Subgroups — C₂×C₆₇×#3366 #30740555

Order 17152.

2198 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#554×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#550×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3366	C₂	1	16	yes	maximal, direct product factor
4	C₂×#490×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#486×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#478×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#476×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#459×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#459×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#442×C₆₇	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₆₇	—	2	1	no	—
14	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
16	#554×C₆₇	—	2	4	no	—
17	#554×C₆₇	C₂²	1	8	yes	complemented
18	#550×C₆₇	—	2	4	no	—
19	#550×C₆₇	C₂²	1	8	yes	complemented
20	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#127×C₆₇	C₂²	1	2	yes	—
23	C₂×#125×C₆₇	C₂²	1	2	yes	—
24	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
25	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
26	C₂×#121×C₆₇	—	2	1	no	—
27	C₂×#121×C₆₇	—	2	2	no	—
28	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
33	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
34	#490×C₆₇	C₂²	1	16	yes	complemented
35	#486×C₆₇	C₂²	1	8	yes	complemented
36	#478×C₆₇	C₂²	1	16	yes	complemented
37	#476×C₆₇	C₂²	1	8	yes	complemented
38	#459×C₆₇	C₂²	1	16	yes	complemented
39	#459×C₆₇	C₂²	1	8	yes	complemented
40	#446×C₆₇	C₂²	1	8	yes	complemented
41	#442×C₆₇	C₂²	1	16	yes	complemented
42	C₄×D₈×C₆₇	C₂²	1	8	yes	complemented
43	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×#106×C₆₇	C₂²	1	2	yes	—
45	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
46	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
47	C₂×#102×C₆₇	C₂²	1	2	yes	—
48	C₂×#102×C₆₇	C₂²	1	2	yes	—
49	C₂×#102×C₆₇	C₂²	1	2	yes	—
50	C₂×#102×C₆₇	C₂²	1	2	yes	—
51	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
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	C₂×#41×C₆₇	—	2	8	no	—
56	C₂×#41×C₆₇	—	4	8	no	—
57	C₂²×Q₈×C₆₇	—	2	1	no	—
58	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
59	C₂²×D₄×C₆₇	—	2	2	no	—
60	C₂²×D₄×C₆₇	—	2	1	no	—
61	C₂²×D₄×C₆₇	—	2	1	no	—
62	C₂²×D₄×C₆₇	—	2	2	no	—
63	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
65	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₆₇	—	2	1	no	—
67	C₂³×C₄×C₆₇	—	2	2	no	—
68	C₂³×C₄×C₆₇	—	2	1	no	—
69	C₂³×C₄×C₆₇	—	2	2	no	—
70	C₂×D₈×C₆₇	—	2	4	no	—
71	C₂×D₈×C₆₇	—	2	8	no	—
72	C₂×D₈×C₆₇	C₂³	1	4	yes	—
73	C₂×D₈×C₆₇	C₂³	1	8	yes	—
74	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
76	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
77	C₄⋊Q₈×C₆₇	—	2	2	no	—
78	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
79	#127×C₆₇	—	2	4	no	—
80	#127×C₆₇	—	2	2	no	—
81	#127×C₆₇	C₂³	1	4	yes	—
82	#127×C₆₇	C₂³	1	8	yes	—
83	#125×C₆₇	—	2	4	no	—
84	#125×C₆₇	C₂³	1	8	yes	—
85	C₂²⋊Q₈×C₆₇	—	2	8	no	—
86	C₂²⋊Q₈×C₆₇	—	2	16	no	—
87	#121×C₆₇	—	2	8	no	—
88	#121×C₆₇	—	2	16	no	—
89	C₄×Q₈×C₆₇	D₄	1	4	yes	—
90	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
91	C₄×D₄×C₆₇	—	2	4	no	—
92	C₄×D₄×C₆₇	—	2	2	no	—
93	C₄×D₄×C₆₇	—	2	4	no	—
94	C₄×D₄×C₆₇	—	2	2	no	—
95	C₄×D₄×C₆₇	C₂³	1	4	yes	—
96	C₄×D₄×C₆₇	C₂³	1	8	yes	—
97	C₄×D₄×C₆₇	C₂³	1	4	yes	—
98	C₄×D₄×C₆₇	C₂³	1	8	yes	—
99	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
100	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
101	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
102	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
103	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
108	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
109	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
110	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
111	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
112	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
113	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
114	#107×C₆₇	C₂³	1	4	yes	—
115	#106×C₆₇	C₂³	1	8	yes	—
116	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
117	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
118	#102×C₆₇	C₂³	1	8	yes	—
119	#102×C₆₇	C₂³	1	8	yes	—
120	#102×C₆₇	C₂³	1	8	yes	—
121	#102×C₆₇	C₂³	1	8	yes	—
122	#102×C₆₇	C₂³	1	4	yes	—
123	#102×C₆₇	C₂³	1	4	yes	—
124	#97×C₆₇	C₂³	1	8	yes	—
125	C₄×C₈×C₆₇	C₂³	1	4	yes	—
126	C₂⁴×C₆₇	—	2	1	no	—
127	C₂⁴×C₆₇	—	2	2	no	—
128	C₂⁴×C₆₇	—	2	1	no	—
129	#41×C₆₇	—	4	32	no	—
130	C₂×Q₈×C₆₇	—	2	4	no	—
131	C₂×Q₈×C₆₇	—	2	4	no	—
132	C₂×Q₈×C₆₇	—	8	2	no	—
133	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
134	C₂×D₄×C₆₇	—	2	1	no	—
135	C₂×D₄×C₆₇	—	2	2	no	—
136	C₂×D₄×C₆₇	—	2	1	no	—
137	C₂×D₄×C₆₇	—	2	8	no	—
138	C₂×D₄×C₆₇	—	2	4	no	—
139	C₂×D₄×C₆₇	—	2	4	no	—
140	C₂×D₄×C₆₇	—	2	8	no	—
141	C₂×D₄×C₆₇	—	2	2	no	—
142	C₂×D₄×C₆₇	—	2	4	no	—
143	C₂×D₄×C₆₇	—	2	2	no	—
144	C₂×D₄×C₆₇	—	2	2	no	—
145	C₂×D₄×C₆₇	—	2	4	no	—
146	C₂×D₄×C₆₇	—	4	8	no	—
147	C₂×D₄×C₆₇	—	4	4	no	—
148	C₂×D₄×C₆₇	—	8	4	no	—
149	C₂×D₄×C₆₇	—	8	2	no	—
150	C₂×D₄×C₆₇	#41	1	4	yes	complemented
151	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
152	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
153	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
154	C₂²×C₄×C₆₇	—	2	1	no	—
155	C₂²×C₄×C₆₇	—	2	1	no	—
156	C₂²×C₄×C₆₇	—	2	2	no	—
157	C₂²×C₄×C₆₇	—	2	4	no	—
158	C₂²×C₄×C₆₇	—	2	8	no	—
159	C₂²×C₄×C₆₇	—	2	4	no	—
160	C₂²×C₄×C₆₇	—	2	8	no	—
161	C₂²×C₄×C₆₇	—	4	8	no	—
162	C₂²×C₄×C₆₇	—	4	4	no	—
163	C₂²×C₄×C₆₇	—	4	4	no	—
164	C₂²×C₄×C₆₇	—	8	4	no	—
165	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
168	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
170	D₈×C₆₇	—	2	8	no	—
171	D₈×C₆₇	—	2	16	no	—
172	C₂×C₈×C₆₇	—	2	2	no	—
173	C₂×C₈×C₆₇	—	2	4	no	—
174	C₂×C₈×C₆₇	—	2	2	no	—
175	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
176	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
177	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
178	C₄⋊C₄×C₆₇	—	2	8	no	—
179	C₄⋊C₄×C₆₇	—	2	4	no	—
180	C₄⋊C₄×C₆₇	—	2	8	no	—
181	C₄⋊C₄×C₆₇	—	2	1	no	—
182	C₄⋊C₄×C₆₇	—	2	1	no	—
183	C₄⋊C₄×C₆₇	—	2	2	no	—
184	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
185	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
186	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
187	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
188	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
189	C₂²⋊C₄×C₆₇	—	2	4	no	—
190	C₂²⋊C₄×C₆₇	—	2	8	no	—
191	C₂²⋊C₄×C₆₇	—	2	4	no	—
192	C₂²⋊C₄×C₆₇	—	2	8	no	—
193	C₄²×C₆₇	C₂×D₄	1	4	yes	—
194	C₄²×C₆₇	C₂×D₄	1	2	yes	—
195	C₄²×C₆₇	C₂⁴	1	4	yes	—
196	C₄²×C₆₇	C₂⁴	1	2	yes	—
197	C₂³×C₆₇	—	2	4	no	—
198	C₂³×C₆₇	—	2	2	no	—
199	C₂³×C₆₇	—	2	4	no	—
200	C₂³×C₆₇	—	2	2	no	—
201	C₂³×C₆₇	—	4	2	no	—
202	C₂³×C₆₇	—	4	2	no	—
203	C₂³×C₆₇	—	4	4	no	—
204	C₂³×C₆₇	—	4	2	no	—
205	C₂³×C₆₇	—	4	4	no	—
206	C₂³×C₆₇	—	4	2	no	—
207	C₂³×C₆₇	—	4	2	no	—
208	C₂³×C₆₇	—	4	4	no	—
209	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
210	Q₈×C₆₇	—	2	8	no	—
211	Q₈×C₆₇	—	8	4	no	—
212	D₄×C₆₇	—	2	4	no	—
213	D₄×C₆₇	—	2	2	no	—
214	D₄×C₆₇	—	2	4	no	—
215	D₄×C₆₇	—	2	2	no	—
216	D₄×C₆₇	—	2	2	no	—
217	D₄×C₆₇	—	2	4	no	—
218	D₄×C₆₇	—	4	16	no	—
219	D₄×C₆₇	—	4	4	no	—
220	D₄×C₆₇	—	4	2	no	—
221	D₄×C₆₇	—	4	8	no	—
222	D₄×C₆₇	—	8	8	no	—
223	D₄×C₆₇	—	8	4	no	—
224	D₄×C₆₇	C₂×#41	1	4	yes	complemented
225	C₂×C₄×C₆₇	—	2	4	no	—
226	C₂×C₄×C₆₇	—	2	4	no	—
227	C₂×C₄×C₆₇	—	2	2	no	—
228	C₂×C₄×C₆₇	—	2	2	no	—
229	C₂×C₄×C₆₇	—	2	2	no	—
230	C₂×C₄×C₆₇	—	2	4	no	—
231	C₂×C₄×C₆₇	—	4	16	no	—
232	C₂×C₄×C₆₇	—	4	2	no	—
233	C₂×C₄×C₆₇	—	4	2	no	—
234	C₂×C₄×C₆₇	—	4	4	no	—
235	C₂×C₄×C₆₇	—	4	8	no	—
236	C₂×C₄×C₆₇	—	4	8	no	—
237	C₂×C₄×C₆₇	—	8	8	no	—
238	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
239	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
240	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
241	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
242	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
244	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
245	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
246	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
247	C₈×C₆₇	—	2	4	no	—
248	C₈×C₆₇	—	2	2	no	—
249	C₈×C₆₇	—	2	2	no	—
250	C₂²×C₆₇	—	2	4	no	—
251	C₂²×C₆₇	—	4	8	no	—
252	C₂²×C₆₇	—	4	2	no	—
253	C₂²×C₆₇	—	4	2	no	—
254	C₂²×C₆₇	—	4	4	no	—
255	C₂²×C₆₇	—	4	2	no	—
256	C₂²×C₆₇	—	4	4	no	—
257	C₂²×C₆₇	—	4	2	no	—
258	C₂²×C₆₇	—	4	2	no	—
259	C₂²×C₆₇	—	4	4	no	—
260	C₂²×C₆₇	—	8	8	no	—
261	C₂²×C₆₇	—	8	4	no	—
262	C₂²×C₆₇	#548	1	2	yes	—
263	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
264	C₂²×C₆₇	#576	1	2	yes	—
265	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
266	C₄×C₆₇	—	2	4	no	—
267	C₄×C₆₇	—	2	4	no	—
268	C₄×C₆₇	—	2	2	no	—
269	C₄×C₆₇	—	4	2	no	—
270	C₄×C₆₇	—	4	2	no	—
271	C₄×C₆₇	—	4	4	no	—
272	C₄×C₆₇	#548	1	2	yes	—
273	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
274	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
275	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
276	C₂×#3366	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
277	C₂×C₆₇	—	4	4	no	—
278	C₂×C₆₇	—	8	4	no	—
279	C₂×C₆₇	—	8	2	no	—
280	C₂×C₆₇	#3366	1	4	yes	direct product factor
281	C₂×C₆₇	C₂×#548	1	1	yes	characteristic
282	C₂×C₆₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
283	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
284	C₂×#554	C₂×C₆₇	1	1	yes	characteristic, complemented
285	C₂×#550	C₂×C₆₇	1	1	yes	characteristic, complemented
286	#3366	C₂×C₆₇	1	16	yes	direct product factor
287	C₂×#490	C₂×C₆₇	1	2	yes	complemented
288	C₂×#486	C₂×C₆₇	1	1	yes	characteristic, complemented
289	C₂×#478	C₂×C₆₇	1	2	yes	complemented
290	C₂×#476	C₂×C₆₇	1	1	yes	characteristic, complemented
291	C₂×#459	C₂×C₆₇	1	1	yes	characteristic, complemented
292	C₂×#459	C₂×C₆₇	1	2	yes	complemented
293	C₂×#446	C₂×C₆₇	1	1	yes	characteristic, complemented
294	C₂×#442	C₂×C₆₇	1	2	yes	complemented
295	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
296	C₆₇	C₂×#3366	1	1	yes	characteristic, direct product factor, Sylow
297	C₂²×#41	—	2	1	no	—
298	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
299	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
300	#554	—	2	4	no	—
301	#554	C₂²×C₆₇	1	8	yes	complemented
302	#550	—	2	4	no	—
303	#550	C₂²×C₆₇	1	8	yes	complemented
304	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
305	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
306	C₂×#127	C₂²×C₆₇	1	2	yes	—
307	C₂×#125	C₂²×C₆₇	1	2	yes	—
308	C₂×C₂²⋊Q₈	—	2	1	no	—
309	C₂×C₂²⋊Q₈	—	2	2	no	—
310	C₂×#121	—	2	1	no	—
311	C₂×#121	—	2	2	no	—
312	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
313	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
314	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
315	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
316	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
317	C₂²×C₄⋊C₄	—	2	1	no	—
318	#490	C₂²×C₆₇	1	16	yes	complemented
319	#486	C₂²×C₆₇	1	8	yes	complemented
320	#478	C₂²×C₆₇	1	16	yes	complemented
321	#476	C₂²×C₆₇	1	8	yes	complemented
322	#459	C₂²×C₆₇	1	16	yes	complemented
323	#459	C₂²×C₆₇	1	8	yes	complemented
324	#446	C₂²×C₆₇	1	8	yes	complemented
325	#442	C₂²×C₆₇	1	16	yes	complemented
326	C₄×D₈	C₂²×C₆₇	1	8	yes	complemented
327	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
328	C₂×#106	C₂²×C₆₇	1	2	yes	—
329	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
330	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
331	C₂×#102	C₂²×C₆₇	1	2	yes	—
332	C₂×#102	C₂²×C₆₇	1	2	yes	—
333	C₂×#102	C₂²×C₆₇	1	2	yes	—
334	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
335	C₂×#102	C₂²×C₆₇	1	2	yes	—
336	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
337	C₂×#97	C₂²×C₆₇	1	2	yes	—
338	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
339	C₂×#41	—	2	8	no	—
340	C₂×#41	—	4	8	no	—
341	C₂²×Q₈	—	2	1	no	—
342	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
343	C₂²×D₄	—	2	2	no	—
344	C₂²×D₄	—	2	1	no	—
345	C₂²×D₄	—	2	1	no	—
346	C₂²×D₄	—	2	2	no	—
347	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
348	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
349	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
350	C₂³×C₄	—	2	1	no	—
351	C₂³×C₄	—	2	2	no	—
352	C₂³×C₄	—	2	1	no	—
353	C₂³×C₄	—	2	2	no	—
354	C₂×D₈	—	2	4	no	—
355	C₂×D₈	—	2	8	no	—
356	C₂×D₈	C₂³×C₆₇	1	4	yes	—
357	C₂×D₈	C₂³×C₆₇	1	8	yes	—
358	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
359	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
360	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
361	C₄⋊Q₈	—	2	2	no	—
362	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
363	#127	—	2	4	no	—
364	#127	—	2	2	no	—
365	#127	C₂³×C₆₇	1	4	yes	—
366	#127	C₂³×C₆₇	1	8	yes	—
367	#125	—	2	4	no	—
368	#125	C₂³×C₆₇	1	8	yes	—
369	C₂²⋊Q₈	—	2	8	no	—
370	C₂²⋊Q₈	—	2	16	no	—
371	#121	—	2	8	no	—
372	#121	—	2	16	no	—
373	C₄×Q₈	D₄×C₆₇	1	4	yes	—
374	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
375	C₄×D₄	—	2	4	no	—
376	C₄×D₄	—	2	2	no	—
377	C₄×D₄	—	2	4	no	—
378	C₄×D₄	—	2	2	no	—
379	C₄×D₄	C₂³×C₆₇	1	4	yes	—
380	C₄×D₄	C₂³×C₆₇	1	8	yes	—
381	C₄×D₄	C₂³×C₆₇	1	4	yes	—
382	C₄×D₄	C₂³×C₆₇	1	8	yes	—
383	C₂×C₄⋊C₄	—	2	2	no	—
384	C₂×C₄⋊C₄	—	2	1	no	—
385	C₂×C₄⋊C₄	—	2	2	no	—
386	C₂×C₄⋊C₄	—	2	8	no	—
387	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
388	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
389	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
390	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
391	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
392	C₂×C₂²⋊C₄	—	2	1	no	—
393	C₂×C₂²⋊C₄	—	2	2	no	—
394	C₂×C₂²⋊C₄	—	2	1	no	—
395	C₂×C₂²⋊C₄	—	2	2	no	—
396	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
397	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
398	#107	C₂³×C₆₇	1	4	yes	—
399	#106	C₂³×C₆₇	1	8	yes	—
400	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
401	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
402	#102	C₂³×C₆₇	1	4	yes	—
403	#102	C₂³×C₆₇	1	4	yes	—
404	#102	C₂³×C₆₇	1	8	yes	—
405	#102	C₂³×C₆₇	1	8	yes	—
406	#102	C₂³×C₆₇	1	8	yes	—
407	#102	C₂³×C₆₇	1	8	yes	—
408	#97	C₂³×C₆₇	1	8	yes	—
409	C₄×C₈	C₂³×C₆₇	1	4	yes	—
410	C₂⁴	—	2	1	no	—
411	C₂⁴	—	2	2	no	—
412	C₂⁴	—	2	1	no	—
413	#41	—	4	32	no	—
414	C₂×Q₈	—	2	4	no	—
415	C₂×Q₈	—	2	4	no	—
416	C₂×Q₈	—	8	2	no	—
417	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
418	C₂×D₄	—	2	1	no	—
419	C₂×D₄	—	2	2	no	—
420	C₂×D₄	—	2	1	no	—
421	C₂×D₄	—	2	8	no	—
422	C₂×D₄	—	2	4	no	—
423	C₂×D₄	—	2	4	no	—
424	C₂×D₄	—	2	8	no	—
425	C₂×D₄	—	2	2	no	—
426	C₂×D₄	—	2	4	no	—
427	C₂×D₄	—	2	2	no	—
428	C₂×D₄	—	2	4	no	—
429	C₂×D₄	—	2	2	no	—
430	C₂×D₄	—	4	8	no	—
431	C₂×D₄	—	4	4	no	—
432	C₂×D₄	—	8	4	no	—
433	C₂×D₄	—	8	2	no	—
434	C₂×D₄	#41×C₆₇	1	4	yes	complemented
435	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
436	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
437	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	2	no	—
441	C₂²×C₄	—	2	4	no	—
442	C₂²×C₄	—	2	8	no	—
443	C₂²×C₄	—	2	4	no	—
444	C₂²×C₄	—	2	8	no	—
445	C₂²×C₄	—	4	8	no	—
446	C₂²×C₄	—	4	4	no	—
447	C₂²×C₄	—	4	4	no	—
448	C₂²×C₄	—	8	4	no	—
449	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
450	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
451	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
452	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
453	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
454	D₈	—	2	8	no	—
455	D₈	—	2	16	no	—
456	C₂×C₈	—	2	2	no	—
457	C₂×C₈	—	2	4	no	—
458	C₂×C₈	—	2	2	no	—
459	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
460	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
461	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
462	C₄⋊C₄	—	2	8	no	—
463	C₄⋊C₄	—	2	4	no	—
464	C₄⋊C₄	—	2	8	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	2	no	—
468	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
469	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
470	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
471	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
472	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
473	C₂²⋊C₄	—	2	4	no	—
474	C₂²⋊C₄	—	2	8	no	—
475	C₂²⋊C₄	—	2	4	no	—
476	C₂²⋊C₄	—	2	8	no	—
477	C₄²	C₂×D₄×C₆₇	1	4	yes	—
478	C₄²	C₂×D₄×C₆₇	1	2	yes	—
479	C₄²	C₂⁴×C₆₇	1	2	yes	—
480	C₄²	C₂⁴×C₆₇	1	4	yes	—
481	C₂³	—	2	4	no	—
482	C₂³	—	2	2	no	—
483	C₂³	—	2	4	no	—
484	C₂³	—	2	2	no	—
485	C₂³	—	4	2	no	—
486	C₂³	—	4	2	no	—
487	C₂³	—	4	4	no	—
488	C₂³	—	4	2	no	—
489	C₂³	—	4	4	no	—
490	C₂³	—	4	2	no	—
491	C₂³	—	4	2	no	—
492	C₂³	—	4	4	no	—
493	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
494	Q₈	—	2	8	no	—
495	Q₈	—	8	4	no	—
496	D₄	—	2	4	no	—
497	D₄	—	2	2	no	—
498	D₄	—	2	4	no	—
499	D₄	—	2	2	no	—
500	D₄	—	2	2	no	—
501	D₄	—	2	4	no	—
502	D₄	—	4	16	no	—
503	D₄	—	4	4	no	—
504	D₄	—	4	2	no	—
505	D₄	—	4	8	no	—
506	D₄	—	8	8	no	—
507	D₄	—	8	4	no	—
508	D₄	C₂×#41×C₆₇	1	4	yes	complemented
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	4	16	no	—
516	C₂×C₄	—	4	2	no	—
517	C₂×C₄	—	4	2	no	—
518	C₂×C₄	—	4	4	no	—
519	C₂×C₄	—	4	8	no	—
520	C₂×C₄	—	4	8	no	—
521	C₂×C₄	—	8	8	no	—
522	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
523	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
524	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
525	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
526	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
527	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
528	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
529	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
530	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
531	C₈	—	2	2	no	—
532	C₈	—	2	4	no	—
533	C₈	—	2	2	no	—
534	C₂²	—	2	4	no	—
535	C₂²	—	4	2	no	—
536	C₂²	—	4	2	no	—
537	C₂²	—	4	8	no	—
538	C₂²	—	4	4	no	—
539	C₂²	—	4	2	no	—
540	C₂²	—	4	2	no	—
541	C₂²	—	4	4	no	—
542	C₂²	—	4	4	no	—
543	C₂²	—	4	2	no	—
544	C₂²	—	8	8	no	—
545	C₂²	—	8	4	no	—
546	C₂²	#548×C₆₇	1	2	yes	—
547	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
548	C₂²	#576×C₆₇	1	2	yes	—
549	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
550	C₄	—	2	4	no	—
551	C₄	—	2	4	no	—
552	C₄	—	2	2	no	—
553	C₄	—	4	2	no	—
554	C₄	—	4	2	no	—
555	C₄	—	4	4	no	—
556	C₄	#548×C₆₇	1	2	yes	—
557	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	4	yes	—
558	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
559	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
560	C₂	—	4	4	no	—
561	C₂	—	8	4	no	—
562	C₂	—	8	2	no	—
563	C₂	C₆₇×#3366	1	4	yes	direct product factor
564	C₂	C₂×#548×C₆₇	1	1	yes	characteristic
565	C₂	C₂²×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
566	C₂	C₂×#576×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
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×#554×C₆₇	C₂	semidirect
2. C₂×#550×C₆₇	C₂	semidirect
3. C₆₇×#3366	C₂	direct (N × H)
4. C₂×#490×C₆₇	C₂	semidirect
5. C₂×#486×C₆₇	C₂	semidirect
6. C₂×#478×C₆₇	C₂	semidirect
7. C₂×#476×C₆₇	C₂	semidirect
8. C₂×#459×C₆₇	C₂	semidirect
9. C₂×#459×C₆₇	C₂	semidirect
10. C₂×#446×C₆₇	C₂	semidirect
11. C₂×#442×C₆₇	C₂	semidirect
12. C₂×C₄×D₈×C₆₇	C₂	semidirect
17. #554×C₆₇	C₂²	semidirect
19. #550×C₆₇	C₂²	semidirect
34. #490×C₆₇	C₂²	semidirect
35. #486×C₆₇	C₂²	semidirect
36. #478×C₆₇	C₂²	semidirect
37. #476×C₆₇	C₂²	semidirect
38. #459×C₆₇	C₂²	semidirect
39. #459×C₆₇	C₂²	semidirect
40. #446×C₆₇	C₂²	semidirect
41. #442×C₆₇	C₂²	semidirect
42. C₄×D₈×C₆₇	C₂²	semidirect
150. C₂×D₄×C₆₇	#41	semidirect
224. D₄×C₆₇	C₂×#41	semidirect
276. C₂×#3366	C₆₇	direct (N × H)
280. C₂×C₆₇	#3366	direct (N × H)
284. C₂×#554	C₂×C₆₇	semidirect
285. C₂×#550	C₂×C₆₇	semidirect
286. #3366	C₂×C₆₇	direct (N × H)
287. C₂×#490	C₂×C₆₇	semidirect
288. C₂×#486	C₂×C₆₇	semidirect
289. C₂×#478	C₂×C₆₇	semidirect
290. C₂×#476	C₂×C₆₇	semidirect
291. C₂×#459	C₂×C₆₇	semidirect
292. C₂×#459	C₂×C₆₇	semidirect
293. C₂×#446	C₂×C₆₇	semidirect
294. C₂×#442	C₂×C₆₇	semidirect
295. C₂×C₄×D₈	C₂×C₆₇	semidirect
296. C₆₇	C₂×#3366	direct (N × H)
301. #554	C₂²×C₆₇	semidirect
303. #550	C₂²×C₆₇	semidirect
318. #490	C₂²×C₆₇	semidirect
319. #486	C₂²×C₆₇	semidirect
320. #478	C₂²×C₆₇	semidirect
321. #476	C₂²×C₆₇	semidirect
322. #459	C₂²×C₆₇	semidirect
323. #459	C₂²×C₆₇	semidirect
324. #446	C₂²×C₆₇	semidirect
325. #442	C₂²×C₆₇	semidirect
326. C₄×D₈	C₂²×C₆₇	semidirect
434. C₂×D₄	#41×C₆₇	semidirect
508. D₄	C₂×#41×C₆₇	semidirect
563. C₂	C₆₇×#3366	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×#554×C₆₇	2
C₂×#550×C₆₇	2
C₆₇×#3366	2
C₂×#490×C₆₇	2
C₂×#486×C₆₇	2
C₂×#478×C₆₇	2
C₂×#476×C₆₇	2
C₂×#459×C₆₇	2
C₂×#446×C₆₇	2
C₂×#442×C₆₇	2
C₂×C₄×D₈×C₆₇	2
C₂×#3366	67

Subgroups — C2×C67×#3366 #30740555

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3366 #30740555