Subgroups — C₆₇×#33542 #30712793

Order 17152.

1084 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3584	C₂	1	1	yes	maximal, characteristic
2	C₂×#510×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#510×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#473×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2195	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2194	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2148	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2143	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2117	C₂	1	1	yes	maximal, characteristic, complemented
10	#585×C₆₇	—	2	1	no	—
11	#578×C₆₇	—	2	1	no	—
12	#578×C₆₇	C₂²	1	2	yes	—
13	#577×C₆₇	—	2	2	no	—
14	#577×C₆₇	—	2	2	no	—
15	#576×C₆₇	—	2	1	no	—
16	#576×C₆₇	C₂²	1	2	yes	—
17	#575×C₆₇	—	2	2	no	—
18	#575×C₆₇	—	2	1	no	—
19	C₂×#137×C₆₇	—	2	1	no	—
20	C₂×#135×C₆₇	—	2	2	no	—
21	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
24	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
25	#510×C₆₇	—	2	2	no	—
26	#510×C₆₇	—	2	2	no	—
27	#510×C₆₇	—	2	2	no	—
28	#510×C₆₇	C₂²	1	4	yes	complemented
29	C₂×Q₃₂×C₆₇	—	2	1	no	—
30	C₂×Q₃₂×C₆₇	C₂²	1	2	yes	complemented
31	C₂×SD₃₂×C₆₇	—	2	1	no	—
32	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	complemented
33	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic, complemented
34	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic, complemented
35	#473×C₆₇	—	2	2	no	—
36	#473×C₆₇	C₂²	1	4	yes	complemented
37	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic, complemented
38	C₂×#101×C₆₇	—	2	1	no	—
39	#362×C₆₇	C₂²	1	2	yes	complemented
40	#361×C₆₇	C₂²	1	2	yes	complemented
41	#359×C₆₇	C₂²	1	4	yes	complemented
42	#350×C₆₇	C₂²	1	4	yes	complemented
43	2₋¹⁺⁴×C₆₇	—	2	2	no	—
44	2₋¹⁺⁴×C₆₇	—	2	2	no	—
45	2₋¹⁺⁴×C₆₇	—	4	1	no	—
46	H(5,𝔽₂)×C₆₇	—	2	2	no	—
47	H(5,𝔽₂)×C₆₇	—	2	2	no	—
48	H(5,𝔽₂)×C₆₇	—	4	2	no	—
49	H(5,𝔽₂)×C₆₇	—	4	1	no	—
50	C₂×#41×C₆₇	—	2	4	no	—
51	C₂×#41×C₆₇	—	2	1	no	—
52	C₂×#41×C₆₇	—	2	1	no	—
53	C₂×#41×C₆₇	—	4	2	no	—
54	C₂×#41×C₆₇	—	4	2	no	—
55	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
56	C₂²×Q₈×C₆₇	—	2	1	no	—
57	#137×C₆₇	—	2	2	no	—
58	#137×C₆₇	—	2	4	no	—
59	#137×C₆₇	—	2	4	no	—
60	#137×C₆₇	—	2	4	no	—
61	#137×C₆₇	—	2	2	no	—
62	#137×C₆₇	—	4	2	no	—
63	AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
64	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
65	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
66	AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
67	#135×C₆₇	—	2	4	no	—
68	#135×C₆₇	—	2	4	no	—
69	#135×C₆₇	—	2	2	no	—
70	#135×C₆₇	—	2	2	no	—
71	#135×C₆₇	—	2	1	no	—
72	#135×C₆₇	—	2	1	no	—
73	#135×C₆₇	—	2	1	no	—
74	#135×C₆₇	—	2	1	no	—
75	#135×C₆₇	D₄	1	2	yes	—
76	#135×C₆₇	D₄	1	2	yes	—
77	#135×C₆₇	D₄	1	2	yes	—
78	#135×C₆₇	C₂³	1	2	yes	—
79	C₂×Q₁₆×C₆₇	—	2	2	no	—
80	C₂×Q₁₆×C₆₇	—	2	1	no	—
81	C₂×Q₁₆×C₆₇	—	2	1	no	—
82	C₂×Q₁₆×C₆₇	—	2	1	no	—
83	C₂×Q₁₆×C₆₇	—	2	1	no	—
84	C₂×Q₁₆×C₆₇	—	2	1	no	—
85	C₂×Q₁₆×C₆₇	—	4	1	no	—
86	C₂×Q₁₆×C₆₇	D₄	1	1	yes	characteristic
87	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
88	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
89	C₂×SD₁₆×C₆₇	—	2	4	no	—
90	C₂×SD₁₆×C₆₇	—	2	1	no	—
91	C₂×SD₁₆×C₆₇	—	2	1	no	—
92	C₂×SD₁₆×C₆₇	—	4	1	no	—
93	C₂×D₈×C₆₇	—	2	2	no	—
94	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
95	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
96	#131×C₆₇	—	2	1	no	—
97	#131×C₆₇	—	2	1	no	—
98	#131×C₆₇	D₄	1	2	yes	—
99	#131×C₆₇	C₂³	1	2	yes	—
100	C₂×M₁₆×C₆₇	—	2	1	no	—
101	C₂×M₁₆×C₆₇	—	2	1	no	—
102	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
103	C₂²×C₈×C₆₇	—	2	1	no	—
104	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
105	Q₃₂×C₆₇	—	2	4	no	—
106	Q₃₂×C₆₇	—	2	4	no	—
107	SD₃₂×C₆₇	—	2	4	no	—
108	SD₃₂×C₆₇	—	2	4	no	—
109	M₃₂×C₆₇	—	2	1	no	—
110	M₃₂×C₆₇	—	2	1	no	—
111	M₃₂×C₆₇	—	2	1	no	—
112	M₃₂×C₆₇	C₂³	1	2	yes	—
113	C₂×C₁₆×C₆₇	—	2	1	no	—
114	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
115	#108×C₆₇	—	2	1	no	—
116	#108×C₆₇	C₂³	1	2	yes	—
117	#101×C₆₇	—	2	4	no	—
118	#41×C₆₇	—	2	1	no	—
119	#41×C₆₇	—	2	1	no	—
120	#41×C₆₇	—	2	4	no	—
121	#41×C₆₇	—	2	4	no	—
122	#41×C₆₇	—	2	2	no	—
123	#41×C₆₇	—	2	2	no	—
124	#41×C₆₇	—	2	2	no	—
125	#41×C₆₇	—	2	2	no	—
126	#41×C₆₇	—	4	4	no	—
127	#41×C₆₇	—	4	4	no	—
128	#41×C₆₇	—	4	4	no	—
129	#41×C₆₇	—	4	4	no	—
130	#41×C₆₇	—	4	2	no	—
131	#41×C₆₇	—	4	2	no	—
132	#41×C₆₇	—	4	2	no	—
133	#41×C₆₇	—	4	2	no	—
134	#41×C₆₇	—	4	1	no	—
135	#41×C₆₇	—	4	1	no	—
136	#41×C₆₇	—	4	1	no	—
137	#41×C₆₇	—	4	1	no	—
138	#41×C₆₇	D₈	1	2	yes	—
139	#41×C₆₇	C₂×D₄	1	2	yes	—
140	C₂×Q₈×C₆₇	—	2	4	no	—
141	C₂×Q₈×C₆₇	—	2	1	no	—
142	C₂×Q₈×C₆₇	—	2	1	no	—
143	C₂×Q₈×C₆₇	—	2	2	no	—
144	C₂×Q₈×C₆₇	—	4	2	no	—
145	C₂×Q₈×C₆₇	—	4	2	no	—
146	C₂×Q₈×C₆₇	—	4	1	no	—
147	C₂×Q₈×C₆₇	—	4	1	no	—
148	C₂×Q₈×C₆₇	—	4	1	no	—
149	C₂×Q₈×C₆₇	—	4	1	no	—
150	C₂×Q₈×C₆₇	—	4	1	no	—
151	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
152	C₂×D₄×C₆₇	—	2	1	no	—
153	C₂×D₄×C₆₇	—	2	4	no	—
154	C₂×D₄×C₆₇	—	2	1	no	—
155	C₂×D₄×C₆₇	—	2	1	no	—
156	C₂×D₄×C₆₇	—	4	4	no	—
157	C₂×D₄×C₆₇	—	4	4	no	—
158	C₂×D₄×C₆₇	—	4	2	no	—
159	C₂×D₄×C₆₇	—	4	4	no	—
160	C₂×D₄×C₆₇	—	4	1	no	—
161	C₂×D₄×C₆₇	—	4	1	no	—
162	C₂×D₄×C₆₇	—	4	2	no	—
163	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
164	C₂²×C₄×C₆₇	—	2	1	no	—
165	C₂²×C₄×C₆₇	—	4	4	no	—
166	C₂²×C₄×C₆₇	—	4	1	no	—
167	C₂²×C₄×C₆₇	—	4	1	no	—
168	C₂²×C₄×C₆₇	—	4	1	no	—
169	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
170	Q₁₆×C₆₇	—	2	4	no	—
171	Q₁₆×C₆₇	—	2	1	no	—
172	Q₁₆×C₆₇	—	2	1	no	—
173	Q₁₆×C₆₇	—	2	2	no	—
174	Q₁₆×C₆₇	—	2	2	no	—
175	Q₁₆×C₆₇	—	4	4	no	—
176	Q₁₆×C₆₇	—	4	1	no	—
177	Q₁₆×C₆₇	—	4	1	no	—
178	Q₁₆×C₆₇	C₂×D₄	1	2	yes	—
179	Q₁₆×C₆₇	C₂×D₄	1	2	yes	—
180	SD₁₆×C₆₇	—	2	4	no	—
181	SD₁₆×C₆₇	—	2	4	no	—
182	SD₁₆×C₆₇	—	2	2	no	—
183	SD₁₆×C₆₇	—	2	2	no	—
184	SD₁₆×C₆₇	—	2	2	no	—
185	SD₁₆×C₆₇	—	2	2	no	—
186	SD₁₆×C₆₇	—	4	4	no	—
187	D₈×C₆₇	—	2	4	no	—
188	D₈×C₆₇	—	2	1	no	—
189	D₈×C₆₇	—	2	1	no	—
190	D₈×C₆₇	C₂×D₄	1	2	yes	—
191	D₈×C₆₇	C₂×D₄	1	2	yes	—
192	M₁₆×C₆₇	—	2	1	no	—
193	M₁₆×C₆₇	—	2	2	no	—
194	M₁₆×C₆₇	—	2	2	no	—
195	M₁₆×C₆₇	—	2	2	no	—
196	M₁₆×C₆₇	—	4	1	no	—
197	M₁₆×C₆₇	C₂×D₄	1	2	yes	—
198	C₂×C₈×C₆₇	—	2	1	no	—
199	C₂×C₈×C₆₇	—	2	2	no	—
200	C₂×C₈×C₆₇	—	2	2	no	—
201	C₂×C₈×C₆₇	—	4	1	no	—
202	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
203	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
204	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
205	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
206	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
207	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
208	C₁₆×C₆₇	—	2	2	no	—
209	C₁₆×C₆₇	—	2	2	no	—
210	C₂³×C₆₇	—	2	1	no	—
211	C₂³×C₆₇	—	4	4	no	—
212	C₂³×C₆₇	—	4	1	no	—
213	C₂³×C₆₇	—	4	1	no	—
214	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
215	Q₈×C₆₇	—	2	1	no	—
216	Q₈×C₆₇	—	2	2	no	—
217	Q₈×C₆₇	—	2	2	no	—
218	Q₈×C₆₇	—	4	4	no	—
219	Q₈×C₆₇	—	4	1	no	—
220	Q₈×C₆₇	—	4	1	no	—
221	Q₈×C₆₇	—	4	2	no	—
222	Q₈×C₆₇	—	4	2	no	—
223	Q₈×C₆₇	—	8	1	no	—
224	Q₈×C₆₇	—	8	1	no	—
225	Q₈×C₆₇	C₂×D₈	1	2	yes	—
226	D₄×C₆₇	—	2	2	no	—
227	D₄×C₆₇	—	2	2	no	—
228	D₄×C₆₇	—	2	2	no	—
229	D₄×C₆₇	—	2	2	no	—
230	D₄×C₆₇	—	2	1	no	—
231	D₄×C₆₇	—	4	4	no	—
232	D₄×C₆₇	—	4	4	no	—
233	D₄×C₆₇	—	4	4	no	—
234	D₄×C₆₇	—	4	2	no	—
235	D₄×C₆₇	—	4	2	no	—
236	D₄×C₆₇	—	4	2	no	—
237	D₄×C₆₇	—	4	2	no	—
238	D₄×C₆₇	—	4	1	no	—
239	D₄×C₆₇	—	4	1	no	—
240	D₄×C₆₇	C₂×D₈	1	2	yes	—
241	C₂×C₄×C₆₇	—	2	1	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	4	4	no	—
245	C₂×C₄×C₆₇	—	4	4	no	—
246	C₂×C₄×C₆₇	—	4	4	no	—
247	C₂×C₄×C₆₇	—	4	1	no	—
248	C₂×C₄×C₆₇	—	4	1	no	—
249	C₂×C₄×C₆₇	—	4	2	no	—
250	C₂×C₄×C₆₇	—	4	2	no	—
251	C₂×C₄×C₆₇	—	4	2	no	—
252	C₂×C₄×C₆₇	—	4	2	no	—
253	C₂×C₄×C₆₇	—	4	2	no	—
254	C₂×C₄×C₆₇	—	8	1	no	—
255	C₂×C₄×C₆₇	—	8	1	no	—
256	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
257	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
258	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
259	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
260	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
261	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
262	C₈×C₆₇	—	2	2	no	—
263	C₈×C₆₇	—	4	2	no	—
264	C₈×C₆₇	#120	1	1	yes	characteristic
265	C₈×C₆₇	#120	1	1	yes	characteristic
266	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
267	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
268	C₂²×C₆₇	—	2	2	no	—
269	C₂²×C₆₇	—	4	1	no	—
270	C₂²×C₆₇	—	4	2	no	—
271	C₂²×C₆₇	—	4	2	no	—
272	C₂²×C₆₇	—	8	1	no	—
273	C₂²×C₆₇	—	16	1	no	—
274	C₂²×C₆₇	—	16	1	no	—
275	C₂²×C₆₇	—	16	4	no	—
276	C₂²×C₆₇	#447	1	1	yes	characteristic
277	C₂²×C₆₇	#447	1	1	yes	characteristic
278	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
279	C₄×C₆₇	—	2	2	no	—
280	C₄×C₆₇	—	4	2	no	—
281	C₄×C₆₇	—	4	2	no	—
282	C₄×C₆₇	—	8	2	no	—
283	C₄×C₆₇	#447	1	1	yes	characteristic
284	C₄×C₆₇	#447	1	1	yes	characteristic
285	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
286	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
287	#33542	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
288	C₂×C₆₇	—	2	1	no	—
289	C₂×C₆₇	—	2	1	no	—
290	C₂×C₆₇	—	2	1	no	—
291	C₂×C₆₇	—	4	2	no	—
292	C₂×C₆₇	—	8	2	no	—
293	C₂×C₆₇	—	8	2	no	—
294	C₂×C₆₇	C₂×#447	1	1	yes	characteristic
295	#3584	C₂×C₆₇	1	1	yes	characteristic
296	C₂×#510	C₂×C₆₇	1	1	yes	characteristic, complemented
297	C₂×#510	C₂×C₆₇	1	1	yes	characteristic, complemented
298	C₂×#473	C₂×C₆₇	1	1	yes	characteristic, complemented
299	#2195	C₂×C₆₇	1	4	yes	complemented
300	#2194	C₂×C₆₇	1	4	yes	complemented
301	#2148	C₂×C₆₇	1	1	yes	characteristic, complemented
302	#2143	C₂×C₆₇	1	1	yes	characteristic, complemented
303	#2117	C₂×C₆₇	1	1	yes	characteristic, complemented
304	C₆₇	#33542	1	1	yes	characteristic, direct product factor, Sylow
305	#585	—	2	1	no	—
306	#578	—	2	1	no	—
307	#578	C₂²×C₆₇	1	2	yes	—
308	#577	—	2	2	no	—
309	#577	—	2	2	no	—
310	#576	—	2	1	no	—
311	#576	C₂²×C₆₇	1	2	yes	—
312	#575	—	2	2	no	—
313	#575	—	2	1	no	—
314	C₂×#137	—	2	1	no	—
315	C₂×#135	—	2	2	no	—
316	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
317	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
318	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
319	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
320	#510	—	2	2	no	—
321	#510	—	2	2	no	—
322	#510	—	2	2	no	—
323	#510	C₂²×C₆₇	1	4	yes	complemented
324	C₂×Q₃₂	—	2	1	no	—
325	C₂×Q₃₂	C₂²×C₆₇	1	2	yes	complemented
326	C₂×SD₃₂	—	2	1	no	—
327	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	complemented
328	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic, complemented
329	C₂×M₃₂	C₂²×C₆₇	1	1	yes	characteristic, complemented
330	#473	—	2	2	no	—
331	#473	C₂²×C₆₇	1	4	yes	complemented
332	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic, complemented
333	C₂×#101	—	2	1	no	—
334	#362	C₂²×C₆₇	1	2	yes	complemented
335	#361	C₂²×C₆₇	1	2	yes	complemented
336	#359	C₂²×C₆₇	1	4	yes	complemented
337	#350	C₂²×C₆₇	1	4	yes	complemented
338	2₋¹⁺⁴	—	2	2	no	—
339	2₋¹⁺⁴	—	2	2	no	—
340	2₋¹⁺⁴	—	4	1	no	—
341	H(5,𝔽₂)	—	2	2	no	—
342	H(5,𝔽₂)	—	2	2	no	—
343	H(5,𝔽₂)	—	4	2	no	—
344	H(5,𝔽₂)	—	4	1	no	—
345	C₂×#41	—	2	4	no	—
346	C₂×#41	—	2	1	no	—
347	C₂×#41	—	2	1	no	—
348	C₂×#41	—	4	2	no	—
349	C₂×#41	—	4	2	no	—
350	C₂×#41	D₄×C₆₇	1	1	yes	characteristic
351	C₂²×Q₈	—	2	1	no	—
352	#137	—	2	2	no	—
353	#137	—	2	4	no	—
354	#137	—	2	4	no	—
355	#137	—	2	4	no	—
356	#137	—	2	2	no	—
357	#137	—	4	2	no	—
358	AGL(1,ℤ/8)	—	2	2	no	—
359	AGL(1,ℤ/8)	—	2	4	no	—
360	AGL(1,ℤ/8)	—	2	4	no	—
361	AGL(1,ℤ/8)	—	2	2	no	—
362	#135	—	2	4	no	—
363	#135	—	2	4	no	—
364	#135	—	2	2	no	—
365	#135	—	2	2	no	—
366	#135	—	2	1	no	—
367	#135	—	2	1	no	—
368	#135	—	2	1	no	—
369	#135	—	2	1	no	—
370	#135	D₄×C₆₇	1	2	yes	—
371	#135	D₄×C₆₇	1	2	yes	—
372	#135	D₄×C₆₇	1	2	yes	—
373	#135	C₂³×C₆₇	1	2	yes	—
374	C₂×Q₁₆	—	2	2	no	—
375	C₂×Q₁₆	—	2	1	no	—
376	C₂×Q₁₆	—	2	1	no	—
377	C₂×Q₁₆	—	2	1	no	—
378	C₂×Q₁₆	—	2	1	no	—
379	C₂×Q₁₆	—	2	1	no	—
380	C₂×Q₁₆	—	4	1	no	—
381	C₂×Q₁₆	D₄×C₆₇	1	1	yes	characteristic
382	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
383	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
384	C₂×SD₁₆	—	2	4	no	—
385	C₂×SD₁₆	—	2	1	no	—
386	C₂×SD₁₆	—	2	1	no	—
387	C₂×SD₁₆	—	4	1	no	—
388	C₂×D₈	—	2	2	no	—
389	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
390	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
391	#131	—	2	1	no	—
392	#131	—	2	1	no	—
393	#131	D₄×C₆₇	1	2	yes	—
394	#131	C₂³×C₆₇	1	2	yes	—
395	C₂×M₁₆	—	2	1	no	—
396	C₂×M₁₆	—	2	1	no	—
397	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
398	C₂²×C₈	—	2	1	no	—
399	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
400	Q₃₂	—	2	4	no	—
401	Q₃₂	—	2	4	no	—
402	SD₃₂	—	2	4	no	—
403	SD₃₂	—	2	4	no	—
404	M₃₂	—	2	1	no	—
405	M₃₂	—	2	1	no	—
406	M₃₂	—	2	1	no	—
407	M₃₂	C₂³×C₆₇	1	2	yes	—
408	C₂×C₁₆	—	2	1	no	—
409	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
410	#108	—	2	1	no	—
411	#108	C₂³×C₆₇	1	2	yes	—
412	#101	—	2	4	no	—
413	#41	—	2	1	no	—
414	#41	—	2	1	no	—
415	#41	—	2	4	no	—
416	#41	—	2	4	no	—
417	#41	—	2	2	no	—
418	#41	—	2	2	no	—
419	#41	—	2	2	no	—
420	#41	—	2	2	no	—
421	#41	—	4	4	no	—
422	#41	—	4	4	no	—
423	#41	—	4	4	no	—
424	#41	—	4	4	no	—
425	#41	—	4	2	no	—
426	#41	—	4	2	no	—
427	#41	—	4	2	no	—
428	#41	—	4	2	no	—
429	#41	—	4	1	no	—
430	#41	—	4	1	no	—
431	#41	—	4	1	no	—
432	#41	—	4	1	no	—
433	#41	D₈×C₆₇	1	2	yes	—
434	#41	C₂×D₄×C₆₇	1	2	yes	—
435	C₂×Q₈	—	2	4	no	—
436	C₂×Q₈	—	2	1	no	—
437	C₂×Q₈	—	2	1	no	—
438	C₂×Q₈	—	2	2	no	—
439	C₂×Q₈	—	4	2	no	—
440	C₂×Q₈	—	4	2	no	—
441	C₂×Q₈	—	4	1	no	—
442	C₂×Q₈	—	4	1	no	—
443	C₂×Q₈	—	4	1	no	—
444	C₂×Q₈	—	4	1	no	—
445	C₂×Q₈	—	4	1	no	—
446	C₂×Q₈	D₈×C₆₇	1	1	yes	characteristic
447	C₂×D₄	—	2	1	no	—
448	C₂×D₄	—	2	4	no	—
449	C₂×D₄	—	2	1	no	—
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	—	4	4	no	—
452	C₂×D₄	—	4	4	no	—
453	C₂×D₄	—	4	2	no	—
454	C₂×D₄	—	4	4	no	—
455	C₂×D₄	—	4	1	no	—
456	C₂×D₄	—	4	1	no	—
457	C₂×D₄	—	4	2	no	—
458	C₂×D₄	D₈×C₆₇	1	1	yes	characteristic
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	4	4	no	—
461	C₂²×C₄	—	4	1	no	—
462	C₂²×C₄	—	4	1	no	—
463	C₂²×C₄	—	4	1	no	—
464	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
465	Q₁₆	—	2	4	no	—
466	Q₁₆	—	2	1	no	—
467	Q₁₆	—	2	1	no	—
468	Q₁₆	—	2	2	no	—
469	Q₁₆	—	2	2	no	—
470	Q₁₆	—	4	4	no	—
471	Q₁₆	—	4	1	no	—
472	Q₁₆	—	4	1	no	—
473	Q₁₆	C₂×D₄×C₆₇	1	2	yes	—
474	Q₁₆	C₂×D₄×C₆₇	1	2	yes	—
475	SD₁₆	—	2	4	no	—
476	SD₁₆	—	2	4	no	—
477	SD₁₆	—	2	2	no	—
478	SD₁₆	—	2	2	no	—
479	SD₁₆	—	2	2	no	—
480	SD₁₆	—	2	2	no	—
481	SD₁₆	—	4	4	no	—
482	D₈	—	2	4	no	—
483	D₈	—	2	1	no	—
484	D₈	—	2	1	no	—
485	D₈	C₂×D₄×C₆₇	1	2	yes	—
486	D₈	C₂×D₄×C₆₇	1	2	yes	—
487	M₁₆	—	2	1	no	—
488	M₁₆	—	2	2	no	—
489	M₁₆	—	2	2	no	—
490	M₁₆	—	2	2	no	—
491	M₁₆	—	4	1	no	—
492	M₁₆	C₂×D₄×C₆₇	1	2	yes	—
493	C₂×C₈	—	2	1	no	—
494	C₂×C₈	—	2	2	no	—
495	C₂×C₈	—	2	2	no	—
496	C₂×C₈	—	4	1	no	—
497	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
498	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
499	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
500	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
501	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
502	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
503	C₁₆	—	2	2	no	—
504	C₁₆	—	2	2	no	—
505	C₂³	—	2	1	no	—
506	C₂³	—	4	4	no	—
507	C₂³	—	4	1	no	—
508	C₂³	—	4	1	no	—
509	C₂³	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
510	Q₈	—	2	1	no	—
511	Q₈	—	2	2	no	—
512	Q₈	—	2	2	no	—
513	Q₈	—	4	4	no	—
514	Q₈	—	4	1	no	—
515	Q₈	—	4	1	no	—
516	Q₈	—	4	2	no	—
517	Q₈	—	4	2	no	—
518	Q₈	—	8	1	no	—
519	Q₈	—	8	1	no	—
520	Q₈	C₂×D₈×C₆₇	1	2	yes	—
521	D₄	—	2	2	no	—
522	D₄	—	2	2	no	—
523	D₄	—	2	2	no	—
524	D₄	—	2	2	no	—
525	D₄	—	2	1	no	—
526	D₄	—	4	4	no	—
527	D₄	—	4	4	no	—
528	D₄	—	4	4	no	—
529	D₄	—	4	2	no	—
530	D₄	—	4	2	no	—
531	D₄	—	4	2	no	—
532	D₄	—	4	2	no	—
533	D₄	—	4	1	no	—
534	D₄	—	4	1	no	—
535	D₄	C₂×D₈×C₆₇	1	2	yes	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	4	4	no	—
540	C₂×C₄	—	4	4	no	—
541	C₂×C₄	—	4	4	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	2	no	—
545	C₂×C₄	—	4	2	no	—
546	C₂×C₄	—	4	2	no	—
547	C₂×C₄	—	4	2	no	—
548	C₂×C₄	—	4	2	no	—
549	C₂×C₄	—	8	1	no	—
550	C₂×C₄	—	8	1	no	—
551	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
552	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
553	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
554	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
555	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
556	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
557	C₈	—	2	2	no	—
558	C₈	—	4	2	no	—
559	C₈	#120×C₆₇	1	1	yes	characteristic
560	C₈	#120×C₆₇	1	1	yes	characteristic
561	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
562	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
563	C₂²	—	2	2	no	—
564	C₂²	—	4	1	no	—
565	C₂²	—	4	2	no	—
566	C₂²	—	4	2	no	—
567	C₂²	—	8	1	no	—
568	C₂²	—	16	1	no	—
569	C₂²	—	16	1	no	—
570	C₂²	—	16	4	no	—
571	C₂²	#447×C₆₇	1	1	yes	characteristic
572	C₂²	#447×C₆₇	1	1	yes	characteristic
573	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
574	C₄	—	2	2	no	—
575	C₄	—	4	2	no	—
576	C₄	—	4	2	no	—
577	C₄	—	8	2	no	—
578	C₄	#447×C₆₇	1	1	yes	characteristic
579	C₄	#447×C₆₇	1	1	yes	characteristic
580	C₄	C₂×#120×C₆₇	1	1	yes	characteristic
581	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
582	C₂	—	2	1	no	—
583	C₂	—	2	1	no	—
584	C₂	—	2	1	no	—
585	C₂	—	4	2	no	—
586	C₂	—	8	2	no	—
587	C₂	—	8	2	no	—
588	C₂	C₂×#447×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	C₂×#447
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₆₇×#3584	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂×#131×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#510×C₆₇	C₂	semidirect
3. C₂×#510×C₆₇	C₂	semidirect
4. C₂×#473×C₆₇	C₂	semidirect
5. C₆₇×#2195	C₂	semidirect
6. C₆₇×#2194	C₂	semidirect
7. C₆₇×#2148	C₂	semidirect
8. C₆₇×#2143	C₂	semidirect
9. C₆₇×#2117	C₂	semidirect
23. C₂²×Q₁₆×C₆₇	C₂²	semidirect
28. #510×C₆₇	C₂²	semidirect
30. C₂×Q₃₂×C₆₇	C₂²	semidirect
32. C₂×SD₃₂×C₆₇	C₂²	semidirect
33. C₂×M₃₂×C₆₇	C₂²	semidirect
34. C₂×M₃₂×C₆₇	C₂²	semidirect
36. #473×C₆₇	C₂²	semidirect
37. C₂×#108×C₆₇	C₂²	semidirect
39. #362×C₆₇	C₂²	semidirect
40. #361×C₆₇	C₂²	semidirect
41. #359×C₆₇	C₂²	semidirect
42. #350×C₆₇	C₂²	semidirect
287. #33542	C₆₇	direct (N × H)
296. C₂×#510	C₂×C₆₇	semidirect
297. C₂×#510	C₂×C₆₇	semidirect
298. C₂×#473	C₂×C₆₇	semidirect
299. #2195	C₂×C₆₇	semidirect
300. #2194	C₂×C₆₇	semidirect
301. #2148	C₂×C₆₇	semidirect
302. #2143	C₂×C₆₇	semidirect
303. #2117	C₂×C₆₇	semidirect
304. C₆₇	#33542	direct (N × H)
318. C₂²×Q₁₆	C₂²×C₆₇	semidirect
323. #510	C₂²×C₆₇	semidirect
325. C₂×Q₃₂	C₂²×C₆₇	semidirect
327. C₂×SD₃₂	C₂²×C₆₇	semidirect
328. C₂×M₃₂	C₂²×C₆₇	semidirect
329. C₂×M₃₂	C₂²×C₆₇	semidirect
331. #473	C₂²×C₆₇	semidirect
332. C₂×#108	C₂²×C₆₇	semidirect
334. #362	C₂²×C₆₇	semidirect
335. #361	C₂²×C₆₇	semidirect
336. #359	C₂²×C₆₇	semidirect
337. #350	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3584	2
C₂×#510×C₆₇	2
C₂×#473×C₆₇	2
C₆₇×#2195	2
C₆₇×#2194	2
C₆₇×#2148	2
C₆₇×#2143	2
C₆₇×#2117	2
#33542	67

Subgroups — C67×#33542 #30712793

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33542 #30712793