Subgroups — C₆₇×#33164 #30712415

Order 17152.

1246 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#463×C₆₇	C₂	1	4	yes	maximal, complemented
2	C₂×#461×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#459×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2858	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄²×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1983	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1982	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1981	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1842	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
11	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#124×C₆₇	C₂²	1	4	yes	—
15	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₆₇	—	2	1	no	—
18	#463×C₆₇	—	2	16	no	—
19	#461×C₆₇	—	2	4	no	—
20	#459×C₆₇	—	2	4	no	—
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
24	C₂×#103×C₆₇	C₂²	1	2	yes	—
25	C₂×#103×C₆₇	C₂²	1	4	yes	—
26	C₂×#102×C₆₇	C₂²	1	2	yes	—
27	C₂×#102×C₆₇	C₂²	1	4	yes	—
28	#390×C₆₇	—	2	1	no	—
29	#390×C₆₇	—	2	1	no	—
30	#390×C₆₇	—	2	2	no	—
31	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
33	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
34	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
35	C₄³×C₆₇	C₂²	1	1	yes	characteristic
36	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
37	C₂²×D₄×C₆₇	—	2	1	no	—
38	C₂²×D₄×C₆₇	—	2	1	no	—
39	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
40	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	—	2	1	no	—
42	C₂³×C₄×C₆₇	—	2	2	no	—
43	C₂×SD₁₆×C₆₇	—	2	4	no	—
44	C₂×SD₁₆×C₆₇	—	2	8	no	—
45	C₂×D₈×C₆₇	—	2	4	no	—
46	C₂×D₈×C₆₇	—	2	8	no	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
49	C₄⋊Q₈×C₆₇	—	2	2	no	—
50	C₄⋊Q₈×C₆₇	—	2	2	no	—
51	#127×C₆₇	—	2	2	no	—
52	#127×C₆₇	—	2	2	no	—
53	#124×C₆₇	—	2	8	no	—
54	#124×C₆₇	—	2	8	no	—
55	C₄×D₄×C₆₇	—	2	4	no	—
56	C₄×D₄×C₆₇	—	2	2	no	—
57	C₄×D₄×C₆₇	D₄	1	4	yes	—
58	C₄×D₄×C₆₇	D₄	1	8	yes	complemented
59	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
61	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
63	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
64	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
67	C₂×C₄²×C₆₇	—	2	2	no	—
68	C₂×C₄²×C₆₇	—	2	1	no	—
69	C₂×C₄²×C₆₇	—	4	4	no	—
70	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₆₇	D₄	1	2	yes	—
73	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₆₇	—	2	2	no	—
76	C₄⋊C₈×C₆₇	—	2	2	no	—
77	C₄⋊C₈×C₆₇	—	2	8	no	—
78	#103×C₆₇	—	2	4	no	—
79	#103×C₆₇	—	2	8	no	—
80	#102×C₆₇	—	2	4	no	—
81	#102×C₆₇	—	2	8	no	—
82	#95×C₆₇	—	2	2	no	—
83	#95×C₆₇	—	2	1	no	—
84	C₂⁴×C₆₇	—	2	1	no	—
85	C₂⁴×C₆₇	—	2	2	no	—
86	C₂×Q₈×C₆₇	—	2	4	no	—
87	C₂×Q₈×C₆₇	—	2	2	no	—
88	C₂×Q₈×C₆₇	—	2	4	no	—
89	C₂×Q₈×C₆₇	—	2	2	no	—
90	C₂×D₄×C₆₇	—	2	4	no	—
91	C₂×D₄×C₆₇	—	2	2	no	—
92	C₂×D₄×C₆₇	—	2	1	no	—
93	C₂×D₄×C₆₇	—	2	2	no	—
94	C₂×D₄×C₆₇	—	2	4	no	—
95	C₂×D₄×C₆₇	—	2	2	no	—
96	C₂×D₄×C₆₇	—	4	4	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	C₂×D₄	1	2	yes	—
102	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
103	C₂²×C₄×C₆₇	—	2	1	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	4	no	—
106	C₂²×C₄×C₆₇	—	2	8	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	4	4	no	—
109	C₂²×C₄×C₆₇	—	4	4	no	—
110	C₂²×C₄×C₆₇	—	4	2	no	—
111	C₂²×C₄×C₆₇	—	4	2	no	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
117	SD₁₆×C₆₇	—	4	8	no	—
118	D₈×C₆₇	—	4	8	no	—
119	C₂×C₈×C₆₇	—	2	2	no	—
120	C₂×C₈×C₆₇	—	2	4	no	—
121	C₂×C₈×C₆₇	—	2	4	no	—
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₄⋊C₄×C₆₇	—	2	1	no	—
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	4	2	no	—
126	C₄⋊C₄×C₆₇	—	4	2	no	—
127	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
128	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
129	C₂²⋊C₄×C₆₇	—	2	4	no	—
130	C₂²⋊C₄×C₆₇	—	2	8	no	—
131	C₂²⋊C₄×C₆₇	—	4	8	no	—
132	C₂²⋊C₄×C₆₇	—	4	8	no	—
133	C₄²×C₆₇	—	2	2	no	—
134	C₄²×C₆₇	—	2	2	no	—
135	C₄²×C₆₇	—	4	2	no	—
136	C₄²×C₆₇	—	4	4	no	—
137	C₄²×C₆₇	—	4	2	no	—
138	C₄²×C₆₇	C₂×D₄	1	2	yes	—
139	C₄²×C₆₇	C₂×D₄	1	4	yes	—
140	C₄²×C₆₇	#41	1	4	yes	—
141	C₄²×C₆₇	#41	1	2	yes	—
142	C₂³×C₆₇	—	2	2	no	—
143	C₂³×C₆₇	—	2	4	no	—
144	C₂³×C₆₇	—	4	1	no	—
145	C₂³×C₆₇	—	4	1	no	—
146	C₂³×C₆₇	—	4	2	no	—
147	C₂³×C₆₇	—	4	4	no	—
148	C₂³×C₆₇	—	4	2	no	—
149	C₂³×C₆₇	—	4	2	no	—
150	C₂³×C₆₇	—	4	2	no	—
151	C₂³×C₆₇	—	4	4	no	—
152	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
153	Q₈×C₆₇	—	4	2	no	—
154	Q₈×C₆₇	—	4	4	no	—
155	Q₈×C₆₇	—	4	2	no	—
156	D₄×C₆₇	—	2	2	no	—
157	D₄×C₆₇	—	2	4	no	—
158	D₄×C₆₇	—	4	2	no	—
159	D₄×C₆₇	—	4	4	no	—
160	D₄×C₆₇	—	4	2	no	—
161	D₄×C₆₇	—	8	4	no	—
162	D₄×C₆₇	—	8	4	no	—
163	D₄×C₆₇	#127	1	4	yes	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	2	2	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	2	4	no	—
171	C₂×C₄×C₆₇	—	4	16	no	—
172	C₂×C₄×C₆₇	—	4	1	no	—
173	C₂×C₄×C₆₇	—	4	1	no	—
174	C₂×C₄×C₆₇	—	4	2	no	—
175	C₂×C₄×C₆₇	—	4	2	no	—
176	C₂×C₄×C₆₇	—	4	8	no	—
177	C₂×C₄×C₆₇	—	4	4	no	—
178	C₂×C₄×C₆₇	—	4	2	no	—
179	C₂×C₄×C₆₇	#121	1	4	yes	—
180	C₂×C₄×C₆₇	#121	1	2	yes	—
181	C₂×C₄×C₆₇	#121	1	4	yes	—
182	C₂×C₄×C₆₇	#121	1	2	yes	—
183	C₂×C₄×C₆₇	#135	1	2	yes	—
184	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
185	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
187	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
188	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
189	C₈×C₆₇	—	4	2	no	—
190	C₈×C₆₇	—	4	2	no	—
191	C₂²×C₆₇	—	2	4	no	—
192	C₂²×C₆₇	—	4	2	no	—
193	C₂²×C₆₇	—	4	4	no	—
194	C₂²×C₆₇	—	4	4	no	—
195	C₂²×C₆₇	—	4	2	no	—
196	C₂²×C₆₇	—	8	2	no	—
197	C₂²×C₆₇	—	8	2	no	—
198	C₂²×C₆₇	—	8	4	no	—
199	C₂²×C₆₇	—	8	4	no	—
200	C₂²×C₆₇	—	8	2	no	—
201	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
202	C₂²×C₆₇	C₂×#121	1	2	yes	—
203	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
204	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
205	C₂²×C₆₇	#575	1	2	yes	—
206	C₄×C₆₇	—	2	4	no	—
207	C₄×C₆₇	—	2	4	no	—
208	C₄×C₆₇	—	2	2	no	—
209	C₄×C₆₇	—	4	2	no	—
210	C₄×C₆₇	—	4	4	no	—
211	C₄×C₆₇	—	8	2	no	—
212	C₄×C₆₇	#463	1	4	yes	complemented
213	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
214	C₄×C₆₇	#532	1	1	yes	characteristic
215	C₄×C₆₇	#532	1	2	yes	—
216	#33164	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
217	C₂×C₆₇	—	4	4	no	—
218	C₂×C₆₇	—	16	2	no	—
219	C₂×C₆₇	#2856	1	1	yes	characteristic
220	C₂×C₆₇	C₂×#463	1	1	yes	characteristic
221	C₂×C₆₇	#3035	1	2	yes	—
222	C₂×C₆₇	#3036	1	2	yes	—
223	C₂×C₆₇	#3040	1	1	yes	characteristic
224	C₂×#463	C₂×C₆₇	1	4	yes	complemented
225	C₂×#461	C₂×C₆₇	1	1	yes	characteristic, complemented
226	C₂×#459	C₂×C₆₇	1	1	yes	characteristic
227	#2858	C₂×C₆₇	1	1	yes	characteristic, complemented
228	C₄²×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
229	#1983	C₂×C₆₇	1	4	yes	complemented
230	#1982	C₂×C₆₇	1	1	yes	characteristic, complemented
231	#1981	C₂×C₆₇	1	1	yes	characteristic, complemented
232	#1842	C₂×C₆₇	1	1	yes	characteristic, complemented
233	C₆₇	#33164	1	1	yes	characteristic, direct product factor, Sylow
234	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
235	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
236	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
237	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
238	C₂×#124	C₂²×C₆₇	1	4	yes	—
239	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
240	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
241	C₂²×C₄²	—	2	1	no	—
242	#463	—	2	16	no	—
243	#461	—	2	4	no	—
244	#459	—	2	4	no	—
245	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
246	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
247	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
248	C₂×#103	C₂²×C₆₇	1	4	yes	—
249	C₂×#103	C₂²×C₆₇	1	2	yes	—
250	C₂×#102	C₂²×C₆₇	1	4	yes	—
251	C₂×#102	C₂²×C₆₇	1	2	yes	—
252	#390	—	2	1	no	—
253	#390	—	2	1	no	—
254	#390	—	2	2	no	—
255	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
256	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
257	C₄×C₂²⋊C₄	—	2	1	no	—
258	C₄×C₂²⋊C₄	—	2	2	no	—
259	C₄³	C₂²×C₆₇	1	1	yes	characteristic
260	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
261	C₂²×D₄	—	2	1	no	—
262	C₂²×D₄	—	2	1	no	—
263	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
264	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
265	C₂³×C₄	—	2	1	no	—
266	C₂³×C₄	—	2	2	no	—
267	C₂×SD₁₆	—	2	4	no	—
268	C₂×SD₁₆	—	2	8	no	—
269	C₂×D₈	—	2	4	no	—
270	C₂×D₈	—	2	8	no	—
271	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
272	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
273	C₄⋊Q₈	—	2	2	no	—
274	C₄⋊Q₈	—	2	2	no	—
275	#127	—	2	2	no	—
276	#127	—	2	2	no	—
277	#124	—	2	8	no	—
278	#124	—	2	8	no	—
279	C₄×D₄	—	2	4	no	—
280	C₄×D₄	—	2	2	no	—
281	C₄×D₄	D₄×C₆₇	1	4	yes	—
282	C₄×D₄	D₄×C₆₇	1	8	yes	complemented
283	C₂×C₄⋊C₄	—	2	1	no	—
284	C₂×C₄⋊C₄	—	2	1	no	—
285	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
286	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
287	C₂×C₂²⋊C₄	—	2	1	no	—
288	C₂×C₂²⋊C₄	—	2	2	no	—
289	C₂×C₂²⋊C₄	—	2	4	no	—
290	C₂×C₂²⋊C₄	—	2	4	no	—
291	C₂×C₄²	—	2	2	no	—
292	C₂×C₄²	—	2	1	no	—
293	C₂×C₄²	—	4	4	no	—
294	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
295	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
296	C₂×C₄²	D₄×C₆₇	1	2	yes	—
297	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
298	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
299	C₄⋊C₈	—	2	2	no	—
300	C₄⋊C₈	—	2	2	no	—
301	C₄⋊C₈	—	2	8	no	—
302	#103	—	2	4	no	—
303	#103	—	2	8	no	—
304	#102	—	2	4	no	—
305	#102	—	2	8	no	—
306	#95	—	2	2	no	—
307	#95	—	2	1	no	—
308	C₂⁴	—	2	1	no	—
309	C₂⁴	—	2	2	no	—
310	C₂×Q₈	—	2	4	no	—
311	C₂×Q₈	—	2	2	no	—
312	C₂×Q₈	—	2	4	no	—
313	C₂×Q₈	—	2	2	no	—
314	C₂×D₄	—	2	4	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	1	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	2	4	no	—
319	C₂×D₄	—	2	2	no	—
320	C₂×D₄	—	4	4	no	—
321	C₂×D₄	—	4	2	no	—
322	C₂×D₄	—	4	2	no	—
323	C₂×D₄	—	4	2	no	—
324	C₂×D₄	—	4	2	no	—
325	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
326	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	4	no	—
330	C₂²×C₄	—	2	8	no	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	4	4	no	—
333	C₂²×C₄	—	4	4	no	—
334	C₂²×C₄	—	4	2	no	—
335	C₂²×C₄	—	4	2	no	—
336	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
337	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
338	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
339	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
340	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
341	SD₁₆	—	4	8	no	—
342	D₈	—	4	8	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	4	no	—
345	C₂×C₈	—	2	4	no	—
346	C₂×C₈	—	2	2	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	4	2	no	—
350	C₄⋊C₄	—	4	2	no	—
351	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
352	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
353	C₂²⋊C₄	—	2	4	no	—
354	C₂²⋊C₄	—	2	8	no	—
355	C₂²⋊C₄	—	4	8	no	—
356	C₂²⋊C₄	—	4	8	no	—
357	C₄²	—	2	2	no	—
358	C₄²	—	2	2	no	—
359	C₄²	—	4	2	no	—
360	C₄²	—	4	4	no	—
361	C₄²	—	4	2	no	—
362	C₄²	C₂×D₄×C₆₇	1	2	yes	—
363	C₄²	C₂×D₄×C₆₇	1	4	yes	—
364	C₄²	#41×C₆₇	1	4	yes	—
365	C₄²	#41×C₆₇	1	2	yes	—
366	C₂³	—	2	2	no	—
367	C₂³	—	2	4	no	—
368	C₂³	—	4	1	no	—
369	C₂³	—	4	1	no	—
370	C₂³	—	4	2	no	—
371	C₂³	—	4	4	no	—
372	C₂³	—	4	2	no	—
373	C₂³	—	4	2	no	—
374	C₂³	—	4	2	no	—
375	C₂³	—	4	4	no	—
376	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
377	Q₈	—	4	2	no	—
378	Q₈	—	4	4	no	—
379	Q₈	—	4	2	no	—
380	D₄	—	2	2	no	—
381	D₄	—	2	4	no	—
382	D₄	—	4	2	no	—
383	D₄	—	4	4	no	—
384	D₄	—	4	2	no	—
385	D₄	—	8	4	no	—
386	D₄	—	8	4	no	—
387	D₄	#127×C₆₇	1	4	yes	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	4	16	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	4	1	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	8	no	—
401	C₂×C₄	—	4	4	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	#121×C₆₇	1	4	yes	—
404	C₂×C₄	#121×C₆₇	1	2	yes	—
405	C₂×C₄	#121×C₆₇	1	4	yes	—
406	C₂×C₄	#121×C₆₇	1	2	yes	—
407	C₂×C₄	#135×C₆₇	1	2	yes	—
408	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
409	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
410	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
411	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
412	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
413	C₈	—	4	2	no	—
414	C₈	—	4	2	no	—
415	C₂²	—	2	4	no	—
416	C₂²	—	4	2	no	—
417	C₂²	—	4	4	no	—
418	C₂²	—	4	4	no	—
419	C₂²	—	4	2	no	—
420	C₂²	—	8	2	no	—
421	C₂²	—	8	4	no	—
422	C₂²	—	8	4	no	—
423	C₂²	—	8	2	no	—
424	C₂²	—	8	2	no	—
425	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
426	C₂²	C₂×#121×C₆₇	1	2	yes	—
427	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
428	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
429	C₂²	#575×C₆₇	1	2	yes	—
430	C₄	—	2	4	no	—
431	C₄	—	2	4	no	—
432	C₄	—	2	2	no	—
433	C₄	—	4	2	no	—
434	C₄	—	4	4	no	—
435	C₄	—	8	2	no	—
436	C₄	#463×C₆₇	1	4	yes	complemented
437	C₄	C₂×#127×C₆₇	1	1	yes	characteristic
438	C₄	#532×C₆₇	1	1	yes	characteristic
439	C₄	#532×C₆₇	1	2	yes	—
440	C₂	—	4	4	no	—
441	C₂	—	16	2	no	—
442	C₂	C₆₇×#2856	1	1	yes	characteristic
443	C₂	C₂×#463×C₆₇	1	1	yes	characteristic
444	C₂	C₆₇×#3035	1	2	yes	—
445	C₂	C₆₇×#3036	1	2	yes	—
446	C₂	C₆₇×#3040	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#459×C₆₇	C₂
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. C₂×#463×C₆₇	C₂	semidirect
2. C₂×#461×C₆₇	C₂	semidirect
4. C₆₇×#2858	C₂	semidirect
5. C₄²×D₄×C₆₇	C₂	semidirect
6. C₆₇×#1983	C₂	semidirect
7. C₆₇×#1982	C₂	semidirect
8. C₆₇×#1981	C₂	semidirect
9. C₆₇×#1842	C₂	semidirect
58. C₄×D₄×C₆₇	D₄	semidirect
212. C₄×C₆₇	#463	semidirect
216. #33164	C₆₇	direct (N × H)
224. C₂×#463	C₂×C₆₇	semidirect
225. C₂×#461	C₂×C₆₇	semidirect
227. #2858	C₂×C₆₇	semidirect
228. C₄²×D₄	C₂×C₆₇	semidirect
229. #1983	C₂×C₆₇	semidirect
230. #1982	C₂×C₆₇	semidirect
231. #1981	C₂×C₆₇	semidirect
232. #1842	C₂×C₆₇	semidirect
233. C₆₇	#33164	direct (N × H)
282. C₄×D₄	D₄×C₆₇	semidirect
436. C₄	#463×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#463×C₆₇	2
C₂×#461×C₆₇	2
C₂×#459×C₆₇	2
C₆₇×#2858	2
C₄²×D₄×C₆₇	2
C₆₇×#1983	2
C₆₇×#1982	2
C₆₇×#1981	2
C₆₇×#1842	2
#33164	67

Subgroups — C67×#33164 #30712415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33164 #30712415