Subgroups — C₆₇×#33048 #30712299

Order 17152.

1054 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#486×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2857	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2856	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1990	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1957	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1936	C₂	1	4	yes	maximal, complemented
8	C₆₇×#1895	C₂	1	4	yes	maximal, complemented
9	C₆₇×#1834	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
14	C₂×#121×C₆₇	—	2	1	no	—
15	C₂×#121×C₆₇	—	2	1	no	—
16	C₂×#121×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	#486×C₆₇	—	2	4	no	—
19	C₂×#102×C₆₇	C₂²	1	4	yes	—
20	C₂×#102×C₆₇	C₂²	1	4	yes	—
21	C₄×M₁₆×C₆₇	—	2	4	no	—
22	C₂×#97×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	#391×C₆₇	—	2	1	no	—
26	#390×C₆₇	—	2	1	no	—
27	#390×C₆₇	—	2	1	no	—
28	#390×C₆₇	C₂²	1	2	yes	—
29	#389×C₆₇	—	2	1	no	—
30	#389×C₆₇	C₂²	1	2	yes	—
31	#384×C₆₇	C₂²	1	1	yes	characteristic
32	#384×C₆₇	C₂²	1	1	yes	characteristic
33	#380×C₆₇	—	2	1	no	—
34	#380×C₆₇	C₂²	1	2	yes	—
35	#340×C₆₇	C₂²	1	4	yes	—
36	#340×C₆₇	C₂²	1	4	yes	—
37	C₂²×Q₈×C₆₇	—	2	1	no	—
38	C₂²×D₄×C₆₇	—	2	2	no	—
39	C₂²×D₄×C₆₇	—	2	1	no	—
40	C₂²×D₄×C₆₇	—	2	1	no	—
41	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
42	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₆₇	—	2	4	no	—
46	C₂×M₁₆×C₆₇	—	2	4	no	—
47	C₂×M₁₆×C₆₇	—	2	4	no	—
48	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₆₇	—	2	2	no	—
51	C₄⋊Q₈×C₆₇	—	2	2	no	—
52	#127×C₆₇	—	2	2	no	—
53	#127×C₆₇	—	2	2	no	—
54	C₂²⋊Q₈×C₆₇	—	4	4	no	—
55	#121×C₆₇	—	2	8	no	—
56	#121×C₆₇	—	4	4	no	—
57	#121×C₆₇	—	4	4	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
67	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
71	C₂×C₄²×C₆₇	—	2	2	no	—
72	C₂×C₄²×C₆₇	—	2	2	no	—
73	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	#102×C₆₇	—	2	8	no	—
78	#102×C₆₇	—	2	8	no	—
79	#97×C₆₇	—	2	4	no	—
80	C₄×C₈×C₆₇	—	2	2	no	—
81	C₄×C₈×C₆₇	D₄	1	4	yes	complemented
82	#95×C₆₇	—	2	2	no	—
83	#95×C₆₇	—	2	2	no	—
84	C₂⁴×C₆₇	—	2	2	no	—
85	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
86	C₂×Q₈×C₆₇	—	4	2	no	—
87	C₂×Q₈×C₆₇	—	4	2	no	—
88	C₂×Q₈×C₆₇	—	4	2	no	—
89	C₂×D₄×C₆₇	—	2	2	no	—
90	C₂×D₄×C₆₇	—	2	4	no	—
91	C₂×D₄×C₆₇	—	2	4	no	—
92	C₂×D₄×C₆₇	—	2	2	no	—
93	C₂×D₄×C₆₇	—	4	4	no	—
94	C₂×D₄×C₆₇	—	4	4	no	—
95	C₂×D₄×C₆₇	—	4	4	no	—
96	C₂×D₄×C₆₇	—	4	2	no	—
97	C₂×D₄×C₆₇	—	4	2	no	—
98	C₂×D₄×C₆₇	—	4	2	no	—
99	C₂×D₄×C₆₇	—	4	2	no	—
100	C₂×D₄×C₆₇	—	4	2	no	—
101	C₂×D₄×C₆₇	—	4	2	no	—
102	C₂²×C₄×C₆₇	—	2	1	no	—
103	C₂²×C₄×C₆₇	—	2	1	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	2	2	no	—
112	C₂²×C₄×C₆₇	—	2	2	no	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	#41	1	4	yes	—
120	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
121	M₁₆×C₆₇	—	2	16	no	—
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
127	C₂×C₈×C₆₇	#41	1	4	yes	—
128	C₄⋊C₄×C₆₇	—	2	2	no	—
129	C₄⋊C₄×C₆₇	—	2	4	no	—
130	C₄⋊C₄×C₆₇	—	2	4	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	4	4	no	—
133	C₄⋊C₄×C₆₇	—	4	2	no	—
134	C₄⋊C₄×C₆₇	—	4	2	no	—
135	C₄⋊C₄×C₆₇	—	4	2	no	—
136	C₄⋊C₄×C₆₇	—	4	2	no	—
137	C₄⋊C₄×C₆₇	—	4	2	no	—
138	C₂²⋊C₄×C₆₇	—	4	4	no	—
139	C₂²⋊C₄×C₆₇	—	4	4	no	—
140	C₂²⋊C₄×C₆₇	—	4	4	no	—
141	C₂²⋊C₄×C₆₇	—	4	4	no	—
142	C₄²×C₆₇	—	2	2	no	—
143	C₄²×C₆₇	—	2	2	no	—
144	C₄²×C₆₇	—	2	1	no	—
145	C₄²×C₆₇	—	2	1	no	—
146	C₄²×C₆₇	C₂×D₄	1	2	yes	—
147	C₄²×C₆₇	#41	1	2	yes	—
148	C₂³×C₆₇	—	2	1	no	—
149	C₂³×C₆₇	—	2	1	no	—
150	C₂³×C₆₇	—	2	1	no	—
151	C₂³×C₆₇	—	2	1	no	—
152	C₂³×C₆₇	—	2	1	no	—
153	C₂³×C₆₇	—	2	1	no	—
154	C₂³×C₆₇	—	4	2	no	—
155	C₂³×C₆₇	—	4	2	no	—
156	C₂³×C₆₇	—	4	2	no	—
157	C₂³×C₆₇	—	4	2	no	—
158	C₂³×C₆₇	—	4	2	no	—
159	C₂³×C₆₇	—	4	2	no	—
160	C₂³×C₆₇	—	4	2	no	—
161	C₂³×C₆₇	#124	1	2	yes	—
162	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
163	Q₈×C₆₇	—	8	4	no	—
164	D₄×C₆₇	—	4	4	no	—
165	D₄×C₆₇	—	4	4	no	—
166	D₄×C₆₇	—	8	8	no	—
167	D₄×C₆₇	—	8	4	no	—
168	D₄×C₆₇	—	8	4	no	—
169	C₂×C₄×C₆₇	—	2	1	no	—
170	C₂×C₄×C₆₇	—	2	1	no	—
171	C₂×C₄×C₆₇	—	2	4	no	—
172	C₂×C₄×C₆₇	—	2	4	no	—
173	C₂×C₄×C₆₇	—	2	1	no	—
174	C₂×C₄×C₆₇	—	2	1	no	—
175	C₂×C₄×C₆₇	—	2	1	no	—
176	C₂×C₄×C₆₇	—	2	1	no	—
177	C₂×C₄×C₆₇	—	2	1	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	4	4	no	—
180	C₂×C₄×C₆₇	—	4	4	no	—
181	C₂×C₄×C₆₇	—	4	2	no	—
182	C₂×C₄×C₆₇	—	4	2	no	—
183	C₂×C₄×C₆₇	—	4	2	no	—
184	C₂×C₄×C₆₇	—	4	2	no	—
185	C₂×C₄×C₆₇	—	4	2	no	—
186	C₂×C₄×C₆₇	—	4	2	no	—
187	C₂×C₄×C₆₇	—	4	2	no	—
188	C₂×C₄×C₆₇	—	4	2	no	—
189	C₂×C₄×C₆₇	—	4	2	no	—
190	C₂×C₄×C₆₇	#121	1	2	yes	—
191	C₂×C₄×C₆₇	#121	1	2	yes	—
192	C₂×C₄×C₆₇	#124	1	2	yes	—
193	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
194	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
195	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
196	C₂×C₄×C₆₇	#137	1	2	yes	—
197	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
199	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
200	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
201	C₈×C₆₇	—	2	4	no	—
202	C₈×C₆₇	—	2	4	no	—
203	C₂²×C₆₇	—	2	2	no	—
204	C₂²×C₆₇	—	2	2	no	—
205	C₂²×C₆₇	—	2	2	no	—
206	C₂²×C₆₇	—	4	1	no	—
207	C₂²×C₆₇	—	4	1	no	—
208	C₂²×C₆₇	—	4	1	no	—
209	C₂²×C₆₇	—	4	1	no	—
210	C₂²×C₆₇	—	8	2	no	—
211	C₂²×C₆₇	—	8	2	no	—
212	C₂²×C₆₇	—	8	2	no	—
213	C₂²×C₆₇	—	8	2	no	—
214	C₂²×C₆₇	—	8	2	no	—
215	C₂²×C₆₇	—	8	2	no	—
216	C₂²×C₆₇	—	8	2	no	—
217	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
218	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
219	C₂²×C₆₇	#532	1	1	yes	characteristic
220	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
221	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
222	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
223	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
224	C₄×C₆₇	—	2	2	no	—
225	C₄×C₆₇	—	4	1	no	—
226	C₄×C₆₇	—	4	1	no	—
227	C₄×C₆₇	—	8	2	no	—
228	C₄×C₆₇	—	8	2	no	—
229	C₄×C₆₇	—	8	2	no	—
230	C₄×C₆₇	#468	1	4	yes	—
231	C₄×C₆₇	#469	1	4	yes	—
232	C₄×C₆₇	#525	1	2	yes	—
233	C₄×C₆₇	#529	1	2	yes	—
234	#33048	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
235	C₂×C₆₇	—	4	2	no	—
236	C₂×C₆₇	—	16	2	no	—
237	C₂×C₆₇	#2850	1	1	yes	characteristic
238	C₂×C₆₇	C₂×#468	1	1	yes	characteristic
239	C₂×C₆₇	C₂×#469	1	1	yes	characteristic
240	C₂×C₆₇	#3138	1	1	yes	characteristic
241	C₂×C₆₇	#3141	1	1	yes	characteristic
242	C₂×C₆₇	#3182	1	1	yes	characteristic
243	C₂×C₆₇	#3184	1	1	yes	characteristic
244	C₂×#486	C₂×C₆₇	1	1	yes	characteristic, complemented
245	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
246	#2857	C₂×C₆₇	1	1	yes	characteristic, complemented
247	#2856	C₂×C₆₇	1	1	yes	characteristic
248	#1990	C₂×C₆₇	1	1	yes	characteristic, complemented
249	#1957	C₂×C₆₇	1	1	yes	characteristic, complemented
250	#1936	C₂×C₆₇	1	4	yes	complemented
251	#1895	C₂×C₆₇	1	4	yes	complemented
252	#1834	C₂×C₆₇	1	1	yes	characteristic, complemented
253	C₆₇	#33048	1	1	yes	characteristic, direct product factor, Sylow
254	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
255	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
256	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
257	C₂×C₂²⋊Q₈	—	2	1	no	—
258	C₂×#121	—	2	1	no	—
259	C₂×#121	—	2	1	no	—
260	C₂×#121	C₂²×C₆₇	1	2	yes	—
261	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
262	#486	—	2	4	no	—
263	C₂×#102	C₂²×C₆₇	1	4	yes	—
264	C₂×#102	C₂²×C₆₇	1	4	yes	—
265	C₄×M₁₆	—	2	4	no	—
266	C₂×#97	C₂²×C₆₇	1	2	yes	—
267	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
268	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
269	#391	—	2	1	no	—
270	#390	—	2	1	no	—
271	#390	—	2	1	no	—
272	#390	C₂²×C₆₇	1	2	yes	—
273	#389	—	2	1	no	—
274	#389	C₂²×C₆₇	1	2	yes	—
275	#384	C₂²×C₆₇	1	1	yes	characteristic
276	#384	C₂²×C₆₇	1	1	yes	characteristic
277	#380	—	2	1	no	—
278	#380	C₂²×C₆₇	1	2	yes	—
279	#340	C₂²×C₆₇	1	4	yes	—
280	#340	C₂²×C₆₇	1	4	yes	—
281	C₂²×Q₈	—	2	1	no	—
282	C₂²×D₄	—	2	2	no	—
283	C₂²×D₄	—	2	1	no	—
284	C₂²×D₄	—	2	1	no	—
285	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
286	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
287	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
288	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
289	C₂×M₁₆	—	2	4	no	—
290	C₂×M₁₆	—	2	4	no	—
291	C₂×M₁₆	—	2	4	no	—
292	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
293	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
294	C₄⋊Q₈	—	2	2	no	—
295	C₄⋊Q₈	—	2	2	no	—
296	#127	—	2	2	no	—
297	#127	—	2	2	no	—
298	C₂²⋊Q₈	—	4	4	no	—
299	#121	—	2	8	no	—
300	#121	—	4	4	no	—
301	#121	—	4	4	no	—
302	C₂×C₄⋊C₄	—	2	1	no	—
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄⋊C₄	—	2	1	no	—
305	C₂×C₄⋊C₄	—	2	2	no	—
306	C₂×C₄⋊C₄	—	2	1	no	—
307	C₂×C₄⋊C₄	—	2	1	no	—
308	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
309	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
310	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
311	C₂×C₂²⋊C₄	—	2	2	no	—
312	C₂×C₂²⋊C₄	—	2	2	no	—
313	C₂×C₂²⋊C₄	—	2	2	no	—
314	C₂×C₂²⋊C₄	—	2	2	no	—
315	C₂×C₄²	—	2	2	no	—
316	C₂×C₄²	—	2	2	no	—
317	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
318	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
319	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
320	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
321	#102	—	2	8	no	—
322	#102	—	2	8	no	—
323	#97	—	2	4	no	—
324	C₄×C₈	—	2	2	no	—
325	C₄×C₈	D₄×C₆₇	1	4	yes	complemented
326	#95	—	2	2	no	—
327	#95	—	2	2	no	—
328	C₂⁴	—	2	2	no	—
329	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
330	C₂×Q₈	—	4	2	no	—
331	C₂×Q₈	—	4	2	no	—
332	C₂×Q₈	—	4	2	no	—
333	C₂×D₄	—	2	2	no	—
334	C₂×D₄	—	2	4	no	—
335	C₂×D₄	—	2	4	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	—	4	4	no	—
338	C₂×D₄	—	4	4	no	—
339	C₂×D₄	—	4	4	no	—
340	C₂×D₄	—	4	2	no	—
341	C₂×D₄	—	4	2	no	—
342	C₂×D₄	—	4	2	no	—
343	C₂×D₄	—	4	2	no	—
344	C₂×D₄	—	4	2	no	—
345	C₂×D₄	—	4	2	no	—
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
359	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
360	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
361	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
362	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
363	C₂²×C₄	#41×C₆₇	1	4	yes	—
364	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
365	M₁₆	—	2	16	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	2	2	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
371	C₂×C₈	#41×C₆₇	1	4	yes	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	2	4	no	—
374	C₄⋊C₄	—	2	4	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	4	4	no	—
377	C₄⋊C₄	—	4	2	no	—
378	C₄⋊C₄	—	4	2	no	—
379	C₄⋊C₄	—	4	2	no	—
380	C₄⋊C₄	—	4	2	no	—
381	C₄⋊C₄	—	4	2	no	—
382	C₂²⋊C₄	—	4	4	no	—
383	C₂²⋊C₄	—	4	4	no	—
384	C₂²⋊C₄	—	4	4	no	—
385	C₂²⋊C₄	—	4	4	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	2	2	no	—
388	C₄²	—	2	1	no	—
389	C₄²	—	2	1	no	—
390	C₄²	C₂×D₄×C₆₇	1	2	yes	—
391	C₄²	#41×C₆₇	1	2	yes	—
392	C₂³	—	2	1	no	—
393	C₂³	—	2	1	no	—
394	C₂³	—	2	1	no	—
395	C₂³	—	2	1	no	—
396	C₂³	—	2	1	no	—
397	C₂³	—	2	1	no	—
398	C₂³	—	4	2	no	—
399	C₂³	—	4	2	no	—
400	C₂³	—	4	2	no	—
401	C₂³	—	4	2	no	—
402	C₂³	—	4	2	no	—
403	C₂³	—	4	2	no	—
404	C₂³	—	4	2	no	—
405	C₂³	#124×C₆₇	1	2	yes	—
406	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
407	Q₈	—	8	4	no	—
408	D₄	—	4	4	no	—
409	D₄	—	4	4	no	—
410	D₄	—	8	8	no	—
411	D₄	—	8	4	no	—
412	D₄	—	8	4	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	4	4	no	—
424	C₂×C₄	—	4	4	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	#121×C₆₇	1	2	yes	—
435	C₂×C₄	#121×C₆₇	1	2	yes	—
436	C₂×C₄	#124×C₆₇	1	2	yes	—
437	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
438	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
439	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
440	C₂×C₄	#137×C₆₇	1	2	yes	—
441	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
442	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
443	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
444	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
445	C₈	—	2	4	no	—
446	C₈	—	2	4	no	—
447	C₂²	—	2	2	no	—
448	C₂²	—	2	2	no	—
449	C₂²	—	2	2	no	—
450	C₂²	—	4	1	no	—
451	C₂²	—	4	1	no	—
452	C₂²	—	4	1	no	—
453	C₂²	—	4	1	no	—
454	C₂²	—	8	2	no	—
455	C₂²	—	8	2	no	—
456	C₂²	—	8	2	no	—
457	C₂²	—	8	2	no	—
458	C₂²	—	8	2	no	—
459	C₂²	—	8	2	no	—
460	C₂²	—	8	2	no	—
461	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
462	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
463	C₂²	#532×C₆₇	1	1	yes	characteristic
464	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
465	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
466	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
467	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
468	C₄	—	2	2	no	—
469	C₄	—	4	1	no	—
470	C₄	—	4	1	no	—
471	C₄	—	8	2	no	—
472	C₄	—	8	2	no	—
473	C₄	—	8	2	no	—
474	C₄	#468×C₆₇	1	4	yes	—
475	C₄	#469×C₆₇	1	4	yes	—
476	C₄	#525×C₆₇	1	2	yes	—
477	C₄	#529×C₆₇	1	2	yes	—
478	C₂	—	4	2	no	—
479	C₂	—	16	2	no	—
480	C₂	C₆₇×#2850	1	1	yes	characteristic
481	C₂	C₂×#468×C₆₇	1	1	yes	characteristic
482	C₂	C₂×#469×C₆₇	1	1	yes	characteristic
483	C₂	C₆₇×#3138	1	1	yes	characteristic
484	C₂	C₆₇×#3141	1	1	yes	characteristic
485	C₂	C₆₇×#3182	1	1	yes	characteristic
486	C₂	C₆₇×#3184	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#2856	C₂
LCS	G ▷ C₂²×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. C₂×#486×C₆₇	C₂	semidirect
2. C₂×C₄×M₁₆×C₆₇	C₂	semidirect
3. C₆₇×#2857	C₂	semidirect
5. C₆₇×#1990	C₂	semidirect
6. C₆₇×#1957	C₂	semidirect
7. C₆₇×#1936	C₂	semidirect
8. C₆₇×#1895	C₂	semidirect
9. C₆₇×#1834	C₂	semidirect
81. C₄×C₈×C₆₇	D₄	semidirect
234. #33048	C₆₇	direct (N × H)
244. C₂×#486	C₂×C₆₇	semidirect
245. C₂×C₄×M₁₆	C₂×C₆₇	semidirect
246. #2857	C₂×C₆₇	semidirect
248. #1990	C₂×C₆₇	semidirect
249. #1957	C₂×C₆₇	semidirect
250. #1936	C₂×C₆₇	semidirect
251. #1895	C₂×C₆₇	semidirect
252. #1834	C₂×C₆₇	semidirect
253. C₆₇	#33048	direct (N × H)
325. C₄×C₈	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#486×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₆₇×#2857	2
C₆₇×#2856	2
C₆₇×#1990	2
C₆₇×#1957	2
C₆₇×#1936	2
C₆₇×#1895	2
C₆₇×#1834	2
#33048	67

Subgroups — C67×#33048 #30712299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33048 #30712299