Subgroups — C₆₇×#33589 #30712840

Order 17152.

726 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×#509×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#508×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3155	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2951	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2211	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2210	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2207	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2206	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2139	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#357×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₆₇	—	2	1	no	—
13	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
16	#535×C₆₇	—	2	1	no	—
17	#520×C₆₇	—	2	1	no	—
18	#509×C₆₇	—	2	4	no	—
19	#508×C₆₇	—	2	4	no	—
20	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic, complemented
21	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	—
22	C₂×SD₃₂×C₆₇	C₂²	1	1	yes	characteristic, complemented
23	C₂×SD₃₂×C₆₇	C₂²	1	1	yes	characteristic, complemented
24	C₂×D₁₆×C₆₇	C₂²	1	2	yes	—
25	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
26	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
28	#496×C₆₇	—	2	2	no	—
29	#495×C₆₇	C₂²	1	2	yes	—
30	#493×C₆₇	C₂²	1	2	yes	—
31	#440×C₆₇	—	2	2	no	—
32	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	complemented
33	C₄×D₈×C₆₇	C₂²	1	2	yes	complemented
34	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
35	#417×C₆₇	—	2	1	no	—
36	#405×C₆₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	complemented
38	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	complemented
39	#358×C₆₇	C₂²	1	2	yes	complemented
40	#357×C₆₇	C₂²	1	2	yes	—
41	#357×C₆₇	C₂²	1	2	yes	—
42	#357×C₆₇	C₂²	1	2	yes	complemented
43	C₂×#41×C₆₇	—	2	1	no	—
44	C₂×#41×C₆₇	—	2	1	no	—
45	C₂²×D₄×C₆₇	—	2	1	no	—
46	AGL(1,ℤ/8)×C₆₇	—	4	4	no	—
47	#135×C₆₇	—	2	2	no	—
48	#135×C₆₇	—	2	2	no	—
49	#135×C₆₇	—	2	2	no	—
50	#135×C₆₇	D₄	1	4	yes	complemented
51	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
52	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
53	C₂×SD₁₆×C₆₇	—	2	1	no	—
54	C₂×SD₁₆×C₆₇	—	2	2	no	—
55	C₂×SD₁₆×C₆₇	—	2	1	no	—
56	C₂×D₈×C₆₇	—	2	2	no	—
57	C₂×D₈×C₆₇	—	2	2	no	—
58	C₂×D₈×C₆₇	—	2	1	no	—
59	C₂×D₈×C₆₇	—	2	2	no	—
60	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
61	C₂×D₈×C₆₇	C₂³	1	2	yes	—
62	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	#127×C₆₇	—	2	2	no	—
66	#124×C₆₇	—	2	2	no	—
67	#121×C₆₇	—	4	1	no	—
68	#121×C₆₇	—	4	1	no	—
69	#120×C₆₇	—	4	2	no	—
70	C₄×Q₈×C₆₇	—	2	2	no	—
71	C₄×D₄×C₆₇	—	2	2	no	—
72	C₄×D₄×C₆₇	—	4	2	no	—
73	#117×C₆₇	—	4	1	no	—
74	#117×C₆₇	D₄	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
76	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
77	Q₃₂×C₆₇	—	2	2	no	—
78	SD₃₂×C₆₇	—	2	4	no	—
79	SD₃₂×C₆₇	—	2	2	no	—
80	SD₃₂×C₆₇	—	2	2	no	—
81	D₁₆×C₆₇	—	2	4	no	—
82	D₁₆×C₆₇	—	2	2	no	—
83	M₃₂×C₆₇	—	2	2	no	—
84	C₂×C₁₆×C₆₇	—	2	2	no	—
85	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
86	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
87	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
88	#107×C₆₇	—	2	1	no	—
89	#107×C₆₇	C₂³	1	2	yes	—
90	#103×C₆₇	—	2	2	no	—
91	#102×C₆₇	—	2	2	no	—
92	#97×C₆₇	D₄	1	2	yes	—
93	C₄×C₈×C₆₇	C₂³	1	2	yes	—
94	C₂⁴×C₆₇	—	4	1	no	—
95	#41×C₆₇	—	2	4	no	—
96	#41×C₆₇	—	4	2	no	—
97	#41×C₆₇	—	4	2	no	—
98	#41×C₆₇	—	4	2	no	—
99	C₂×Q₈×C₆₇	—	2	1	no	—
100	C₂×Q₈×C₆₇	—	2	1	no	—
101	C₂×D₄×C₆₇	—	2	1	no	—
102	C₂×D₄×C₆₇	—	2	2	no	—
103	C₂×D₄×C₆₇	—	2	1	no	—
104	C₂×D₄×C₆₇	—	4	1	no	—
105	C₂×D₄×C₆₇	—	4	1	no	—
106	C₂×D₄×C₆₇	—	4	2	no	—
107	C₂×D₄×C₆₇	—	4	1	no	—
108	C₂×D₄×C₆₇	—	4	2	no	—
109	C₂×D₄×C₆₇	—	4	2	no	—
110	C₂×D₄×C₆₇	—	4	2	no	—
111	C₂²×C₄×C₆₇	—	4	1	no	—
112	C₂²×C₄×C₆₇	—	4	1	no	—
113	C₂²×C₄×C₆₇	—	4	1	no	—
114	C₂²×C₄×C₆₇	—	4	2	no	—
115	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
116	Q₁₆×C₆₇	—	2	1	no	—
117	Q₁₆×C₆₇	—	2	1	no	—
118	Q₁₆×C₆₇	—	2	1	no	—
119	Q₁₆×C₆₇	C₂×D₄	1	2	yes	—
120	SD₁₆×C₆₇	—	2	4	no	—
121	SD₁₆×C₆₇	—	4	4	no	—
122	SD₁₆×C₆₇	—	4	2	no	—
123	D₈×C₆₇	—	2	2	no	—
124	D₈×C₆₇	—	2	1	no	—
125	D₈×C₆₇	—	2	1	no	—
126	D₈×C₆₇	—	2	2	no	—
127	D₈×C₆₇	—	2	1	no	—
128	D₈×C₆₇	—	4	4	no	—
129	D₈×C₆₇	—	4	2	no	—
130	D₈×C₆₇	C₂×D₄	1	2	yes	—
131	M₁₆×C₆₇	—	2	2	no	—
132	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
133	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
135	C₂×C₈×C₆₇	#41	1	2	yes	—
136	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
137	C₄⋊C₄×C₆₇	—	2	2	no	—
138	C₄⋊C₄×C₆₇	—	4	1	no	—
139	C₄⋊C₄×C₆₇	—	4	2	no	—
140	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
141	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
142	C₂²⋊C₄×C₆₇	—	4	2	no	—
143	C₂²⋊C₄×C₆₇	—	4	2	no	—
144	C₂²⋊C₄×C₆₇	—	4	2	no	—
145	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
146	C₄²×C₆₇	—	4	2	no	—
147	C₄²×C₆₇	C₂×D₄	1	2	yes	—
148	C₁₆×C₆₇	—	2	1	no	—
149	C₁₆×C₆₇	—	2	1	no	—
150	C₁₆×C₆₇	—	2	2	no	—
151	C₂³×C₆₇	—	4	1	no	—
152	C₂³×C₆₇	—	4	2	no	—
153	C₂³×C₆₇	—	4	1	no	—
154	C₂³×C₆₇	—	8	2	no	—
155	C₂³×C₆₇	—	16	1	no	—
156	C₂³×C₆₇	—	16	1	no	—
157	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
158	Q₈×C₆₇	—	2	2	no	—
159	Q₈×C₆₇	—	4	1	no	—
160	Q₈×C₆₇	—	4	1	no	—
161	Q₈×C₆₇	—	4	1	no	—
162	D₄×C₆₇	—	2	2	no	—
163	D₄×C₆₇	—	4	4	no	—
164	D₄×C₆₇	—	4	2	no	—
165	D₄×C₆₇	—	4	1	no	—
166	D₄×C₆₇	—	4	1	no	—
167	D₄×C₆₇	—	4	1	no	—
168	D₄×C₆₇	—	4	2	no	—
169	D₄×C₆₇	—	8	2	no	—
170	D₄×C₆₇	—	8	2	no	—
171	D₄×C₆₇	—	16	2	no	—
172	D₄×C₆₇	—	16	2	no	—
173	C₂×C₄×C₆₇	—	4	1	no	—
174	C₂×C₄×C₆₇	—	4	4	no	—
175	C₂×C₄×C₆₇	—	4	2	no	—
176	C₂×C₄×C₆₇	—	4	1	no	—
177	C₂×C₄×C₆₇	—	8	2	no	—
178	C₂×C₄×C₆₇	—	16	2	no	—
179	C₂×C₄×C₆₇	—	16	1	no	—
180	C₂×C₄×C₆₇	#121	1	2	yes	—
181	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
182	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
183	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
184	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
185	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
186	C₈×C₆₇	—	2	2	no	—
187	C₈×C₆₇	#121	1	2	yes	—
188	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
189	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
190	C₂²×C₆₇	—	2	1	no	—
191	C₂²×C₆₇	—	2	1	no	—
192	C₂²×C₆₇	—	4	2	no	—
193	C₂²×C₆₇	—	8	1	no	—
194	C₂²×C₆₇	—	8	1	no	—
195	C₂²×C₆₇	—	8	1	no	—
196	C₂²×C₆₇	—	8	2	no	—
197	C₂²×C₆₇	—	8	1	no	—
198	C₂²×C₆₇	—	8	2	no	—
199	C₂²×C₆₇	—	8	1	no	—
200	C₂²×C₆₇	—	8	2	no	—
201	C₂²×C₆₇	—	16	4	no	—
202	C₂²×C₆₇	#459	1	2	yes	—
203	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
204	C₄×C₆₇	—	2	2	no	—
205	C₄×C₆₇	—	2	2	no	—
206	C₄×C₆₇	—	4	2	no	—
207	C₄×C₆₇	—	8	2	no	—
208	C₄×C₆₇	—	8	1	no	—
209	C₄×C₆₇	#459	1	2	yes	—
210	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
211	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
212	#33589	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂×C₆₇	—	2	2	no	—
214	C₂×C₆₇	—	8	2	no	—
215	C₂×C₆₇	—	16	1	no	—
216	C₂×C₆₇	—	16	2	no	—
217	C₂×C₆₇	C₂×#459	1	1	yes	characteristic
218	C₂×C₆₇	#3415	1	1	yes	characteristic
219	C₂×C₆₇	#3416	1	1	yes	characteristic
220	C₂×#509	C₂×C₆₇	1	1	yes	characteristic, complemented
221	C₂×#508	C₂×C₆₇	1	1	yes	characteristic, complemented
222	#3155	C₂×C₆₇	1	1	yes	characteristic, complemented
223	#2951	C₂×C₆₇	1	1	yes	characteristic, complemented
224	#2211	C₂×C₆₇	1	2	yes	complemented
225	#2210	C₂×C₆₇	1	2	yes	complemented
226	#2207	C₂×C₆₇	1	2	yes	complemented
227	#2206	C₂×C₆₇	1	2	yes	complemented
228	#2151	C₂×C₆₇	1	1	yes	characteristic, complemented
229	#2139	C₂×C₆₇	1	1	yes	characteristic, complemented
230	C₂×#357	C₂×C₆₇	1	1	yes	characteristic, complemented
231	C₆₇	#33589	1	1	yes	characteristic, direct product factor, Sylow
232	C₂×AGL(1,ℤ/8)	—	2	1	no	—
233	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
234	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
235	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
236	#535	—	2	1	no	—
237	#520	—	2	1	no	—
238	#509	—	2	4	no	—
239	#508	—	2	4	no	—
240	C₂×Q₃₂	C₂²×C₆₇	1	1	yes	characteristic, complemented
241	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	—
242	C₂×SD₃₂	C₂²×C₆₇	1	1	yes	characteristic, complemented
243	C₂×SD₃₂	C₂²×C₆₇	1	1	yes	characteristic, complemented
244	C₂×D₁₆	C₂²×C₆₇	1	2	yes	—
245	C₂×D₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
246	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic
247	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
248	#496	—	2	2	no	—
249	#495	C₂²×C₆₇	1	2	yes	—
250	#493	C₂²×C₆₇	1	2	yes	—
251	#440	—	2	2	no	—
252	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	complemented
253	C₄×D₈	C₂²×C₆₇	1	2	yes	complemented
254	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
255	#417	—	2	1	no	—
256	#405	C₂²×C₆₇	1	1	yes	characteristic
257	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	complemented
258	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	complemented
259	#358	C₂²×C₆₇	1	2	yes	complemented
260	#357	C₂²×C₆₇	1	2	yes	—
261	#357	C₂²×C₆₇	1	2	yes	—
262	#357	C₂²×C₆₇	1	2	yes	complemented
263	C₂×#41	—	2	1	no	—
264	C₂×#41	—	2	1	no	—
265	C₂²×D₄	—	2	1	no	—
266	AGL(1,ℤ/8)	—	4	4	no	—
267	#135	—	2	2	no	—
268	#135	—	2	2	no	—
269	#135	—	2	2	no	—
270	#135	D₄×C₆₇	1	4	yes	complemented
271	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
272	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
273	C₂×SD₁₆	—	2	1	no	—
274	C₂×SD₁₆	—	2	1	no	—
275	C₂×SD₁₆	—	2	2	no	—
276	C₂×D₈	—	2	2	no	—
277	C₂×D₈	—	2	1	no	—
278	C₂×D₈	—	2	2	no	—
279	C₂×D₈	—	2	2	no	—
280	C₂×D₈	C₂³×C₆₇	1	2	yes	—
281	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
282	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
283	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
284	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
285	#127	—	2	2	no	—
286	#124	—	2	2	no	—
287	#121	—	4	1	no	—
288	#121	—	4	1	no	—
289	#120	—	4	2	no	—
290	C₄×Q₈	—	2	2	no	—
291	C₄×D₄	—	2	2	no	—
292	C₄×D₄	—	4	2	no	—
293	#117	—	4	1	no	—
294	#117	D₄×C₆₇	1	1	yes	characteristic
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄⋊C₄	—	4	1	no	—
297	Q₃₂	—	2	2	no	—
298	SD₃₂	—	2	4	no	—
299	SD₃₂	—	2	2	no	—
300	SD₃₂	—	2	2	no	—
301	D₁₆	—	2	4	no	—
302	D₁₆	—	2	2	no	—
303	M₃₂	—	2	2	no	—
304	C₂×C₁₆	—	2	2	no	—
305	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
306	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
307	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
308	#107	—	2	1	no	—
309	#107	C₂³×C₆₇	1	2	yes	—
310	#103	—	2	2	no	—
311	#102	—	2	2	no	—
312	#97	D₄×C₆₇	1	2	yes	—
313	C₄×C₈	C₂³×C₆₇	1	2	yes	—
314	C₂⁴	—	4	1	no	—
315	#41	—	2	4	no	—
316	#41	—	4	2	no	—
317	#41	—	4	2	no	—
318	#41	—	4	2	no	—
319	C₂×Q₈	—	2	1	no	—
320	C₂×Q₈	—	2	1	no	—
321	C₂×D₄	—	2	1	no	—
322	C₂×D₄	—	2	2	no	—
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	—	4	1	no	—
325	C₂×D₄	—	4	1	no	—
326	C₂×D₄	—	4	2	no	—
327	C₂×D₄	—	4	1	no	—
328	C₂×D₄	—	4	2	no	—
329	C₂×D₄	—	4	2	no	—
330	C₂×D₄	—	4	2	no	—
331	C₂²×C₄	—	4	1	no	—
332	C₂²×C₄	—	4	1	no	—
333	C₂²×C₄	—	4	1	no	—
334	C₂²×C₄	—	4	2	no	—
335	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
336	Q₁₆	—	2	1	no	—
337	Q₁₆	—	2	1	no	—
338	Q₁₆	—	2	1	no	—
339	Q₁₆	C₂×D₄×C₆₇	1	2	yes	—
340	SD₁₆	—	2	4	no	—
341	SD₁₆	—	4	4	no	—
342	SD₁₆	—	4	2	no	—
343	D₈	—	2	2	no	—
344	D₈	—	2	1	no	—
345	D₈	—	2	1	no	—
346	D₈	—	2	2	no	—
347	D₈	—	2	1	no	—
348	D₈	—	4	4	no	—
349	D₈	—	4	2	no	—
350	D₈	C₂×D₄×C₆₇	1	2	yes	—
351	M₁₆	—	2	2	no	—
352	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
353	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
354	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
355	C₂×C₈	#41×C₆₇	1	2	yes	—
356	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	4	1	no	—
359	C₄⋊C₄	—	4	2	no	—
360	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
361	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
362	C₂²⋊C₄	—	4	2	no	—
363	C₂²⋊C₄	—	4	2	no	—
364	C₂²⋊C₄	—	4	2	no	—
365	C₂²⋊C₄	D₈×C₆₇	1	2	yes	—
366	C₄²	—	4	2	no	—
367	C₄²	C₂×D₄×C₆₇	1	2	yes	—
368	C₁₆	—	2	1	no	—
369	C₁₆	—	2	1	no	—
370	C₁₆	—	2	2	no	—
371	C₂³	—	4	1	no	—
372	C₂³	—	4	2	no	—
373	C₂³	—	4	1	no	—
374	C₂³	—	8	2	no	—
375	C₂³	—	16	1	no	—
376	C₂³	—	16	1	no	—
377	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
378	Q₈	—	2	2	no	—
379	Q₈	—	4	1	no	—
380	Q₈	—	4	1	no	—
381	Q₈	—	4	1	no	—
382	D₄	—	2	2	no	—
383	D₄	—	4	4	no	—
384	D₄	—	4	2	no	—
385	D₄	—	4	1	no	—
386	D₄	—	4	1	no	—
387	D₄	—	4	1	no	—
388	D₄	—	4	2	no	—
389	D₄	—	8	2	no	—
390	D₄	—	8	2	no	—
391	D₄	—	16	2	no	—
392	D₄	—	16	2	no	—
393	C₂×C₄	—	4	1	no	—
394	C₂×C₄	—	4	4	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	8	2	no	—
398	C₂×C₄	—	16	2	no	—
399	C₂×C₄	—	16	1	no	—
400	C₂×C₄	#121×C₆₇	1	2	yes	—
401	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
402	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
403	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
404	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
405	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
406	C₈	—	2	2	no	—
407	C₈	#121×C₆₇	1	2	yes	—
408	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
409	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
410	C₂²	—	2	1	no	—
411	C₂²	—	2	1	no	—
412	C₂²	—	4	2	no	—
413	C₂²	—	8	1	no	—
414	C₂²	—	8	1	no	—
415	C₂²	—	8	1	no	—
416	C₂²	—	8	2	no	—
417	C₂²	—	8	1	no	—
418	C₂²	—	8	2	no	—
419	C₂²	—	8	1	no	—
420	C₂²	—	8	2	no	—
421	C₂²	—	16	4	no	—
422	C₂²	#459×C₆₇	1	2	yes	—
423	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
424	C₄	—	2	2	no	—
425	C₄	—	2	2	no	—
426	C₄	—	4	2	no	—
427	C₄	—	8	1	no	—
428	C₄	—	8	2	no	—
429	C₄	#459×C₆₇	1	2	yes	—
430	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
431	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
432	C₂	—	2	2	no	—
433	C₂	—	8	2	no	—
434	C₂	—	16	2	no	—
435	C₂	—	16	1	no	—
436	C₂	C₂×#459×C₆₇	1	1	yes	characteristic
437	C₂	C₆₇×#3415	1	1	yes	characteristic
438	C₂	C₆₇×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ #117×C₆₇ ◁ #405×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#509×C₆₇	C₂	semidirect
2. C₂×#508×C₆₇	C₂	semidirect
3. C₆₇×#3155	C₂	semidirect
4. C₆₇×#2951	C₂	semidirect
5. C₆₇×#2211	C₂	semidirect
6. C₆₇×#2210	C₂	semidirect
7. C₆₇×#2207	C₂	semidirect
8. C₆₇×#2206	C₂	semidirect
9. C₆₇×#2151	C₂	semidirect
10. C₆₇×#2139	C₂	semidirect
11. C₂×#357×C₆₇	C₂	semidirect
20. C₂×Q₃₂×C₆₇	C₂²	semidirect
22. C₂×SD₃₂×C₆₇	C₂²	semidirect
23. C₂×SD₃₂×C₆₇	C₂²	semidirect
25. C₂×D₁₆×C₆₇	C₂²	semidirect
32. C₄×Q₁₆×C₆₇	C₂²	semidirect
33. C₄×D₈×C₆₇	C₂²	semidirect
37. C₄⋊C₁₆×C₆₇	C₂²	semidirect
38. C₄⋊C₁₆×C₆₇	C₂²	semidirect
39. #358×C₆₇	C₂²	semidirect
42. #357×C₆₇	C₂²	semidirect
50. #135×C₆₇	D₄	semidirect
212. #33589	C₆₇	direct (N × H)
220. C₂×#509	C₂×C₆₇	semidirect
221. C₂×#508	C₂×C₆₇	semidirect
222. #3155	C₂×C₆₇	semidirect
223. #2951	C₂×C₆₇	semidirect
224. #2211	C₂×C₆₇	semidirect
225. #2210	C₂×C₆₇	semidirect
226. #2207	C₂×C₆₇	semidirect
227. #2206	C₂×C₆₇	semidirect
228. #2151	C₂×C₆₇	semidirect
229. #2139	C₂×C₆₇	semidirect
230. C₂×#357	C₂×C₆₇	semidirect
231. C₆₇	#33589	direct (N × H)
240. C₂×Q₃₂	C₂²×C₆₇	semidirect
242. C₂×SD₃₂	C₂²×C₆₇	semidirect
243. C₂×SD₃₂	C₂²×C₆₇	semidirect
245. C₂×D₁₆	C₂²×C₆₇	semidirect
252. C₄×Q₁₆	C₂²×C₆₇	semidirect
253. C₄×D₈	C₂²×C₆₇	semidirect
257. C₄⋊C₁₆	C₂²×C₆₇	semidirect
258. C₄⋊C₁₆	C₂²×C₆₇	semidirect
259. #358	C₂²×C₆₇	semidirect
262. #357	C₂²×C₆₇	semidirect
270. #135	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×#509×C₆₇	2
C₂×#508×C₆₇	2
C₆₇×#3155	2
C₆₇×#2951	2
C₆₇×#2211	2
C₆₇×#2210	2
C₆₇×#2207	2
C₆₇×#2206	2
C₆₇×#2151	2
C₆₇×#2139	2
C₂×#357×C₆₇	2
#33589	67

Subgroups — C67×#33589 #30712840

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33589 #30712840