Subgroups — C₆₇×#33044 #30712295

Order 17152.

948 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×#488×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2851	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2850	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1991	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1958	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1934	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1933	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1893	C₂	1	2	yes	maximal
10	C₆₇×#1892	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1835	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
13	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	#488×C₆₇	—	2	4	no	—
19	C₂×#103×C₆₇	C₂²	1	2	yes	—
20	C₂×#103×C₆₇	C₂²	1	2	yes	—
21	C₂×#102×C₆₇	C₂²	1	2	yes	—
22	C₂×#102×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₆₇	—	2	4	no	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	#391×C₆₇	C₂²	1	1	yes	characteristic
28	#390×C₆₇	C₂²	1	1	yes	characteristic
29	#389×C₆₇	C₂²	1	2	yes	—
30	#388×C₆₇	—	2	2	no	—
31	#387×C₆₇	—	2	2	no	—
32	#386×C₆₇	—	2	2	no	—
33	#385×C₆₇	—	2	2	no	—
34	#382×C₆₇	C₂²	1	1	yes	characteristic
35	#382×C₆₇	C₂²	1	1	yes	characteristic
36	#380×C₆₇	C₂²	1	2	yes	—
37	#340×C₆₇	C₂²	1	2	yes	—
38	#340×C₆₇	C₂²	1	2	yes	—
39	#340×C₆₇	C₂²	1	2	yes	—
40	#340×C₆₇	C₂²	1	2	yes	—
41	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
42	C₂²×D₄×C₆₇	—	2	1	no	—
43	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	D₄	1	2	yes	—
45	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	—	2	4	no	—
47	C₂²×C₈×C₆₇	—	2	4	no	—
48	C₂²×C₈×C₆₇	—	2	4	no	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
51	#125×C₆₇	—	2	2	no	—
52	#125×C₆₇	—	2	2	no	—
53	#124×C₆₇	—	2	2	no	—
54	#124×C₆₇	—	2	2	no	—
55	C₂²⋊Q₈×C₆₇	—	2	4	no	—
56	#121×C₆₇	—	2	4	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
64	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₄²×C₆₇	—	2	2	no	—
70	C₂×C₄²×C₆₇	—	2	2	no	—
71	C₂×C₄²×C₆₇	D₄	1	2	yes	—
72	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	#103×C₆₇	—	2	4	no	—
75	#103×C₆₇	—	2	4	no	—
76	#102×C₆₇	—	2	4	no	—
77	#102×C₆₇	—	2	4	no	—
78	C₄×C₈×C₆₇	—	2	4	no	—
79	C₄×C₈×C₆₇	—	2	2	no	—
80	C₄×C₈×C₆₇	D₄	1	4	yes	complemented
81	#95×C₆₇	—	2	2	no	—
82	#95×C₆₇	—	2	2	no	—
83	#95×C₆₇	—	2	2	no	—
84	#95×C₆₇	—	2	2	no	—
85	#95×C₆₇	—	2	2	no	—
86	C₂⁴×C₆₇	—	2	1	no	—
87	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₂×Q₈×C₆₇	—	2	1	no	—
89	C₂×Q₈×C₆₇	—	2	2	no	—
90	C₂×Q₈×C₆₇	—	2	2	no	—
91	C₂×Q₈×C₆₇	—	2	1	no	—
92	C₂×D₄×C₆₇	—	2	1	no	—
93	C₂×D₄×C₆₇	—	2	2	no	—
94	C₂×D₄×C₆₇	—	2	2	no	—
95	C₂×D₄×C₆₇	—	2	1	no	—
96	C₂×D₄×C₆₇	—	4	2	no	—
97	C₂×D₄×C₆₇	—	4	4	no	—
98	C₂²×C₄×C₆₇	—	2	4	no	—
99	C₂²×C₄×C₆₇	—	2	4	no	—
100	C₂²×C₄×C₆₇	—	2	4	no	—
101	C₂²×C₄×C₆₇	—	2	1	no	—
102	C₂²×C₄×C₆₇	—	2	1	no	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	2	1	no	—
105	C₂²×C₄×C₆₇	—	2	1	no	—
106	C₂²×C₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	1	no	—
109	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₆₇	#41	1	4	yes	—
113	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
114	C₂×C₈×C₆₇	—	2	16	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	—	2	2	no	—
117	C₂×C₈×C₆₇	—	2	2	no	—
118	C₂×C₈×C₆₇	—	2	2	no	—
119	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
120	C₂×C₈×C₆₇	#41	1	4	yes	—
121	C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	—	2	2	no	—
123	C₄⋊C₄×C₆₇	—	2	2	no	—
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	2	2	no	—
127	C₄⋊C₄×C₆₇	—	4	4	no	—
128	C₄⋊C₄×C₆₇	—	4	2	no	—
129	C₄⋊C₄×C₆₇	—	4	4	no	—
130	C₂²⋊C₄×C₆₇	—	4	2	no	—
131	C₂²⋊C₄×C₆₇	—	4	4	no	—
132	C₂²⋊C₄×C₆₇	—	4	4	no	—
133	C₂²⋊C₄×C₆₇	—	4	2	no	—
134	C₂²⋊C₄×C₆₇	—	4	4	no	—
135	C₄²×C₆₇	—	2	4	no	—
136	C₄²×C₆₇	—	2	1	no	—
137	C₄²×C₆₇	—	2	1	no	—
138	C₄²×C₆₇	C₂×D₄	1	2	yes	—
139	C₄²×C₆₇	#41	1	2	yes	—
140	C₂³×C₆₇	—	2	1	no	—
141	C₂³×C₆₇	—	2	1	no	—
142	C₂³×C₆₇	—	2	2	no	—
143	C₂³×C₆₇	—	4	1	no	—
144	C₂³×C₆₇	—	4	1	no	—
145	C₂³×C₆₇	—	4	1	no	—
146	C₂³×C₆₇	—	4	2	no	—
147	C₂³×C₆₇	—	4	2	no	—
148	C₂³×C₆₇	#124	1	2	yes	—
149	C₂³×C₆₇	#135	1	4	yes	—
150	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
151	Q₈×C₆₇	—	4	2	no	—
152	Q₈×C₆₇	—	4	2	no	—
153	D₄×C₆₇	—	4	2	no	—
154	D₄×C₆₇	—	4	2	no	—
155	D₄×C₆₇	—	8	4	no	—
156	C₂×C₄×C₆₇	—	2	16	no	—
157	C₂×C₄×C₆₇	—	2	1	no	—
158	C₂×C₄×C₆₇	—	2	2	no	—
159	C₂×C₄×C₆₇	—	2	1	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	—	2	2	no	—
163	C₂×C₄×C₆₇	—	2	4	no	—
164	C₂×C₄×C₆₇	—	2	4	no	—
165	C₂×C₄×C₆₇	—	4	1	no	—
166	C₂×C₄×C₆₇	—	4	2	no	—
167	C₂×C₄×C₆₇	—	4	1	no	—
168	C₂×C₄×C₆₇	—	4	1	no	—
169	C₂×C₄×C₆₇	—	4	1	no	—
170	C₂×C₄×C₆₇	—	4	2	no	—
171	C₂×C₄×C₆₇	—	4	1	no	—
172	C₂×C₄×C₆₇	—	4	2	no	—
173	C₂×C₄×C₆₇	—	4	2	no	—
174	C₂×C₄×C₆₇	—	4	2	no	—
175	C₂×C₄×C₆₇	—	4	2	no	—
176	C₂×C₄×C₆₇	—	4	2	no	—
177	C₂×C₄×C₆₇	—	4	2	no	—
178	C₂×C₄×C₆₇	#121	1	4	yes	—
179	C₂×C₄×C₆₇	#124	1	2	yes	—
180	C₂×C₄×C₆₇	#135	1	4	yes	—
181	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
183	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
184	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
185	C₈×C₆₇	—	2	4	no	—
186	C₈×C₆₇	—	2	4	no	—
187	C₂²×C₆₇	—	2	2	no	—
188	C₂²×C₆₇	—	2	2	no	—
189	C₂²×C₆₇	—	2	4	no	—
190	C₂²×C₆₇	—	2	4	no	—
191	C₂²×C₆₇	—	8	1	no	—
192	C₂²×C₆₇	—	8	1	no	—
193	C₂²×C₆₇	—	8	1	no	—
194	C₂²×C₆₇	—	8	2	no	—
195	C₂²×C₆₇	—	8	2	no	—
196	C₂²×C₆₇	#488	1	4	yes	complemented
197	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
198	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
199	C₂²×C₆₇	#532	1	1	yes	characteristic
200	C₂²×C₆₇	C₂×#135	1	2	yes	—
201	C₂²×C₆₇	C₂×#135	1	2	yes	—
202	C₄×C₆₇	—	2	2	no	—
203	C₄×C₆₇	—	2	4	no	—
204	C₄×C₆₇	—	2	4	no	—
205	C₄×C₆₇	—	8	1	no	—
206	C₄×C₆₇	—	8	1	no	—
207	C₄×C₆₇	—	8	1	no	—
208	C₄×C₆₇	—	8	2	no	—
209	C₄×C₆₇	—	8	2	no	—
210	C₄×C₆₇	#525	1	2	yes	—
211	C₄×C₆₇	#529	1	2	yes	—
212	#33044	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂×C₆₇	—	2	4	no	—
214	C₂×C₆₇	—	16	1	no	—
215	C₂×C₆₇	#2850	1	1	yes	characteristic
216	C₂×C₆₇	#3050	1	2	yes	—
217	C₂×C₆₇	C₂×#488	1	1	yes	characteristic
218	C₂×C₆₇	#3131	1	1	yes	characteristic
219	C₂×C₆₇	#3168	1	2	yes	—
220	C₂×#488	C₂×C₆₇	1	1	yes	characteristic, complemented
221	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
222	#2851	C₂×C₆₇	1	1	yes	characteristic, complemented
223	#2850	C₂×C₆₇	1	1	yes	characteristic
224	#1991	C₂×C₆₇	1	1	yes	characteristic, complemented
225	#1958	C₂×C₆₇	1	1	yes	characteristic, complemented
226	#1934	C₂×C₆₇	1	2	yes	complemented
227	#1933	C₂×C₆₇	1	2	yes	complemented
228	#1893	C₂×C₆₇	1	2	yes	—
229	#1892	C₂×C₆₇	1	2	yes	complemented
230	#1835	C₂×C₆₇	1	1	yes	characteristic, complemented
231	C₆₇	#33044	1	1	yes	characteristic, direct product factor, Sylow
232	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
233	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
234	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
235	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
236	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
237	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
238	#488	—	2	4	no	—
239	C₂×#103	C₂²×C₆₇	1	2	yes	—
240	C₂×#103	C₂²×C₆₇	1	2	yes	—
241	C₂×#102	C₂²×C₆₇	1	2	yes	—
242	C₂×#102	C₂²×C₆₇	1	2	yes	—
243	C₂×C₄×C₈	—	2	4	no	—
244	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
245	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
246	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
247	#391	C₂²×C₆₇	1	1	yes	characteristic
248	#390	C₂²×C₆₇	1	1	yes	characteristic
249	#389	C₂²×C₆₇	1	2	yes	—
250	#388	—	2	2	no	—
251	#387	—	2	2	no	—
252	#386	—	2	2	no	—
253	#385	—	2	2	no	—
254	#382	C₂²×C₆₇	1	1	yes	characteristic
255	#382	C₂²×C₆₇	1	1	yes	characteristic
256	#380	C₂²×C₆₇	1	2	yes	—
257	#340	C₂²×C₆₇	1	2	yes	—
258	#340	C₂²×C₆₇	1	2	yes	—
259	#340	C₂²×C₆₇	1	2	yes	—
260	#340	C₂²×C₆₇	1	2	yes	—
261	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
262	C₂²×D₄	—	2	1	no	—
263	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
264	C₂³×C₄	D₄×C₆₇	1	2	yes	—
265	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
266	C₂²×C₈	—	2	4	no	—
267	C₂²×C₈	—	2	4	no	—
268	C₂²×C₈	—	2	4	no	—
269	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
270	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
271	#125	—	2	2	no	—
272	#125	—	2	2	no	—
273	#124	—	2	2	no	—
274	#124	—	2	2	no	—
275	C₂²⋊Q₈	—	2	4	no	—
276	#121	—	2	4	no	—
277	C₂×C₄⋊C₄	—	2	1	no	—
278	C₂×C₄⋊C₄	—	2	2	no	—
279	C₂×C₄⋊C₄	—	2	2	no	—
280	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
281	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
282	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
283	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
284	C₂×C₂²⋊C₄	—	2	1	no	—
285	C₂×C₂²⋊C₄	—	2	2	no	—
286	C₂×C₂²⋊C₄	—	2	1	no	—
287	C₂×C₂²⋊C₄	—	2	2	no	—
288	C₂×C₂²⋊C₄	—	2	2	no	—
289	C₂×C₄²	—	2	2	no	—
290	C₂×C₄²	—	2	2	no	—
291	C₂×C₄²	D₄×C₆₇	1	2	yes	—
292	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
293	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
294	#103	—	2	4	no	—
295	#103	—	2	4	no	—
296	#102	—	2	4	no	—
297	#102	—	2	4	no	—
298	C₄×C₈	—	2	4	no	—
299	C₄×C₈	—	2	2	no	—
300	C₄×C₈	D₄×C₆₇	1	4	yes	complemented
301	#95	—	2	2	no	—
302	#95	—	2	2	no	—
303	#95	—	2	2	no	—
304	#95	—	2	2	no	—
305	#95	—	2	2	no	—
306	C₂⁴	—	2	1	no	—
307	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
308	C₂×Q₈	—	2	1	no	—
309	C₂×Q₈	—	2	2	no	—
310	C₂×Q₈	—	2	2	no	—
311	C₂×Q₈	—	2	1	no	—
312	C₂×D₄	—	2	1	no	—
313	C₂×D₄	—	2	2	no	—
314	C₂×D₄	—	2	2	no	—
315	C₂×D₄	—	2	1	no	—
316	C₂×D₄	—	4	2	no	—
317	C₂×D₄	—	4	4	no	—
318	C₂²×C₄	—	2	4	no	—
319	C₂²×C₄	—	2	4	no	—
320	C₂²×C₄	—	2	4	no	—
321	C₂²×C₄	—	2	1	no	—
322	C₂²×C₄	—	2	1	no	—
323	C₂²×C₄	—	2	2	no	—
324	C₂²×C₄	—	2	2	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
330	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
331	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
332	C₂²×C₄	#41×C₆₇	1	4	yes	—
333	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
334	C₂×C₈	—	2	16	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
340	C₂×C₈	#41×C₆₇	1	4	yes	—
341	C₄⋊C₄	—	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	—	4	4	no	—
348	C₄⋊C₄	—	4	2	no	—
349	C₄⋊C₄	—	4	4	no	—
350	C₂²⋊C₄	—	4	2	no	—
351	C₂²⋊C₄	—	4	4	no	—
352	C₂²⋊C₄	—	4	4	no	—
353	C₂²⋊C₄	—	4	2	no	—
354	C₂²⋊C₄	—	4	4	no	—
355	C₄²	—	2	4	no	—
356	C₄²	—	2	1	no	—
357	C₄²	—	2	1	no	—
358	C₄²	C₂×D₄×C₆₇	1	2	yes	—
359	C₄²	#41×C₆₇	1	2	yes	—
360	C₂³	—	2	1	no	—
361	C₂³	—	2	1	no	—
362	C₂³	—	2	2	no	—
363	C₂³	—	4	1	no	—
364	C₂³	—	4	1	no	—
365	C₂³	—	4	1	no	—
366	C₂³	—	4	2	no	—
367	C₂³	—	4	2	no	—
368	C₂³	#124×C₆₇	1	2	yes	—
369	C₂³	#135×C₆₇	1	4	yes	—
370	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
371	Q₈	—	4	2	no	—
372	Q₈	—	4	2	no	—
373	D₄	—	4	2	no	—
374	D₄	—	4	2	no	—
375	D₄	—	8	4	no	—
376	C₂×C₄	—	2	16	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	4	no	—
384	C₂×C₄	—	2	4	no	—
385	C₂×C₄	—	4	1	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	—	4	1	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	#121×C₆₇	1	4	yes	—
399	C₂×C₄	#124×C₆₇	1	2	yes	—
400	C₂×C₄	#135×C₆₇	1	4	yes	—
401	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
402	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
403	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
404	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
405	C₈	—	2	4	no	—
406	C₈	—	2	4	no	—
407	C₂²	—	2	4	no	—
408	C₂²	—	2	2	no	—
409	C₂²	—	2	2	no	—
410	C₂²	—	2	4	no	—
411	C₂²	—	8	2	no	—
412	C₂²	—	8	1	no	—
413	C₂²	—	8	1	no	—
414	C₂²	—	8	1	no	—
415	C₂²	—	8	2	no	—
416	C₂²	#488×C₆₇	1	4	yes	complemented
417	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
418	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
419	C₂²	#532×C₆₇	1	1	yes	characteristic
420	C₂²	C₂×#135×C₆₇	1	2	yes	—
421	C₂²	C₂×#135×C₆₇	1	2	yes	—
422	C₄	—	2	4	no	—
423	C₄	—	2	4	no	—
424	C₄	—	2	2	no	—
425	C₄	—	8	2	no	—
426	C₄	—	8	1	no	—
427	C₄	—	8	1	no	—
428	C₄	—	8	1	no	—
429	C₄	—	8	2	no	—
430	C₄	#525×C₆₇	1	2	yes	—
431	C₄	#529×C₆₇	1	2	yes	—
432	C₂	—	2	4	no	—
433	C₂	—	16	1	no	—
434	C₂	C₆₇×#2850	1	1	yes	characteristic
435	C₂	C₆₇×#3050	1	2	yes	—
436	C₂	C₂×#488×C₆₇	1	1	yes	characteristic
437	C₂	C₆₇×#3131	1	1	yes	characteristic
438	C₂	C₆₇×#3168	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×#488×C₆₇	C₂	semidirect
2. C₂²×C₄×C₈×C₆₇	C₂	semidirect
3. C₆₇×#2851	C₂	semidirect
5. C₆₇×#1991	C₂	semidirect
6. C₆₇×#1958	C₂	semidirect
7. C₆₇×#1934	C₂	semidirect
8. C₆₇×#1933	C₂	semidirect
10. C₆₇×#1892	C₂	semidirect
11. C₆₇×#1835	C₂	semidirect
80. C₄×C₈×C₆₇	D₄	semidirect
196. C₂²×C₆₇	#488	semidirect
212. #33044	C₆₇	direct (N × H)
220. C₂×#488	C₂×C₆₇	semidirect
221. C₂²×C₄×C₈	C₂×C₆₇	semidirect
222. #2851	C₂×C₆₇	semidirect
224. #1991	C₂×C₆₇	semidirect
225. #1958	C₂×C₆₇	semidirect
226. #1934	C₂×C₆₇	semidirect
227. #1933	C₂×C₆₇	semidirect
229. #1892	C₂×C₆₇	semidirect
230. #1835	C₂×C₆₇	semidirect
231. C₆₇	#33044	direct (N × H)
300. C₄×C₈	D₄×C₆₇	semidirect
416. C₂²	#488×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#488×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₆₇×#2851	2
C₆₇×#2850	2
C₆₇×#1991	2
C₆₇×#1958	2
C₆₇×#1934	2
C₆₇×#1933	2
C₆₇×#1893	2
C₆₇×#1892	2
C₆₇×#1835	2
#33044	67

Subgroups — C67×#33044 #30712295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33044 #30712295