Subgroups — C₂×C₆₇×#3386 #30740611

Order 17152.

2006 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#557×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3386	C₂	1	16	yes	maximal, direct product factor
4	C₂×#501×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#477×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#475×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#478×C₆₇	C₂	1	2	yes	maximal
9	C₂×#474×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#440×C₆₇	C₂	1	2	yes	maximal
12	C₂×C₄×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
13	C₂³×Q₈×C₆₇	—	2	1	no	—
14	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	#557×C₆₇	—	2	4	no	—
16	#557×C₆₇	C₂²	1	8	yes	complemented
17	D₄×Q₈×C₆₇	—	2	4	no	—
18	D₄×Q₈×C₆₇	C₂²	1	8	yes	—
19	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
20	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
23	C₂×#125×C₆₇	—	2	2	no	—
24	C₂×#125×C₆₇	C₂²	1	2	yes	—
25	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
26	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
27	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
28	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
33	#501×C₆₇	C₂²	1	16	yes	complemented
34	#500×C₆₇	C₂²	1	8	yes	complemented
35	#478×C₆₇	C₂²	1	16	yes	—
36	#477×C₆₇	C₂²	1	16	yes	complemented
37	#475×C₆₇	C₂²	1	8	yes	complemented
38	#474×C₆₇	C₂²	1	8	yes	—
39	#446×C₆₇	C₂²	1	8	yes	complemented
40	#440×C₆₇	C₂²	1	16	yes	—
41	C₄×SD₁₆×C₆₇	C₂²	1	8	yes	—
42	C₂×#107×C₆₇	C₂²	1	2	yes	—
43	C₂×#107×C₆₇	C₂²	1	2	yes	—
44	C₂×#107×C₆₇	C₂²	1	2	yes	—
45	C₂×#106×C₆₇	C₂²	1	2	yes	—
46	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
48	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
49	C₂×#103×C₆₇	C₂²	1	2	yes	—
50	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
51	C₂×#102×C₆₇	C₂²	1	2	yes	—
52	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
53	C₂×#97×C₆₇	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
55	C₂²×Q₈×C₆₇	—	2	8	no	—
56	C₂²×Q₈×C₆₇	—	2	1	no	—
57	C₂²×Q₈×C₆₇	—	2	2	no	—
58	C₂²×Q₈×C₆₇	—	2	1	no	—
59	C₂²×Q₈×C₆₇	—	4	8	no	—
60	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
61	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₆₇	—	2	2	no	—
64	C₂³×C₄×C₆₇	—	2	1	no	—
65	C₂×SD₁₆×C₆₇	Q₈	1	8	yes	complemented
66	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
67	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
69	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₄⋊Q₈×C₆₇	—	2	8	no	—
71	C₄⋊Q₈×C₆₇	—	2	4	no	—
72	C₄⋊Q₈×C₆₇	—	2	2	no	—
73	C₄⋊Q₈×C₆₇	—	2	2	no	—
74	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
75	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
76	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	—
77	#125×C₆₇	—	2	16	no	—
78	#125×C₆₇	—	2	4	no	—
79	#125×C₆₇	C₂³	1	8	yes	—
80	C₂²⋊Q₈×C₆₇	—	2	8	no	—
81	C₂²⋊Q₈×C₆₇	—	2	16	no	—
82	C₄×Q₈×C₆₇	—	2	8	no	—
83	C₄×Q₈×C₆₇	—	2	4	no	—
84	C₄×Q₈×C₆₇	—	2	2	no	—
85	C₄×Q₈×C₆₇	D₄	1	4	yes	—
86	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
87	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
88	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
89	C₄×D₄×C₆₇	—	2	4	no	—
90	C₄×D₄×C₆₇	—	2	2	no	—
91	C₄×D₄×C₆₇	C₂³	1	4	yes	—
92	C₄×D₄×C₆₇	C₂³	1	8	yes	—
93	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
94	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
95	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
96	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
97	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
98	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
99	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
100	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
106	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
107	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
108	C₂×C₄²×C₆₇	—	2	2	no	—
109	C₂×C₄²×C₆₇	—	2	1	no	—
110	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
111	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
112	#107×C₆₇	C₂³	1	8	yes	—
113	#107×C₆₇	C₂³	1	8	yes	—
114	#107×C₆₇	C₂³	1	8	yes	—
115	#106×C₆₇	C₂³	1	8	yes	—
116	#106×C₆₇	C₂³	1	4	yes	—
117	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
118	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
119	#103×C₆₇	C₂³	1	8	yes	—
120	#103×C₆₇	C₂³	1	4	yes	—
121	#102×C₆₇	C₂³	1	8	yes	—
122	#102×C₆₇	C₂³	1	4	yes	—
123	#97×C₆₇	C₂³	1	8	yes	—
124	C₄×C₈×C₆₇	C₂³	1	4	yes	—
125	C₂⁴×C₆₇	—	2	1	no	—
126	C₂×Q₈×C₆₇	—	2	4	no	—
127	C₂×Q₈×C₆₇	—	2	1	no	—
128	C₂×Q₈×C₆₇	—	2	4	no	—
129	C₂×Q₈×C₆₇	—	2	8	no	—
130	C₂×Q₈×C₆₇	—	2	4	no	—
131	C₂×Q₈×C₆₇	—	2	2	no	—
132	C₂×Q₈×C₆₇	—	4	8	no	—
133	C₂×Q₈×C₆₇	—	4	32	no	—
134	C₂×Q₈×C₆₇	—	4	4	no	—
135	C₂×Q₈×C₆₇	—	4	4	no	—
136	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
137	C₂×Q₈×C₆₇	C₂×Q₈	1	4	yes	complemented
138	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
139	C₂×D₄×C₆₇	—	2	1	no	—
140	C₂×D₄×C₆₇	—	2	2	no	—
141	C₂×D₄×C₆₇	C₂×Q₈	1	4	yes	—
142	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
143	C₂²×C₄×C₆₇	—	2	1	no	—
144	C₂²×C₄×C₆₇	—	2	1	no	—
145	C₂²×C₄×C₆₇	—	2	2	no	—
146	C₂²×C₄×C₆₇	—	2	2	no	—
147	C₂²×C₄×C₆₇	—	2	1	no	—
148	C₂²×C₄×C₆₇	—	2	8	no	—
149	C₂²×C₄×C₆₇	—	2	4	no	—
150	C₂²×C₄×C₆₇	—	4	8	no	—
151	C₂²×C₄×C₆₇	—	4	4	no	—
152	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
155	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
157	SD₁₆×C₆₇	C₂×Q₈	1	16	yes	complemented
158	C₂×C₈×C₆₇	—	2	2	no	—
159	C₂×C₈×C₆₇	—	2	4	no	—
160	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	complemented
161	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
162	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
163	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
164	C₄⋊C₄×C₆₇	—	2	8	no	—
165	C₄⋊C₄×C₆₇	—	2	4	no	—
166	C₄⋊C₄×C₆₇	—	2	8	no	—
167	C₄⋊C₄×C₆₇	—	2	8	no	—
168	C₄⋊C₄×C₆₇	—	2	4	no	—
169	C₄⋊C₄×C₆₇	—	2	4	no	—
170	C₄⋊C₄×C₆₇	—	2	8	no	—
171	C₄⋊C₄×C₆₇	—	2	1	no	—
172	C₄⋊C₄×C₆₇	—	2	2	no	—
173	C₄⋊C₄×C₆₇	—	2	1	no	—
174	C₄⋊C₄×C₆₇	—	2	2	no	—
175	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
176	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
177	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
178	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
179	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
180	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
181	C₂²⋊C₄×C₆₇	—	2	8	no	—
182	C₂²⋊C₄×C₆₇	—	2	4	no	—
183	C₄²×C₆₇	—	2	8	no	—
184	C₄²×C₆₇	—	2	4	no	—
185	C₄²×C₆₇	C₂×D₄	1	4	yes	—
186	C₄²×C₆₇	C₂×D₄	1	2	yes	—
187	C₄²×C₆₇	C₂⁴	1	4	yes	—
188	C₄²×C₆₇	C₂⁴	1	2	yes	—
189	C₂³×C₆₇	—	2	4	no	—
190	C₂³×C₆₇	—	2	2	no	—
191	C₂³×C₆₇	—	4	2	no	—
192	C₂³×C₆₇	—	4	2	no	—
193	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
194	Q₈×C₆₇	—	2	4	no	—
195	Q₈×C₆₇	—	2	8	no	—
196	Q₈×C₆₇	—	2	2	no	—
197	Q₈×C₆₇	—	4	16	no	—
198	Q₈×C₆₇	—	4	8	no	—
199	Q₈×C₆₇	—	4	8	no	—
200	Q₈×C₆₇	C₂²×Q₈	1	4	yes	complemented
201	D₄×C₆₇	—	2	4	no	—
202	D₄×C₆₇	—	2	2	no	—
203	D₄×C₆₇	C₂²×Q₈	1	4	yes	—
204	C₂×C₄×C₆₇	—	2	4	no	—
205	C₂×C₄×C₆₇	—	2	4	no	—
206	C₂×C₄×C₆₇	—	2	4	no	—
207	C₂×C₄×C₆₇	—	2	2	no	—
208	C₂×C₄×C₆₇	—	2	2	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	4	no	—
211	C₂×C₄×C₆₇	—	2	4	no	—
212	C₂×C₄×C₆₇	—	2	2	no	—
213	C₂×C₄×C₆₇	—	4	16	no	—
214	C₂×C₄×C₆₇	—	4	4	no	—
215	C₂×C₄×C₆₇	—	4	2	no	—
216	C₂×C₄×C₆₇	—	4	4	no	—
217	C₂×C₄×C₆₇	—	4	2	no	—
218	C₂×C₄×C₆₇	—	4	8	no	—
219	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
220	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
221	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
222	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
223	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
225	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
226	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
227	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
228	C₈×C₆₇	—	2	4	no	—
229	C₈×C₆₇	—	2	2	no	—
230	C₈×C₆₇	C₂²×Q₈	1	4	yes	complemented
231	C₂²×C₆₇	—	2	4	no	—
232	C₂²×C₆₇	—	4	8	no	—
233	C₂²×C₆₇	—	4	2	no	—
234	C₂²×C₆₇	—	4	2	no	—
235	C₂²×C₆₇	D₄×Q₈	1	2	yes	—
236	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
237	C₂²×C₆₇	#578	1	2	yes	—
238	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
239	C₄×C₆₇	—	2	4	no	—
240	C₄×C₆₇	—	2	4	no	—
241	C₄×C₆₇	—	2	4	no	—
242	C₄×C₆₇	—	2	2	no	—
243	C₄×C₆₇	—	4	4	no	—
244	C₄×C₆₇	—	4	2	no	—
245	C₄×C₆₇	—	4	4	no	—
246	C₄×C₆₇	—	4	2	no	—
247	C₄×C₆₇	D₄×Q₈	1	2	yes	—
248	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
249	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
250	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
251	C₂×#3386	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
252	C₂×C₆₇	—	4	4	no	—
253	C₂×C₆₇	#3386	1	4	yes	direct product factor
254	C₂×C₆₇	C₂×D₄×Q₈	1	1	yes	characteristic
255	C₂×C₆₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
256	C₂×C₆₇	C₂×#578	1	1	yes	characteristic
257	C₂×#557	C₂×C₆₇	1	1	yes	characteristic, complemented
258	C₂×D₄×Q₈	C₂×C₆₇	1	1	yes	characteristic
259	#3386	C₂×C₆₇	1	16	yes	direct product factor
260	C₂×#501	C₂×C₆₇	1	2	yes	complemented
261	C₂×#500	C₂×C₆₇	1	1	yes	characteristic, complemented
262	C₂×#477	C₂×C₆₇	1	2	yes	complemented
263	C₂×#475	C₂×C₆₇	1	1	yes	characteristic, complemented
264	C₂×#478	C₂×C₆₇	1	2	yes	—
265	C₂×#474	C₂×C₆₇	1	1	yes	characteristic
266	C₂×#446	C₂×C₆₇	1	1	yes	characteristic, complemented
267	C₂×#440	C₂×C₆₇	1	2	yes	—
268	C₂×C₄×SD₁₆	C₂×C₆₇	1	1	yes	characteristic
269	C₆₇	C₂×#3386	1	1	yes	characteristic, direct product factor, Sylow
270	C₂³×Q₈	—	2	1	no	—
271	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
272	#557	—	2	4	no	—
273	#557	C₂²×C₆₇	1	8	yes	complemented
274	D₄×Q₈	—	2	4	no	—
275	D₄×Q₈	C₂²×C₆₇	1	8	yes	—
276	C₂×C₄⋊Q₈	—	2	1	no	—
277	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
278	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
279	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
280	C₂×#125	—	2	2	no	—
281	C₂×#125	C₂²×C₆₇	1	2	yes	—
282	C₂×C₂²⋊Q₈	—	2	1	no	—
283	C₂×C₂²⋊Q₈	—	2	2	no	—
284	C₂×C₄×Q₈	—	2	1	no	—
285	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
286	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
287	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
288	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
289	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
290	#501	C₂²×C₆₇	1	16	yes	complemented
291	#500	C₂²×C₆₇	1	8	yes	complemented
292	#478	C₂²×C₆₇	1	16	yes	—
293	#477	C₂²×C₆₇	1	16	yes	complemented
294	#475	C₂²×C₆₇	1	8	yes	complemented
295	#474	C₂²×C₆₇	1	8	yes	—
296	#446	C₂²×C₆₇	1	8	yes	complemented
297	#440	C₂²×C₆₇	1	16	yes	—
298	C₄×SD₁₆	C₂²×C₆₇	1	8	yes	—
299	C₂×#107	C₂²×C₆₇	1	2	yes	—
300	C₂×#107	C₂²×C₆₇	1	2	yes	—
301	C₂×#107	C₂²×C₆₇	1	2	yes	—
302	C₂×#106	C₂²×C₆₇	1	2	yes	—
303	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
304	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
305	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
306	C₂×#103	C₂²×C₆₇	1	2	yes	—
307	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
308	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
309	C₂×#102	C₂²×C₆₇	1	2	yes	—
310	C₂×#97	C₂²×C₆₇	1	2	yes	—
311	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
312	C₂²×Q₈	—	2	8	no	—
313	C₂²×Q₈	—	2	1	no	—
314	C₂²×Q₈	—	2	2	no	—
315	C₂²×Q₈	—	2	1	no	—
316	C₂²×Q₈	—	4	8	no	—
317	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
318	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
319	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
320	C₂³×C₄	—	2	2	no	—
321	C₂³×C₄	—	2	1	no	—
322	C₂×SD₁₆	Q₈×C₆₇	1	8	yes	complemented
323	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
324	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
325	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
326	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
327	C₄⋊Q₈	—	2	8	no	—
328	C₄⋊Q₈	—	2	4	no	—
329	C₄⋊Q₈	—	2	2	no	—
330	C₄⋊Q₈	—	2	2	no	—
331	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
332	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	—
333	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
334	#125	—	2	16	no	—
335	#125	—	2	4	no	—
336	#125	C₂³×C₆₇	1	8	yes	—
337	C₂²⋊Q₈	—	2	8	no	—
338	C₂²⋊Q₈	—	2	16	no	—
339	C₄×Q₈	—	2	8	no	—
340	C₄×Q₈	—	2	4	no	—
341	C₄×Q₈	—	2	2	no	—
342	C₄×Q₈	D₄×C₆₇	1	4	yes	—
343	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
344	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
345	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
346	C₄×D₄	—	2	4	no	—
347	C₄×D₄	—	2	2	no	—
348	C₄×D₄	C₂³×C₆₇	1	4	yes	—
349	C₄×D₄	C₂³×C₆₇	1	8	yes	—
350	C₂×C₄⋊C₄	—	2	2	no	—
351	C₂×C₄⋊C₄	—	2	1	no	—
352	C₂×C₄⋊C₄	—	2	2	no	—
353	C₂×C₄⋊C₄	—	2	2	no	—
354	C₂×C₄⋊C₄	—	2	1	no	—
355	C₂×C₄⋊C₄	—	2	1	no	—
356	C₂×C₄⋊C₄	—	2	2	no	—
357	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
358	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
359	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
360	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
361	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
362	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
363	C₂×C₂²⋊C₄	—	2	2	no	—
364	C₂×C₂²⋊C₄	—	2	1	no	—
365	C₂×C₄²	—	2	2	no	—
366	C₂×C₄²	—	2	1	no	—
367	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
368	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
369	#107	C₂³×C₆₇	1	8	yes	—
370	#107	C₂³×C₆₇	1	8	yes	—
371	#107	C₂³×C₆₇	1	8	yes	—
372	#106	C₂³×C₆₇	1	4	yes	—
373	#106	C₂³×C₆₇	1	8	yes	—
374	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
375	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
376	#103	C₂³×C₆₇	1	4	yes	—
377	#103	C₂³×C₆₇	1	8	yes	—
378	#102	C₂³×C₆₇	1	4	yes	—
379	#102	C₂³×C₆₇	1	8	yes	—
380	#97	C₂³×C₆₇	1	8	yes	—
381	C₄×C₈	C₂³×C₆₇	1	4	yes	—
382	C₂⁴	—	2	1	no	—
383	C₂×Q₈	—	2	4	no	—
384	C₂×Q₈	—	2	1	no	—
385	C₂×Q₈	—	2	4	no	—
386	C₂×Q₈	—	2	8	no	—
387	C₂×Q₈	—	2	4	no	—
388	C₂×Q₈	—	2	2	no	—
389	C₂×Q₈	—	4	8	no	—
390	C₂×Q₈	—	4	32	no	—
391	C₂×Q₈	—	4	4	no	—
392	C₂×Q₈	—	4	4	no	—
393	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
394	C₂×Q₈	C₂×Q₈×C₆₇	1	4	yes	complemented
395	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
396	C₂×D₄	—	2	1	no	—
397	C₂×D₄	—	2	2	no	—
398	C₂×D₄	C₂×Q₈×C₆₇	1	4	yes	—
399	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
400	C₂²×C₄	—	2	1	no	—
401	C₂²×C₄	—	2	1	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	2	no	—
404	C₂²×C₄	—	2	1	no	—
405	C₂²×C₄	—	2	8	no	—
406	C₂²×C₄	—	2	4	no	—
407	C₂²×C₄	—	4	8	no	—
408	C₂²×C₄	—	4	4	no	—
409	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
410	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
411	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
412	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
413	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
414	SD₁₆	C₂×Q₈×C₆₇	1	16	yes	complemented
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	4	no	—
417	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	complemented
418	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
419	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
420	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
421	C₄⋊C₄	—	2	8	no	—
422	C₄⋊C₄	—	2	4	no	—
423	C₄⋊C₄	—	2	8	no	—
424	C₄⋊C₄	—	2	8	no	—
425	C₄⋊C₄	—	2	4	no	—
426	C₄⋊C₄	—	2	4	no	—
427	C₄⋊C₄	—	2	8	no	—
428	C₄⋊C₄	—	2	2	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	—	2	1	no	—
431	C₄⋊C₄	—	2	2	no	—
432	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
433	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
434	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
435	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
436	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
437	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
438	C₂²⋊C₄	—	2	8	no	—
439	C₂²⋊C₄	—	2	4	no	—
440	C₄²	—	2	8	no	—
441	C₄²	—	2	4	no	—
442	C₄²	C₂×D₄×C₆₇	1	4	yes	—
443	C₄²	C₂×D₄×C₆₇	1	2	yes	—
444	C₄²	C₂⁴×C₆₇	1	2	yes	—
445	C₄²	C₂⁴×C₆₇	1	4	yes	—
446	C₂³	—	2	4	no	—
447	C₂³	—	2	2	no	—
448	C₂³	—	4	2	no	—
449	C₂³	—	4	2	no	—
450	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
451	Q₈	—	2	4	no	—
452	Q₈	—	2	8	no	—
453	Q₈	—	2	2	no	—
454	Q₈	—	4	16	no	—
455	Q₈	—	4	8	no	—
456	Q₈	—	4	8	no	—
457	Q₈	C₂²×Q₈×C₆₇	1	4	yes	complemented
458	D₄	—	2	4	no	—
459	D₄	—	2	2	no	—
460	D₄	C₂²×Q₈×C₆₇	1	4	yes	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	4	16	no	—
471	C₂×C₄	—	4	4	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	4	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	8	no	—
476	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
477	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
478	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
479	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
480	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
481	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
482	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
483	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
484	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
485	C₈	—	2	2	no	—
486	C₈	—	2	4	no	—
487	C₈	C₂²×Q₈×C₆₇	1	4	yes	complemented
488	C₂²	—	2	4	no	—
489	C₂²	—	4	2	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	4	8	no	—
492	C₂²	D₄×Q₈×C₆₇	1	2	yes	—
493	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
494	C₂²	#578×C₆₇	1	2	yes	—
495	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
496	C₄	—	2	4	no	—
497	C₄	—	2	4	no	—
498	C₄	—	2	2	no	—
499	C₄	—	2	4	no	—
500	C₄	—	4	4	no	—
501	C₄	—	4	2	no	—
502	C₄	—	4	4	no	—
503	C₄	—	4	2	no	—
504	C₄	D₄×Q₈×C₆₇	1	2	yes	—
505	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	4	yes	—
506	C₄	C₂³×Q₈×C₆₇	1	1	yes	characteristic
507	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
508	C₂	—	4	4	no	—
509	C₂	C₆₇×#3386	1	4	yes	direct product factor
510	C₂	C₂×D₄×Q₈×C₆₇	1	1	yes	characteristic
511	C₂	C₂²×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
512	C₂	C₂×#578×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#557×C₆₇	C₂	semidirect
3. C₆₇×#3386	C₂	direct (N × H)
4. C₂×#501×C₆₇	C₂	semidirect
5. C₂×#500×C₆₇	C₂	semidirect
6. C₂×#477×C₆₇	C₂	semidirect
7. C₂×#475×C₆₇	C₂	semidirect
10. C₂×#446×C₆₇	C₂	semidirect
16. #557×C₆₇	C₂²	semidirect
33. #501×C₆₇	C₂²	semidirect
34. #500×C₆₇	C₂²	semidirect
36. #477×C₆₇	C₂²	semidirect
37. #475×C₆₇	C₂²	semidirect
39. #446×C₆₇	C₂²	semidirect
65. C₂×SD₁₆×C₆₇	Q₈	semidirect
137. C₂×Q₈×C₆₇	C₂×Q₈	semidirect
157. SD₁₆×C₆₇	C₂×Q₈	semidirect
160. C₂×C₈×C₆₇	C₂×Q₈	semidirect
200. Q₈×C₆₇	C₂²×Q₈	semidirect
230. C₈×C₆₇	C₂²×Q₈	semidirect
251. C₂×#3386	C₆₇	direct (N × H)
253. C₂×C₆₇	#3386	direct (N × H)
257. C₂×#557	C₂×C₆₇	semidirect
259. #3386	C₂×C₆₇	direct (N × H)
260. C₂×#501	C₂×C₆₇	semidirect
261. C₂×#500	C₂×C₆₇	semidirect
262. C₂×#477	C₂×C₆₇	semidirect
263. C₂×#475	C₂×C₆₇	semidirect
266. C₂×#446	C₂×C₆₇	semidirect
269. C₆₇	C₂×#3386	direct (N × H)
273. #557	C₂²×C₆₇	semidirect
290. #501	C₂²×C₆₇	semidirect
291. #500	C₂²×C₆₇	semidirect
293. #477	C₂²×C₆₇	semidirect
294. #475	C₂²×C₆₇	semidirect
296. #446	C₂²×C₆₇	semidirect
322. C₂×SD₁₆	Q₈×C₆₇	semidirect
394. C₂×Q₈	C₂×Q₈×C₆₇	semidirect
414. SD₁₆	C₂×Q₈×C₆₇	semidirect
417. C₂×C₈	C₂×Q₈×C₆₇	semidirect
457. Q₈	C₂²×Q₈×C₆₇	semidirect
487. C₈	C₂²×Q₈×C₆₇	semidirect
509. C₂	C₆₇×#3386	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×#557×C₆₇	2
C₂×D₄×Q₈×C₆₇	2
C₆₇×#3386	2
C₂×#501×C₆₇	2
C₂×#500×C₆₇	2
C₂×#477×C₆₇	2
C₂×#475×C₆₇	2
C₂×#478×C₆₇	2
C₂×#474×C₆₇	2
C₂×#446×C₆₇	2
C₂×#440×C₆₇	2
C₂×C₄×SD₁₆×C₆₇	2
C₂×#3386	67

Subgroups — C2×C67×#3386 #30740611

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3386 #30740611