Subgroups — C₂×C₆₇×#3361 #30740549

Order 17152.

1970 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#559×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#554×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3361	C₂	1	16	yes	maximal, direct product factor
4	C₂×#488×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#486×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#477×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#478×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#463×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×#459×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×D₈×C₆₇	C₂	1	2	yes	maximal, complemented
13	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
15	#559×C₆₇	—	2	4	no	—
16	#559×C₆₇	C₂²	1	8	yes	complemented
17	#554×C₆₇	—	2	4	no	—
18	#554×C₆₇	C₂²	1	8	yes	complemented
19	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#125×C₆₇	C₂²	1	2	yes	—
22	C₂×#124×C₆₇	—	2	1	no	—
23	C₂×#124×C₆₇	C₂²	1	2	yes	—
24	C₂×#123×C₆₇	—	2	2	no	—
25	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
26	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
27	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₄×C₆₇	—	2	1	no	—
30	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
32	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
33	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
34	#488×C₆₇	C₂²	1	16	yes	complemented
35	#486×C₆₇	C₂²	1	8	yes	complemented
36	#478×C₆₇	C₂²	1	16	yes	complemented
37	#477×C₆₇	C₂²	1	8	yes	complemented
38	#463×C₆₇	C₂²	1	16	yes	complemented
39	#459×C₆₇	C₂²	1	8	yes	—
40	C₈×Q₈×C₆₇	C₂²	1	8	yes	complemented
41	C₄×SD₁₆×C₆₇	C₂²	1	8	yes	complemented
42	C₄×D₈×C₆₇	C₂²	1	16	yes	complemented
43	C₂×#107×C₆₇	C₂²	1	2	yes	—
44	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
46	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
47	C₂×#103×C₆₇	C₂²	1	2	yes	—
48	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
49	C₂×#102×C₆₇	C₂²	1	2	yes	—
50	C₂×#102×C₆₇	C₂²	1	2	yes	—
51	C₂×#102×C₆₇	C₂²	1	2	yes	—
52	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
53	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
54	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
55	C₂²×Q₈×C₆₇	—	2	1	no	—
56	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
57	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
58	C₂²×D₄×C₆₇	—	2	1	no	—
59	C₂²×D₄×C₆₇	—	2	1	no	—
60	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
61	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₆₇	—	2	1	no	—
63	C₂³×C₄×C₆₇	—	2	2	no	—
64	C₂³×C₄×C₆₇	—	2	2	no	—
65	C₂×SD₁₆×C₆₇	—	2	4	no	—
66	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
67	C₂×D₈×C₆₇	—	2	8	no	—
68	C₂×D₈×C₆₇	C₂³	1	8	yes	—
69	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
70	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₄⋊Q₈×C₆₇	—	2	2	no	—
73	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
74	#127×C₆₇	—	2	2	no	—
75	#127×C₆₇	C₂³	1	4	yes	—
76	#125×C₆₇	—	2	4	no	—
77	#125×C₆₇	C₂³	1	8	yes	—
78	#124×C₆₇	—	2	8	no	—
79	#124×C₆₇	—	2	4	no	—
80	#124×C₆₇	C₂³	1	8	yes	—
81	#123×C₆₇	—	2	16	no	—
82	C₂²⋊Q₈×C₆₇	—	2	8	no	—
83	C₂²⋊Q₈×C₆₇	—	2	16	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	4	yes	—
88	C₄×D₄×C₆₇	—	2	8	no	—
89	C₄×D₄×C₆₇	—	2	4	no	—
90	C₄×D₄×C₆₇	—	2	2	no	—
91	C₄×D₄×C₆₇	—	2	4	no	—
92	C₄×D₄×C₆₇	C₂³	1	4	yes	—
93	C₄×D₄×C₆₇	C₂³	1	8	yes	—
94	C₄×D₄×C₆₇	C₂³	1	8	yes	—
95	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
96	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
97	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
98	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₂×C₄⋊C₄×C₆₇	—	2	8	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	1	yes	characteristic
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
105	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
106	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
107	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
108	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
109	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
110	C₂×C₄²×C₆₇	—	2	1	no	—
111	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
112	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
113	#107×C₆₇	C₂³	1	8	yes	—
114	#106×C₆₇	C₂³	1	4	yes	—
115	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
116	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
117	#103×C₆₇	C₂³	1	8	yes	—
118	#103×C₆₇	C₂³	1	4	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	C₄×C₈×C₆₇	C₂³	1	4	yes	—
124	C₄×C₈×C₆₇	C₂³	1	8	yes	—
125	C₂⁴×C₆₇	—	2	1	no	—
126	C₂⁴×C₆₇	—	2	2	no	—
127	C₂×Q₈×C₆₇	—	2	4	no	—
128	C₂×Q₈×C₆₇	—	2	1	no	—
129	C₂×Q₈×C₆₇	—	2	4	no	—
130	C₂×Q₈×C₆₇	—	2	2	no	—
131	C₂×Q₈×C₆₇	—	2	2	no	—
132	C₂×Q₈×C₆₇	—	8	2	no	—
133	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
134	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
135	C₂×D₄×C₆₇	—	2	1	no	—
136	C₂×D₄×C₆₇	—	2	2	no	—
137	C₂×D₄×C₆₇	—	2	4	no	—
138	C₂×D₄×C₆₇	—	2	4	no	—
139	C₂×D₄×C₆₇	—	2	4	no	—
140	C₂×D₄×C₆₇	—	2	2	no	—
141	C₂×D₄×C₆₇	—	2	4	no	—
142	C₂×D₄×C₆₇	—	4	4	no	—
143	C₂×D₄×C₆₇	—	8	2	no	—
144	C₂×D₄×C₆₇	#41	1	4	yes	—
145	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
146	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₆₇	—	2	1	no	—
148	C₂²×C₄×C₆₇	—	2	1	no	—
149	C₂²×C₄×C₆₇	—	2	1	no	—
150	C₂²×C₄×C₆₇	—	2	2	no	—
151	C₂²×C₄×C₆₇	—	2	4	no	—
152	C₂²×C₄×C₆₇	—	2	8	no	—
153	C₂²×C₄×C₆₇	—	2	8	no	—
154	C₂²×C₄×C₆₇	—	4	4	no	—
155	C₂²×C₄×C₆₇	—	8	4	no	—
156	C₂²×C₄×C₆₇	—	8	4	no	—
157	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
158	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
162	SD₁₆×C₆₇	—	2	8	no	—
163	D₈×C₆₇	—	2	16	no	—
164	C₂×C₈×C₆₇	—	2	4	no	—
165	C₂×C₈×C₆₇	—	2	2	no	—
166	C₂×C₈×C₆₇	—	2	2	no	—
167	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
168	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
169	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
170	C₄⋊C₄×C₆₇	—	2	8	no	—
171	C₄⋊C₄×C₆₇	—	2	4	no	—
172	C₄⋊C₄×C₆₇	—	2	4	no	—
173	C₄⋊C₄×C₆₇	—	2	8	no	—
174	C₄⋊C₄×C₆₇	—	2	1	no	—
175	C₄⋊C₄×C₆₇	—	2	1	no	—
176	C₄⋊C₄×C₆₇	—	2	2	no	—
177	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
178	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
179	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
180	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
181	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
182	C₂²⋊C₄×C₆₇	—	2	8	no	—
183	C₂²⋊C₄×C₆₇	—	2	4	no	—
184	C₂²⋊C₄×C₆₇	—	2	8	no	—
185	C₂²⋊C₄×C₆₇	—	2	8	no	—
186	C₂²⋊C₄×C₆₇	—	2	8	no	—
187	C₄²×C₆₇	—	2	4	no	—
188	C₄²×C₆₇	C₂×D₄	1	4	yes	—
189	C₄²×C₆₇	C₂×D₄	1	2	yes	—
190	C₄²×C₆₇	C₂⁴	1	4	yes	—
191	C₄²×C₆₇	C₂⁴	1	2	yes	—
192	C₂³×C₆₇	—	2	4	no	—
193	C₂³×C₆₇	—	2	2	no	—
194	C₂³×C₆₇	—	2	4	no	—
195	C₂³×C₆₇	—	4	2	no	—
196	C₂³×C₆₇	—	4	2	no	—
197	C₂³×C₆₇	—	4	4	no	—
198	C₂³×C₆₇	—	4	4	no	—
199	C₂³×C₆₇	—	4	4	no	—
200	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
201	Q₈×C₆₇	—	2	8	no	—
202	Q₈×C₆₇	—	2	2	no	—
203	Q₈×C₆₇	—	2	2	no	—
204	Q₈×C₆₇	—	4	2	no	—
205	Q₈×C₆₇	—	8	4	no	—
206	D₄×C₆₇	—	2	4	no	—
207	D₄×C₆₇	—	2	4	no	—
208	D₄×C₆₇	—	2	2	no	—
209	D₄×C₆₇	—	2	4	no	—
210	D₄×C₆₇	—	4	4	no	—
211	D₄×C₆₇	—	4	8	no	—
212	D₄×C₆₇	—	8	4	no	—
213	D₄×C₆₇	C₂×#41	1	4	yes	—
214	C₂×C₄×C₆₇	—	2	4	no	—
215	C₂×C₄×C₆₇	—	2	4	no	—
216	C₂×C₄×C₆₇	—	2	2	no	—
217	C₂×C₄×C₆₇	—	2	2	no	—
218	C₂×C₄×C₆₇	—	2	2	no	—
219	C₂×C₄×C₆₇	—	2	2	no	—
220	C₂×C₄×C₆₇	—	2	4	no	—
221	C₂×C₄×C₆₇	—	4	2	no	—
222	C₂×C₄×C₆₇	—	4	2	no	—
223	C₂×C₄×C₆₇	—	4	2	no	—
224	C₂×C₄×C₆₇	—	4	4	no	—
225	C₂×C₄×C₆₇	—	4	8	no	—
226	C₂×C₄×C₆₇	—	8	8	no	—
227	C₂×C₄×C₆₇	—	8	8	no	—
228	C₂×C₄×C₆₇	#135	1	4	yes	—
229	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
230	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
231	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
232	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
233	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
234	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
235	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
236	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
237	C₈×C₆₇	—	2	4	no	—
238	C₈×C₆₇	—	2	2	no	—
239	C₈×C₆₇	—	2	2	no	—
240	C₂²×C₆₇	—	2	4	no	—
241	C₂²×C₆₇	—	4	8	no	—
242	C₂²×C₆₇	—	4	2	no	—
243	C₂²×C₆₇	—	4	2	no	—
244	C₂²×C₆₇	—	4	4	no	—
245	C₂²×C₆₇	—	4	4	no	—
246	C₂²×C₆₇	—	4	4	no	—
247	C₂²×C₆₇	—	8	8	no	—
248	C₂²×C₆₇	#548	1	2	yes	—
249	C₂²×C₆₇	C₂×#135	1	2	yes	—
250	C₂²×C₆₇	#576	1	2	yes	—
251	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
252	C₄×C₆₇	—	2	4	no	—
253	C₄×C₆₇	—	2	4	no	—
254	C₄×C₆₇	—	2	2	no	—
255	C₄×C₆₇	—	4	2	no	—
256	C₄×C₆₇	—	4	2	no	—
257	C₄×C₆₇	—	4	2	no	—
258	C₄×C₆₇	—	4	4	no	—
259	C₄×C₆₇	#548	1	2	yes	—
260	C₄×C₆₇	C₂×#135	1	4	yes	—
261	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
262	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
263	C₂×#3361	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
264	C₂×C₆₇	—	4	4	no	—
265	C₂×C₆₇	—	8	4	no	—
266	C₂×C₆₇	#3361	1	4	yes	direct product factor
267	C₂×C₆₇	C₂×#548	1	1	yes	characteristic
268	C₂×C₆₇	C₂²×#135	1	1	yes	characteristic
269	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
270	C₂×#559	C₂×C₆₇	1	1	yes	characteristic, complemented
271	C₂×#554	C₂×C₆₇	1	1	yes	characteristic, complemented
272	#3361	C₂×C₆₇	1	16	yes	direct product factor
273	C₂×#488	C₂×C₆₇	1	2	yes	complemented
274	C₂×#486	C₂×C₆₇	1	1	yes	characteristic, complemented
275	C₂×#477	C₂×C₆₇	1	1	yes	characteristic, complemented
276	C₂×#478	C₂×C₆₇	1	2	yes	complemented
277	C₂×#463	C₂×C₆₇	1	2	yes	complemented
278	C₂×#459	C₂×C₆₇	1	1	yes	characteristic
279	C₂×C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
280	C₂×C₄×SD₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
281	C₂×C₄×D₈	C₂×C₆₇	1	2	yes	complemented
282	C₆₇	C₂×#3361	1	1	yes	characteristic, direct product factor, Sylow
283	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
284	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
285	#559	—	2	4	no	—
286	#559	C₂²×C₆₇	1	8	yes	complemented
287	#554	—	2	4	no	—
288	#554	C₂²×C₆₇	1	8	yes	complemented
289	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
290	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
291	C₂×#125	C₂²×C₆₇	1	2	yes	—
292	C₂×#124	—	2	1	no	—
293	C₂×#124	C₂²×C₆₇	1	2	yes	—
294	C₂×#123	—	2	2	no	—
295	C₂×C₂²⋊Q₈	—	2	1	no	—
296	C₂×C₂²⋊Q₈	—	2	2	no	—
297	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
298	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
299	C₂×C₄×D₄	—	2	1	no	—
300	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
301	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
302	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
303	C₂²×C₄⋊C₄	—	2	1	no	—
304	#488	C₂²×C₆₇	1	16	yes	complemented
305	#486	C₂²×C₆₇	1	8	yes	complemented
306	#478	C₂²×C₆₇	1	16	yes	complemented
307	#477	C₂²×C₆₇	1	8	yes	complemented
308	#463	C₂²×C₆₇	1	16	yes	complemented
309	#459	C₂²×C₆₇	1	8	yes	—
310	C₈×Q₈	C₂²×C₆₇	1	8	yes	complemented
311	C₄×SD₁₆	C₂²×C₆₇	1	8	yes	complemented
312	C₄×D₈	C₂²×C₆₇	1	16	yes	complemented
313	C₂×#107	C₂²×C₆₇	1	2	yes	—
314	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
315	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
316	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
317	C₂×#103	C₂²×C₆₇	1	2	yes	—
318	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
319	C₂×#102	C₂²×C₆₇	1	2	yes	—
320	C₂×#102	C₂²×C₆₇	1	2	yes	—
321	C₂×#102	C₂²×C₆₇	1	2	yes	—
322	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
323	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
324	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
325	C₂²×Q₈	—	2	1	no	—
326	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
327	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
328	C₂²×D₄	—	2	1	no	—
329	C₂²×D₄	—	2	1	no	—
330	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
331	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
332	C₂³×C₄	—	2	1	no	—
333	C₂³×C₄	—	2	2	no	—
334	C₂³×C₄	—	2	2	no	—
335	C₂×SD₁₆	—	2	4	no	—
336	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
337	C₂×D₈	—	2	8	no	—
338	C₂×D₈	C₂³×C₆₇	1	8	yes	—
339	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
340	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
341	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
342	C₄⋊Q₈	—	2	2	no	—
343	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
344	#127	—	2	2	no	—
345	#127	C₂³×C₆₇	1	4	yes	—
346	#125	—	2	4	no	—
347	#125	C₂³×C₆₇	1	8	yes	—
348	#124	—	2	8	no	—
349	#124	—	2	4	no	—
350	#124	C₂³×C₆₇	1	8	yes	—
351	#123	—	2	16	no	—
352	C₂²⋊Q₈	—	2	8	no	—
353	C₂²⋊Q₈	—	2	16	no	—
354	C₄×Q₈	—	2	2	no	—
355	C₄×Q₈	D₄×C₆₇	1	4	yes	—
356	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
357	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
358	C₄×D₄	—	2	8	no	—
359	C₄×D₄	—	2	4	no	—
360	C₄×D₄	—	2	2	no	—
361	C₄×D₄	—	2	4	no	—
362	C₄×D₄	C₂³×C₆₇	1	8	yes	—
363	C₄×D₄	C₂³×C₆₇	1	4	yes	—
364	C₄×D₄	C₂³×C₆₇	1	8	yes	—
365	C₂×C₄⋊C₄	—	2	2	no	—
366	C₂×C₄⋊C₄	—	2	1	no	—
367	C₂×C₄⋊C₄	—	2	1	no	—
368	C₂×C₄⋊C₄	—	2	2	no	—
369	C₂×C₄⋊C₄	—	2	8	no	—
370	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
371	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
372	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
373	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
374	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
375	C₂×C₂²⋊C₄	—	2	2	no	—
376	C₂×C₂²⋊C₄	—	2	1	no	—
377	C₂×C₂²⋊C₄	—	2	2	no	—
378	C₂×C₂²⋊C₄	—	2	2	no	—
379	C₂×C₂²⋊C₄	—	2	2	no	—
380	C₂×C₄²	—	2	1	no	—
381	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
382	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
383	#107	C₂³×C₆₇	1	8	yes	—
384	#106	C₂³×C₆₇	1	4	yes	—
385	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
386	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
387	#103	C₂³×C₆₇	1	4	yes	—
388	#103	C₂³×C₆₇	1	8	yes	—
389	#102	C₂³×C₆₇	1	4	yes	—
390	#102	C₂³×C₆₇	1	8	yes	—
391	#102	C₂³×C₆₇	1	8	yes	—
392	#102	C₂³×C₆₇	1	8	yes	—
393	C₄×C₈	C₂³×C₆₇	1	8	yes	—
394	C₄×C₈	C₂³×C₆₇	1	4	yes	—
395	C₂⁴	—	2	1	no	—
396	C₂⁴	—	2	2	no	—
397	C₂×Q₈	—	2	4	no	—
398	C₂×Q₈	—	2	1	no	—
399	C₂×Q₈	—	2	4	no	—
400	C₂×Q₈	—	2	2	no	—
401	C₂×Q₈	—	2	2	no	—
402	C₂×Q₈	—	8	2	no	—
403	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
404	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
405	C₂×D₄	—	2	1	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	4	no	—
408	C₂×D₄	—	2	4	no	—
409	C₂×D₄	—	2	4	no	—
410	C₂×D₄	—	2	2	no	—
411	C₂×D₄	—	2	4	no	—
412	C₂×D₄	—	4	4	no	—
413	C₂×D₄	—	8	2	no	—
414	C₂×D₄	#41×C₆₇	1	4	yes	—
415	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
416	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
417	C₂²×C₄	—	2	1	no	—
418	C₂²×C₄	—	2	1	no	—
419	C₂²×C₄	—	2	1	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	8	no	—
423	C₂²×C₄	—	2	8	no	—
424	C₂²×C₄	—	4	4	no	—
425	C₂²×C₄	—	8	4	no	—
426	C₂²×C₄	—	8	4	no	—
427	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
428	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
429	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
430	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
431	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
432	SD₁₆	—	2	8	no	—
433	D₈	—	2	16	no	—
434	C₂×C₈	—	2	4	no	—
435	C₂×C₈	—	2	2	no	—
436	C₂×C₈	—	2	2	no	—
437	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
438	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
439	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
440	C₄⋊C₄	—	2	8	no	—
441	C₄⋊C₄	—	2	4	no	—
442	C₄⋊C₄	—	2	4	no	—
443	C₄⋊C₄	—	2	8	no	—
444	C₄⋊C₄	—	2	1	no	—
445	C₄⋊C₄	—	2	1	no	—
446	C₄⋊C₄	—	2	2	no	—
447	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
448	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
449	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
450	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
451	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
452	C₂²⋊C₄	—	2	8	no	—
453	C₂²⋊C₄	—	2	4	no	—
454	C₂²⋊C₄	—	2	8	no	—
455	C₂²⋊C₄	—	2	8	no	—
456	C₂²⋊C₄	—	2	8	no	—
457	C₄²	—	2	4	no	—
458	C₄²	C₂×D₄×C₆₇	1	4	yes	—
459	C₄²	C₂×D₄×C₆₇	1	2	yes	—
460	C₄²	C₂⁴×C₆₇	1	4	yes	—
461	C₄²	C₂⁴×C₆₇	1	2	yes	—
462	C₂³	—	2	4	no	—
463	C₂³	—	2	2	no	—
464	C₂³	—	2	4	no	—
465	C₂³	—	4	2	no	—
466	C₂³	—	4	2	no	—
467	C₂³	—	4	4	no	—
468	C₂³	—	4	4	no	—
469	C₂³	—	4	4	no	—
470	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
471	Q₈	—	2	8	no	—
472	Q₈	—	2	2	no	—
473	Q₈	—	2	2	no	—
474	Q₈	—	4	2	no	—
475	Q₈	—	8	4	no	—
476	D₄	—	2	4	no	—
477	D₄	—	2	4	no	—
478	D₄	—	2	2	no	—
479	D₄	—	2	4	no	—
480	D₄	—	4	4	no	—
481	D₄	—	4	8	no	—
482	D₄	—	8	4	no	—
483	D₄	C₂×#41×C₆₇	1	4	yes	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	4	no	—
491	C₂×C₄	—	4	2	no	—
492	C₂×C₄	—	4	2	no	—
493	C₂×C₄	—	4	2	no	—
494	C₂×C₄	—	4	4	no	—
495	C₂×C₄	—	4	8	no	—
496	C₂×C₄	—	8	8	no	—
497	C₂×C₄	—	8	8	no	—
498	C₂×C₄	#135×C₆₇	1	4	yes	—
499	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
500	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
501	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
502	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
503	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
504	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
505	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
506	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
507	C₈	—	2	4	no	—
508	C₈	—	2	2	no	—
509	C₈	—	2	2	no	—
510	C₂²	—	2	4	no	—
511	C₂²	—	4	2	no	—
512	C₂²	—	4	2	no	—
513	C₂²	—	4	8	no	—
514	C₂²	—	4	4	no	—
515	C₂²	—	4	4	no	—
516	C₂²	—	4	4	no	—
517	C₂²	—	8	8	no	—
518	C₂²	#548×C₆₇	1	2	yes	—
519	C₂²	C₂×#135×C₆₇	1	2	yes	—
520	C₂²	#576×C₆₇	1	2	yes	—
521	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
522	C₄	—	2	4	no	—
523	C₄	—	2	4	no	—
524	C₄	—	2	2	no	—
525	C₄	—	4	2	no	—
526	C₄	—	4	2	no	—
527	C₄	—	4	4	no	—
528	C₄	—	4	2	no	—
529	C₄	#548×C₆₇	1	2	yes	—
530	C₄	C₂×#135×C₆₇	1	4	yes	—
531	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
532	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
533	C₂	—	4	4	no	—
534	C₂	—	8	4	no	—
535	C₂	C₆₇×#3361	1	4	yes	direct product factor
536	C₂	C₂×#548×C₆₇	1	1	yes	characteristic
537	C₂	C₂²×#135×C₆₇	1	1	yes	characteristic
538	C₂	C₂×#576×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂×#459×C₆₇	C₂
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×#559×C₆₇	C₂	semidirect
2. C₂×#554×C₆₇	C₂	semidirect
3. C₆₇×#3361	C₂	direct (N × H)
4. C₂×#488×C₆₇	C₂	semidirect
5. C₂×#486×C₆₇	C₂	semidirect
6. C₂×#477×C₆₇	C₂	semidirect
7. C₂×#478×C₆₇	C₂	semidirect
8. C₂×#463×C₆₇	C₂	semidirect
10. C₂×C₈×Q₈×C₆₇	C₂	semidirect
11. C₂×C₄×SD₁₆×C₆₇	C₂	semidirect
12. C₂×C₄×D₈×C₆₇	C₂	semidirect
16. #559×C₆₇	C₂²	semidirect
18. #554×C₆₇	C₂²	semidirect
34. #488×C₆₇	C₂²	semidirect
35. #486×C₆₇	C₂²	semidirect
36. #478×C₆₇	C₂²	semidirect
37. #477×C₆₇	C₂²	semidirect
38. #463×C₆₇	C₂²	semidirect
40. C₈×Q₈×C₆₇	C₂²	semidirect
41. C₄×SD₁₆×C₆₇	C₂²	semidirect
42. C₄×D₈×C₆₇	C₂²	semidirect
263. C₂×#3361	C₆₇	direct (N × H)
266. C₂×C₆₇	#3361	direct (N × H)
270. C₂×#559	C₂×C₆₇	semidirect
271. C₂×#554	C₂×C₆₇	semidirect
272. #3361	C₂×C₆₇	direct (N × H)
273. C₂×#488	C₂×C₆₇	semidirect
274. C₂×#486	C₂×C₆₇	semidirect
275. C₂×#477	C₂×C₆₇	semidirect
276. C₂×#478	C₂×C₆₇	semidirect
277. C₂×#463	C₂×C₆₇	semidirect
279. C₂×C₈×Q₈	C₂×C₆₇	semidirect
280. C₂×C₄×SD₁₆	C₂×C₆₇	semidirect
281. C₂×C₄×D₈	C₂×C₆₇	semidirect
282. C₆₇	C₂×#3361	direct (N × H)
286. #559	C₂²×C₆₇	semidirect
288. #554	C₂²×C₆₇	semidirect
304. #488	C₂²×C₆₇	semidirect
305. #486	C₂²×C₆₇	semidirect
306. #478	C₂²×C₆₇	semidirect
307. #477	C₂²×C₆₇	semidirect
308. #463	C₂²×C₆₇	semidirect
310. C₈×Q₈	C₂²×C₆₇	semidirect
311. C₄×SD₁₆	C₂²×C₆₇	semidirect
312. C₄×D₈	C₂²×C₆₇	semidirect
535. C₂	C₆₇×#3361	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×#559×C₆₇	2
C₂×#554×C₆₇	2
C₆₇×#3361	2
C₂×#488×C₆₇	2
C₂×#486×C₆₇	2
C₂×#477×C₆₇	2
C₂×#478×C₆₇	2
C₂×#463×C₆₇	2
C₂×#459×C₆₇	2
C₂×C₈×Q₈×C₆₇	2
C₂×C₄×SD₁₆×C₆₇	2
C₂×C₄×D₈×C₆₇	2
C₂×#3361	67

Subgroups — C2×C67×#3361 #30740549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3361 #30740549