Subgroups — C₆₇×#33228 #30712479

Order 17152.

1004 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#490×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#489×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#488×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#486×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2660	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2657	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1961	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1959	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1958	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1957	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#127×C₆₇	C₂²	1	2	yes	—
15	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#124×C₆₇	C₂²	1	2	yes	—
18	#490×C₆₇	—	2	4	no	—
19	#489×C₆₇	—	2	4	no	—
20	#488×C₆₇	—	2	4	no	—
21	#486×C₆₇	—	2	4	no	—
22	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#103×C₆₇	C₂²	1	2	yes	—
25	C₂×#102×C₆₇	C₂²	1	2	yes	—
26	C₂×#102×C₆₇	C₂²	1	2	yes	—
27	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#102×C₆₇	C₂²	1	2	yes	—
30	C₂×#102×C₆₇	C₂²	1	2	yes	—
31	C₂×#102×C₆₇	C₂²	1	2	yes	—
32	C₂×#97×C₆₇	C₂²	1	2	yes	—
33	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
35	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
36	#398×C₆₇	—	2	2	no	—
37	#395×C₆₇	—	2	2	no	—
38	#392×C₆₇	—	2	2	no	—
39	#392×C₆₇	—	2	2	no	—
40	#384×C₆₇	C₂²	1	2	yes	—
41	#382×C₆₇	C₂²	1	2	yes	—
42	#376×C₆₇	C₂²	1	1	yes	characteristic
43	C₂²×Q₈×C₆₇	—	2	1	no	—
44	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₆₇	—	2	4	no	—
46	C₂²×D₄×C₆₇	—	2	1	no	—
47	C₂²×D₄×C₆₇	—	2	1	no	—
48	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₆₇	—	2	2	no	—
54	C₄⋊Q₈×C₆₇	—	2	2	no	—
55	#127×C₆₇	—	2	2	no	—
56	#127×C₆₇	—	2	2	no	—
57	#127×C₆₇	—	2	4	no	—
58	#127×C₆₇	—	2	4	no	—
59	#125×C₆₇	—	2	2	no	—
60	#125×C₆₇	—	2	2	no	—
61	#124×C₆₇	—	2	2	no	—
62	#124×C₆₇	—	2	2	no	—
63	#124×C₆₇	—	2	4	no	—
64	#124×C₆₇	—	2	4	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
68	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
72	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
77	#103×C₆₇	—	2	2	no	—
78	#103×C₆₇	—	2	2	no	—
79	#103×C₆₇	—	2	4	no	—
80	#102×C₆₇	—	2	4	no	—
81	#102×C₆₇	—	2	4	no	—
82	#102×C₆₇	—	2	2	no	—
83	#102×C₆₇	—	2	2	no	—
84	#102×C₆₇	—	2	4	no	—
85	#102×C₆₇	—	2	4	no	—
86	#102×C₆₇	—	2	4	no	—
87	#97×C₆₇	—	2	4	no	—
88	#97×C₆₇	D₄	1	4	yes	complemented
89	C₄×C₈×C₆₇	—	2	4	no	—
90	C₄×C₈×C₆₇	D₄	1	4	yes	complemented
91	#95×C₆₇	—	2	4	no	—
92	#95×C₆₇	D₄	1	2	yes	—
93	#95×C₆₇	D₄	1	2	yes	—
94	C₂⁴×C₆₇	—	2	2	no	—
95	C₂⁴×C₆₇	—	2	1	no	—
96	C₂×Q₈×C₆₇	—	2	1	no	—
97	C₂×Q₈×C₆₇	—	2	1	no	—
98	C₂×Q₈×C₆₇	—	2	2	no	—
99	C₂×Q₈×C₆₇	—	2	2	no	—
100	C₂×Q₈×C₆₇	—	4	2	no	—
101	C₂×Q₈×C₆₇	—	8	2	no	—
102	C₂×D₄×C₆₇	—	2	1	no	—
103	C₂×D₄×C₆₇	—	2	1	no	—
104	C₂×D₄×C₆₇	—	2	2	no	—
105	C₂×D₄×C₆₇	—	2	2	no	—
106	C₂×D₄×C₆₇	—	2	2	no	—
107	C₂×D₄×C₆₇	—	2	2	no	—
108	C₂×D₄×C₆₇	—	2	2	no	—
109	C₂×D₄×C₆₇	—	2	2	no	—
110	C₂×D₄×C₆₇	—	2	2	no	—
111	C₂×D₄×C₆₇	—	2	2	no	—
112	C₂×D₄×C₆₇	—	4	8	no	—
113	C₂×D₄×C₆₇	—	4	2	no	—
114	C₂×D₄×C₆₇	—	4	2	no	—
115	C₂×D₄×C₆₇	—	8	4	no	—
116	C₂×D₄×C₆₇	—	8	2	no	—
117	C₂×D₄×C₆₇	—	8	4	no	—
118	C₂×D₄×C₆₇	—	8	2	no	—
119	C₂²×C₄×C₆₇	—	2	1	no	—
120	C₂²×C₄×C₆₇	—	2	1	no	—
121	C₂²×C₄×C₆₇	—	2	2	no	—
122	C₂²×C₄×C₆₇	—	2	1	no	—
123	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
124	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
127	C₂×C₈×C₆₇	—	2	2	no	—
128	C₂×C₈×C₆₇	—	2	2	no	—
129	C₂×C₈×C₆₇	—	2	2	no	—
130	C₂×C₈×C₆₇	—	2	2	no	—
131	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
132	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
133	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
134	C₄⋊C₄×C₆₇	—	2	1	no	—
135	C₄⋊C₄×C₆₇	—	2	1	no	—
136	C₄⋊C₄×C₆₇	—	2	2	no	—
137	C₄⋊C₄×C₆₇	—	2	1	no	—
138	C₄⋊C₄×C₆₇	—	2	1	no	—
139	C₄⋊C₄×C₆₇	—	2	2	no	—
140	C₄⋊C₄×C₆₇	—	4	2	no	—
141	C₄⋊C₄×C₆₇	—	4	2	no	—
142	C₄⋊C₄×C₆₇	—	4	2	no	—
143	C₄⋊C₄×C₆₇	—	8	4	no	—
144	C₂²⋊C₄×C₆₇	—	4	8	no	—
145	C₂²⋊C₄×C₆₇	—	4	2	no	—
146	C₂²⋊C₄×C₆₇	—	4	2	no	—
147	C₄²×C₆₇	C₂×D₄	1	2	yes	—
148	C₄²×C₆₇	C₂×D₄	1	2	yes	—
149	C₄²×C₆₇	#41	1	4	yes	—
150	C₄²×C₆₇	#41	1	4	yes	—
151	C₂³×C₆₇	—	4	2	no	—
152	C₂³×C₆₇	—	4	2	no	—
153	C₂³×C₆₇	—	4	2	no	—
154	C₂³×C₆₇	—	4	1	no	—
155	C₂³×C₆₇	—	4	2	no	—
156	C₂³×C₆₇	—	4	2	no	—
157	C₂³×C₆₇	—	4	4	no	—
158	C₂³×C₆₇	—	4	2	no	—
159	C₂³×C₆₇	—	4	4	no	—
160	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
161	Q₈×C₆₇	—	4	1	no	—
162	Q₈×C₆₇	—	4	1	no	—
163	Q₈×C₆₇	—	4	2	no	—
164	Q₈×C₆₇	—	8	4	no	—
165	D₄×C₆₇	—	4	1	no	—
166	D₄×C₆₇	—	4	1	no	—
167	D₄×C₆₇	—	4	2	no	—
168	D₄×C₆₇	—	4	2	no	—
169	D₄×C₆₇	—	4	2	no	—
170	D₄×C₆₇	—	4	2	no	—
171	D₄×C₆₇	—	4	2	no	—
172	D₄×C₆₇	—	8	16	no	—
173	D₄×C₆₇	—	8	4	no	—
174	D₄×C₆₇	—	8	4	no	—
175	C₂×C₄×C₆₇	—	2	4	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	4	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	—	4	2	no	—
180	C₂×C₄×C₆₇	—	4	1	no	—
181	C₂×C₄×C₆₇	—	4	1	no	—
182	C₂×C₄×C₆₇	—	4	1	no	—
183	C₂×C₄×C₆₇	—	4	2	no	—
184	C₂×C₄×C₆₇	—	4	2	no	—
185	C₂×C₄×C₆₇	—	4	4	no	—
186	C₂×C₄×C₆₇	—	4	2	no	—
187	C₂×C₄×C₆₇	#124	1	8	yes	—
188	C₂×C₄×C₆₇	#127	1	2	yes	—
189	C₂×C₄×C₆₇	#127	1	2	yes	—
190	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
193	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
194	C₈×C₆₇	—	2	2	no	—
195	C₈×C₆₇	—	2	4	no	—
196	C₈×C₆₇	—	2	2	no	—
197	C₂²×C₆₇	—	8	2	no	—
198	C₂²×C₆₇	—	8	1	no	—
199	C₂²×C₆₇	—	8	4	no	—
200	C₂²×C₆₇	—	8	4	no	—
201	C₂²×C₆₇	—	8	4	no	—
202	C₂²×C₆₇	—	8	2	no	—
203	C₂²×C₆₇	—	8	1	no	—
204	C₂²×C₆₇	—	8	2	no	—
205	C₂²×C₆₇	—	8	1	no	—
206	C₂²×C₆₇	C₂×#124	1	2	yes	—
207	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
208	C₂²×C₆₇	#576	1	2	yes	—
209	C₂²×C₆₇	#577	1	2	yes	—
210	C₄×C₆₇	—	2	8	no	—
211	C₄×C₆₇	—	2	2	no	—
212	C₄×C₆₇	—	2	2	no	—
213	C₄×C₆₇	—	8	1	no	—
214	C₄×C₆₇	—	8	1	no	—
215	C₄×C₆₇	—	8	1	no	—
216	C₄×C₆₇	—	8	2	no	—
217	C₄×C₆₇	#532	1	1	yes	characteristic
218	C₄×C₆₇	#532	1	2	yes	—
219	C₄×C₆₇	#532	1	1	yes	characteristic
220	#33228	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
221	C₂×C₆₇	—	16	1	no	—
222	C₂×C₆₇	—	16	2	no	—
223	C₂×C₆₇	#2858	1	1	yes	characteristic
224	C₂×C₆₇	#3136	1	4	yes	—
225	C₂×C₆₇	#3154	1	1	yes	characteristic
226	C₂×C₆₇	#3155	1	1	yes	characteristic
227	C₂×#490	C₂×C₆₇	1	1	yes	characteristic, complemented
228	C₂×#489	C₂×C₆₇	1	1	yes	characteristic, complemented
229	C₂×#488	C₂×C₆₇	1	1	yes	characteristic, complemented
230	C₂×#486	C₂×C₆₇	1	1	yes	characteristic, complemented
231	#2660	C₂×C₆₇	1	1	yes	characteristic, complemented
232	#2657	C₂×C₆₇	1	1	yes	characteristic
233	#1961	C₂×C₆₇	1	2	yes	complemented
234	#1959	C₂×C₆₇	1	2	yes	complemented
235	#1958	C₂×C₆₇	1	2	yes	complemented
236	#1957	C₂×C₆₇	1	2	yes	complemented
237	#1726	C₂×C₆₇	1	1	yes	characteristic, complemented
238	C₆₇	#33228	1	1	yes	characteristic, direct product factor, Sylow
239	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
240	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
241	C₂×#127	C₂²×C₆₇	1	2	yes	—
242	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
243	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
244	C₂×#124	C₂²×C₆₇	1	2	yes	—
245	#490	—	2	4	no	—
246	#489	—	2	4	no	—
247	#488	—	2	4	no	—
248	#486	—	2	4	no	—
249	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
250	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
251	C₂×#103	C₂²×C₆₇	1	2	yes	—
252	C₂×#102	C₂²×C₆₇	1	2	yes	—
253	C₂×#102	C₂²×C₆₇	1	2	yes	—
254	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
255	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
256	C₂×#102	C₂²×C₆₇	1	2	yes	—
257	C₂×#102	C₂²×C₆₇	1	2	yes	—
258	C₂×#102	C₂²×C₆₇	1	2	yes	—
259	C₂×#97	C₂²×C₆₇	1	2	yes	—
260	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
261	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
262	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
263	#398	—	2	2	no	—
264	#395	—	2	2	no	—
265	#392	—	2	2	no	—
266	#392	—	2	2	no	—
267	#384	C₂²×C₆₇	1	2	yes	—
268	#382	C₂²×C₆₇	1	2	yes	—
269	#376	C₂²×C₆₇	1	1	yes	characteristic
270	C₂²×Q₈	—	2	1	no	—
271	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
272	C₂²×D₄	—	2	4	no	—
273	C₂²×D₄	—	2	1	no	—
274	C₂²×D₄	—	2	1	no	—
275	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
276	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
277	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
278	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
279	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
280	C₄⋊Q₈	—	2	2	no	—
281	C₄⋊Q₈	—	2	2	no	—
282	#127	—	2	2	no	—
283	#127	—	2	2	no	—
284	#127	—	2	4	no	—
285	#127	—	2	4	no	—
286	#125	—	2	2	no	—
287	#125	—	2	2	no	—
288	#124	—	2	2	no	—
289	#124	—	2	2	no	—
290	#124	—	2	4	no	—
291	#124	—	2	4	no	—
292	C₂×C₄⋊C₄	—	2	1	no	—
293	C₂×C₄⋊C₄	—	2	1	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	—	2	4	no	—
296	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
297	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
298	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
299	C₂×C₂²⋊C₄	—	2	4	no	—
300	C₂×C₂²⋊C₄	—	2	1	no	—
301	C₂×C₂²⋊C₄	—	2	1	no	—
302	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
303	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
304	#103	—	2	2	no	—
305	#103	—	2	2	no	—
306	#103	—	2	4	no	—
307	#102	—	2	4	no	—
308	#102	—	2	4	no	—
309	#102	—	2	2	no	—
310	#102	—	2	2	no	—
311	#102	—	2	4	no	—
312	#102	—	2	4	no	—
313	#102	—	2	4	no	—
314	#97	—	2	4	no	—
315	#97	D₄×C₆₇	1	4	yes	complemented
316	C₄×C₈	—	2	4	no	—
317	C₄×C₈	D₄×C₆₇	1	4	yes	complemented
318	#95	—	2	4	no	—
319	#95	D₄×C₆₇	1	2	yes	—
320	#95	D₄×C₆₇	1	2	yes	—
321	C₂⁴	—	2	2	no	—
322	C₂⁴	—	2	1	no	—
323	C₂×Q₈	—	2	1	no	—
324	C₂×Q₈	—	2	1	no	—
325	C₂×Q₈	—	2	2	no	—
326	C₂×Q₈	—	2	2	no	—
327	C₂×Q₈	—	4	2	no	—
328	C₂×Q₈	—	8	2	no	—
329	C₂×D₄	—	2	1	no	—
330	C₂×D₄	—	2	1	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	—	2	2	no	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	4	8	no	—
340	C₂×D₄	—	4	2	no	—
341	C₂×D₄	—	4	2	no	—
342	C₂×D₄	—	8	4	no	—
343	C₂×D₄	—	8	2	no	—
344	C₂×D₄	—	8	4	no	—
345	C₂×D₄	—	8	2	no	—
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
351	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
353	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
359	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
360	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	4	2	no	—
368	C₄⋊C₄	—	4	2	no	—
369	C₄⋊C₄	—	4	2	no	—
370	C₄⋊C₄	—	8	4	no	—
371	C₂²⋊C₄	—	4	8	no	—
372	C₂²⋊C₄	—	4	2	no	—
373	C₂²⋊C₄	—	4	2	no	—
374	C₄²	C₂×D₄×C₆₇	1	2	yes	—
375	C₄²	C₂×D₄×C₆₇	1	2	yes	—
376	C₄²	#41×C₆₇	1	4	yes	—
377	C₄²	#41×C₆₇	1	4	yes	—
378	C₂³	—	4	2	no	—
379	C₂³	—	4	2	no	—
380	C₂³	—	4	2	no	—
381	C₂³	—	4	1	no	—
382	C₂³	—	4	2	no	—
383	C₂³	—	4	2	no	—
384	C₂³	—	4	4	no	—
385	C₂³	—	4	2	no	—
386	C₂³	—	4	4	no	—
387	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
388	Q₈	—	4	1	no	—
389	Q₈	—	4	1	no	—
390	Q₈	—	4	2	no	—
391	Q₈	—	8	4	no	—
392	D₄	—	4	1	no	—
393	D₄	—	4	1	no	—
394	D₄	—	4	2	no	—
395	D₄	—	4	2	no	—
396	D₄	—	4	2	no	—
397	D₄	—	4	2	no	—
398	D₄	—	4	2	no	—
399	D₄	—	8	16	no	—
400	D₄	—	8	4	no	—
401	D₄	—	8	4	no	—
402	C₂×C₄	—	2	4	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	4	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	4	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	#124×C₆₇	1	8	yes	—
415	C₂×C₄	#127×C₆₇	1	2	yes	—
416	C₂×C₄	#127×C₆₇	1	2	yes	—
417	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
418	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
419	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
420	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
421	C₈	—	2	2	no	—
422	C₈	—	2	4	no	—
423	C₈	—	2	2	no	—
424	C₂²	—	8	2	no	—
425	C₂²	—	8	1	no	—
426	C₂²	—	8	4	no	—
427	C₂²	—	8	4	no	—
428	C₂²	—	8	4	no	—
429	C₂²	—	8	2	no	—
430	C₂²	—	8	1	no	—
431	C₂²	—	8	2	no	—
432	C₂²	—	8	1	no	—
433	C₂²	C₂×#124×C₆₇	1	2	yes	—
434	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
435	C₂²	#576×C₆₇	1	2	yes	—
436	C₂²	#577×C₆₇	1	2	yes	—
437	C₄	—	2	8	no	—
438	C₄	—	2	2	no	—
439	C₄	—	2	2	no	—
440	C₄	—	8	1	no	—
441	C₄	—	8	1	no	—
442	C₄	—	8	2	no	—
443	C₄	—	8	1	no	—
444	C₄	#532×C₆₇	1	1	yes	characteristic
445	C₄	#532×C₆₇	1	2	yes	—
446	C₄	#532×C₆₇	1	1	yes	characteristic
447	C₂	—	16	1	no	—
448	C₂	—	16	2	no	—
449	C₂	C₆₇×#2858	1	1	yes	characteristic
450	C₂	C₆₇×#3136	1	4	yes	—
451	C₂	C₆₇×#3154	1	1	yes	characteristic
452	C₂	C₆₇×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#2657	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ #376×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#490×C₆₇	C₂	semidirect
2. C₂×#489×C₆₇	C₂	semidirect
3. C₂×#488×C₆₇	C₂	semidirect
4. C₂×#486×C₆₇	C₂	semidirect
5. C₆₇×#2660	C₂	semidirect
7. C₆₇×#1961	C₂	semidirect
8. C₆₇×#1959	C₂	semidirect
9. C₆₇×#1958	C₂	semidirect
10. C₆₇×#1957	C₂	semidirect
11. C₆₇×#1726	C₂	semidirect
88. #97×C₆₇	D₄	semidirect
90. C₄×C₈×C₆₇	D₄	semidirect
220. #33228	C₆₇	direct (N × H)
227. C₂×#490	C₂×C₆₇	semidirect
228. C₂×#489	C₂×C₆₇	semidirect
229. C₂×#488	C₂×C₆₇	semidirect
230. C₂×#486	C₂×C₆₇	semidirect
231. #2660	C₂×C₆₇	semidirect
233. #1961	C₂×C₆₇	semidirect
234. #1959	C₂×C₆₇	semidirect
235. #1958	C₂×C₆₇	semidirect
236. #1957	C₂×C₆₇	semidirect
237. #1726	C₂×C₆₇	semidirect
238. C₆₇	#33228	direct (N × H)
315. #97	D₄×C₆₇	semidirect
317. C₄×C₈	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#490×C₆₇	2
C₂×#489×C₆₇	2
C₂×#488×C₆₇	2
C₂×#486×C₆₇	2
C₆₇×#2660	2
C₆₇×#2657	2
C₆₇×#1961	2
C₆₇×#1959	2
C₆₇×#1958	2
C₆₇×#1957	2
C₆₇×#1726	2
#33228	67

Subgroups — C67×#33228 #30712479

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33228 #30712479