Subgroups — C₂×C₆₇×#3168 #30739921

Order 17152.

2070 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#529×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3168	C₂	1	16	yes	maximal, direct product factor
3	C₂×#499×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#495×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#488×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#467×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#466×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#465×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×SD₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#529×C₆₇	—	2	8	no	—
19	#529×C₆₇	C₂²	1	16	yes	complemented
20	C₂×#126×C₆₇	—	2	2	no	—
21	C₂×#125×C₆₇	C₂²	1	2	yes	—
22	C₂×#124×C₆₇	C₂²	1	2	yes	—
23	C₂×#123×C₆₇	—	2	2	no	—
24	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
25	C₂×#121×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₆₇	—	2	2	no	—
28	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
29	C₂×#117×C₆₇	—	2	2	no	—
30	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
31	#499×C₆₇	C₂²	1	8	yes	complemented
32	#495×C₆₇	C₂²	1	8	yes	complemented
33	#488×C₆₇	C₂²	1	16	yes	complemented
34	#467×C₆₇	C₂²	1	8	yes	complemented
35	#466×C₆₇	C₂²	1	8	yes	—
36	#465×C₆₇	C₂²	1	16	yes	complemented
37	C₄×Q₁₆×C₆₇	C₂²	1	8	yes	complemented
38	C₄×SD₁₆×C₆₇	C₂²	1	16	yes	complemented
39	C₄×D₈×C₆₇	C₂²	1	8	yes	complemented
40	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×#106×C₆₇	C₂²	1	2	yes	—
43	C₂×#103×C₆₇	C₂²	1	2	yes	—
44	C₂×#103×C₆₇	C₂²	1	2	yes	—
45	C₂×#102×C₆₇	C₂²	1	2	yes	—
46	C₂×#102×C₆₇	C₂²	1	2	yes	—
47	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
48	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
49	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
50	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
51	C₂×C₄×C₈×C₆₇	C₂²	1	8	yes	complemented
52	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
53	C₂²×D₄×C₆₇	—	2	2	no	—
54	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
55	C₂³×C₄×C₆₇	—	2	2	no	—
56	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
57	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
58	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
59	C₂×Q₁₆×C₆₇	—	2	4	no	—
60	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
61	C₂×SD₁₆×C₆₇	—	2	8	no	—
62	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
63	C₂×D₈×C₆₇	—	2	4	no	—
64	C₂×D₈×C₆₇	C₂³	1	4	yes	—
65	C₂²×C₈×C₆₇	—	2	4	no	—
66	C₂²×C₈×C₆₇	—	2	4	no	—
67	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
68	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
69	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
73	#126×C₆₇	—	2	16	no	—
74	#125×C₆₇	—	2	4	no	—
75	#125×C₆₇	C₂³	1	8	yes	—
76	#124×C₆₇	—	2	4	no	—
77	#124×C₆₇	C₂³	1	8	yes	—
78	#123×C₆₇	—	2	16	no	—
79	C₂²⋊Q₈×C₆₇	—	2	4	no	—
80	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
81	#121×C₆₇	—	2	4	no	—
82	#121×C₆₇	C₂³	1	8	yes	—
83	C₄×Q₈×C₆₇	—	2	4	no	—
84	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
85	C₄×D₄×C₆₇	—	2	16	no	—
86	C₄×D₄×C₆₇	—	2	4	no	—
87	C₄×D₄×C₆₇	C₂³	1	8	yes	—
88	#117×C₆₇	—	2	16	no	—
89	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
90	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
91	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
92	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
93	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
95	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
96	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
97	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
98	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
99	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
100	C₂×C₄²×C₆₇	—	2	2	no	—
101	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
102	C₂×C₄²×C₆₇	D₄	1	4	yes	—
103	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
104	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
106	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
107	#107×C₆₇	C₂³	1	4	yes	—
108	#107×C₆₇	C₂³	1	4	yes	—
109	#106×C₆₇	C₂³	1	8	yes	—
110	#103×C₆₇	C₂³	1	8	yes	—
111	#103×C₆₇	C₂³	1	8	yes	—
112	#102×C₆₇	C₂³	1	8	yes	—
113	#102×C₆₇	C₂³	1	8	yes	—
114	C₄×C₈×C₆₇	C₂³	1	4	yes	—
115	C₄×C₈×C₆₇	C₂³	1	4	yes	—
116	C₄×C₈×C₆₇	C₂³	1	4	yes	—
117	C₄×C₈×C₆₇	C₂³	1	4	yes	—
118	C₂⁴×C₆₇	—	2	2	no	—
119	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂×Q₈×C₆₇	—	2	2	no	—
121	C₂×Q₈×C₆₇	—	2	4	no	—
122	C₂×Q₈×C₆₇	—	2	4	no	—
123	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
124	C₂×D₄×C₆₇	—	2	8	no	—
125	C₂×D₄×C₆₇	—	2	2	no	—
126	C₂×D₄×C₆₇	—	2	4	no	—
127	C₂×D₄×C₆₇	—	2	4	no	—
128	C₂×D₄×C₆₇	—	4	8	no	—
129	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₆₇	—	2	4	no	—
131	C₂²×C₄×C₆₇	—	2	4	no	—
132	C₂²×C₄×C₆₇	—	2	2	no	—
133	C₂²×C₄×C₆₇	—	2	2	no	—
134	C₂²×C₄×C₆₇	—	2	2	no	—
135	C₂²×C₄×C₆₇	—	2	2	no	—
136	C₂²×C₄×C₆₇	—	2	2	no	—
137	C₂²×C₄×C₆₇	—	2	8	no	—
138	C₂²×C₄×C₆₇	—	4	8	no	—
139	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
141	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
145	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
146	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
148	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
149	Q₁₆×C₆₇	—	2	8	no	—
150	SD₁₆×C₆₇	—	2	16	no	—
151	D₈×C₆₇	—	2	8	no	—
152	C₂×C₈×C₆₇	—	2	2	no	—
153	C₂×C₈×C₆₇	—	2	2	no	—
154	C₂×C₈×C₆₇	—	2	8	no	—
155	C₂×C₈×C₆₇	—	2	8	no	—
156	C₂×C₈×C₆₇	#41	1	4	yes	—
157	C₂×C₈×C₆₇	#41	1	4	yes	—
158	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
159	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
160	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
161	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
162	C₄⋊C₄×C₆₇	—	2	8	no	—
163	C₄⋊C₄×C₆₇	—	2	8	no	—
164	C₄⋊C₄×C₆₇	—	2	8	no	—
165	C₄⋊C₄×C₆₇	—	2	8	no	—
166	C₄⋊C₄×C₆₇	—	2	2	no	—
167	C₄⋊C₄×C₆₇	—	2	2	no	—
168	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
169	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
170	C₂²⋊C₄×C₆₇	—	2	8	no	—
171	C₂²⋊C₄×C₆₇	—	2	8	no	—
172	C₂²⋊C₄×C₆₇	—	2	8	no	—
173	C₂²⋊C₄×C₆₇	—	2	8	no	—
174	C₂²⋊C₄×C₆₇	—	2	8	no	—
175	C₄²×C₆₇	—	2	8	no	—
176	C₄²×C₆₇	C₂×D₄	1	4	yes	—
177	C₄²×C₆₇	C₂×D₄	1	2	yes	—
178	C₄²×C₆₇	C₂×D₄	1	2	yes	—
179	C₄²×C₆₇	C₂×D₄	1	4	yes	—
180	C₄²×C₆₇	C₂⁴	1	2	yes	—
181	C₄²×C₆₇	C₂⁴	1	2	yes	—
182	C₂³×C₆₇	—	2	2	no	—
183	C₂³×C₆₇	—	2	2	no	—
184	C₂³×C₆₇	—	2	4	no	—
185	C₂³×C₆₇	—	4	4	no	—
186	C₂³×C₆₇	—	4	4	no	—
187	C₂³×C₆₇	—	4	4	no	—
188	C₂³×C₆₇	#135	1	4	yes	—
189	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
190	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
191	Q₈×C₆₇	—	2	4	no	—
192	Q₈×C₆₇	—	2	4	no	—
193	Q₈×C₆₇	—	4	4	no	—
194	D₄×C₆₇	—	2	4	no	—
195	D₄×C₆₇	—	2	4	no	—
196	D₄×C₆₇	—	4	16	no	—
197	D₄×C₆₇	—	4	4	no	—
198	C₂×C₄×C₆₇	—	2	2	no	—
199	C₂×C₄×C₆₇	—	2	8	no	—
200	C₂×C₄×C₆₇	—	2	2	no	—
201	C₂×C₄×C₆₇	—	2	2	no	—
202	C₂×C₄×C₆₇	—	2	2	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	8	no	—
205	C₂×C₄×C₆₇	—	2	2	no	—
206	C₂×C₄×C₆₇	—	2	2	no	—
207	C₂×C₄×C₆₇	—	2	4	no	—
208	C₂×C₄×C₆₇	—	2	4	no	—
209	C₂×C₄×C₆₇	—	2	4	no	—
210	C₂×C₄×C₆₇	—	4	4	no	—
211	C₂×C₄×C₆₇	—	4	4	no	—
212	C₂×C₄×C₆₇	—	4	4	no	—
213	C₂×C₄×C₆₇	—	4	16	no	—
214	C₂×C₄×C₆₇	#135	1	4	yes	—
215	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
216	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
217	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
218	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
219	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
221	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
222	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
223	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
224	C₈×C₆₇	—	2	2	no	—
225	C₈×C₆₇	—	2	2	no	—
226	C₈×C₆₇	C₂×#41	1	4	yes	—
227	C₈×C₆₇	C₂×#41	1	4	yes	—
228	C₂²×C₆₇	—	2	8	no	—
229	C₂²×C₆₇	—	2	2	no	—
230	C₂²×C₆₇	—	2	2	no	—
231	C₂²×C₆₇	—	4	4	no	—
232	C₂²×C₆₇	—	4	4	no	—
233	C₂²×C₆₇	—	4	4	no	—
234	C₂²×C₆₇	—	8	8	no	—
235	C₂²×C₆₇	#532	1	2	yes	—
236	C₂²×C₆₇	C₂×#135	1	4	yes	—
237	C₂²×C₆₇	C₂×#135	1	2	yes	—
238	C₂²×C₆₇	C₂×#135	1	2	yes	—
239	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
240	C₄×C₆₇	—	2	2	no	—
241	C₄×C₆₇	—	2	2	no	—
242	C₄×C₆₇	—	2	2	no	—
243	C₄×C₆₇	—	2	2	no	—
244	C₄×C₆₇	—	4	4	no	—
245	C₄×C₆₇	—	4	4	no	—
246	C₄×C₆₇	—	4	4	no	—
247	C₄×C₆₇	#532	1	2	yes	—
248	C₄×C₆₇	C₂×#135	1	4	yes	—
249	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
250	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
251	C₂×#3168	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
252	C₂×C₆₇	—	2	4	no	—
253	C₂×C₆₇	—	8	4	no	—
254	C₂×C₆₇	#3168	1	4	yes	direct product factor
255	C₂×C₆₇	C₂×#532	1	1	yes	characteristic
256	C₂×C₆₇	C₂²×#135	1	1	yes	characteristic
257	C₂×C₆₇	C₂²×#135	1	1	yes	characteristic
258	C₂×#529	C₂×C₆₇	1	2	yes	complemented
259	#3168	C₂×C₆₇	1	16	yes	direct product factor
260	C₂×#499	C₂×C₆₇	1	1	yes	characteristic, complemented
261	C₂×#495	C₂×C₆₇	1	1	yes	characteristic, complemented
262	C₂×#488	C₂×C₆₇	1	2	yes	complemented
263	C₂×#467	C₂×C₆₇	1	1	yes	characteristic, complemented
264	C₂×#466	C₂×C₆₇	1	1	yes	characteristic
265	C₂×#465	C₂×C₆₇	1	2	yes	complemented
266	C₂×C₄×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
267	C₂×C₄×SD₁₆	C₂×C₆₇	1	2	yes	complemented
268	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
269	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
270	C₆₇	C₂×#3168	1	1	yes	characteristic, direct product factor, Sylow
271	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
272	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
273	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
274	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
275	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
276	#529	—	2	8	no	—
277	#529	C₂²×C₆₇	1	16	yes	complemented
278	C₂×#126	—	2	2	no	—
279	C₂×#125	C₂²×C₆₇	1	2	yes	—
280	C₂×#124	C₂²×C₆₇	1	2	yes	—
281	C₂×#123	—	2	2	no	—
282	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
283	C₂×#121	C₂²×C₆₇	1	2	yes	—
284	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
285	C₂×C₄×D₄	—	2	2	no	—
286	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
287	C₂×#117	—	2	2	no	—
288	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
289	#499	C₂²×C₆₇	1	8	yes	complemented
290	#495	C₂²×C₆₇	1	8	yes	complemented
291	#488	C₂²×C₆₇	1	16	yes	complemented
292	#467	C₂²×C₆₇	1	8	yes	complemented
293	#466	C₂²×C₆₇	1	8	yes	—
294	#465	C₂²×C₆₇	1	16	yes	complemented
295	C₄×Q₁₆	C₂²×C₆₇	1	8	yes	complemented
296	C₄×SD₁₆	C₂²×C₆₇	1	16	yes	complemented
297	C₄×D₈	C₂²×C₆₇	1	8	yes	complemented
298	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
299	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
300	C₂×#106	C₂²×C₆₇	1	2	yes	—
301	C₂×#103	C₂²×C₆₇	1	2	yes	—
302	C₂×#103	C₂²×C₆₇	1	2	yes	—
303	C₂×#102	C₂²×C₆₇	1	2	yes	—
304	C₂×#102	C₂²×C₆₇	1	2	yes	—
305	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
306	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
307	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
308	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
309	C₂×C₄×C₈	C₂²×C₆₇	1	8	yes	complemented
310	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
311	C₂²×D₄	—	2	2	no	—
312	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
313	C₂³×C₄	—	2	2	no	—
314	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
315	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
316	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
317	C₂×Q₁₆	—	2	4	no	—
318	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
319	C₂×SD₁₆	—	2	8	no	—
320	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
321	C₂×D₈	—	2	4	no	—
322	C₂×D₈	C₂³×C₆₇	1	4	yes	—
323	C₂²×C₈	—	2	4	no	—
324	C₂²×C₈	—	2	4	no	—
325	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
326	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
327	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
328	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
329	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
330	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
331	#126	—	2	16	no	—
332	#125	—	2	4	no	—
333	#125	C₂³×C₆₇	1	8	yes	—
334	#124	—	2	4	no	—
335	#124	C₂³×C₆₇	1	8	yes	—
336	#123	—	2	16	no	—
337	C₂²⋊Q₈	—	2	4	no	—
338	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
339	#121	—	2	4	no	—
340	#121	C₂³×C₆₇	1	8	yes	—
341	C₄×Q₈	—	2	4	no	—
342	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
343	C₄×D₄	—	2	16	no	—
344	C₄×D₄	—	2	4	no	—
345	C₄×D₄	C₂³×C₆₇	1	8	yes	—
346	#117	—	2	16	no	—
347	C₂×C₄⋊C₄	—	2	2	no	—
348	C₂×C₄⋊C₄	—	2	2	no	—
349	C₂×C₄⋊C₄	—	2	2	no	—
350	C₂×C₄⋊C₄	—	2	2	no	—
351	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
352	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
353	C₂×C₂²⋊C₄	—	2	2	no	—
354	C₂×C₂²⋊C₄	—	2	2	no	—
355	C₂×C₂²⋊C₄	—	2	2	no	—
356	C₂×C₂²⋊C₄	—	2	2	no	—
357	C₂×C₂²⋊C₄	—	2	2	no	—
358	C₂×C₄²	—	2	2	no	—
359	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
360	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
361	C₂×C₄²	D₄×C₆₇	1	4	yes	—
362	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
363	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
364	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
365	#107	C₂³×C₆₇	1	4	yes	—
366	#107	C₂³×C₆₇	1	4	yes	—
367	#106	C₂³×C₆₇	1	8	yes	—
368	#103	C₂³×C₆₇	1	8	yes	—
369	#103	C₂³×C₆₇	1	8	yes	—
370	#102	C₂³×C₆₇	1	8	yes	—
371	#102	C₂³×C₆₇	1	8	yes	—
372	C₄×C₈	C₂³×C₆₇	1	4	yes	—
373	C₄×C₈	C₂³×C₆₇	1	4	yes	—
374	C₄×C₈	C₂³×C₆₇	1	4	yes	—
375	C₄×C₈	C₂³×C₆₇	1	4	yes	—
376	C₂⁴	—	2	2	no	—
377	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
378	C₂×Q₈	—	2	2	no	—
379	C₂×Q₈	—	2	4	no	—
380	C₂×Q₈	—	2	4	no	—
381	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
382	C₂×D₄	—	2	8	no	—
383	C₂×D₄	—	2	2	no	—
384	C₂×D₄	—	2	4	no	—
385	C₂×D₄	—	2	4	no	—
386	C₂×D₄	—	4	8	no	—
387	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	2	4	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	2	8	no	—
396	C₂²×C₄	—	4	8	no	—
397	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
398	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
399	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
400	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
401	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
402	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
403	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
404	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
405	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
406	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
407	Q₁₆	—	2	8	no	—
408	SD₁₆	—	2	16	no	—
409	D₈	—	2	8	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	8	no	—
413	C₂×C₈	—	2	8	no	—
414	C₂×C₈	#41×C₆₇	1	4	yes	—
415	C₂×C₈	#41×C₆₇	1	4	yes	—
416	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
417	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
418	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
419	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
420	C₄⋊C₄	—	2	8	no	—
421	C₄⋊C₄	—	2	8	no	—
422	C₄⋊C₄	—	2	8	no	—
423	C₄⋊C₄	—	2	8	no	—
424	C₄⋊C₄	—	2	2	no	—
425	C₄⋊C₄	—	2	2	no	—
426	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
427	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
428	C₂²⋊C₄	—	2	8	no	—
429	C₂²⋊C₄	—	2	8	no	—
430	C₂²⋊C₄	—	2	8	no	—
431	C₂²⋊C₄	—	2	8	no	—
432	C₂²⋊C₄	—	2	8	no	—
433	C₄²	—	2	8	no	—
434	C₄²	C₂×D₄×C₆₇	1	4	yes	—
435	C₄²	C₂×D₄×C₆₇	1	2	yes	—
436	C₄²	C₂×D₄×C₆₇	1	2	yes	—
437	C₄²	C₂×D₄×C₆₇	1	4	yes	—
438	C₄²	C₂⁴×C₆₇	1	2	yes	—
439	C₄²	C₂⁴×C₆₇	1	2	yes	—
440	C₂³	—	2	2	no	—
441	C₂³	—	2	2	no	—
442	C₂³	—	2	4	no	—
443	C₂³	—	4	4	no	—
444	C₂³	—	4	4	no	—
445	C₂³	—	4	4	no	—
446	C₂³	#135×C₆₇	1	4	yes	—
447	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
448	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
449	Q₈	—	2	4	no	—
450	Q₈	—	2	4	no	—
451	Q₈	—	4	4	no	—
452	D₄	—	2	4	no	—
453	D₄	—	2	4	no	—
454	D₄	—	4	16	no	—
455	D₄	—	4	4	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	8	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	8	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	4	4	no	—
469	C₂×C₄	—	4	4	no	—
470	C₂×C₄	—	4	4	no	—
471	C₂×C₄	—	4	16	no	—
472	C₂×C₄	#135×C₆₇	1	4	yes	—
473	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
474	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
475	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
476	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
477	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
478	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
479	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
480	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
481	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
482	C₈	—	2	2	no	—
483	C₈	—	2	2	no	—
484	C₈	C₂×#41×C₆₇	1	4	yes	—
485	C₈	C₂×#41×C₆₇	1	4	yes	—
486	C₂²	—	2	2	no	—
487	C₂²	—	2	8	no	—
488	C₂²	—	2	2	no	—
489	C₂²	—	4	4	no	—
490	C₂²	—	4	4	no	—
491	C₂²	—	4	4	no	—
492	C₂²	—	8	8	no	—
493	C₂²	#532×C₆₇	1	2	yes	—
494	C₂²	C₂×#135×C₆₇	1	4	yes	—
495	C₂²	C₂×#135×C₆₇	1	2	yes	—
496	C₂²	C₂×#135×C₆₇	1	2	yes	—
497	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
498	C₄	—	2	2	no	—
499	C₄	—	2	2	no	—
500	C₄	—	2	2	no	—
501	C₄	—	2	2	no	—
502	C₄	—	4	4	no	—
503	C₄	—	4	4	no	—
504	C₄	—	4	4	no	—
505	C₄	#532×C₆₇	1	2	yes	—
506	C₄	C₂×#135×C₆₇	1	4	yes	—
507	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
508	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
509	C₂	—	2	4	no	—
510	C₂	—	8	4	no	—
511	C₂	C₆₇×#3168	1	4	yes	direct product factor
512	C₂	C₂×#532×C₆₇	1	1	yes	characteristic
513	C₂	C₂²×#135×C₆₇	1	1	yes	characteristic
514	C₂	C₂²×#135×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂×#466×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#529×C₆₇	C₂	semidirect
2. C₆₇×#3168	C₂	direct (N × H)
3. C₂×#499×C₆₇	C₂	semidirect
4. C₂×#495×C₆₇	C₂	semidirect
5. C₂×#488×C₆₇	C₂	semidirect
6. C₂×#467×C₆₇	C₂	semidirect
8. C₂×#465×C₆₇	C₂	semidirect
9. C₂×C₄×Q₁₆×C₆₇	C₂	semidirect
10. C₂×C₄×SD₁₆×C₆₇	C₂	semidirect
11. C₂×C₄×D₈×C₆₇	C₂	semidirect
12. C₂²×C₄×C₈×C₆₇	C₂	semidirect
19. #529×C₆₇	C₂²	semidirect
31. #499×C₆₇	C₂²	semidirect
32. #495×C₆₇	C₂²	semidirect
33. #488×C₆₇	C₂²	semidirect
34. #467×C₆₇	C₂²	semidirect
36. #465×C₆₇	C₂²	semidirect
37. C₄×Q₁₆×C₆₇	C₂²	semidirect
38. C₄×SD₁₆×C₆₇	C₂²	semidirect
39. C₄×D₈×C₆₇	C₂²	semidirect
51. C₂×C₄×C₈×C₆₇	C₂²	semidirect
251. C₂×#3168	C₆₇	direct (N × H)
254. C₂×C₆₇	#3168	direct (N × H)
258. C₂×#529	C₂×C₆₇	semidirect
259. #3168	C₂×C₆₇	direct (N × H)
260. C₂×#499	C₂×C₆₇	semidirect
261. C₂×#495	C₂×C₆₇	semidirect
262. C₂×#488	C₂×C₆₇	semidirect
263. C₂×#467	C₂×C₆₇	semidirect
265. C₂×#465	C₂×C₆₇	semidirect
266. C₂×C₄×Q₁₆	C₂×C₆₇	semidirect
267. C₂×C₄×SD₁₆	C₂×C₆₇	semidirect
268. C₂×C₄×D₈	C₂×C₆₇	semidirect
269. C₂²×C₄×C₈	C₂×C₆₇	semidirect
270. C₆₇	C₂×#3168	direct (N × H)
277. #529	C₂²×C₆₇	semidirect
289. #499	C₂²×C₆₇	semidirect
290. #495	C₂²×C₆₇	semidirect
291. #488	C₂²×C₆₇	semidirect
292. #467	C₂²×C₆₇	semidirect
294. #465	C₂²×C₆₇	semidirect
295. C₄×Q₁₆	C₂²×C₆₇	semidirect
296. C₄×SD₁₆	C₂²×C₆₇	semidirect
297. C₄×D₈	C₂²×C₆₇	semidirect
309. C₂×C₄×C₈	C₂²×C₆₇	semidirect
511. C₂	C₆₇×#3168	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×#529×C₆₇	2
C₆₇×#3168	2
C₂×#499×C₆₇	2
C₂×#495×C₆₇	2
C₂×#488×C₆₇	2
C₂×#467×C₆₇	2
C₂×#466×C₆₇	2
C₂×#465×C₆₇	2
C₂×C₄×Q₁₆×C₆₇	2
C₂×C₄×SD₁₆×C₆₇	2
C₂×C₄×D₈×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₂×#3168	67

Subgroups — C2×C67×#3168 #30739921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3168 #30739921