Subgroups — C₆₇×#33043 #30712294

Order 17152.

966 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#487×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2857	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2857	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1990	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1956	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1933	C₂	1	4	yes	maximal, complemented
8	C₆₇×#1892	C₂	1	4	yes	maximal
9	C₆₇×#1834	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
14	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	#487×C₆₇	—	2	4	no	—
17	C₂×#103×C₆₇	C₂²	1	4	yes	—
18	C₂×#103×C₆₇	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₆₇	—	2	4	no	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	#391×C₆₇	—	2	1	no	—
24	#391×C₆₇	C₂²	1	2	yes	—
25	#389×C₆₇	—	2	1	no	—
26	#389×C₆₇	—	2	1	no	—
27	#389×C₆₇	—	2	1	no	—
28	#389×C₆₇	C₂²	1	2	yes	—
29	#384×C₆₇	C₂²	1	1	yes	characteristic
30	#384×C₆₇	C₂²	1	1	yes	characteristic
31	#380×C₆₇	—	2	1	no	—
32	#380×C₆₇	—	2	1	no	—
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×Q₈×C₆₇	C₂³	1	2	yes	—
39	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	—	2	4	no	—
43	C₂²×C₈×C₆₇	—	2	4	no	—
44	C₂²×C₈×C₆₇	—	2	4	no	—
45	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₆₇	—	2	2	no	—
48	C₄⋊Q₈×C₆₇	—	2	2	no	—
49	C₄⋊Q₈×C₆₇	—	2	2	no	—
50	C₄⋊Q₈×C₆₇	—	2	2	no	—
51	C₂²⋊Q₈×C₆₇	—	2	8	no	—
52	C₂²⋊Q₈×C₆₇	—	4	4	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	1	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	1	no	—
62	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
65	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
67	C₂×C₄²×C₆₇	—	2	2	no	—
68	C₂×C₄²×C₆₇	—	2	2	no	—
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	#103×C₆₇	—	2	8	no	—
74	#103×C₆₇	—	2	8	no	—
75	C₄×C₈×C₆₇	—	2	4	no	—
76	C₄×C₈×C₆₇	—	2	2	no	—
77	C₄×C₈×C₆₇	D₄	1	4	yes	—
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	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₂×Q₈×C₆₇	—	2	2	no	—
84	C₂×Q₈×C₆₇	—	2	4	no	—
85	C₂×Q₈×C₆₇	—	2	4	no	—
86	C₂×Q₈×C₆₇	—	2	2	no	—
87	C₂×Q₈×C₆₇	—	4	2	no	—
88	C₂×Q₈×C₆₇	—	4	2	no	—
89	C₂×Q₈×C₆₇	—	4	2	no	—
90	C₂²×C₄×C₆₇	—	2	2	no	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	2	2	no	—
93	C₂²×C₄×C₆₇	—	2	2	no	—
94	C₂²×C₄×C₆₇	—	2	1	no	—
95	C₂²×C₄×C₆₇	—	2	1	no	—
96	C₂²×C₄×C₆₇	—	2	2	no	—
97	C₂²×C₄×C₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	—
102	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
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	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	#41	1	4	yes	—
109	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
110	C₂×C₈×C₆₇	—	2	16	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
116	C₂×C₈×C₆₇	#41	1	4	yes	—
117	C₄⋊C₄×C₆₇	—	2	2	no	—
118	C₄⋊C₄×C₆₇	—	2	4	no	—
119	C₄⋊C₄×C₆₇	—	2	4	no	—
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	—	4	2	no	—
122	C₄⋊C₄×C₆₇	—	4	2	no	—
123	C₄⋊C₄×C₆₇	—	4	2	no	—
124	C₄⋊C₄×C₆₇	—	4	4	no	—
125	C₄⋊C₄×C₆₇	—	4	2	no	—
126	C₄⋊C₄×C₆₇	—	4	4	no	—
127	C₄⋊C₄×C₆₇	—	4	2	no	—
128	C₄⋊C₄×C₆₇	—	4	2	no	—
129	C₄⋊C₄×C₆₇	—	4	2	no	—
130	C₂²⋊C₄×C₆₇	—	4	4	no	—
131	C₂²⋊C₄×C₆₇	—	4	4	no	—
132	C₄²×C₆₇	—	2	2	no	—
133	C₄²×C₆₇	—	2	2	no	—
134	C₄²×C₆₇	—	2	1	no	—
135	C₄²×C₆₇	—	2	1	no	—
136	C₄²×C₆₇	C₂×D₄	1	2	yes	—
137	C₄²×C₆₇	#41	1	2	yes	—
138	C₂³×C₆₇	—	2	1	no	—
139	C₂³×C₆₇	—	2	1	no	—
140	C₂³×C₆₇	—	2	1	no	—
141	C₂³×C₆₇	—	2	1	no	—
142	C₂³×C₆₇	#124	1	2	yes	—
143	C₂³×C₆₇	C₂×SD₁₆	1	2	yes	—
144	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
145	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
146	Q₈×C₆₇	—	4	4	no	—
147	Q₈×C₆₇	—	4	4	no	—
148	Q₈×C₆₇	—	8	4	no	—
149	C₂×C₄×C₆₇	—	2	1	no	—
150	C₂×C₄×C₆₇	—	2	1	no	—
151	C₂×C₄×C₆₇	—	2	1	no	—
152	C₂×C₄×C₆₇	—	2	1	no	—
153	C₂×C₄×C₆₇	—	2	1	no	—
154	C₂×C₄×C₆₇	—	2	1	no	—
155	C₂×C₄×C₆₇	—	2	4	no	—
156	C₂×C₄×C₆₇	—	2	4	no	—
157	C₂×C₄×C₆₇	—	2	8	no	—
158	C₂×C₄×C₆₇	—	2	8	no	—
159	C₂×C₄×C₆₇	—	4	2	no	—
160	C₂×C₄×C₆₇	—	4	2	no	—
161	C₂×C₄×C₆₇	—	4	2	no	—
162	C₂×C₄×C₆₇	—	4	2	no	—
163	C₂×C₄×C₆₇	—	4	2	no	—
164	C₂×C₄×C₆₇	—	4	2	no	—
165	C₂×C₄×C₆₇	—	4	2	no	—
166	C₂×C₄×C₆₇	—	4	2	no	—
167	C₂×C₄×C₆₇	—	4	2	no	—
168	C₂×C₄×C₆₇	—	4	2	no	—
169	C₂×C₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	4	2	no	—
171	C₂×C₄×C₆₇	#121	1	2	yes	—
172	C₂×C₄×C₆₇	#121	1	2	yes	—
173	C₂×C₄×C₆₇	#124	1	2	yes	—
174	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
175	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
176	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
177	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
178	C₂×C₄×C₆₇	#135	1	2	yes	—
179	C₂×C₄×C₆₇	#135	1	2	yes	—
180	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
182	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
183	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
184	C₈×C₆₇	—	2	4	no	—
185	C₈×C₆₇	—	2	4	no	—
186	C₂²×C₆₇	—	2	2	no	—
187	C₂²×C₆₇	—	2	2	no	—
188	C₂²×C₆₇	—	2	2	no	—
189	C₂²×C₆₇	—	2	2	no	—
190	C₂²×C₆₇	—	2	2	no	—
191	C₂²×C₆₇	—	2	2	no	—
192	C₂²×C₆₇	#487	1	4	yes	complemented
193	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
194	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
195	C₂²×C₆₇	#532	1	1	yes	characteristic
196	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
197	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
198	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
199	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
200	C₄×C₆₇	—	2	2	no	—
201	C₄×C₆₇	—	2	2	no	—
202	C₄×C₆₇	—	2	2	no	—
203	C₄×C₆₇	—	8	2	no	—
204	C₄×C₆₇	—	8	2	no	—
205	C₄×C₆₇	—	8	2	no	—
206	C₄×C₆₇	—	8	2	no	—
207	C₄×C₆₇	#465	1	4	yes	—
208	C₄×C₆₇	#467	1	4	yes	—
209	C₄×C₆₇	#525	1	2	yes	—
210	C₄×C₆₇	#529	1	2	yes	—
211	#33043	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
212	C₂×C₆₇	—	2	4	no	—
213	C₂×C₆₇	#2850	1	1	yes	characteristic
214	C₂×C₆₇	C₂×#465	1	1	yes	characteristic
215	C₂×C₆₇	C₂×#467	1	1	yes	characteristic
216	C₂×C₆₇	C₂×#487	1	1	yes	characteristic
217	C₂×C₆₇	#3131	1	1	yes	characteristic
218	C₂×C₆₇	#3167	1	1	yes	characteristic
219	C₂×C₆₇	#3170	1	1	yes	characteristic
220	C₂×#487	C₂×C₆₇	1	1	yes	characteristic, complemented
221	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic
222	#2857	C₂×C₆₇	1	1	yes	characteristic
223	#2857	C₂×C₆₇	1	1	yes	characteristic
224	#1990	C₂×C₆₇	1	1	yes	characteristic, complemented
225	#1956	C₂×C₆₇	1	1	yes	characteristic, complemented
226	#1933	C₂×C₆₇	1	4	yes	complemented
227	#1892	C₂×C₆₇	1	4	yes	—
228	#1834	C₂×C₆₇	1	1	yes	characteristic, complemented
229	C₆₇	#33043	1	1	yes	characteristic, direct product factor, Sylow
230	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
231	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
232	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
233	C₂×C₂²⋊Q₈	—	2	1	no	—
234	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
235	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
236	#487	—	2	4	no	—
237	C₂×#103	C₂²×C₆₇	1	4	yes	—
238	C₂×#103	C₂²×C₆₇	1	4	yes	—
239	C₂×C₄×C₈	—	2	4	no	—
240	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
241	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
242	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
243	#391	—	2	1	no	—
244	#391	C₂²×C₆₇	1	2	yes	—
245	#389	—	2	1	no	—
246	#389	—	2	1	no	—
247	#389	—	2	1	no	—
248	#389	C₂²×C₆₇	1	2	yes	—
249	#384	C₂²×C₆₇	1	1	yes	characteristic
250	#384	C₂²×C₆₇	1	1	yes	characteristic
251	#380	—	2	1	no	—
252	#380	—	2	1	no	—
253	#380	—	2	1	no	—
254	#380	C₂²×C₆₇	1	2	yes	—
255	#340	C₂²×C₆₇	1	4	yes	—
256	#340	C₂²×C₆₇	1	4	yes	—
257	C₂²×Q₈	—	2	1	no	—
258	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
259	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
260	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
261	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
262	C₂²×C₈	—	2	4	no	—
263	C₂²×C₈	—	2	4	no	—
264	C₂²×C₈	—	2	4	no	—
265	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
266	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
267	C₄⋊Q₈	—	2	2	no	—
268	C₄⋊Q₈	—	2	2	no	—
269	C₄⋊Q₈	—	2	2	no	—
270	C₄⋊Q₈	—	2	2	no	—
271	C₂²⋊Q₈	—	2	8	no	—
272	C₂²⋊Q₈	—	4	4	no	—
273	C₂×C₄⋊C₄	—	2	1	no	—
274	C₂×C₄⋊C₄	—	2	2	no	—
275	C₂×C₄⋊C₄	—	2	2	no	—
276	C₂×C₄⋊C₄	—	2	1	no	—
277	C₂×C₄⋊C₄	—	2	1	no	—
278	C₂×C₄⋊C₄	—	2	1	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	C₂³×C₆₇	1	2	yes	—
283	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
284	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
285	C₂×C₂²⋊C₄	—	2	2	no	—
286	C₂×C₂²⋊C₄	—	2	2	no	—
287	C₂×C₄²	—	2	2	no	—
288	C₂×C₄²	—	2	2	no	—
289	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
290	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
291	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
292	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
293	#103	—	2	8	no	—
294	#103	—	2	8	no	—
295	C₄×C₈	—	2	4	no	—
296	C₄×C₈	—	2	2	no	—
297	C₄×C₈	D₄×C₆₇	1	4	yes	—
298	#95	—	2	2	no	—
299	#95	—	2	2	no	—
300	#95	—	2	2	no	—
301	#95	—	2	2	no	—
302	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
303	C₂×Q₈	—	2	2	no	—
304	C₂×Q₈	—	2	4	no	—
305	C₂×Q₈	—	2	4	no	—
306	C₂×Q₈	—	2	2	no	—
307	C₂×Q₈	—	4	2	no	—
308	C₂×Q₈	—	4	2	no	—
309	C₂×Q₈	—	4	2	no	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	—	2	2	no	—
321	C₂²×C₄	SD₁₆×C₆₇	1	4	yes	—
322	C₂²×C₄	Q₁₆×C₆₇	1	4	yes	—
323	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
324	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
325	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
326	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
327	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
328	C₂²×C₄	#41×C₆₇	1	4	yes	—
329	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
330	C₂×C₈	—	2	16	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
336	C₂×C₈	#41×C₆₇	1	4	yes	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	2	4	no	—
339	C₄⋊C₄	—	2	4	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	4	2	no	—
342	C₄⋊C₄	—	4	2	no	—
343	C₄⋊C₄	—	4	2	no	—
344	C₄⋊C₄	—	4	4	no	—
345	C₄⋊C₄	—	4	2	no	—
346	C₄⋊C₄	—	4	4	no	—
347	C₄⋊C₄	—	4	2	no	—
348	C₄⋊C₄	—	4	2	no	—
349	C₄⋊C₄	—	4	2	no	—
350	C₂²⋊C₄	—	4	4	no	—
351	C₂²⋊C₄	—	4	4	no	—
352	C₄²	—	2	2	no	—
353	C₄²	—	2	2	no	—
354	C₄²	—	2	1	no	—
355	C₄²	—	2	1	no	—
356	C₄²	C₂×D₄×C₆₇	1	2	yes	—
357	C₄²	#41×C₆₇	1	2	yes	—
358	C₂³	—	2	1	no	—
359	C₂³	—	2	1	no	—
360	C₂³	—	2	1	no	—
361	C₂³	—	2	1	no	—
362	C₂³	#124×C₆₇	1	2	yes	—
363	C₂³	C₂×SD₁₆×C₆₇	1	2	yes	—
364	C₂³	C₂×Q₁₆×C₆₇	1	2	yes	—
365	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
366	Q₈	—	4	4	no	—
367	Q₈	—	4	4	no	—
368	Q₈	—	8	4	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	8	no	—
376	C₂×C₄	—	2	8	no	—
377	C₂×C₄	—	2	4	no	—
378	C₂×C₄	—	2	4	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	#121×C₆₇	1	2	yes	—
392	C₂×C₄	#121×C₆₇	1	2	yes	—
393	C₂×C₄	#124×C₆₇	1	2	yes	—
394	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
395	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
396	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
397	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
398	C₂×C₄	#135×C₆₇	1	2	yes	—
399	C₂×C₄	#135×C₆₇	1	2	yes	—
400	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
401	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
402	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
403	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
404	C₈	—	2	4	no	—
405	C₈	—	2	4	no	—
406	C₂²	—	2	2	no	—
407	C₂²	—	2	2	no	—
408	C₂²	—	2	2	no	—
409	C₂²	—	2	2	no	—
410	C₂²	—	2	2	no	—
411	C₂²	—	2	2	no	—
412	C₂²	#487×C₆₇	1	4	yes	complemented
413	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
414	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
415	C₂²	#532×C₆₇	1	1	yes	characteristic
416	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
417	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
418	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
419	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
420	C₄	—	2	2	no	—
421	C₄	—	2	2	no	—
422	C₄	—	2	2	no	—
423	C₄	—	8	2	no	—
424	C₄	—	8	2	no	—
425	C₄	—	8	2	no	—
426	C₄	—	8	2	no	—
427	C₄	#465×C₆₇	1	4	yes	—
428	C₄	#467×C₆₇	1	4	yes	—
429	C₄	#525×C₆₇	1	2	yes	—
430	C₄	#529×C₆₇	1	2	yes	—
431	C₂	—	2	4	no	—
432	C₂	C₆₇×#2850	1	1	yes	characteristic
433	C₂	C₂×#465×C₆₇	1	1	yes	characteristic
434	C₂	C₂×#467×C₆₇	1	1	yes	characteristic
435	C₂	C₂×#487×C₆₇	1	1	yes	characteristic
436	C₂	C₆₇×#3131	1	1	yes	characteristic
437	C₂	C₆₇×#3167	1	1	yes	characteristic
438	C₂	C₆₇×#3170	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
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×#487×C₆₇	C₂	semidirect
5. C₆₇×#1990	C₂	semidirect
6. C₆₇×#1956	C₂	semidirect
7. C₆₇×#1933	C₂	semidirect
9. C₆₇×#1834	C₂	semidirect
192. C₂²×C₆₇	#487	semidirect
211. #33043	C₆₇	direct (N × H)
220. C₂×#487	C₂×C₆₇	semidirect
224. #1990	C₂×C₆₇	semidirect
225. #1956	C₂×C₆₇	semidirect
226. #1933	C₂×C₆₇	semidirect
228. #1834	C₂×C₆₇	semidirect
229. C₆₇	#33043	direct (N × H)
412. C₂²	#487×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#487×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₆₇×#2857	2
C₆₇×#1990	2
C₆₇×#1956	2
C₆₇×#1933	2
C₆₇×#1892	2
C₆₇×#1834	2
#33043	67

Subgroups — C67×#33043 #30712294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33043 #30712294