Subgroups — C₆₇×#32840 #30712091

Order 17152.

648 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2862	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#2860	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2101	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2092	C₂	1	2	yes	maximal
5	C₆₇×#2085	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2073	C₂	1	2	yes	maximal
7	C₆₇×#2058	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2057	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2023	C₂	1	1	yes	maximal, characteristic
10	C₆₇×#2023	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1839	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#126×C₆₇	—	2	1	no	—
15	C₂×#107×C₆₇	C₂²	1	2	yes	—
16	C₂×#106×C₆₇	C₂²	1	2	yes	—
17	C₂×#102×C₆₇	C₂²	1	2	yes	—
18	C₂×#102×C₆₇	C₂²	1	2	yes	—
19	#391×C₆₇	—	2	1	no	—
20	#391×C₆₇	—	2	1	no	—
21	#391×C₆₇	—	2	1	no	—
22	#391×C₆₇	—	2	1	no	—
23	#390×C₆₇	C₂²	1	1	yes	characteristic
24	#390×C₆₇	C₂²	1	1	yes	characteristic
25	#390×C₆₇	C₂²	1	1	yes	characteristic
26	#389×C₆₇	C₂²	1	1	yes	characteristic
27	#389×C₆₇	C₂²	1	1	yes	characteristic
28	#389×C₆₇	C₂²	1	1	yes	characteristic
29	#388×C₆₇	—	2	1	no	—
30	#388×C₆₇	—	2	1	no	—
31	#388×C₆₇	—	2	1	no	—
32	#382×C₆₇	C₂²	1	2	yes	—
33	#382×C₆₇	C₂²	1	2	yes	—
34	C₄³×C₆₇	C₂²	1	1	yes	characteristic
35	#340×C₆₇	C₂²	1	2	yes	—
36	#340×C₆₇	C₂²	1	2	yes	—
37	#340×C₆₇	C₂²	1	2	yes	—
38	#340×C₆₇	C₂²	1	2	yes	—
39	#336×C₆₇	C₂²	1	1	yes	characteristic
40	#336×C₆₇	C₂²	1	1	yes	characteristic
41	#336×C₆₇	C₂²	1	2	yes	—
42	#336×C₆₇	C₂²	1	2	yes	—
43	C₂²×Q₈×C₆₇	—	2	1	no	—
44	C₂²×Q₈×C₆₇	—	2	1	no	—
45	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₄⋊Q₈×C₆₇	—	4	2	no	—
49	C₄⋊Q₈×C₆₇	—	4	2	no	—
50	#126×C₆₇	—	4	4	no	—
51	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
56	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
62	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
63	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
64	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	#107×C₆₇	—	4	2	no	—
75	#106×C₆₇	—	4	2	no	—
76	#102×C₆₇	—	4	2	no	—
77	#102×C₆₇	—	4	2	no	—
78	#95×C₆₇	—	2	2	no	—
79	#95×C₆₇	—	2	2	no	—
80	#95×C₆₇	—	2	2	no	—
81	#95×C₆₇	—	2	2	no	—
82	#95×C₆₇	—	2	1	no	—
83	#95×C₆₇	—	2	1	no	—
84	#95×C₆₇	—	2	1	no	—
85	C₂⁴×C₆₇	—	2	1	no	—
86	C₂×Q₈×C₆₇	—	4	2	no	—
87	C₂×Q₈×C₆₇	—	4	2	no	—
88	C₂×Q₈×C₆₇	—	4	2	no	—
89	C₂×Q₈×C₆₇	—	4	2	no	—
90	C₂×Q₈×C₆₇	—	4	2	no	—
91	C₂×Q₈×C₆₇	—	4	2	no	—
92	C₂×D₄×C₆₇	—	2	2	no	—
93	C₂×D₄×C₆₇	—	2	2	no	—
94	C₂×D₄×C₆₇	—	2	1	no	—
95	C₂×D₄×C₆₇	—	2	1	no	—
96	C₂²×C₄×C₆₇	—	2	1	no	—
97	C₂²×C₄×C₆₇	—	2	1	no	—
98	C₂²×C₄×C₆₇	—	2	1	no	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
108	C₂×C₈×C₆₇	—	2	2	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	2	2	no	—
118	C₄⋊C₄×C₆₇	—	2	2	no	—
119	C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	2	1	no	—
121	C₄⋊C₄×C₆₇	—	2	1	no	—
122	C₄⋊C₄×C₆₇	—	4	2	no	—
123	C₄⋊C₄×C₆₇	—	4	2	no	—
124	C₄⋊C₄×C₆₇	—	4	2	no	—
125	C₄⋊C₄×C₆₇	—	4	2	no	—
126	C₄⋊C₄×C₆₇	—	4	2	no	—
127	C₂²⋊C₄×C₆₇	—	4	2	no	—
128	C₂²⋊C₄×C₆₇	—	4	2	no	—
129	C₂²⋊C₄×C₆₇	—	4	2	no	—
130	C₂²⋊C₄×C₆₇	—	4	2	no	—
131	C₂²⋊C₄×C₆₇	—	4	2	no	—
132	C₂²⋊C₄×C₆₇	—	4	2	no	—
133	C₄²×C₆₇	—	2	2	no	—
134	C₄²×C₆₇	—	2	2	no	—
135	C₄²×C₆₇	—	2	2	no	—
136	C₄²×C₆₇	—	2	2	no	—
137	C₄²×C₆₇	—	2	1	no	—
138	C₄²×C₆₇	—	2	1	no	—
139	C₄²×C₆₇	—	4	1	no	—
140	C₄²×C₆₇	—	4	1	no	—
141	C₂³×C₆₇	—	4	1	no	—
142	C₂³×C₆₇	—	4	1	no	—
143	C₂³×C₆₇	—	4	1	no	—
144	C₂³×C₆₇	—	4	1	no	—
145	C₂³×C₆₇	—	4	1	no	—
146	C₂³×C₆₇	—	4	1	no	—
147	C₂³×C₆₇	—	4	1	no	—
148	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
149	Q₈×C₆₇	—	8	4	no	—
150	Q₈×C₆₇	—	8	4	no	—
151	D₄×C₆₇	—	4	2	no	—
152	D₄×C₆₇	—	4	2	no	—
153	C₂×C₄×C₆₇	—	2	1	no	—
154	C₂×C₄×C₆₇	—	2	1	no	—
155	C₂×C₄×C₆₇	—	2	1	no	—
156	C₂×C₄×C₆₇	—	2	1	no	—
157	C₂×C₄×C₆₇	—	2	1	no	—
158	C₂×C₄×C₆₇	—	2	1	no	—
159	C₂×C₄×C₆₇	—	2	1	no	—
160	C₂×C₄×C₆₇	—	2	1	no	—
161	C₂×C₄×C₆₇	—	2	1	no	—
162	C₂×C₄×C₆₇	—	2	1	no	—
163	C₂×C₄×C₆₇	—	2	1	no	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	2	1	no	—
166	C₂×C₄×C₆₇	—	2	1	no	—
167	C₂×C₄×C₆₇	—	2	1	no	—
168	C₂×C₄×C₆₇	—	2	1	no	—
169	C₂×C₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	4	2	no	—
171	C₂×C₄×C₆₇	—	4	1	no	—
172	C₂×C₄×C₆₇	—	4	1	no	—
173	C₂×C₄×C₆₇	—	4	1	no	—
174	C₂×C₄×C₆₇	—	4	1	no	—
175	C₂×C₄×C₆₇	—	4	2	no	—
176	C₂×C₄×C₆₇	—	4	1	no	—
177	C₂×C₄×C₆₇	—	4	1	no	—
178	C₂×C₄×C₆₇	—	4	1	no	—
179	C₂×C₄×C₆₇	—	4	2	no	—
180	C₂×C₄×C₆₇	—	4	1	no	—
181	C₂×C₄×C₆₇	—	4	1	no	—
182	C₂×C₄×C₆₇	—	4	2	no	—
183	C₂×C₄×C₆₇	—	4	2	no	—
184	C₂×C₄×C₆₇	#135	1	2	yes	—
185	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
186	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
187	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
188	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
189	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
190	C₈×C₆₇	—	4	2	no	—
191	C₈×C₆₇	—	4	2	no	—
192	C₂²×C₆₇	—	8	1	no	—
193	C₂²×C₆₇	—	8	1	no	—
194	C₂²×C₆₇	—	8	1	no	—
195	C₂²×C₆₇	—	8	1	no	—
196	C₂²×C₆₇	—	8	1	no	—
197	C₂²×C₆₇	—	8	1	no	—
198	C₂²×C₆₇	—	8	1	no	—
199	C₂²×C₆₇	#535	1	1	yes	characteristic
200	C₂²×C₆₇	#536	1	1	yes	characteristic
201	C₂²×C₆₇	#536	1	1	yes	characteristic
202	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
203	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
204	C₂²×C₆₇	#575	1	1	yes	characteristic
205	C₂²×C₆₇	#575	1	1	yes	characteristic
206	C₄×C₆₇	—	2	2	no	—
207	C₄×C₆₇	—	2	2	no	—
208	C₄×C₆₇	—	4	1	no	—
209	C₄×C₆₇	—	4	1	no	—
210	C₄×C₆₇	—	4	1	no	—
211	C₄×C₆₇	—	4	1	no	—
212	C₄×C₆₇	—	8	2	no	—
213	C₄×C₆₇	—	8	1	no	—
214	C₄×C₆₇	—	8	2	no	—
215	C₄×C₆₇	—	8	1	no	—
216	C₄×C₆₇	—	8	1	no	—
217	C₄×C₆₇	#563	1	2	yes	—
218	C₄×C₆₇	#563	1	2	yes	—
219	#32840	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
220	C₂×C₆₇	—	16	1	no	—
221	C₂×C₆₇	#2821	1	1	yes	characteristic
222	C₂×C₆₇	#3211	1	1	yes	characteristic
223	C₂×C₆₇	#3213	1	1	yes	characteristic
224	C₂×C₆₇	#3228	1	1	yes	characteristic
225	C₂×C₆₇	#3233	1	1	yes	characteristic
226	C₂×C₆₇	#3235	1	1	yes	characteristic
227	C₂×C₆₇	#3240	1	1	yes	characteristic
228	#2862	C₂×C₆₇	1	1	yes	characteristic, complemented
229	#2860	C₂×C₆₇	1	1	yes	characteristic
230	#2101	C₂×C₆₇	1	2	yes	complemented
231	#2092	C₂×C₆₇	1	2	yes	—
232	#2085	C₂×C₆₇	1	2	yes	complemented
233	#2073	C₂×C₆₇	1	2	yes	—
234	#2058	C₂×C₆₇	1	1	yes	characteristic, complemented
235	#2057	C₂×C₆₇	1	1	yes	characteristic, complemented
236	#2023	C₂×C₆₇	1	1	yes	characteristic
237	#2023	C₂×C₆₇	1	1	yes	characteristic
238	#1839	C₂×C₆₇	1	1	yes	characteristic, complemented
239	C₆₇	#32840	1	1	yes	characteristic, direct product factor, Sylow
240	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
241	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
242	C₂×#126	—	2	1	no	—
243	C₂×#107	C₂²×C₆₇	1	2	yes	—
244	C₂×#106	C₂²×C₆₇	1	2	yes	—
245	C₂×#102	C₂²×C₆₇	1	2	yes	—
246	C₂×#102	C₂²×C₆₇	1	2	yes	—
247	#391	—	2	1	no	—
248	#391	—	2	1	no	—
249	#391	—	2	1	no	—
250	#391	—	2	1	no	—
251	#390	C₂²×C₆₇	1	1	yes	characteristic
252	#390	C₂²×C₆₇	1	1	yes	characteristic
253	#390	C₂²×C₆₇	1	1	yes	characteristic
254	#389	C₂²×C₆₇	1	1	yes	characteristic
255	#389	C₂²×C₆₇	1	1	yes	characteristic
256	#389	C₂²×C₆₇	1	1	yes	characteristic
257	#388	—	2	1	no	—
258	#388	—	2	1	no	—
259	#388	—	2	1	no	—
260	#382	C₂²×C₆₇	1	2	yes	—
261	#382	C₂²×C₆₇	1	2	yes	—
262	C₄³	C₂²×C₆₇	1	1	yes	characteristic
263	#340	C₂²×C₆₇	1	2	yes	—
264	#340	C₂²×C₆₇	1	2	yes	—
265	#340	C₂²×C₆₇	1	2	yes	—
266	#340	C₂²×C₆₇	1	2	yes	—
267	#336	C₂²×C₆₇	1	1	yes	characteristic
268	#336	C₂²×C₆₇	1	1	yes	characteristic
269	#336	C₂²×C₆₇	1	2	yes	—
270	#336	C₂²×C₆₇	1	2	yes	—
271	C₂²×Q₈	—	2	1	no	—
272	C₂²×Q₈	—	2	1	no	—
273	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
274	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
275	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
276	C₄⋊Q₈	—	4	2	no	—
277	C₄⋊Q₈	—	4	2	no	—
278	#126	—	4	4	no	—
279	C₂×C₄⋊C₄	—	2	1	no	—
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄⋊C₄	—	2	1	no	—
283	C₂×C₄⋊C₄	—	2	1	no	—
284	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
285	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
286	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
287	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
288	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
289	C₂×C₂²⋊C₄	—	2	1	no	—
290	C₂×C₂²⋊C₄	—	2	1	no	—
291	C₂×C₂²⋊C₄	—	2	1	no	—
292	C₂×C₂²⋊C₄	—	2	1	no	—
293	C₂×C₂²⋊C₄	—	2	1	no	—
294	C₂×C₂²⋊C₄	—	2	1	no	—
295	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
296	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
297	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
298	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
299	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
300	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
301	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
302	#107	—	4	2	no	—
303	#106	—	4	2	no	—
304	#102	—	4	2	no	—
305	#102	—	4	2	no	—
306	#95	—	2	2	no	—
307	#95	—	2	2	no	—
308	#95	—	2	2	no	—
309	#95	—	2	2	no	—
310	#95	—	2	1	no	—
311	#95	—	2	1	no	—
312	#95	—	2	1	no	—
313	C₂⁴	—	2	1	no	—
314	C₂×Q₈	—	4	2	no	—
315	C₂×Q₈	—	4	2	no	—
316	C₂×Q₈	—	4	2	no	—
317	C₂×Q₈	—	4	2	no	—
318	C₂×Q₈	—	4	2	no	—
319	C₂×Q₈	—	4	2	no	—
320	C₂×D₄	—	2	2	no	—
321	C₂×D₄	—	2	2	no	—
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	—	2	1	no	—
324	C₂²×C₄	—	2	2	no	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
330	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
331	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
332	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
333	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
334	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
335	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	4	2	no	—
351	C₄⋊C₄	—	4	2	no	—
352	C₄⋊C₄	—	4	2	no	—
353	C₄⋊C₄	—	4	2	no	—
354	C₄⋊C₄	—	4	2	no	—
355	C₂²⋊C₄	—	4	2	no	—
356	C₂²⋊C₄	—	4	2	no	—
357	C₂²⋊C₄	—	4	2	no	—
358	C₂²⋊C₄	—	4	2	no	—
359	C₂²⋊C₄	—	4	2	no	—
360	C₂²⋊C₄	—	4	2	no	—
361	C₄²	—	2	2	no	—
362	C₄²	—	2	2	no	—
363	C₄²	—	2	2	no	—
364	C₄²	—	2	2	no	—
365	C₄²	—	2	1	no	—
366	C₄²	—	2	1	no	—
367	C₄²	—	4	1	no	—
368	C₄²	—	4	1	no	—
369	C₂³	—	4	1	no	—
370	C₂³	—	4	1	no	—
371	C₂³	—	4	1	no	—
372	C₂³	—	4	1	no	—
373	C₂³	—	4	1	no	—
374	C₂³	—	4	1	no	—
375	C₂³	—	4	1	no	—
376	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
377	Q₈	—	8	4	no	—
378	Q₈	—	8	4	no	—
379	D₄	—	4	2	no	—
380	D₄	—	4	2	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	—	4	1	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	1	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	#135×C₆₇	1	2	yes	—
413	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
414	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
415	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
416	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
417	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
418	C₈	—	4	2	no	—
419	C₈	—	4	2	no	—
420	C₂²	—	8	1	no	—
421	C₂²	—	8	1	no	—
422	C₂²	—	8	1	no	—
423	C₂²	—	8	1	no	—
424	C₂²	—	8	1	no	—
425	C₂²	—	8	1	no	—
426	C₂²	—	8	1	no	—
427	C₂²	#535×C₆₇	1	1	yes	characteristic
428	C₂²	#536×C₆₇	1	1	yes	characteristic
429	C₂²	#536×C₆₇	1	1	yes	characteristic
430	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
431	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
432	C₂²	#575×C₆₇	1	1	yes	characteristic
433	C₂²	#575×C₆₇	1	1	yes	characteristic
434	C₄	—	2	2	no	—
435	C₄	—	2	2	no	—
436	C₄	—	4	1	no	—
437	C₄	—	4	1	no	—
438	C₄	—	4	1	no	—
439	C₄	—	4	1	no	—
440	C₄	—	8	2	no	—
441	C₄	—	8	2	no	—
442	C₄	—	8	1	no	—
443	C₄	—	8	1	no	—
444	C₄	—	8	1	no	—
445	C₄	#563×C₆₇	1	2	yes	—
446	C₄	#563×C₆₇	1	2	yes	—
447	C₂	—	16	1	no	—
448	C₂	C₆₇×#2821	1	1	yes	characteristic
449	C₂	C₆₇×#3211	1	1	yes	characteristic
450	C₂	C₆₇×#3213	1	1	yes	characteristic
451	C₂	C₆₇×#3228	1	1	yes	characteristic
452	C₂	C₆₇×#3233	1	1	yes	characteristic
453	C₂	C₆₇×#3235	1	1	yes	characteristic
454	C₂	C₆₇×#3240	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2862	C₂	semidirect
3. C₆₇×#2101	C₂	semidirect
5. C₆₇×#2085	C₂	semidirect
7. C₆₇×#2058	C₂	semidirect
8. C₆₇×#2057	C₂	semidirect
11. C₆₇×#1839	C₂	semidirect
219. #32840	C₆₇	direct (N × H)
228. #2862	C₂×C₆₇	semidirect
230. #2101	C₂×C₆₇	semidirect
232. #2085	C₂×C₆₇	semidirect
234. #2058	C₂×C₆₇	semidirect
235. #2057	C₂×C₆₇	semidirect
238. #1839	C₂×C₆₇	semidirect
239. C₆₇	#32840	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2862	2
C₆₇×#2860	2
C₆₇×#2101	2
C₆₇×#2092	2
C₆₇×#2085	2
C₆₇×#2073	2
C₆₇×#2058	2
C₆₇×#2057	2
C₆₇×#2023	2
C₆₇×#1839	2
#32840	67

Subgroups — C67×#32840 #30712091

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32840 #30712091