Subgroups — C₆₇×#33243 #30712494

Order 17152.

964 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#501×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#496×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#488×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#486×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2641	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2640	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1973	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1972	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1970	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1968	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
13	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
18	#501×C₆₇	—	2	4	no	—
19	#496×C₆₇	—	2	4	no	—
20	#488×C₆₇	—	2	4	no	—
21	#486×C₆₇	—	2	4	no	—
22	C₂×#107×C₆₇	C₂²	1	2	yes	—
23	C₂×#106×C₆₇	C₂²	1	2	yes	—
24	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#102×C₆₇	C₂²	1	2	yes	—
27	C₂×#102×C₆₇	C₂²	1	2	yes	—
28	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#97×C₆₇	C₂²	1	2	yes	—
31	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
34	#397×C₆₇	—	2	2	no	—
35	#394×C₆₇	—	2	2	no	—
36	#394×C₆₇	—	2	2	no	—
37	#393×C₆₇	—	2	2	no	—
38	#391×C₆₇	C₂²	1	2	yes	—
39	#390×C₆₇	C₂²	1	2	yes	—
40	#390×C₆₇	C₂²	1	2	yes	—
41	#390×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	1	no	—
46	C₂²×D₄×C₆₇	—	2	1	no	—
47	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
48	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
49	C₂×SD₁₆×C₆₇	—	2	4	no	—
50	C₂×SD₁₆×C₆₇	—	2	4	no	—
51	C₂×SD₁₆×C₆₇	—	2	4	no	—
52	C₂×D₈×C₆₇	—	2	4	no	—
53	C₂×D₈×C₆₇	—	2	4	no	—
54	C₂×D₈×C₆₇	—	2	4	no	—
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
58	C₄⋊Q₈×C₆₇	—	2	2	no	—
59	C₄⋊Q₈×C₆₇	—	2	2	no	—
60	#127×C₆₇	—	2	2	no	—
61	#127×C₆₇	—	2	2	no	—
62	#125×C₆₇	—	2	2	no	—
63	#125×C₆₇	—	2	2	no	—
64	#124×C₆₇	—	2	2	no	—
65	#124×C₆₇	—	2	2	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
68	C₂×C₄⋊C₄×C₆₇	—	2	1	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	1	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
75	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
76	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
77	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
78	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
79	#107×C₆₇	—	2	4	no	—
80	#106×C₆₇	—	2	4	no	—
81	#103×C₆₇	—	2	2	no	—
82	#103×C₆₇	—	2	2	no	—
83	#102×C₆₇	—	2	4	no	—
84	#102×C₆₇	—	2	4	no	—
85	#102×C₆₇	—	2	2	no	—
86	#102×C₆₇	—	2	2	no	—
87	#97×C₆₇	—	2	4	no	—
88	#97×C₆₇	D₄	1	4	yes	—
89	C₄×C₈×C₆₇	—	2	4	no	—
90	C₄×C₈×C₆₇	D₄	1	4	yes	—
91	#95×C₆₇	—	2	4	no	—
92	#95×C₆₇	D₄	1	2	yes	—
93	#95×C₆₇	D₄	1	2	yes	—
94	C₂⁴×C₆₇	—	2	1	no	—
95	C₂⁴×C₆₇	—	2	2	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	2	no	—
104	C₂×D₄×C₆₇	—	2	1	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	2	no	—
113	C₂×D₄×C₆₇	—	4	2	no	—
114	C₂×D₄×C₆₇	—	8	2	no	—
115	C₂×D₄×C₆₇	—	8	2	no	—
116	C₂²×C₄×C₆₇	—	2	1	no	—
117	C₂²×C₄×C₆₇	—	2	1	no	—
118	C₂²×C₄×C₆₇	—	2	2	no	—
119	C₂²×C₄×C₆₇	—	2	1	no	—
120	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
121	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
124	SD₁₆×C₆₇	—	4	8	no	—
125	D₈×C₆₇	—	4	8	no	—
126	C₂×C₈×C₆₇	—	2	2	no	—
127	C₂×C₈×C₆₇	—	2	2	no	—
128	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
129	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
130	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
131	C₂×C₈×C₆₇	#41	1	4	yes	—
132	C₂×C₈×C₆₇	#41	1	4	yes	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	—	2	1	no	—
135	C₄⋊C₄×C₆₇	—	2	1	no	—
136	C₄⋊C₄×C₆₇	—	2	1	no	—
137	C₄⋊C₄×C₆₇	—	2	2	no	—
138	C₄⋊C₄×C₆₇	—	2	1	no	—
139	C₄⋊C₄×C₆₇	—	4	2	no	—
140	C₄⋊C₄×C₆₇	—	4	2	no	—
141	C₄⋊C₄×C₆₇	—	4	2	no	—
142	C₂²⋊C₄×C₆₇	—	4	2	no	—
143	C₂²⋊C₄×C₆₇	—	4	2	no	—
144	C₂²⋊C₄×C₆₇	—	8	4	no	—
145	C₂²⋊C₄×C₆₇	—	8	4	no	—
146	C₂²⋊C₄×C₆₇	—	8	4	no	—
147	C₄²×C₆₇	—	2	2	no	—
148	C₄²×C₆₇	—	2	2	no	—
149	C₄²×C₆₇	C₂×D₄	1	2	yes	—
150	C₄²×C₆₇	C₂×D₄	1	2	yes	—
151	C₂³×C₆₇	—	4	1	no	—
152	C₂³×C₆₇	—	4	2	no	—
153	C₂³×C₆₇	—	4	2	no	—
154	C₂³×C₆₇	—	4	2	no	—
155	C₂³×C₆₇	—	4	2	no	—
156	C₂³×C₆₇	—	4	2	no	—
157	C₂³×C₆₇	—	4	4	no	—
158	C₂³×C₆₇	—	4	4	no	—
159	C₂³×C₆₇	—	4	2	no	—
160	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
161	Q₈×C₆₇	—	4	2	no	—
162	Q₈×C₆₇	—	4	1	no	—
163	Q₈×C₆₇	—	4	1	no	—
164	Q₈×C₆₇	—	8	4	no	—
165	D₄×C₆₇	—	4	2	no	—
166	D₄×C₆₇	—	4	2	no	—
167	D₄×C₆₇	—	4	2	no	—
168	D₄×C₆₇	—	4	1	no	—
169	D₄×C₆₇	—	4	1	no	—
170	D₄×C₆₇	—	4	2	no	—
171	D₄×C₆₇	—	4	2	no	—
172	D₄×C₆₇	—	8	4	no	—
173	D₄×C₆₇	—	8	4	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	4	no	—
177	C₂×C₄×C₆₇	—	2	4	no	—
178	C₂×C₄×C₆₇	—	2	4	no	—
179	C₂×C₄×C₆₇	—	4	1	no	—
180	C₂×C₄×C₆₇	—	4	1	no	—
181	C₂×C₄×C₆₇	—	4	1	no	—
182	C₂×C₄×C₆₇	—	4	2	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	#127	1	2	yes	—
188	C₂×C₄×C₆₇	#127	1	2	yes	—
189	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
192	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
193	C₈×C₆₇	—	2	2	no	—
194	C₈×C₆₇	—	2	2	no	—
195	C₈×C₆₇	#124	1	8	yes	—
196	C₂²×C₆₇	—	8	1	no	—
197	C₂²×C₆₇	—	8	2	no	—
198	C₂²×C₆₇	—	8	2	no	—
199	C₂²×C₆₇	—	8	1	no	—
200	C₂²×C₆₇	—	8	2	no	—
201	C₂²×C₆₇	—	8	1	no	—
202	C₂²×C₆₇	—	8	4	no	—
203	C₂²×C₆₇	—	8	4	no	—
204	C₂²×C₆₇	—	8	4	no	—
205	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
206	C₂²×C₆₇	#532	1	2	yes	—
207	C₂²×C₆₇	#576	1	2	yes	—
208	C₂²×C₆₇	#577	1	2	yes	—
209	C₄×C₆₇	—	2	2	no	—
210	C₄×C₆₇	—	2	2	no	—
211	C₄×C₆₇	—	4	4	no	—
212	C₄×C₆₇	—	8	1	no	—
213	C₄×C₆₇	—	8	1	no	—
214	C₄×C₆₇	—	8	1	no	—
215	C₄×C₆₇	—	8	2	no	—
216	C₄×C₆₇	C₂×#124	1	2	yes	—
217	C₄×C₆₇	#532	1	1	yes	characteristic
218	C₄×C₆₇	#532	1	1	yes	characteristic
219	#33243	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
220	C₂×C₆₇	—	16	2	no	—
221	C₂×C₆₇	—	16	1	no	—
222	C₂×C₆₇	#2858	1	1	yes	characteristic
223	C₂×C₆₇	#3154	1	1	yes	characteristic
224	C₂×C₆₇	#3155	1	1	yes	characteristic
225	C₂×C₆₇	#3179	1	4	yes	—
226	C₂×#501	C₂×C₆₇	1	1	yes	characteristic, complemented
227	C₂×#496	C₂×C₆₇	1	1	yes	characteristic
228	C₂×#488	C₂×C₆₇	1	1	yes	characteristic, complemented
229	C₂×#486	C₂×C₆₇	1	1	yes	characteristic, complemented
230	#2641	C₂×C₆₇	1	1	yes	characteristic, complemented
231	#2640	C₂×C₆₇	1	1	yes	characteristic, complemented
232	#1973	C₂×C₆₇	1	2	yes	complemented
233	#1972	C₂×C₆₇	1	2	yes	complemented
234	#1970	C₂×C₆₇	1	2	yes	complemented
235	#1968	C₂×C₆₇	1	2	yes	complemented
236	#1726	C₂×C₆₇	1	1	yes	characteristic, complemented
237	C₆₇	#33243	1	1	yes	characteristic, direct product factor, Sylow
238	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
239	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
240	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
241	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
242	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
243	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
244	#501	—	2	4	no	—
245	#496	—	2	4	no	—
246	#488	—	2	4	no	—
247	#486	—	2	4	no	—
248	C₂×#107	C₂²×C₆₇	1	2	yes	—
249	C₂×#106	C₂²×C₆₇	1	2	yes	—
250	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
251	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
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×#97	C₂²×C₆₇	1	2	yes	—
257	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
258	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
259	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
260	#397	—	2	2	no	—
261	#394	—	2	2	no	—
262	#394	—	2	2	no	—
263	#393	—	2	2	no	—
264	#391	C₂²×C₆₇	1	2	yes	—
265	#390	C₂²×C₆₇	1	2	yes	—
266	#390	C₂²×C₆₇	1	2	yes	—
267	#390	C₂²×C₆₇	1	2	yes	—
268	#376	C₂²×C₆₇	1	1	yes	characteristic
269	C₂²×Q₈	—	2	1	no	—
270	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
271	C₂²×D₄	—	2	1	no	—
272	C₂²×D₄	—	2	1	no	—
273	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
274	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
275	C₂×SD₁₆	—	2	4	no	—
276	C₂×SD₁₆	—	2	4	no	—
277	C₂×SD₁₆	—	2	4	no	—
278	C₂×D₈	—	2	4	no	—
279	C₂×D₈	—	2	4	no	—
280	C₂×D₈	—	2	4	no	—
281	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
282	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
283	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
284	C₄⋊Q₈	—	2	2	no	—
285	C₄⋊Q₈	—	2	2	no	—
286	#127	—	2	2	no	—
287	#127	—	2	2	no	—
288	#125	—	2	2	no	—
289	#125	—	2	2	no	—
290	#124	—	2	2	no	—
291	#124	—	2	2	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	C₂³×C₆₇	1	1	yes	characteristic
296	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
297	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
298	C₂×C₂²⋊C₄	—	2	1	no	—
299	C₂×C₂²⋊C₄	—	2	1	no	—
300	C₂×C₂²⋊C₄	—	2	4	no	—
301	C₂×C₂²⋊C₄	—	2	4	no	—
302	C₂×C₂²⋊C₄	—	2	4	no	—
303	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
304	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
305	#107	—	2	4	no	—
306	#106	—	2	4	no	—
307	#103	—	2	2	no	—
308	#103	—	2	2	no	—
309	#102	—	2	4	no	—
310	#102	—	2	4	no	—
311	#102	—	2	2	no	—
312	#102	—	2	2	no	—
313	#97	—	2	4	no	—
314	#97	D₄×C₆₇	1	4	yes	—
315	C₄×C₈	—	2	4	no	—
316	C₄×C₈	D₄×C₆₇	1	4	yes	—
317	#95	—	2	4	no	—
318	#95	D₄×C₆₇	1	2	yes	—
319	#95	D₄×C₆₇	1	2	yes	—
320	C₂⁴	—	2	1	no	—
321	C₂⁴	—	2	2	no	—
322	C₂×Q₈	—	2	1	no	—
323	C₂×Q₈	—	2	1	no	—
324	C₂×Q₈	—	2	2	no	—
325	C₂×Q₈	—	2	2	no	—
326	C₂×Q₈	—	4	2	no	—
327	C₂×Q₈	—	8	2	no	—
328	C₂×D₄	—	2	1	no	—
329	C₂×D₄	—	2	2	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	—	4	2	no	—
339	C₂×D₄	—	4	2	no	—
340	C₂×D₄	—	8	2	no	—
341	C₂×D₄	—	8	2	no	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
347	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
348	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
349	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
350	SD₁₆	—	4	8	no	—
351	D₈	—	4	8	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
355	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
356	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
357	C₂×C₈	#41×C₆₇	1	4	yes	—
358	C₂×C₈	#41×C₆₇	1	4	yes	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	2	1	no	—
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	—	4	2	no	—
366	C₄⋊C₄	—	4	2	no	—
367	C₄⋊C₄	—	4	2	no	—
368	C₂²⋊C₄	—	4	2	no	—
369	C₂²⋊C₄	—	4	2	no	—
370	C₂²⋊C₄	—	8	4	no	—
371	C₂²⋊C₄	—	8	4	no	—
372	C₂²⋊C₄	—	8	4	no	—
373	C₄²	—	2	2	no	—
374	C₄²	—	2	2	no	—
375	C₄²	C₂×D₄×C₆₇	1	2	yes	—
376	C₄²	C₂×D₄×C₆₇	1	2	yes	—
377	C₂³	—	4	1	no	—
378	C₂³	—	4	2	no	—
379	C₂³	—	4	2	no	—
380	C₂³	—	4	2	no	—
381	C₂³	—	4	2	no	—
382	C₂³	—	4	2	no	—
383	C₂³	—	4	4	no	—
384	C₂³	—	4	4	no	—
385	C₂³	—	4	2	no	—
386	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
387	Q₈	—	4	2	no	—
388	Q₈	—	4	1	no	—
389	Q₈	—	4	1	no	—
390	Q₈	—	8	4	no	—
391	D₄	—	4	2	no	—
392	D₄	—	4	2	no	—
393	D₄	—	4	2	no	—
394	D₄	—	4	1	no	—
395	D₄	—	4	1	no	—
396	D₄	—	4	2	no	—
397	D₄	—	4	2	no	—
398	D₄	—	8	4	no	—
399	D₄	—	8	4	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	4	no	—
403	C₂×C₄	—	2	4	no	—
404	C₂×C₄	—	2	4	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	2	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	4	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	#127×C₆₇	1	2	yes	—
414	C₂×C₄	#127×C₆₇	1	2	yes	—
415	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
416	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
417	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
418	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
419	C₈	—	2	2	no	—
420	C₈	—	2	2	no	—
421	C₈	#124×C₆₇	1	8	yes	—
422	C₂²	—	8	1	no	—
423	C₂²	—	8	2	no	—
424	C₂²	—	8	2	no	—
425	C₂²	—	8	1	no	—
426	C₂²	—	8	2	no	—
427	C₂²	—	8	1	no	—
428	C₂²	—	8	4	no	—
429	C₂²	—	8	4	no	—
430	C₂²	—	8	4	no	—
431	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
432	C₂²	#532×C₆₇	1	2	yes	—
433	C₂²	#576×C₆₇	1	2	yes	—
434	C₂²	#577×C₆₇	1	2	yes	—
435	C₄	—	2	2	no	—
436	C₄	—	2	2	no	—
437	C₄	—	4	4	no	—
438	C₄	—	8	1	no	—
439	C₄	—	8	1	no	—
440	C₄	—	8	2	no	—
441	C₄	—	8	1	no	—
442	C₄	C₂×#124×C₆₇	1	2	yes	—
443	C₄	#532×C₆₇	1	1	yes	characteristic
444	C₄	#532×C₆₇	1	1	yes	characteristic
445	C₂	—	16	2	no	—
446	C₂	—	16	1	no	—
447	C₂	C₆₇×#2858	1	1	yes	characteristic
448	C₂	C₆₇×#3154	1	1	yes	characteristic
449	C₂	C₆₇×#3155	1	1	yes	characteristic
450	C₂	C₆₇×#3179	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#496×C₆₇	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×#501×C₆₇	C₂	semidirect
3. C₂×#488×C₆₇	C₂	semidirect
4. C₂×#486×C₆₇	C₂	semidirect
5. C₆₇×#2641	C₂	semidirect
6. C₆₇×#2640	C₂	semidirect
7. C₆₇×#1973	C₂	semidirect
8. C₆₇×#1972	C₂	semidirect
9. C₆₇×#1970	C₂	semidirect
10. C₆₇×#1968	C₂	semidirect
11. C₆₇×#1726	C₂	semidirect
219. #33243	C₆₇	direct (N × H)
226. C₂×#501	C₂×C₆₇	semidirect
228. C₂×#488	C₂×C₆₇	semidirect
229. C₂×#486	C₂×C₆₇	semidirect
230. #2641	C₂×C₆₇	semidirect
231. #2640	C₂×C₆₇	semidirect
232. #1973	C₂×C₆₇	semidirect
233. #1972	C₂×C₆₇	semidirect
234. #1970	C₂×C₆₇	semidirect
235. #1968	C₂×C₆₇	semidirect
236. #1726	C₂×C₆₇	semidirect
237. C₆₇	#33243	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×#501×C₆₇	2
C₂×#496×C₆₇	2
C₂×#488×C₆₇	2
C₂×#486×C₆₇	2
C₆₇×#2641	2
C₆₇×#2640	2
C₆₇×#1973	2
C₆₇×#1972	2
C₆₇×#1970	2
C₆₇×#1968	2
C₆₇×#1726	2
#33243	67

Subgroups — C67×#33243 #30712494

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33243 #30712494