C₂×C₂²⋊C₄	—	2	1	no	—
385	C₂×C₂²⋊C₄	—	2	1	no	—
386	C₂×C₂²⋊C₄	—	2	1	no	—
387	C₂×C₂²⋊C₄	—	2	1	no	—
388	C₂×C₄²	—	2	8	no	—
389	C₂×C₄²	—	2	2	no	—
390	C₂×C₄²	—	2	2	no	—
391	C₂×C₄²	—	2	4	no	—
392	C₂×C₄²	—	2	1	no	—
393	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
394	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
395	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
396	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
397	#107	—	2	4	no	—
398	#103	—	2	4	no	—
399	#103	—	2	4	no	—
400	#103	C₂×C₄×C₆₇	1	8	yes	complemented
401	#102	—	2	4	no	—
402	C₂²⋊C₈	—	2	2	no	—
403	C₂²⋊C₈	C₂×C₄×C₆₇	1	4	yes	complemented
404	#97	—	2	4	no	—
405	#95	—	2	1	no	—
406	#95	—	2	1	no	—
407	#95	—	2	1	no	—
408	C₂⁴	—	2	1	no	—
409	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
410	C₂×Q₈	—	2	4	no	—
411	C₂×Q₈	—	2	16	no	—
412	C₂×Q₈	—	2	4	no	—
413	C₂×Q₈	—	2	16	no	—
414	C₂×Q₈	—	2	4	no	—
415	C₂×Q₈	—	2	2	no	—
416	C₂×Q₈	—	2	2	no	—
417	C₂×Q₈	—	2	2	no	—
418	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
419	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
420	C₂×Q₈	#41×C₆₇	1	4	yes	—
421	C₂×D₄	—	2	4	no	—
422	C₂×D₄	—	2	2	no	—
423	C₂×D₄	—	2	2	no	—
424	C₂×D₄	—	2	1	no	—
425	C₂×D₄	—	4	2	no	—
426	C₂×D₄	—	4	2	no	—
427	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	2	no	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	2	no	—
435	C₂²×C₄	—	2	2	no	—
436	C₂²×C₄	—	2	2	no	—
437	C₂²×C₄	—	2	4	no	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	2	no	—
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	4	1	no	—
445	C₂²×C₄	—	4	2	no	—
446	C₂²×C₄	—	4	2	no	—
447	C₂²×C₄	—	4	2	no	—
448	C₂²×C₄	—	4	2	no	—
449	C₂²×C₄	—	4	2	no	—
450	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
451	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
452	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
453	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
454	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
455	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
456	C₂²×C₄	#41×C₆₇	1	2	yes	—
457	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
458	SD₁₆	—	2	16	no	—
459	C₂×C₈	—	2	2	no	—
460	C₂×C₈	—	2	2	no	—
461	C₂×C₈	—	2	2	no	—
462	C₂×C₈	—	2	2	no	—
463	C₂×C₈	—	2	2	no	—
464	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
465	C₄⋊C₄	—	2	8	no	—
466	C₄⋊C₄	—	2	4	no	—
467	C₄⋊C₄	—	2	8	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	2	no	—
471	C₄⋊C₄	—	2	2	no	—
472	C₄⋊C₄	—	4	2	no	—
473	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
474	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
475	C₂²⋊C₄	—	2	4	no	—
476	C₂²⋊C₄	—	4	2	no	—
477	C₂²⋊C₄	—	4	2	no	—
478	C₂²⋊C₄	—	4	2	no	—
479	C₂²⋊C₄	—	4	2	no	—
480	C₄²	—	2	8	no	—
481	C₄²	—	2	8	no	—
482	C₄²	—	2	2	no	—
483	C₄²	—	2	2	no	—
484	C₄²	—	2	2	no	—
485	C₄²	—	4	2	no	—
486	C₄²	C₂×D₄×C₆₇	1	4	yes	—
487	C₂³	—	2	1	no	—
488	C₂³	—	2	1	no	—
489	C₂³	—	2	1	no	—
490	C₂³	—	2	1	no	—
491	C₂³	—	2	2	no	—
492	C₂³	—	4	1	no	—
493	C₂³	—	4	1	no	—
494	C₂³	—	4	1	no	—
495	C₂³	—	4	1	no	—
496	C₂³	—	4	1	no	—
497	C₂³	—	4	1	no	—
498	C₂³	C₄×D₄×C₆₇	1	2	yes	—
499	C₂³	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
500	C₂³	#137×C₆₇	1	2	yes	—
501	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
502	Q₈	—	2	16	no	—
503	Q₈	—	2	4	no	—
504	Q₈	—	2	4	no	—
505	Q₈	—	2	4	no	—
506	Q₈	C₄×D₄×C₆₇	1	8	yes	complemented
507	D₄	—	2	4	no	—
508	D₄	—	4	8	no	—
509	D₄	—	4	2	no	—
510	C₂×C₄	—	2	16	no	—
511	C₂×C₄	—	2	8	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	4	4	no	—
531	C₂×C₄	—	4	8	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	2	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	2	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	8	4	no	—
543	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
544	C₂×C₄	#120×C₆₇	1	2	yes	—
545	C₂×C₄	#120×C₆₇	1	2	yes	—
546	C₂×C₄	#137×C₆₇	1	2	yes	—
547	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
548	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
549	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
550	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
551	C₈	—	2	4	no	—
552	C₈	—	4	2	no	—
553	C₂²	—	2	2	no	—
554	C₂²	—	2	2	no	—
555	C₂²	—	2	2	no	—
556	C₂²	—	2	2	no	—
557	C₂²	—	2	2	no	—
558	C₂²	—	2	2	no	—
559	C₂²	—	4	2	no	—
560	C₂²	—	4	2	no	—
561	C₂²	—	4	2	no	—
562	C₂²	—	8	1	no	—
563	C₂²	—	8	1	no	—
564	C₂²	—	8	1	no	—
565	C₂²	—	8	1	no	—
566	C₂²	#440×C₆₇	1	4	yes	complemented
567	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
568	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
569	C₂²	#525×C₆₇	1	1	yes	characteristic
570	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
571	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
572	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
573	C₂²	#575×C₆₇	1	1	yes	characteristic
574	C₄	—	2	8	no	—
575	C₄	—	2	2	no	—
576	C₄	—	2	2	no	—
577	C₄	—	2	2	no	—
578	C₄	—	2	2	no	—
579	C₄	—	4	2	no	—
580	C₄	—	4	1	no	—
581	C₄	—	4	4	no	—
582	C₄	—	8	1	no	—
583	C₄	—	8	1	no	—
584	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
585	C₄	#525×C₆₇	1	2	yes	—
586	C₂	—	2	4	no	—
587	C₂	—	8	2	no	—
588	C₂	C₄×#120×C₆₇	1	1	yes	characteristic
589	C₂	C₂×#440×C₆₇	1	1	yes	characteristic
590	C₂	C₄×#137×C₆₇	1	1	yes	characteristic
591	C₂	C₆₇×#3004	1	1	yes	characteristic
592	C₂	C₆₇×#3006	1	1	yes	characteristic
593	C₂	C₆₇×#3112	1	1	yes	characteristic
594	C₂	C₆₇×#3117	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#121×C₆₇	C₂²
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
1. C₂²×C₄×Q₈×C₆₇	C₂	semidirect
3. C₂×#440×C₆₇	C₂	semidirect
5. C₆₇×#1880	C₂	semidirect
7. C₆₇×#1863	C₂	semidirect
8. C₆₇×#1852	C₂	semidirect
9. C₆₇×#1787	C₂	semidirect
10. C₆₇×#1763	C₂	semidirect
11. C₆₇×#1749	C₂	semidirect
26. #448×C₆₇	C₄	semidirect
50. C₂²×Q₈×C₆₇	C₂×C₄	semidirect
62. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
72. C₄×Q₈×C₆₇	D₄	semidirect
102. #103×C₆₇	C₂×C₄	semidirect
105. C₂²⋊C₈×C₆₇	C₂×C₄	semidirect
208. Q₈×C₆₇	C₄×D₄	semidirect
268. C₂²×C₆₇	#440	semidirect
288. #21454	C₆₇	direct (N × H)
298. C₂²×C₄×Q₈	C₂×C₆₇	semidirect
300. C₂×#440	C₂×C₆₇	semidirect
302. #1880	C₂×C₆₇	semidirect
304. #1863	C₂×C₆₇	semidirect
305. #1852	C₂×C₆₇	semidirect
306. #1787	C₂×C₆₇	semidirect
307. #1763	C₂×C₆₇	semidirect
308. #1749	C₂×C₆₇	semidirect
309. C₆₇	#21454	direct (N × H)
324. #448	C₄×C₆₇	semidirect
348. C₂²×Q₈	C₂×C₄×C₆₇	semidirect
360. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
370. C₄×Q₈	D₄×C₆₇	semidirect
400. #103	C₂×C₄×C₆₇	semidirect
403. C₂²⋊C₈	C₂×C₄×C₆₇	semidirect
506. Q₈	C₄×D₄×C₆₇	semidirect
566. C₂²	#440×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₆₇	2
C₂×#448×C₆₇	2
C₂×#440×C₆₇	2
C₄×#121×C₆₇	2
C₆₇×#1880	2
C₆₇×#1874	2
C₆₇×#1863	2
C₆₇×#1852	2
C₆₇×#1787	2
C₆₇×#1763	2
C₆₇×#1749	2
#21454	67

Subgroups — C67×#21454 #30700705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21454 #30700705