Subgroups — C₆₇×#33184 #30712435

Order 17152.

1128 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#497×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#467×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#465×C₆₇	C₂	1	2	yes	maximal
4	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2857	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2850	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1973	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1971	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1960	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1959	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1847	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
13	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	#497×C₆₇	—	2	4	no	—
22	#467×C₆₇	—	2	8	no	—
23	#465×C₆₇	—	2	8	no	—
24	C₂×#107×C₆₇	C₂²	1	2	yes	—
25	C₂×#106×C₆₇	C₂²	1	2	yes	—
26	C₂×#103×C₆₇	C₂²	1	2	yes	—
27	C₂×#103×C₆₇	C₂²	1	2	yes	—
28	C₂×#103×C₆₇	C₂²	1	2	yes	—
29	C₂×#102×C₆₇	C₂²	1	2	yes	—
30	C₂×#97×C₆₇	—	2	4	no	—
31	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#97×C₆₇	C₂²	1	2	yes	—
33	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
34	#391×C₆₇	C₂²	1	1	yes	characteristic
35	#391×C₆₇	C₂²	1	2	yes	—
36	#390×C₆₇	C₂²	1	1	yes	characteristic
37	#389×C₆₇	—	2	2	no	—
38	#388×C₆₇	—	2	2	no	—
39	#386×C₆₇	—	2	2	no	—
40	#384×C₆₇	C₂²	1	1	yes	characteristic
41	#382×C₆₇	C₂²	1	1	yes	characteristic
42	#380×C₆₇	—	2	2	no	—
43	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
44	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
45	C₂²×D₄×C₆₇	—	2	1	no	—
46	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₆₇	D₄	1	2	yes	—
48	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×Q₁₆×C₆₇	—	2	4	no	—
50	C₂×Q₁₆×C₆₇	—	2	4	no	—
51	C₂×Q₁₆×C₆₇	—	2	4	no	—
52	C₂×SD₁₆×C₆₇	—	2	4	no	—
53	C₂×SD₁₆×C₆₇	—	2	4	no	—
54	C₂×SD₁₆×C₆₇	—	2	4	no	—
55	C₂²×C₈×C₆₇	—	2	4	no	—
56	C₂²×C₈×C₆₇	—	2	4	no	—
57	C₂²×C₈×C₆₇	D₄	1	8	yes	—
58	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
59	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
60	C₄⋊Q₈×C₆₇	—	2	2	no	—
61	C₄⋊Q₈×C₆₇	—	2	2	no	—
62	#124×C₆₇	—	2	2	no	—
63	#124×C₆₇	—	2	2	no	—
64	C₂²⋊Q₈×C₆₇	—	2	4	no	—
65	C₂²⋊Q₈×C₆₇	—	2	4	no	—
66	C₂²⋊Q₈×C₆₇	—	2	4	no	—
67	#121×C₆₇	—	2	4	no	—
68	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
69	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
70	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
71	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
72	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
75	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
77	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
78	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
80	C₂×C₄²×C₆₇	—	2	2	no	—
81	C₂×C₄²×C₆₇	—	2	2	no	—
82	C₂×C₄²×C₆₇	D₄	1	2	yes	—
83	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
85	#107×C₆₇	—	2	4	no	—
86	#106×C₆₇	—	2	4	no	—
87	#103×C₆₇	—	2	4	no	—
88	#103×C₆₇	—	2	4	no	—
89	#103×C₆₇	—	2	4	no	—
90	#102×C₆₇	—	2	4	no	—
91	#97×C₆₇	—	2	4	no	—
92	#97×C₆₇	—	2	2	no	—
93	#97×C₆₇	D₄	1	4	yes	complemented
94	#95×C₆₇	—	2	4	no	—
95	#95×C₆₇	—	2	2	no	—
96	#95×C₆₇	—	2	2	no	—
97	C₂⁴×C₆₇	—	2	1	no	—
98	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂×Q₈×C₆₇	—	2	2	no	—
100	C₂×Q₈×C₆₇	—	2	2	no	—
101	C₂×Q₈×C₆₇	—	2	2	no	—
102	C₂×Q₈×C₆₇	—	2	1	no	—
103	C₂×Q₈×C₆₇	—	2	2	no	—
104	C₂×Q₈×C₆₇	—	2	1	no	—
105	C₂×Q₈×C₆₇	—	2	2	no	—
106	C₂×Q₈×C₆₇	—	2	2	no	—
107	C₂×Q₈×C₆₇	—	2	2	no	—
108	C₂×Q₈×C₆₇	—	2	2	no	—
109	C₂×D₄×C₆₇	—	2	2	no	—
110	C₂×D₄×C₆₇	—	2	1	no	—
111	C₂×D₄×C₆₇	—	2	1	no	—
112	C₂×D₄×C₆₇	—	2	2	no	—
113	C₂×D₄×C₆₇	—	4	2	no	—
114	C₂×D₄×C₆₇	—	4	4	no	—
115	C₂²×C₄×C₆₇	—	2	1	no	—
116	C₂²×C₄×C₆₇	—	2	1	no	—
117	C₂²×C₄×C₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	—	2	4	no	—
119	C₂²×C₄×C₆₇	—	2	4	no	—
120	C₂²×C₄×C₆₇	—	2	1	no	—
121	C₂²×C₄×C₆₇	—	2	1	no	—
122	C₂²×C₄×C₆₇	—	2	2	no	—
123	C₂²×C₄×C₆₇	—	2	2	no	—
124	C₂²×C₄×C₆₇	—	2	1	no	—
125	C₂²×C₄×C₆₇	—	4	2	no	—
126	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
130	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
131	Q₁₆×C₆₇	—	4	8	no	—
132	SD₁₆×C₆₇	—	4	8	no	—
133	C₂×C₈×C₆₇	—	2	2	no	—
134	C₂×C₈×C₆₇	—	2	2	no	—
135	C₂×C₈×C₆₇	—	2	16	no	—
136	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
137	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
138	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
139	C₂×C₈×C₆₇	#41	1	4	yes	—
140	C₄⋊C₄×C₆₇	—	2	2	no	—
141	C₄⋊C₄×C₆₇	—	2	2	no	—
142	C₄⋊C₄×C₆₇	—	2	2	no	—
143	C₄⋊C₄×C₆₇	—	2	2	no	—
144	C₄⋊C₄×C₆₇	—	4	4	no	—
145	C₄⋊C₄×C₆₇	—	4	4	no	—
146	C₄⋊C₄×C₆₇	—	4	2	no	—
147	C₄⋊C₄×C₆₇	—	4	4	no	—
148	C₂²⋊C₄×C₆₇	—	4	4	no	—
149	C₂²⋊C₄×C₆₇	—	4	4	no	—
150	C₂²⋊C₄×C₆₇	—	4	2	no	—
151	C₂²⋊C₄×C₆₇	—	4	4	no	—
152	C₂²⋊C₄×C₆₇	—	4	2	no	—
153	C₄²×C₆₇	—	2	4	no	—
154	C₄²×C₆₇	—	2	1	no	—
155	C₄²×C₆₇	—	2	1	no	—
156	C₄²×C₆₇	C₂×D₄	1	2	yes	—
157	C₄²×C₆₇	#41	1	2	yes	—
158	C₂³×C₆₇	—	2	1	no	—
159	C₂³×C₆₇	—	2	1	no	—
160	C₂³×C₆₇	—	2	2	no	—
161	C₂³×C₆₇	—	4	1	no	—
162	C₂³×C₆₇	—	4	1	no	—
163	C₂³×C₆₇	—	4	1	no	—
164	C₂³×C₆₇	—	4	2	no	—
165	C₂³×C₆₇	—	4	2	no	—
166	C₂³×C₆₇	#127	1	2	yes	—
167	C₂³×C₆₇	#137	1	4	yes	—
168	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
169	Q₈×C₆₇	—	4	2	no	—
170	Q₈×C₆₇	—	4	2	no	—
171	Q₈×C₆₇	—	4	2	no	—
172	Q₈×C₆₇	—	4	2	no	—
173	Q₈×C₆₇	—	4	2	no	—
174	Q₈×C₆₇	—	4	2	no	—
175	D₄×C₆₇	—	4	2	no	—
176	D₄×C₆₇	—	4	2	no	—
177	D₄×C₆₇	—	8	4	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	1	no	—
180	C₂×C₄×C₆₇	—	2	2	no	—
181	C₂×C₄×C₆₇	—	2	2	no	—
182	C₂×C₄×C₆₇	—	2	4	no	—
183	C₂×C₄×C₆₇	—	2	4	no	—
184	C₂×C₄×C₆₇	—	2	2	no	—
185	C₂×C₄×C₆₇	—	2	2	no	—
186	C₂×C₄×C₆₇	—	2	2	no	—
187	C₂×C₄×C₆₇	—	4	8	no	—
188	C₂×C₄×C₆₇	—	4	1	no	—
189	C₂×C₄×C₆₇	—	4	2	no	—
190	C₂×C₄×C₆₇	—	4	2	no	—
191	C₂×C₄×C₆₇	—	4	1	no	—
192	C₂×C₄×C₆₇	—	4	1	no	—
193	C₂×C₄×C₆₇	—	4	1	no	—
194	C₂×C₄×C₆₇	—	4	2	no	—
195	C₂×C₄×C₆₇	—	4	1	no	—
196	C₂×C₄×C₆₇	—	4	2	no	—
197	C₂×C₄×C₆₇	—	4	2	no	—
198	C₂×C₄×C₆₇	—	4	2	no	—
199	C₂×C₄×C₆₇	—	4	4	no	—
200	C₂×C₄×C₆₇	#121	1	4	yes	—
201	C₂×C₄×C₆₇	#127	1	2	yes	—
202	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
203	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
204	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
205	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
206	C₈×C₆₇	—	2	4	no	—
207	C₈×C₆₇	#121	1	8	yes	—
208	C₂²×C₆₇	—	2	2	no	—
209	C₂²×C₆₇	—	2	2	no	—
210	C₂²×C₆₇	—	2	4	no	—
211	C₂²×C₆₇	—	2	4	no	—
212	C₂²×C₆₇	—	8	1	no	—
213	C₂²×C₆₇	—	8	1	no	—
214	C₂²×C₆₇	—	8	1	no	—
215	C₂²×C₆₇	—	8	2	no	—
216	C₂²×C₆₇	—	8	2	no	—
217	C₂²×C₆₇	#497	1	4	yes	complemented
218	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
219	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
220	C₂²×C₆₇	#532	1	1	yes	characteristic
221	C₂²×C₆₇	C₂×#137	1	2	yes	—
222	C₂²×C₆₇	#575	1	2	yes	—
223	C₄×C₆₇	—	2	2	no	—
224	C₄×C₆₇	—	2	4	no	—
225	C₄×C₆₇	—	4	2	no	—
226	C₄×C₆₇	—	8	1	no	—
227	C₄×C₆₇	—	8	2	no	—
228	C₄×C₆₇	—	8	1	no	—
229	C₄×C₆₇	—	8	1	no	—
230	C₄×C₆₇	—	8	2	no	—
231	C₄×C₆₇	C₂×#121	1	2	yes	—
232	C₄×C₆₇	#532	1	2	yes	—
233	#33184	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
234	C₂×C₆₇	—	2	4	no	—
235	C₂×C₆₇	—	16	1	no	—
236	C₂×C₆₇	#2856	1	1	yes	characteristic
237	C₂×C₆₇	#3056	1	2	yes	—
238	C₂×C₆₇	C₂×#497	1	1	yes	characteristic
239	C₂×C₆₇	#3150	1	1	yes	characteristic
240	C₂×C₆₇	#3181	1	2	yes	—
241	C₂×#497	C₂×C₆₇	1	1	yes	characteristic, complemented
242	C₂×#467	C₂×C₆₇	1	2	yes	complemented
243	C₂×#465	C₂×C₆₇	1	2	yes	—
244	C₂²×#97	C₂×C₆₇	1	1	yes	characteristic, complemented
245	#2857	C₂×C₆₇	1	1	yes	characteristic, complemented
246	#2850	C₂×C₆₇	1	1	yes	characteristic
247	#1973	C₂×C₆₇	1	2	yes	complemented
248	#1971	C₂×C₆₇	1	2	yes	complemented
249	#1960	C₂×C₆₇	1	1	yes	characteristic, complemented
250	#1959	C₂×C₆₇	1	1	yes	characteristic, complemented
251	#1847	C₂×C₆₇	1	1	yes	characteristic, complemented
252	C₆₇	#33184	1	1	yes	characteristic, direct product factor, Sylow
253	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
254	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
255	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
256	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
257	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
258	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
259	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
260	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
261	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
262	#497	—	2	4	no	—
263	#467	—	2	8	no	—
264	#465	—	2	8	no	—
265	C₂×#107	C₂²×C₆₇	1	2	yes	—
266	C₂×#106	C₂²×C₆₇	1	2	yes	—
267	C₂×#103	C₂²×C₆₇	1	2	yes	—
268	C₂×#103	C₂²×C₆₇	1	2	yes	—
269	C₂×#103	C₂²×C₆₇	1	2	yes	—
270	C₂×#102	C₂²×C₆₇	1	2	yes	—
271	C₂×#97	—	2	4	no	—
272	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
273	C₂×#97	C₂²×C₆₇	1	2	yes	—
274	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
275	#391	C₂²×C₆₇	1	1	yes	characteristic
276	#391	C₂²×C₆₇	1	2	yes	—
277	#390	C₂²×C₆₇	1	1	yes	characteristic
278	#389	—	2	2	no	—
279	#388	—	2	2	no	—
280	#386	—	2	2	no	—
281	#384	C₂²×C₆₇	1	1	yes	characteristic
282	#382	C₂²×C₆₇	1	1	yes	characteristic
283	#380	—	2	2	no	—
284	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
285	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
286	C₂²×D₄	—	2	1	no	—
287	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
288	C₂³×C₄	D₄×C₆₇	1	2	yes	—
289	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
290	C₂×Q₁₆	—	2	4	no	—
291	C₂×Q₁₆	—	2	4	no	—
292	C₂×Q₁₆	—	2	4	no	—
293	C₂×SD₁₆	—	2	4	no	—
294	C₂×SD₁₆	—	2	4	no	—
295	C₂×SD₁₆	—	2	4	no	—
296	C₂²×C₈	—	2	4	no	—
297	C₂²×C₈	—	2	4	no	—
298	C₂²×C₈	D₄×C₆₇	1	8	yes	—
299	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
300	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
301	C₄⋊Q₈	—	2	2	no	—
302	C₄⋊Q₈	—	2	2	no	—
303	#124	—	2	2	no	—
304	#124	—	2	2	no	—
305	C₂²⋊Q₈	—	2	4	no	—
306	C₂²⋊Q₈	—	2	4	no	—
307	C₂²⋊Q₈	—	2	4	no	—
308	#121	—	2	4	no	—
309	C₂×C₄⋊C₄	—	2	1	no	—
310	C₂×C₄⋊C₄	—	2	2	no	—
311	C₂×C₄⋊C₄	—	2	2	no	—
312	C₂×C₄⋊C₄	—	2	2	no	—
313	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
314	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
315	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
316	C₂×C₂²⋊C₄	—	2	2	no	—
317	C₂×C₂²⋊C₄	—	2	1	no	—
318	C₂×C₂²⋊C₄	—	2	1	no	—
319	C₂×C₂²⋊C₄	—	2	2	no	—
320	C₂×C₂²⋊C₄	—	2	2	no	—
321	C₂×C₄²	—	2	2	no	—
322	C₂×C₄²	—	2	2	no	—
323	C₂×C₄²	D₄×C₆₇	1	2	yes	—
324	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
325	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
326	#107	—	2	4	no	—
327	#106	—	2	4	no	—
328	#103	—	2	4	no	—
329	#103	—	2	4	no	—
330	#103	—	2	4	no	—
331	#102	—	2	4	no	—
332	#97	—	2	4	no	—
333	#97	—	2	2	no	—
334	#97	D₄×C₆₇	1	4	yes	complemented
335	#95	—	2	4	no	—
336	#95	—	2	2	no	—
337	#95	—	2	2	no	—
338	C₂⁴	—	2	1	no	—
339	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
340	C₂×Q₈	—	2	2	no	—
341	C₂×Q₈	—	2	2	no	—
342	C₂×Q₈	—	2	2	no	—
343	C₂×Q₈	—	2	1	no	—
344	C₂×Q₈	—	2	2	no	—
345	C₂×Q₈	—	2	1	no	—
346	C₂×Q₈	—	2	2	no	—
347	C₂×Q₈	—	2	2	no	—
348	C₂×Q₈	—	2	2	no	—
349	C₂×Q₈	—	2	2	no	—
350	C₂×D₄	—	2	2	no	—
351	C₂×D₄	—	2	1	no	—
352	C₂×D₄	—	2	1	no	—
353	C₂×D₄	—	2	2	no	—
354	C₂×D₄	—	4	2	no	—
355	C₂×D₄	—	4	4	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	4	no	—
360	C₂²×C₄	—	2	4	no	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	4	2	no	—
367	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
368	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
369	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
370	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
371	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
372	Q₁₆	—	4	8	no	—
373	SD₁₆	—	4	8	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	16	no	—
377	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
378	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
379	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
380	C₂×C₈	#41×C₆₇	1	4	yes	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	4	4	no	—
386	C₄⋊C₄	—	4	4	no	—
387	C₄⋊C₄	—	4	2	no	—
388	C₄⋊C₄	—	4	4	no	—
389	C₂²⋊C₄	—	4	4	no	—
390	C₂²⋊C₄	—	4	4	no	—
391	C₂²⋊C₄	—	4	2	no	—
392	C₂²⋊C₄	—	4	4	no	—
393	C₂²⋊C₄	—	4	2	no	—
394	C₄²	—	2	4	no	—
395	C₄²	—	2	1	no	—
396	C₄²	—	2	1	no	—
397	C₄²	C₂×D₄×C₆₇	1	2	yes	—
398	C₄²	#41×C₆₇	1	2	yes	—
399	C₂³	—	2	1	no	—
400	C₂³	—	2	1	no	—
401	C₂³	—	2	2	no	—
402	C₂³	—	4	1	no	—
403	C₂³	—	4	1	no	—
404	C₂³	—	4	1	no	—
405	C₂³	—	4	2	no	—
406	C₂³	—	4	2	no	—
407	C₂³	#127×C₆₇	1	2	yes	—
408	C₂³	#137×C₆₇	1	4	yes	—
409	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
410	Q₈	—	4	2	no	—
411	Q₈	—	4	2	no	—
412	Q₈	—	4	2	no	—
413	Q₈	—	4	2	no	—
414	Q₈	—	4	2	no	—
415	Q₈	—	4	2	no	—
416	D₄	—	4	2	no	—
417	D₄	—	4	2	no	—
418	D₄	—	8	4	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	2	4	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	4	8	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	4	no	—
441	C₂×C₄	#121×C₆₇	1	4	yes	—
442	C₂×C₄	#127×C₆₇	1	2	yes	—
443	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
444	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
445	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
446	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
447	C₈	—	2	4	no	—
448	C₈	#121×C₆₇	1	8	yes	—
449	C₂²	—	2	4	no	—
450	C₂²	—	2	2	no	—
451	C₂²	—	2	2	no	—
452	C₂²	—	2	4	no	—
453	C₂²	—	8	2	no	—
454	C₂²	—	8	1	no	—
455	C₂²	—	8	1	no	—
456	C₂²	—	8	1	no	—
457	C₂²	—	8	2	no	—
458	C₂²	#497×C₆₇	1	4	yes	complemented
459	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
460	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
461	C₂²	#532×C₆₇	1	1	yes	characteristic
462	C₂²	C₂×#137×C₆₇	1	2	yes	—
463	C₂²	#575×C₆₇	1	2	yes	—
464	C₄	—	2	4	no	—
465	C₄	—	2	2	no	—
466	C₄	—	4	2	no	—
467	C₄	—	8	1	no	—
468	C₄	—	8	2	no	—
469	C₄	—	8	1	no	—
470	C₄	—	8	1	no	—
471	C₄	—	8	2	no	—
472	C₄	C₂×#121×C₆₇	1	2	yes	—
473	C₄	#532×C₆₇	1	2	yes	—
474	C₂	—	2	4	no	—
475	C₂	—	16	1	no	—
476	C₂	C₆₇×#2856	1	1	yes	characteristic
477	C₂	C₆₇×#3056	1	2	yes	—
478	C₂	C₂×#497×C₆₇	1	1	yes	characteristic
479	C₂	C₆₇×#3150	1	1	yes	characteristic
480	C₂	C₆₇×#3181	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#121×C₆₇	C₂²
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×#497×C₆₇	C₂	semidirect
2. C₂×#467×C₆₇	C₂	semidirect
4. C₂²×#97×C₆₇	C₂	semidirect
5. C₆₇×#2857	C₂	semidirect
7. C₆₇×#1973	C₂	semidirect
8. C₆₇×#1971	C₂	semidirect
9. C₆₇×#1960	C₂	semidirect
10. C₆₇×#1959	C₂	semidirect
11. C₆₇×#1847	C₂	semidirect
93. #97×C₆₇	D₄	semidirect
217. C₂²×C₆₇	#497	semidirect
233. #33184	C₆₇	direct (N × H)
241. C₂×#497	C₂×C₆₇	semidirect
242. C₂×#467	C₂×C₆₇	semidirect
244. C₂²×#97	C₂×C₆₇	semidirect
245. #2857	C₂×C₆₇	semidirect
247. #1973	C₂×C₆₇	semidirect
248. #1971	C₂×C₆₇	semidirect
249. #1960	C₂×C₆₇	semidirect
250. #1959	C₂×C₆₇	semidirect
251. #1847	C₂×C₆₇	semidirect
252. C₆₇	#33184	direct (N × H)
334. #97	D₄×C₆₇	semidirect
458. C₂²	#497×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#497×C₆₇	2
C₂×#467×C₆₇	2
C₂×#465×C₆₇	2
C₂²×#97×C₆₇	2
C₆₇×#2857	2
C₆₇×#2850	2
C₆₇×#1973	2
C₆₇×#1971	2
C₆₇×#1960	2
C₆₇×#1959	2
C₆₇×#1847	2
#33184	67

Subgroups — C67×#33184 #30712435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33184 #30712435