Subgroups — C₆₇×#33143 #30712394

Order 17152.

982 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#469×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2854	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1972	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1970	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1943	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1941	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1936	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1934	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1895	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1809	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#123×C₆₇	—	2	2	no	—
16	C₂×#121×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	#469×C₆₇	—	2	4	no	—
19	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#102×C₆₇	C₂²	1	2	yes	—
23	C₂×#102×C₆₇	C₂²	1	2	yes	—
24	C₂×#102×C₆₇	C₂²	1	2	yes	—
25	C₄×M₁₆×C₆₇	—	2	4	no	—
26	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	#390×C₆₇	C₂²	1	2	yes	—
31	#390×C₆₇	C₂²	1	2	yes	—
32	#389×C₆₇	C₂²	1	2	yes	—
33	#389×C₆₇	C₂²	1	2	yes	—
34	#388×C₆₇	—	2	2	no	—
35	#388×C₆₇	—	2	2	no	—
36	#386×C₆₇	—	2	2	no	—
37	#380×C₆₇	C₂²	1	2	yes	—
38	#340×C₆₇	C₂²	1	2	yes	—
39	#340×C₆₇	C₂²	1	2	yes	—
40	#340×C₆₇	C₂²	1	2	yes	—
41	C₂²×D₄×C₆₇	—	2	2	no	—
42	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
43	C₂³×C₄×C₆₇	D₄	1	2	yes	—
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×D₈×C₆₇	—	2	2	no	—
46	C₂×D₈×C₆₇	—	2	2	no	—
47	C₂×D₈×C₆₇	—	2	2	no	—
48	C₂×M₁₆×C₆₇	—	2	2	no	—
49	C₂×M₁₆×C₆₇	—	2	2	no	—
50	C₂×M₁₆×C₆₇	—	2	2	no	—
51	C₂×M₁₆×C₆₇	—	2	2	no	—
52	C₂×M₁₆×C₆₇	—	2	2	no	—
53	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
54	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
58	#123×C₆₇	—	4	8	no	—
59	#121×C₆₇	—	2	4	no	—
60	#121×C₆₇	—	2	4	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
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×C₂²⋊C₄×C₆₇	—	2	2	no	—
72	C₂×C₄²×C₆₇	—	2	2	no	—
73	C₂×C₄²×C₆₇	—	2	2	no	—
74	C₂×C₄²×C₆₇	D₄	1	2	yes	—
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	#107×C₆₇	—	2	2	no	—
78	#106×C₆₇	—	2	2	no	—
79	#106×C₆₇	—	2	2	no	—
80	#102×C₆₇	—	2	4	no	—
81	#102×C₆₇	—	2	4	no	—
82	#102×C₆₇	—	2	4	no	—
83	#97×C₆₇	—	2	2	no	—
84	#97×C₆₇	—	2	2	no	—
85	C₄×C₈×C₆₇	—	2	2	no	—
86	C₄×C₈×C₆₇	—	2	2	no	—
87	#95×C₆₇	—	2	2	no	—
88	#95×C₆₇	—	2	2	no	—
89	#95×C₆₇	—	2	2	no	—
90	C₂⁴×C₆₇	—	2	2	no	—
91	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂×D₄×C₆₇	—	2	2	no	—
93	C₂×D₄×C₆₇	—	2	2	no	—
94	C₂×D₄×C₆₇	—	2	2	no	—
95	C₂×D₄×C₆₇	—	2	2	no	—
96	C₂×D₄×C₆₇	—	2	2	no	—
97	C₂×D₄×C₆₇	—	2	2	no	—
98	C₂×D₄×C₆₇	—	4	4	no	—
99	C₂×D₄×C₆₇	—	4	4	no	—
100	C₂×D₄×C₆₇	—	4	4	no	—
101	C₂²×C₄×C₆₇	—	2	1	no	—
102	C₂²×C₄×C₆₇	—	2	1	no	—
103	C₂²×C₄×C₆₇	—	2	1	no	—
104	C₂²×C₄×C₆₇	—	2	1	no	—
105	C₂²×C₄×C₆₇	—	2	4	no	—
106	C₂²×C₄×C₆₇	—	2	4	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	—	4	2	no	—
111	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₆₇	#41	1	2	yes	—
116	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
117	D₈×C₆₇	—	4	4	no	—
118	M₁₆×C₆₇	—	2	8	no	—
119	M₁₆×C₆₇	—	2	8	no	—
120	C₂×C₈×C₆₇	—	2	1	no	—
121	C₂×C₈×C₆₇	—	2	1	no	—
122	C₂×C₈×C₆₇	—	2	1	no	—
123	C₂×C₈×C₆₇	—	2	1	no	—
124	C₂×C₈×C₆₇	—	2	1	no	—
125	C₂×C₈×C₆₇	—	2	1	no	—
126	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
127	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
128	C₂×C₈×C₆₇	#41	1	2	yes	—
129	C₂×C₈×C₆₇	#41	1	2	yes	—
130	C₂×C₈×C₆₇	#41	1	2	yes	—
131	C₂×C₈×C₆₇	#41	1	2	yes	—
132	C₄⋊C₄×C₆₇	—	2	2	no	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	—	2	2	no	—
135	C₄⋊C₄×C₆₇	—	2	2	no	—
136	C₄⋊C₄×C₆₇	—	2	2	no	—
137	C₄⋊C₄×C₆₇	—	2	2	no	—
138	C₄⋊C₄×C₆₇	—	4	4	no	—
139	C₄⋊C₄×C₆₇	—	4	4	no	—
140	C₂²⋊C₄×C₆₇	—	4	4	no	—
141	C₂²⋊C₄×C₆₇	—	4	4	no	—
142	C₂²⋊C₄×C₆₇	—	4	4	no	—
143	C₂²⋊C₄×C₆₇	—	4	4	no	—
144	C₂²⋊C₄×C₆₇	—	4	4	no	—
145	C₂²⋊C₄×C₆₇	—	4	4	no	—
146	C₄²×C₆₇	—	2	4	no	—
147	C₄²×C₆₇	—	2	1	no	—
148	C₄²×C₆₇	—	2	1	no	—
149	C₄²×C₆₇	—	2	1	no	—
150	C₄²×C₆₇	—	2	1	no	—
151	C₂³×C₆₇	—	2	1	no	—
152	C₂³×C₆₇	—	2	1	no	—
153	C₂³×C₆₇	—	2	2	no	—
154	C₂³×C₆₇	—	2	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	—	4	2	no	—
162	C₂³×C₆₇	#121	1	2	yes	—
163	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
164	D₄×C₆₇	—	4	2	no	—
165	D₄×C₆₇	—	4	2	no	—
166	D₄×C₆₇	—	4	2	no	—
167	D₄×C₆₇	—	4	2	no	—
168	D₄×C₆₇	—	8	8	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	2	1	no	—
171	C₂×C₄×C₆₇	—	2	1	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	2	2	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	4	8	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	AGL(1,ℤ/8)	1	4	yes	—
192	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
198	C₈×C₆₇	—	2	2	no	—
199	C₈×C₆₇	—	2	2	no	—
200	C₈×C₆₇	—	2	2	no	—
201	C₈×C₆₇	#123	1	4	yes	complemented
202	C₂²×C₆₇	—	2	2	no	—
203	C₂²×C₆₇	—	2	4	no	—
204	C₂²×C₆₇	—	4	2	no	—
205	C₂²×C₆₇	—	4	1	no	—
206	C₂²×C₆₇	—	4	1	no	—
207	C₂²×C₆₇	—	8	2	no	—
208	C₂²×C₆₇	—	8	2	no	—
209	C₂²×C₆₇	—	8	2	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	C₂×#121	1	1	yes	characteristic
215	C₂²×C₆₇	#532	1	1	yes	characteristic
216	C₂²×C₆₇	#532	1	1	yes	characteristic
217	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
218	C₂²×C₆₇	#575	1	2	yes	—
219	C₄×C₆₇	—	2	2	no	—
220	C₄×C₆₇	—	2	4	no	—
221	C₄×C₆₇	—	4	2	no	—
222	C₄×C₆₇	—	8	2	no	—
223	C₄×C₆₇	—	8	2	no	—
224	C₄×C₆₇	—	8	2	no	—
225	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
226	C₄×C₆₇	#525	1	1	yes	characteristic
227	C₄×C₆₇	#529	1	1	yes	characteristic
228	C₄×C₆₇	#529	1	1	yes	characteristic
229	#33143	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
230	C₂×C₆₇	—	4	2	no	—
231	C₂×C₆₇	—	16	2	no	—
232	C₂×C₆₇	#2854	1	1	yes	characteristic
233	C₂×C₆₇	#3055	1	2	yes	—
234	C₂×C₆₇	#3171	1	2	yes	—
235	C₂×C₆₇	#3183	1	1	yes	characteristic
236	C₂×C₆₇	#3184	1	1	yes	characteristic
237	C₂×#469	C₂×C₆₇	1	1	yes	characteristic
238	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
239	#2854	C₂×C₆₇	1	2	yes	complemented
240	#1972	C₂×C₆₇	1	1	yes	characteristic, complemented
241	#1970	C₂×C₆₇	1	1	yes	characteristic, complemented
242	#1943	C₂×C₆₇	1	1	yes	characteristic, complemented
243	#1941	C₂×C₆₇	1	1	yes	characteristic, complemented
244	#1936	C₂×C₆₇	1	2	yes	complemented
245	#1934	C₂×C₆₇	1	2	yes	complemented
246	#1895	C₂×C₆₇	1	2	yes	complemented
247	#1809	C₂×C₆₇	1	1	yes	characteristic, complemented
248	C₆₇	#33143	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
250	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
251	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
252	C₂×#123	—	2	2	no	—
253	C₂×#121	C₂²×C₆₇	1	2	yes	—
254	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
255	#469	—	2	4	no	—
256	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
257	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
258	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
259	C₂×#102	C₂²×C₆₇	1	2	yes	—
260	C₂×#102	C₂²×C₆₇	1	2	yes	—
261	C₂×#102	C₂²×C₆₇	1	2	yes	—
262	C₄×M₁₆	—	2	4	no	—
263	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
264	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
265	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
266	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
267	#390	C₂²×C₆₇	1	2	yes	—
268	#390	C₂²×C₆₇	1	2	yes	—
269	#389	C₂²×C₆₇	1	2	yes	—
270	#389	C₂²×C₆₇	1	2	yes	—
271	#388	—	2	2	no	—
272	#388	—	2	2	no	—
273	#386	—	2	2	no	—
274	#380	C₂²×C₆₇	1	2	yes	—
275	#340	C₂²×C₆₇	1	2	yes	—
276	#340	C₂²×C₆₇	1	2	yes	—
277	#340	C₂²×C₆₇	1	2	yes	—
278	C₂²×D₄	—	2	2	no	—
279	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
280	C₂³×C₄	D₄×C₆₇	1	2	yes	—
281	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
282	C₂×D₈	—	2	2	no	—
283	C₂×D₈	—	2	2	no	—
284	C₂×D₈	—	2	2	no	—
285	C₂×M₁₆	—	2	2	no	—
286	C₂×M₁₆	—	2	2	no	—
287	C₂×M₁₆	—	2	2	no	—
288	C₂×M₁₆	—	2	2	no	—
289	C₂×M₁₆	—	2	2	no	—
290	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
291	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
292	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
293	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
294	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
295	#123	—	4	8	no	—
296	#121	—	2	4	no	—
297	#121	—	2	4	no	—
298	C₂×C₄⋊C₄	—	2	2	no	—
299	C₂×C₄⋊C₄	—	2	2	no	—
300	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
301	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
302	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
303	C₂×C₂²⋊C₄	—	2	2	no	—
304	C₂×C₂²⋊C₄	—	2	2	no	—
305	C₂×C₂²⋊C₄	—	2	2	no	—
306	C₂×C₂²⋊C₄	—	2	2	no	—
307	C₂×C₂²⋊C₄	—	2	2	no	—
308	C₂×C₂²⋊C₄	—	2	2	no	—
309	C₂×C₄²	—	2	2	no	—
310	C₂×C₄²	—	2	2	no	—
311	C₂×C₄²	D₄×C₆₇	1	2	yes	—
312	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
313	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
314	#107	—	2	2	no	—
315	#106	—	2	2	no	—
316	#106	—	2	2	no	—
317	#102	—	2	4	no	—
318	#102	—	2	4	no	—
319	#102	—	2	4	no	—
320	#97	—	2	2	no	—
321	#97	—	2	2	no	—
322	C₄×C₈	—	2	2	no	—
323	C₄×C₈	—	2	2	no	—
324	#95	—	2	2	no	—
325	#95	—	2	2	no	—
326	#95	—	2	2	no	—
327	C₂⁴	—	2	2	no	—
328	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	2	2	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	—	2	2	no	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	—	4	4	no	—
336	C₂×D₄	—	4	4	no	—
337	C₂×D₄	—	4	4	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	4	no	—
343	C₂²×C₄	—	2	4	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	4	2	no	—
348	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
349	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
350	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
351	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
352	C₂²×C₄	#41×C₆₇	1	2	yes	—
353	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
354	D₈	—	4	4	no	—
355	M₁₆	—	2	8	no	—
356	M₁₆	—	2	8	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	—	2	1	no	—
362	C₂×C₈	—	2	1	no	—
363	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
364	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
365	C₂×C₈	#41×C₆₇	1	2	yes	—
366	C₂×C₈	#41×C₆₇	1	2	yes	—
367	C₂×C₈	#41×C₆₇	1	2	yes	—
368	C₂×C₈	#41×C₆₇	1	2	yes	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	2	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	4	4	no	—
376	C₄⋊C₄	—	4	4	no	—
377	C₂²⋊C₄	—	4	4	no	—
378	C₂²⋊C₄	—	4	4	no	—
379	C₂²⋊C₄	—	4	4	no	—
380	C₂²⋊C₄	—	4	4	no	—
381	C₂²⋊C₄	—	4	4	no	—
382	C₂²⋊C₄	—	4	4	no	—
383	C₄²	—	2	4	no	—
384	C₄²	—	2	1	no	—
385	C₄²	—	2	1	no	—
386	C₄²	—	2	1	no	—
387	C₄²	—	2	1	no	—
388	C₂³	—	2	1	no	—
389	C₂³	—	2	1	no	—
390	C₂³	—	2	2	no	—
391	C₂³	—	2	2	no	—
392	C₂³	—	4	2	no	—
393	C₂³	—	4	2	no	—
394	C₂³	—	4	2	no	—
395	C₂³	—	4	2	no	—
396	C₂³	—	4	2	no	—
397	C₂³	—	4	2	no	—
398	C₂³	—	4	2	no	—
399	C₂³	#121×C₆₇	1	2	yes	—
400	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
401	D₄	—	4	2	no	—
402	D₄	—	4	2	no	—
403	D₄	—	4	2	no	—
404	D₄	—	4	2	no	—
405	D₄	—	8	8	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	4	8	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	#121×C₆₇	1	2	yes	—
428	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
429	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
430	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
431	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
432	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
433	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
434	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
435	C₈	—	2	2	no	—
436	C₈	—	2	2	no	—
437	C₈	—	2	2	no	—
438	C₈	#123×C₆₇	1	4	yes	complemented
439	C₂²	—	2	4	no	—
440	C₂²	—	2	2	no	—
441	C₂²	—	4	2	no	—
442	C₂²	—	4	1	no	—
443	C₂²	—	4	1	no	—
444	C₂²	—	8	2	no	—
445	C₂²	—	8	2	no	—
446	C₂²	—	8	2	no	—
447	C₂²	—	8	2	no	—
448	C₂²	—	8	2	no	—
449	C₂²	—	8	2	no	—
450	C₂²	—	8	2	no	—
451	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
452	C₂²	#532×C₆₇	1	1	yes	characteristic
453	C₂²	#532×C₆₇	1	1	yes	characteristic
454	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
455	C₂²	#575×C₆₇	1	2	yes	—
456	C₄	—	2	4	no	—
457	C₄	—	2	2	no	—
458	C₄	—	4	2	no	—
459	C₄	—	8	2	no	—
460	C₄	—	8	2	no	—
461	C₄	—	8	2	no	—
462	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
463	C₄	#525×C₆₇	1	1	yes	characteristic
464	C₄	#529×C₆₇	1	1	yes	characteristic
465	C₄	#529×C₆₇	1	1	yes	characteristic
466	C₂	—	4	2	no	—
467	C₂	—	16	2	no	—
468	C₂	C₆₇×#2854	1	1	yes	characteristic
469	C₂	C₆₇×#3055	1	2	yes	—
470	C₂	C₆₇×#3171	1	2	yes	—
471	C₂	C₆₇×#3183	1	1	yes	characteristic
472	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	C₂×#469×C₆₇	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
2. C₂×C₄×M₁₆×C₆₇	C₂	semidirect
3. C₆₇×#2854	C₂	semidirect
4. C₆₇×#1972	C₂	semidirect
5. C₆₇×#1970	C₂	semidirect
6. C₆₇×#1943	C₂	semidirect
7. C₆₇×#1941	C₂	semidirect
8. C₆₇×#1936	C₂	semidirect
9. C₆₇×#1934	C₂	semidirect
10. C₆₇×#1895	C₂	semidirect
11. C₆₇×#1809	C₂	semidirect
201. C₈×C₆₇	#123	semidirect
229. #33143	C₆₇	direct (N × H)
238. C₂×C₄×M₁₆	C₂×C₆₇	semidirect
239. #2854	C₂×C₆₇	semidirect
240. #1972	C₂×C₆₇	semidirect
241. #1970	C₂×C₆₇	semidirect
242. #1943	C₂×C₆₇	semidirect
243. #1941	C₂×C₆₇	semidirect
244. #1936	C₂×C₆₇	semidirect
245. #1934	C₂×C₆₇	semidirect
246. #1895	C₂×C₆₇	semidirect
247. #1809	C₂×C₆₇	semidirect
248. C₆₇	#33143	direct (N × H)
438. C₈	#123×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#469×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₆₇×#2854	2
C₆₇×#1972	2
C₆₇×#1970	2
C₆₇×#1943	2
C₆₇×#1941	2
C₆₇×#1936	2
C₆₇×#1934	2
C₆₇×#1895	2
C₆₇×#1809	2
#33143	67

Subgroups — C67×#33143 #30712394

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33143 #30712394