Subgroups — C₆₇×#33016 #30712267

Order 17152.

1292 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#520×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#483×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#452×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2625	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1878	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1877	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1865	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1864	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1824	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1818	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1788	C₂	1	1	yes	maximal, characteristic
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	#520×C₆₇	—	2	4	no	—
17	#520×C₆₇	—	2	4	no	—
18	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×D₄×C₆₇	—	2	2	no	—
20	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
21	C₂×#117×C₆₇	—	2	1	no	—
22	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
25	#483×C₆₇	—	2	4	no	—
26	#452×C₆₇	—	2	4	no	—
27	C₂×#106×C₆₇	C₂²	1	2	yes	—
28	C₂×#103×C₆₇	C₂²	1	2	yes	—
29	C₂×#103×C₆₇	C₂²	1	2	yes	—
30	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#102×C₆₇	C₂²	1	2	yes	—
32	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
35	#400×C₆₇	—	2	1	no	—
36	#398×C₆₇	—	2	1	no	—
37	#397×C₆₇	—	2	2	no	—
38	#391×C₆₇	C₂²	1	2	yes	—
39	#389×C₆₇	C₂²	1	2	yes	—
40	#380×C₆₇	C₂²	1	1	yes	characteristic
41	#340×C₆₇	C₂²	1	2	yes	—
42	#340×C₆₇	C₂²	1	2	yes	—
43	#336×C₆₇	C₂²	1	2	yes	—
44	#336×C₆₇	C₂²	1	2	yes	—
45	C₂×#41×C₆₇	—	2	4	no	—
46	C₂×#41×C₆₇	—	2	4	no	—
47	C₂×#41×C₆₇	—	2	2	no	—
48	C₂×#41×C₆₇	D₄	1	4	yes	—
49	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
50	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×D₄×C₆₇	—	2	1	no	—
52	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
53	C₂³×C₄×C₆₇	—	2	1	no	—
54	C₂³×C₄×C₆₇	—	2	1	no	—
55	C₂³×C₄×C₆₇	—	2	2	no	—
56	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂×Q₁₆×C₆₇	—	2	2	no	—
58	C₂×Q₁₆×C₆₇	—	2	4	no	—
59	C₂×SD₁₆×C₆₇	—	2	2	no	—
60	C₂×SD₁₆×C₆₇	—	2	4	no	—
61	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
64	C₂²⋊Q₈×C₆₇	—	2	2	no	—
65	C₂²⋊Q₈×C₆₇	—	2	2	no	—
66	C₄×Q₈×C₆₇	—	2	4	no	—
67	C₄×Q₈×C₆₇	—	2	4	no	—
68	C₄×D₄×C₆₇	—	2	16	no	—
69	C₄×D₄×C₆₇	—	2	4	no	—
70	C₄×D₄×C₆₇	—	2	4	no	—
71	#117×C₆₇	—	2	8	no	—
72	#117×C₆₇	—	2	2	no	—
73	#117×C₆₇	—	2	2	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
76	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
77	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
78	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
79	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
80	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
83	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
85	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
86	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
88	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
89	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
90	C₂×C₄²×C₆₇	—	2	2	no	—
91	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
92	#106×C₆₇	—	2	4	no	—
93	#103×C₆₇	—	2	2	no	—
94	#103×C₆₇	—	2	4	no	—
95	#103×C₆₇	—	2	4	no	—
96	#102×C₆₇	—	2	2	no	—
97	#102×C₆₇	—	2	4	no	—
98	C₂²⋊C₈×C₆₇	—	2	2	no	—
99	C₂²⋊C₈×C₆₇	—	2	2	no	—
100	#95×C₆₇	—	2	1	no	—
101	#95×C₆₇	—	2	2	no	—
102	#95×C₆₇	—	2	2	no	—
103	#95×C₆₇	—	2	2	no	—
104	C₂⁴×C₆₇	—	2	1	no	—
105	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
106	#41×C₆₇	—	2	16	no	—
107	#41×C₆₇	—	2	8	no	—
108	#41×C₆₇	—	2	8	no	—
109	C₂×Q₈×C₆₇	—	2	1	no	—
110	C₂×Q₈×C₆₇	—	2	2	no	—
111	C₂×Q₈×C₆₇	—	2	1	no	—
112	C₂×Q₈×C₆₇	—	2	1	no	—
113	C₂×Q₈×C₆₇	—	2	2	no	—
114	C₂×Q₈×C₆₇	—	2	2	no	—
115	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
116	C₂×Q₈×C₆₇	#41	1	4	yes	—
117	C₂×D₄×C₆₇	—	2	8	no	—
118	C₂×D₄×C₆₇	—	2	4	no	—
119	C₂×D₄×C₆₇	—	2	1	no	—
120	C₂×D₄×C₆₇	—	2	2	no	—
121	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
122	C₂×D₄×C₆₇	#41	1	4	yes	—
123	C₂²×C₄×C₆₇	—	2	8	no	—
124	C₂²×C₄×C₆₇	—	2	1	no	—
125	C₂²×C₄×C₆₇	—	2	4	no	—
126	C₂²×C₄×C₆₇	—	2	2	no	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	—	2	2	no	—
129	C₂²×C₄×C₆₇	—	2	8	no	—
130	C₂²×C₄×C₆₇	—	2	4	no	—
131	C₂²×C₄×C₆₇	—	2	1	no	—
132	C₂²×C₄×C₆₇	—	2	2	no	—
133	C₂²×C₄×C₆₇	—	2	1	no	—
134	C₂²×C₄×C₆₇	—	8	4	no	—
135	C₂²×C₄×C₆₇	—	8	2	no	—
136	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
138	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
140	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
141	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
142	Q₁₆×C₆₇	—	4	4	no	—
143	SD₁₆×C₆₇	—	4	4	no	—
144	C₂×C₈×C₆₇	—	2	1	no	—
145	C₂×C₈×C₆₇	—	2	2	no	—
146	C₂×C₈×C₆₇	—	2	1	no	—
147	C₂×C₈×C₆₇	—	2	2	no	—
148	C₂×C₈×C₆₇	—	2	2	no	—
149	C₂×C₈×C₆₇	—	2	2	no	—
150	C₂×C₈×C₆₇	—	2	2	no	—
151	C₄⋊C₄×C₆₇	—	2	8	no	—
152	C₄⋊C₄×C₆₇	—	2	2	no	—
153	C₄⋊C₄×C₆₇	—	2	4	no	—
154	C₄⋊C₄×C₆₇	—	2	2	no	—
155	C₄⋊C₄×C₆₇	—	2	2	no	—
156	C₄⋊C₄×C₆₇	—	2	1	no	—
157	C₄⋊C₄×C₆₇	—	2	1	no	—
158	C₄⋊C₄×C₆₇	—	2	2	no	—
159	C₄⋊C₄×C₆₇	—	2	2	no	—
160	C₄⋊C₄×C₆₇	—	4	2	no	—
161	C₄⋊C₄×C₆₇	—	8	2	no	—
162	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
163	C₂²⋊C₄×C₆₇	—	2	8	no	—
164	C₂²⋊C₄×C₆₇	—	2	4	no	—
165	C₂²⋊C₄×C₆₇	—	4	2	no	—
166	C₂²⋊C₄×C₆₇	—	8	1	no	—
167	C₄²×C₆₇	—	2	8	no	—
168	C₄²×C₆₇	—	2	2	no	—
169	C₄²×C₆₇	—	2	2	no	—
170	C₂³×C₆₇	—	2	2	no	—
171	C₂³×C₆₇	—	2	2	no	—
172	C₂³×C₆₇	—	2	4	no	—
173	C₂³×C₆₇	—	2	1	no	—
174	C₂³×C₆₇	—	2	2	no	—
175	C₂³×C₆₇	—	2	1	no	—
176	C₂³×C₆₇	—	4	2	no	—
177	C₂³×C₆₇	—	4	1	no	—
178	C₂³×C₆₇	—	4	1	no	—
179	C₂³×C₆₇	#120	1	2	yes	—
180	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
181	Q₈×C₆₇	—	2	2	no	—
182	Q₈×C₆₇	—	2	4	no	—
183	Q₈×C₆₇	—	4	1	no	—
184	Q₈×C₆₇	—	4	1	no	—
185	Q₈×C₆₇	—	4	2	no	—
186	Q₈×C₆₇	#123	1	4	yes	—
187	D₄×C₆₇	—	2	16	no	—
188	D₄×C₆₇	—	2	2	no	—
189	D₄×C₆₇	—	2	4	no	—
190	D₄×C₆₇	#123	1	4	yes	—
191	C₂×C₄×C₆₇	—	2	16	no	—
192	C₂×C₄×C₆₇	—	2	2	no	—
193	C₂×C₄×C₆₇	—	2	2	no	—
194	C₂×C₄×C₆₇	—	2	4	no	—
195	C₂×C₄×C₆₇	—	2	2	no	—
196	C₂×C₄×C₆₇	—	2	2	no	—
197	C₂×C₄×C₆₇	—	2	4	no	—
198	C₂×C₄×C₆₇	—	2	2	no	—
199	C₂×C₄×C₆₇	—	2	2	no	—
200	C₂×C₄×C₆₇	—	2	4	no	—
201	C₂×C₄×C₆₇	—	2	2	no	—
202	C₂×C₄×C₆₇	—	2	4	no	—
203	C₂×C₄×C₆₇	—	2	2	no	—
204	C₂×C₄×C₆₇	—	4	4	no	—
205	C₂×C₄×C₆₇	—	4	1	no	—
206	C₂×C₄×C₆₇	—	4	1	no	—
207	C₂×C₄×C₆₇	—	4	2	no	—
208	C₂×C₄×C₆₇	—	4	2	no	—
209	C₂×C₄×C₆₇	—	4	2	no	—
210	C₂×C₄×C₆₇	—	4	1	no	—
211	C₂×C₄×C₆₇	—	4	2	no	—
212	C₂×C₄×C₆₇	—	4	1	no	—
213	C₂×C₄×C₆₇	—	8	4	no	—
214	C₂×C₄×C₆₇	—	8	8	no	—
215	C₂×C₄×C₆₇	#120	1	2	yes	—
216	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
219	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
220	C₈×C₆₇	—	4	1	no	—
221	C₈×C₆₇	—	4	1	no	—
222	C₈×C₆₇	—	4	2	no	—
223	C₂²×C₆₇	—	2	2	no	—
224	C₂²×C₆₇	—	2	4	no	—
225	C₂²×C₆₇	—	2	2	no	—
226	C₂²×C₆₇	—	2	2	no	—
227	C₂²×C₆₇	—	4	2	no	—
228	C₂²×C₆₇	—	4	4	no	—
229	C₂²×C₆₇	—	4	2	no	—
230	C₂²×C₆₇	—	4	4	no	—
231	C₂²×C₆₇	—	4	2	no	—
232	C₂²×C₆₇	—	4	2	no	—
233	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
234	C₂²×C₆₇	#525	1	2	yes	—
235	C₂²×C₆₇	#577	1	2	yes	—
236	C₂²×C₆₇	#577	1	1	yes	characteristic
237	C₂²×C₆₇	#578	1	1	yes	characteristic
238	C₄×C₆₇	—	2	2	no	—
239	C₄×C₆₇	—	2	4	no	—
240	C₄×C₆₇	—	4	2	no	—
241	C₄×C₆₇	—	4	4	no	—
242	C₄×C₆₇	—	4	2	no	—
243	C₄×C₆₇	—	8	1	no	—
244	C₄×C₆₇	—	8	2	no	—
245	C₄×C₆₇	—	8	1	no	—
246	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
247	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
248	C₄×C₆₇	#525	1	2	yes	—
249	#33016	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
250	C₂×C₆₇	—	4	2	no	—
251	C₂×C₆₇	—	4	4	no	—
252	C₂×C₆₇	#2849	1	1	yes	characteristic
253	C₂×C₆₇	#3009	1	1	yes	characteristic
254	C₂×C₆₇	#3013	1	1	yes	characteristic
255	C₂×C₆₇	#3120	1	2	yes	—
256	C₂×C₆₇	#3125	1	2	yes	—
257	C₂×#520	C₂×C₆₇	1	1	yes	characteristic
258	C₂×#483	C₂×C₆₇	1	1	yes	characteristic, complemented
259	C₂×#452	C₂×C₆₇	1	1	yes	characteristic
260	#2625	C₂×C₆₇	1	1	yes	characteristic, complemented
261	#1878	C₂×C₆₇	1	2	yes	complemented
262	#1877	C₂×C₆₇	1	2	yes	complemented
263	#1865	C₂×C₆₇	1	2	yes	complemented
264	#1864	C₂×C₆₇	1	2	yes	complemented
265	#1824	C₂×C₆₇	1	1	yes	characteristic, complemented
266	#1818	C₂×C₆₇	1	1	yes	characteristic, complemented
267	#1788	C₂×C₆₇	1	1	yes	characteristic
268	C₆₇	#33016	1	1	yes	characteristic, direct product factor, Sylow
269	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
270	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
271	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
272	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
273	#520	—	2	4	no	—
274	#520	—	2	4	no	—
275	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
276	C₂×C₄×D₄	—	2	2	no	—
277	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
278	C₂×#117	—	2	1	no	—
279	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
280	C₂²×C₄⋊C₄	—	2	1	no	—
281	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
282	#483	—	2	4	no	—
283	#452	—	2	4	no	—
284	C₂×#106	C₂²×C₆₇	1	2	yes	—
285	C₂×#103	C₂²×C₆₇	1	2	yes	—
286	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
287	C₂×#103	C₂²×C₆₇	1	2	yes	—
288	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
289	C₂×#102	C₂²×C₆₇	1	2	yes	—
290	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
291	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
292	#400	—	2	1	no	—
293	#398	—	2	1	no	—
294	#397	—	2	2	no	—
295	#391	C₂²×C₆₇	1	2	yes	—
296	#389	C₂²×C₆₇	1	2	yes	—
297	#380	C₂²×C₆₇	1	1	yes	characteristic
298	#340	C₂²×C₆₇	1	2	yes	—
299	#340	C₂²×C₆₇	1	2	yes	—
300	#336	C₂²×C₆₇	1	2	yes	—
301	#336	C₂²×C₆₇	1	2	yes	—
302	C₂×#41	—	2	4	no	—
303	C₂×#41	—	2	4	no	—
304	C₂×#41	—	2	2	no	—
305	C₂×#41	D₄×C₆₇	1	4	yes	—
306	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
307	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
308	C₂²×D₄	—	2	1	no	—
309	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
310	C₂³×C₄	—	2	1	no	—
311	C₂³×C₄	—	2	1	no	—
312	C₂³×C₄	—	2	2	no	—
313	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
314	C₂×Q₁₆	—	2	2	no	—
315	C₂×Q₁₆	—	2	4	no	—
316	C₂×SD₁₆	—	2	2	no	—
317	C₂×SD₁₆	—	2	4	no	—
318	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
319	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
320	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
321	C₂²⋊Q₈	—	2	2	no	—
322	C₂²⋊Q₈	—	2	2	no	—
323	C₄×Q₈	—	2	4	no	—
324	C₄×Q₈	—	2	4	no	—
325	C₄×D₄	—	2	16	no	—
326	C₄×D₄	—	2	4	no	—
327	C₄×D₄	—	2	4	no	—
328	#117	—	2	8	no	—
329	#117	—	2	2	no	—
330	#117	—	2	2	no	—
331	C₂×C₄⋊C₄	—	2	8	no	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	—	2	2	no	—
334	C₂×C₄⋊C₄	—	2	2	no	—
335	C₂×C₄⋊C₄	—	2	1	no	—
336	C₂×C₄⋊C₄	—	2	2	no	—
337	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
338	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
339	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
340	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
341	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
342	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
343	C₂×C₂²⋊C₄	—	2	2	no	—
344	C₂×C₂²⋊C₄	—	2	1	no	—
345	C₂×C₂²⋊C₄	—	2	1	no	—
346	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
347	C₂×C₄²	—	2	2	no	—
348	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
349	#106	—	2	4	no	—
350	#103	—	2	2	no	—
351	#103	—	2	4	no	—
352	#103	—	2	4	no	—
353	#102	—	2	2	no	—
354	#102	—	2	4	no	—
355	C₂²⋊C₈	—	2	2	no	—
356	C₂²⋊C₈	—	2	2	no	—
357	#95	—	2	1	no	—
358	#95	—	2	2	no	—
359	#95	—	2	2	no	—
360	#95	—	2	2	no	—
361	C₂⁴	—	2	1	no	—
362	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
363	#41	—	2	16	no	—
364	#41	—	2	8	no	—
365	#41	—	2	8	no	—
366	C₂×Q₈	—	2	1	no	—
367	C₂×Q₈	—	2	2	no	—
368	C₂×Q₈	—	2	1	no	—
369	C₂×Q₈	—	2	1	no	—
370	C₂×Q₈	—	2	2	no	—
371	C₂×Q₈	—	2	2	no	—
372	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
373	C₂×Q₈	#41×C₆₇	1	4	yes	—
374	C₂×D₄	—	2	8	no	—
375	C₂×D₄	—	2	4	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
379	C₂×D₄	#41×C₆₇	1	4	yes	—
380	C₂²×C₄	—	2	8	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	8	no	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	8	4	no	—
392	C₂²×C₄	—	8	2	no	—
393	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
394	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
395	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
396	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
397	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
398	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
399	Q₁₆	—	4	4	no	—
400	SD₁₆	—	4	4	no	—
401	C₂×C₈	—	2	1	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	1	no	—
404	C₂×C₈	—	2	2	no	—
405	C₂×C₈	—	2	2	no	—
406	C₂×C₈	—	2	2	no	—
407	C₂×C₈	—	2	2	no	—
408	C₄⋊C₄	—	2	8	no	—
409	C₄⋊C₄	—	2	2	no	—
410	C₄⋊C₄	—	2	4	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	—	2	2	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	—	4	2	no	—
418	C₄⋊C₄	—	8	2	no	—
419	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
420	C₂²⋊C₄	—	2	8	no	—
421	C₂²⋊C₄	—	2	4	no	—
422	C₂²⋊C₄	—	4	2	no	—
423	C₂²⋊C₄	—	8	1	no	—
424	C₄²	—	2	8	no	—
425	C₄²	—	2	2	no	—
426	C₄²	—	2	2	no	—
427	C₂³	—	2	2	no	—
428	C₂³	—	2	2	no	—
429	C₂³	—	2	4	no	—
430	C₂³	—	2	1	no	—
431	C₂³	—	2	2	no	—
432	C₂³	—	2	1	no	—
433	C₂³	—	4	2	no	—
434	C₂³	—	4	1	no	—
435	C₂³	—	4	1	no	—
436	C₂³	#120×C₆₇	1	2	yes	—
437	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
438	Q₈	—	2	2	no	—
439	Q₈	—	2	4	no	—
440	Q₈	—	4	1	no	—
441	Q₈	—	4	1	no	—
442	Q₈	—	4	2	no	—
443	Q₈	#123×C₆₇	1	4	yes	—
444	D₄	—	2	16	no	—
445	D₄	—	2	2	no	—
446	D₄	—	2	4	no	—
447	D₄	#123×C₆₇	1	4	yes	—
448	C₂×C₄	—	2	16	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	4	4	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	4	1	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	4	1	no	—
470	C₂×C₄	—	8	4	no	—
471	C₂×C₄	—	8	8	no	—
472	C₂×C₄	#120×C₆₇	1	2	yes	—
473	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
474	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
475	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
476	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
477	C₈	—	4	1	no	—
478	C₈	—	4	1	no	—
479	C₈	—	4	2	no	—
480	C₂²	—	2	2	no	—
481	C₂²	—	2	2	no	—
482	C₂²	—	2	2	no	—
483	C₂²	—	2	4	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	4	4	no	—
488	C₂²	—	4	2	no	—
489	C₂²	—	4	4	no	—
490	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
491	C₂²	#525×C₆₇	1	2	yes	—
492	C₂²	#577×C₆₇	1	2	yes	—
493	C₂²	#577×C₆₇	1	1	yes	characteristic
494	C₂²	#578×C₆₇	1	1	yes	characteristic
495	C₄	—	2	2	no	—
496	C₄	—	2	4	no	—
497	C₄	—	4	2	no	—
498	C₄	—	4	4	no	—
499	C₄	—	4	2	no	—
500	C₄	—	8	1	no	—
501	C₄	—	8	1	no	—
502	C₄	—	8	2	no	—
503	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
504	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
505	C₄	#525×C₆₇	1	2	yes	—
506	C₂	—	4	2	no	—
507	C₂	—	4	4	no	—
508	C₂	C₆₇×#2849	1	1	yes	characteristic
509	C₂	C₆₇×#3009	1	1	yes	characteristic
510	C₂	C₆₇×#3013	1	1	yes	characteristic
511	C₂	C₆₇×#3120	1	2	yes	—
512	C₂	C₆₇×#3125	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#483×C₆₇	C₂	semidirect
4. C₆₇×#2625	C₂	semidirect
5. C₆₇×#1878	C₂	semidirect
6. C₆₇×#1877	C₂	semidirect
7. C₆₇×#1865	C₂	semidirect
8. C₆₇×#1864	C₂	semidirect
9. C₆₇×#1824	C₂	semidirect
10. C₆₇×#1818	C₂	semidirect
249. #33016	C₆₇	direct (N × H)
258. C₂×#483	C₂×C₆₇	semidirect
260. #2625	C₂×C₆₇	semidirect
261. #1878	C₂×C₆₇	semidirect
262. #1877	C₂×C₆₇	semidirect
263. #1865	C₂×C₆₇	semidirect
264. #1864	C₂×C₆₇	semidirect
265. #1824	C₂×C₆₇	semidirect
266. #1818	C₂×C₆₇	semidirect
268. C₆₇	#33016	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#520×C₆₇	2
C₂×#483×C₆₇	2
C₂×#452×C₆₇	2
C₆₇×#2625	2
C₆₇×#1878	2
C₆₇×#1877	2
C₆₇×#1865	2
C₆₇×#1864	2
C₆₇×#1824	2
C₆₇×#1818	2
C₆₇×#1788	2
#33016	67

Subgroups — C67×#33016 #30712267

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33016 #30712267