Subgroups — C₂×C₆₇×#1860 #30700113

Order 17152.

1478 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#520×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#428×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#104×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#417×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#382×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1860	C₂	1	8	yes	maximal, direct product factor
7	C₂×#343×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#339×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	#520×C₆₇	—	2	2	no	—
13	#520×C₆₇	—	2	2	no	—
14	#520×C₆₇	—	2	2	no	—
15	#520×C₆₇	C₂²	1	4	yes	—
16	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
17	C₂×C₄×D₄×C₆₇	—	2	1	no	—
18	C₂×C₄×D₄×C₆₇	—	2	2	no	—
19	C₂×#117×C₆₇	—	2	1	no	—
20	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
22	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
25	#428×C₆₇	—	2	2	no	—
26	#428×C₆₇	C₂²	1	4	yes	complemented
27	C₂×#107×C₆₇	—	2	1	no	—
28	C₂×#106×C₆₇	—	2	1	no	—
29	C₂×#104×C₆₇	C₄	1	8	yes	complemented
30	C₂×#104×C₆₇	C₂²	1	4	yes	—
31	#417×C₆₇	—	2	2	no	—
32	#417×C₆₇	C₂²	1	4	yes	—
33	C₂×#103×C₆₇	—	2	1	no	—
34	C₂×#102×C₆₇	—	2	1	no	—
35	#382×C₆₇	—	2	2	no	—
36	#382×C₆₇	C₂²	1	4	yes	complemented
37	C₂×#95×C₆₇	—	2	2	no	—
38	#343×C₆₇	C₂²	1	4	yes	complemented
39	#339×C₆₇	C₂²	1	4	yes	complemented
40	C₂×#41×C₆₇	—	2	1	no	—
41	C₂×#41×C₆₇	—	2	1	no	—
42	C₂×#41×C₆₇	—	2	2	no	—
43	C₂×#41×C₆₇	—	2	2	no	—
44	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
45	C₂×#41×C₆₇	D₄	1	2	yes	—
46	C₂×#41×C₆₇	Q₈	1	4	yes	—
47	C₂×#41×C₆₇	C₂³	1	2	yes	—
48	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
49	C₂²×D₄×C₆₇	—	2	1	no	—
50	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
51	C₂³×C₄×C₆₇	—	2	1	no	—
52	C₂³×C₄×C₆₇	—	2	1	no	—
53	C₂³×C₄×C₆₇	—	2	2	no	—
54	C₂³×C₄×C₆₇	—	2	1	no	—
55	C₂³×C₄×C₆₇	D₄	1	2	yes	—
56	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂×M₁₆×C₆₇	—	2	1	no	—
58	C₂×M₁₆×C₆₇	—	2	2	no	—
59	C₂×M₁₆×C₆₇	—	2	1	no	—
60	C₂×M₁₆×C₆₇	—	2	2	no	—
61	C₂×M₁₆×C₆₇	—	2	2	no	—
62	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
63	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
64	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
65	C₂²×C₈×C₆₇	—	2	1	no	—
66	C₂²×C₈×C₆₇	—	2	1	no	—
67	C₄×Q₈×C₆₇	—	2	4	no	—
68	C₄×Q₈×C₆₇	—	2	4	no	—
69	C₄×D₄×C₆₇	—	2	8	no	—
70	C₄×D₄×C₆₇	—	2	4	no	—
71	C₄×D₄×C₆₇	—	2	4	no	—
72	C₄×D₄×C₆₇	—	2	8	no	—
73	#117×C₆₇	—	2	4	no	—
74	#117×C₆₇	—	2	1	no	—
75	#117×C₆₇	—	2	1	no	—
76	#117×C₆₇	—	2	4	no	—
77	#117×C₆₇	D₄	1	2	yes	—
78	#117×C₆₇	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
80	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
81	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
82	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
83	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
84	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
85	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
86	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
87	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
88	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
89	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
90	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
91	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
92	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
96	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
97	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
98	C₂×C₄²×C₆₇	—	2	2	no	—
99	C₂×C₄²×C₆₇	—	2	1	no	—
100	C₂×C₄²×C₆₇	—	2	1	no	—
101	C₂×C₄²×C₆₇	—	2	2	no	—
102	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	complemented
103	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
104	#107×C₆₇	—	2	4	no	—
105	#106×C₆₇	—	2	4	no	—
106	#104×C₆₇	C₂×C₄	1	16	yes	complemented
107	#103×C₆₇	—	2	4	no	—
108	#102×C₆₇	—	2	4	no	—
109	#95×C₆₇	—	2	8	no	—
110	C₂⁴×C₆₇	—	2	1	no	—
111	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
112	#41×C₆₇	—	2	4	no	—
113	#41×C₆₇	—	2	4	no	—
114	#41×C₆₇	—	2	4	no	—
115	#41×C₆₇	—	2	4	no	—
116	#41×C₆₇	—	2	4	no	—
117	#41×C₆₇	—	2	2	no	—
118	#41×C₆₇	—	2	2	no	—
119	#41×C₆₇	C₄⋊C₄	1	8	yes	—
120	#41×C₆₇	C₂²×C₄	1	4	yes	—
121	#41×C₆₇	C₂×Q₈	1	4	yes	—
122	C₂×Q₈×C₆₇	—	2	1	no	—
123	C₂×Q₈×C₆₇	—	2	2	no	—
124	C₂×Q₈×C₆₇	C₄⋊C₄	1	4	yes	—
125	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
126	C₂×D₄×C₆₇	—	2	4	no	—
127	C₂×D₄×C₆₇	—	2	2	no	—
128	C₂×D₄×C₆₇	—	2	4	no	—
129	C₂×D₄×C₆₇	—	2	2	no	—
130	C₂×D₄×C₆₇	—	2	1	no	—
131	C₂×D₄×C₆₇	—	2	2	no	—
132	C₂×D₄×C₆₇	C₄⋊C₄	1	4	yes	—
133	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₆₇	—	2	1	no	—
135	C₂²×C₄×C₆₇	—	2	4	no	—
136	C₂²×C₄×C₆₇	—	2	2	no	—
137	C₂²×C₄×C₆₇	—	2	4	no	—
138	C₂²×C₄×C₆₇	—	2	2	no	—
139	C₂²×C₄×C₆₇	—	2	4	no	—
140	C₂²×C₄×C₆₇	—	2	1	no	—
141	C₂²×C₄×C₆₇	—	2	2	no	—
142	C₂²×C₄×C₆₇	—	2	1	no	—
143	C₂²×C₄×C₆₇	—	2	2	no	—
144	C₂²×C₄×C₆₇	—	2	8	no	—
145	C₂²×C₄×C₆₇	—	2	2	no	—
146	C₂²×C₄×C₆₇	—	2	2	no	—
147	C₂²×C₄×C₆₇	—	4	2	no	—
148	C₂²×C₄×C₆₇	—	4	2	no	—
149	C₂²×C₄×C₆₇	—	4	4	no	—
150	C₂²×C₄×C₆₇	—	4	4	no	—
151	C₂²×C₄×C₆₇	—	4	1	no	—
152	C₂²×C₄×C₆₇	—	4	1	no	—
153	C₂²×C₄×C₆₇	—	8	4	no	—
154	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
155	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
156	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
162	C₂²×C₄×C₆₇	#41	1	2	yes	—
163	C₂²×C₄×C₆₇	#41	1	2	yes	—
164	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
165	M₁₆×C₆₇	—	2	4	no	—
166	M₁₆×C₆₇	—	2	2	no	—
167	M₁₆×C₆₇	—	2	2	no	—
168	M₁₆×C₆₇	—	2	2	no	—
169	M₁₆×C₆₇	—	4	2	no	—
170	M₁₆×C₆₇	C₂²×C₄	1	4	yes	complemented
171	C₂×C₈×C₆₇	—	2	4	no	—
172	C₂×C₈×C₆₇	—	2	2	no	—
173	C₂×C₈×C₆₇	—	2	2	no	—
174	C₂×C₈×C₆₇	—	4	2	no	—
175	C₄⋊C₄×C₆₇	—	2	8	no	—
176	C₄⋊C₄×C₆₇	—	2	2	no	—
177	C₄⋊C₄×C₆₇	—	2	1	no	—
178	C₄⋊C₄×C₆₇	—	2	2	no	—
179	C₄⋊C₄×C₆₇	—	2	1	no	—
180	C₄⋊C₄×C₆₇	—	2	2	no	—
181	C₄⋊C₄×C₆₇	—	2	2	no	—
182	C₄⋊C₄×C₆₇	—	2	2	no	—
183	C₄⋊C₄×C₆₇	—	2	2	no	—
184	C₄⋊C₄×C₆₇	—	4	1	no	—
185	C₄⋊C₄×C₆₇	—	4	1	no	—
186	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
187	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
188	C₂²⋊C₄×C₆₇	—	2	8	no	—
189	C₂²⋊C₄×C₆₇	—	2	2	no	—
190	C₂²⋊C₄×C₆₇	—	2	2	no	—
191	C₄²×C₆₇	—	2	8	no	—
192	C₄²×C₆₇	—	2	4	no	—
193	C₄²×C₆₇	—	2	2	no	—
194	C₄²×C₆₇	—	2	2	no	—
195	C₄²×C₆₇	—	2	2	no	—
196	C₄²×C₆₇	C₂²×C₄	1	4	yes	complemented
197	C₂³×C₆₇	—	2	2	no	—
198	C₂³×C₆₇	—	2	2	no	—
199	C₂³×C₆₇	—	2	2	no	—
200	C₂³×C₆₇	—	2	4	no	—
201	C₂³×C₆₇	—	4	2	no	—
202	C₂³×C₆₇	—	4	2	no	—
203	C₂³×C₆₇	C₄×D₄	1	2	yes	—
204	C₂³×C₆₇	#120	1	1	yes	characteristic
205	C₂³×C₆₇	#120	1	1	yes	characteristic
206	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
207	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
208	Q₈×C₆₇	—	2	2	no	—
209	Q₈×C₆₇	—	2	4	no	—
210	Q₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
211	D₄×C₆₇	—	2	4	no	—
212	D₄×C₆₇	—	2	4	no	—
213	D₄×C₆₇	—	2	8	no	—
214	D₄×C₆₇	—	2	2	no	—
215	D₄×C₆₇	—	2	4	no	—
216	D₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
217	C₂×C₄×C₆₇	—	2	2	no	—
218	C₂×C₄×C₆₇	—	2	4	no	—
219	C₂×C₄×C₆₇	—	2	4	no	—
220	C₂×C₄×C₆₇	—	2	4	no	—
221	C₂×C₄×C₆₇	—	2	8	no	—
222	C₂×C₄×C₆₇	—	2	4	no	—
223	C₂×C₄×C₆₇	—	2	2	no	—
224	C₂×C₄×C₆₇	—	2	4	no	—
225	C₂×C₄×C₆₇	—	2	2	no	—
226	C₂×C₄×C₆₇	—	4	8	no	—
227	C₂×C₄×C₆₇	—	4	2	no	—
228	C₂×C₄×C₆₇	—	4	2	no	—
229	C₂×C₄×C₆₇	—	4	2	no	—
230	C₂×C₄×C₆₇	—	4	4	no	—
231	C₂×C₄×C₆₇	—	4	2	no	—
232	C₂×C₄×C₆₇	—	4	4	no	—
233	C₂×C₄×C₆₇	—	4	4	no	—
234	C₂×C₄×C₆₇	—	4	2	no	—
235	C₂×C₄×C₆₇	—	8	2	no	—
236	C₂×C₄×C₆₇	—	8	8	no	—
237	C₂×C₄×C₆₇	—	8	2	no	—
238	C₂×C₄×C₆₇	—	8	2	no	—
239	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
240	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
241	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
242	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
243	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
244	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
245	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
246	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
247	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
248	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
249	C₂×C₄×C₆₇	#123	1	2	yes	—
250	C₂×C₄×C₆₇	#123	1	2	yes	—
251	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
252	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
253	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
254	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
255	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
256	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
257	C₈×C₆₇	—	2	4	no	—
258	C₈×C₆₇	—	4	2	no	—
259	C₂²×C₆₇	—	2	4	no	—
260	C₂²×C₆₇	—	2	4	no	—
261	C₂²×C₆₇	—	2	1	no	—
262	C₂²×C₆₇	—	2	1	no	—
263	C₂²×C₆₇	—	2	1	no	—
264	C₂²×C₆₇	—	2	1	no	—
265	C₂²×C₆₇	—	2	4	no	—
266	C₂²×C₆₇	—	4	8	no	—
267	C₂²×C₆₇	—	4	2	no	—
268	C₂²×C₆₇	—	4	2	no	—
269	C₂²×C₆₇	#385	1	2	yes	—
270	C₂²×C₆₇	#385	1	2	yes	—
271	C₂²×C₆₇	#454	1	2	yes	—
272	C₂²×C₆₇	#455	1	2	yes	—
273	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
274	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
275	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
276	C₄×C₆₇	—	2	4	no	—
277	C₄×C₆₇	—	4	4	no	—
278	C₄×C₆₇	—	4	4	no	—
279	C₄×C₆₇	—	4	2	no	—
280	C₄×C₆₇	—	4	2	no	—
281	C₄×C₆₇	—	8	2	no	—
282	C₄×C₆₇	#385	1	2	yes	—
283	C₄×C₆₇	#385	1	2	yes	—
284	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
285	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
286	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
287	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
288	C₂×#1860	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
289	C₂×C₆₇	—	2	1	no	—
290	C₂×C₆₇	—	2	1	no	—
291	C₂×C₆₇	—	2	2	no	—
292	C₂×C₆₇	—	4	4	no	—
293	C₂×C₆₇	#1860	1	4	yes	direct product factor
294	C₂×C₆₇	C₂×#385	1	1	yes	characteristic
295	C₂×C₆₇	C₂×#455	1	1	yes	characteristic
296	C₂×C₆₇	C₂×#454	1	1	yes	characteristic
297	C₂×#520	C₂×C₆₇	1	1	yes	characteristic
298	C₂×#428	C₂×C₆₇	1	1	yes	characteristic, complemented
299	C₂²×#104	C₂×C₆₇	1	1	yes	characteristic
300	C₂×#417	C₂×C₆₇	1	1	yes	characteristic
301	C₂×#382	C₂×C₆₇	1	1	yes	characteristic, complemented
302	#1860	C₂×C₆₇	1	8	yes	direct product factor
303	C₂×#343	C₂×C₆₇	1	1	yes	characteristic, complemented
304	C₂×#339	C₂×C₆₇	1	1	yes	characteristic, complemented
305	C₆₇	C₂×#1860	1	1	yes	characteristic, direct product factor, Sylow
306	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
307	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
308	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
309	#520	—	2	2	no	—
310	#520	—	2	2	no	—
311	#520	—	2	2	no	—
312	#520	C₂²×C₆₇	1	4	yes	—
313	C₂×C₄×Q₈	—	2	1	no	—
314	C₂×C₄×D₄	—	2	1	no	—
315	C₂×C₄×D₄	—	2	2	no	—
316	C₂×#117	—	2	1	no	—
317	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
318	C₂²×C₄⋊C₄	—	2	1	no	—
319	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
320	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
321	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
322	#428	—	2	2	no	—
323	#428	C₂²×C₆₇	1	4	yes	complemented
324	C₂×#107	—	2	1	no	—
325	C₂×#106	—	2	1	no	—
326	C₂×#104	C₄×C₆₇	1	8	yes	complemented
327	C₂×#104	C₂²×C₆₇	1	4	yes	—
328	#417	—	2	2	no	—
329	#417	C₂²×C₆₇	1	4	yes	—
330	C₂×#103	—	2	1	no	—
331	C₂×#102	—	2	1	no	—
332	#382	—	2	2	no	—
333	#382	C₂²×C₆₇	1	4	yes	complemented
334	C₂×#95	—	2	2	no	—
335	#343	C₂²×C₆₇	1	4	yes	complemented
336	#339	C₂²×C₆₇	1	4	yes	complemented
337	C₂×#41	—	2	1	no	—
338	C₂×#41	—	2	1	no	—
339	C₂×#41	—	2	2	no	—
340	C₂×#41	—	2	2	no	—
341	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
342	C₂×#41	D₄×C₆₇	1	2	yes	—
343	C₂×#41	Q₈×C₆₇	1	4	yes	—
344	C₂×#41	C₂³×C₆₇	1	2	yes	—
345	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
346	C₂²×D₄	—	2	1	no	—
347	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
348	C₂³×C₄	—	2	1	no	—
349	C₂³×C₄	—	2	1	no	—
350	C₂³×C₄	—	2	2	no	—
351	C₂³×C₄	—	2	1	no	—
352	C₂³×C₄	D₄×C₆₇	1	2	yes	—
353	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
354	C₂×M₁₆	—	2	1	no	—
355	C₂×M₁₆	—	2	2	no	—
356	C₂×M₁₆	—	2	1	no	—
357	C₂×M₁₆	—	2	2	no	—
358	C₂×M₁₆	—	2	2	no	—
359	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
360	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
361	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
362	C₂²×C₈	—	2	1	no	—
363	C₂²×C₈	—	2	1	no	—
364	C₄×Q₈	—	2	4	no	—
365	C₄×Q₈	—	2	4	no	—
366	C₄×D₄	—	2	8	no	—
367	C₄×D₄	—	2	4	no	—
368	C₄×D₄	—	2	4	no	—
369	C₄×D₄	—	2	8	no	—
370	#117	—	2	4	no	—
371	#117	—	2	1	no	—
372	#117	—	2	1	no	—
373	#117	—	2	4	no	—
374	#117	D₄×C₆₇	1	2	yes	—
375	#117	C₂³×C₆₇	1	2	yes	—
376	C₂×C₄⋊C₄	—	2	2	no	—
377	C₂×C₄⋊C₄	—	2	4	no	—
378	C₂×C₄⋊C₄	—	2	1	no	—
379	C₂×C₄⋊C₄	—	2	1	no	—
380	C₂×C₄⋊C₄	—	2	1	no	—
381	C₂×C₄⋊C₄	—	2	1	no	—
382	C₂×C₄⋊C₄	—	2	4	no	—
383	C₂×C₄⋊C₄	—	2	1	no	—
384	C₂×C₄⋊C₄	—	2	1	no	—
385	C₂×C₄⋊C₄	—	2	1	no	—
386	C₂×C₄⋊C₄	—	2	1	no	—
387	C₂×C₄⋊C₄	—	2	1	no	—
388	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
389	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
390	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
391	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
392	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
393	C₂×C₂²⋊C₄	—	2	2	no	—
394	C₂×C₂²⋊C₄	—	2	1	no	—
395	C₂×C₄²	—	2	2	no	—
396	C₂×C₄²	—	2	1	no	—
397	C₂×C₄²	—	2	1	no	—
398	C₂×C₄²	—	2	2	no	—
399	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	complemented
400	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
401	#107	—	2	4	no	—
402	#106	—	2	4	no	—
403	#104	C₂×C₄×C₆₇	1	16	yes	complemented
404	#103	—	2	4	no	—
405	#102	—	2	4	no	—
406	#95	—	2	8	no	—
407	C₂⁴	—	2	1	no	—
408	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
409	#41	—	2	4	no	—
410	#41	—	2	4	no	—
411	#41	—	2	4	no	—
412	#41	—	2	4	no	—
413	#41	—	2	4	no	—
414	#41	—	2	2	no	—
415	#41	—	2	2	no	—
416	#41	C₄⋊C₄×C₆₇	1	8	yes	—
417	#41	C₂²×C₄×C₆₇	1	4	yes	—
418	#41	C₂×Q₈×C₆₇	1	4	yes	—
419	C₂×Q₈	—	2	1	no	—
420	C₂×Q₈	—	2	2	no	—
421	C₂×Q₈	C₄⋊C₄×C₆₇	1	4	yes	—
422	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
423	C₂×D₄	—	2	4	no	—
424	C₂×D₄	—	2	2	no	—
425	C₂×D₄	—	2	4	no	—
426	C₂×D₄	—	2	2	no	—
427	C₂×D₄	—	2	1	no	—
428	C₂×D₄	—	2	2	no	—
429	C₂×D₄	C₄⋊C₄×C₆₇	1	4	yes	—
430	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
431	C₂²×C₄	—	2	1	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	2	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	2	no	—
441	C₂²×C₄	—	2	8	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	—	2	2	no	—
444	C₂²×C₄	—	4	2	no	—
445	C₂²×C₄	—	4	2	no	—
446	C₂²×C₄	—	4	4	no	—
447	C₂²×C₄	—	4	4	no	—
448	C₂²×C₄	—	4	1	no	—
449	C₂²×C₄	—	4	1	no	—
450	C₂²×C₄	—	8	4	no	—
451	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
452	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
453	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
454	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
455	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
456	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
457	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
458	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
459	C₂²×C₄	#41×C₆₇	1	2	yes	—
460	C₂²×C₄	#41×C₆₇	1	2	yes	—
461	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
462	M₁₆	—	2	4	no	—
463	M₁₆	—	2	2	no	—
464	M₁₆	—	2	2	no	—
465	M₁₆	—	2	2	no	—
466	M₁₆	—	4	2	no	—
467	M₁₆	C₂²×C₄×C₆₇	1	4	yes	complemented
468	C₂×C₈	—	2	4	no	—
469	C₂×C₈	—	2	2	no	—
470	C₂×C₈	—	2	2	no	—
471	C₂×C₈	—	4	2	no	—
472	C₄⋊C₄	—	2	8	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	2	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	2	no	—
478	C₄⋊C₄	—	2	2	no	—
479	C₄⋊C₄	—	2	2	no	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	4	1	no	—
482	C₄⋊C₄	—	4	1	no	—
483	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
484	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
485	C₂²⋊C₄	—	2	8	no	—
486	C₂²⋊C₄	—	2	2	no	—
487	C₂²⋊C₄	—	2	2	no	—
488	C₄²	—	2	8	no	—
489	C₄²	—	2	4	no	—
490	C₄²	—	2	2	no	—
491	C₄²	—	2	2	no	—
492	C₄²	—	2	2	no	—
493	C₄²	C₂²×C₄×C₆₇	1	4	yes	complemented
494	C₂³	—	2	2	no	—
495	C₂³	—	2	2	no	—
496	C₂³	—	2	2	no	—
497	C₂³	—	2	4	no	—
498	C₂³	—	4	2	no	—
499	C₂³	—	4	2	no	—
500	C₂³	C₄×D₄×C₆₇	1	2	yes	—
501	C₂³	#120×C₆₇	1	1	yes	characteristic
502	C₂³	#120×C₆₇	1	1	yes	characteristic
503	C₂³	C₂²⋊Q₈×C₆₇	1	2	yes	—
504	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
505	Q₈	—	2	2	no	—
506	Q₈	—	2	4	no	—
507	Q₈	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
508	D₄	—	2	4	no	—
509	D₄	—	2	4	no	—
510	D₄	—	2	8	no	—
511	D₄	—	2	2	no	—
512	D₄	—	2	4	no	—
513	D₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	8	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	4	2	no	—
524	C₂×C₄	—	4	2	no	—
525	C₂×C₄	—	4	8	no	—
526	C₂×C₄	—	4	2	no	—
527	C₂×C₄	—	4	4	no	—
528	C₂×C₄	—	4	2	no	—
529	C₂×C₄	—	4	4	no	—
530	C₂×C₄	—	4	4	no	—
531	C₂×C₄	—	4	2	no	—
532	C₂×C₄	—	8	2	no	—
533	C₂×C₄	—	8	8	no	—
534	C₂×C₄	—	8	2	no	—
535	C₂×C₄	—	8	2	no	—
536	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
537	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
538	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
539	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
540	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
541	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
542	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
543	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
544	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
545	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
546	C₂×C₄	#123×C₆₇	1	2	yes	—
547	C₂×C₄	#123×C₆₇	1	2	yes	—
548	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
549	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
550	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
551	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
552	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
553	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
554	C₈	—	2	4	no	—
555	C₈	—	4	2	no	—
556	C₂²	—	2	1	no	—
557	C₂²	—	2	1	no	—
558	C₂²	—	2	1	no	—
559	C₂²	—	2	1	no	—
560	C₂²	—	2	4	no	—
561	C₂²	—	2	4	no	—
562	C₂²	—	2	4	no	—
563	C₂²	—	4	2	no	—
564	C₂²	—	4	2	no	—
565	C₂²	—	4	8	no	—
566	C₂²	#385×C₆₇	1	2	yes	—
567	C₂²	#385×C₆₇	1	2	yes	—
568	C₂²	#454×C₆₇	1	2	yes	—
569	C₂²	#455×C₆₇	1	2	yes	—
570	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
571	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
572	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
573	C₄	—	2	4	no	—
574	C₄	—	4	4	no	—
575	C₄	—	4	4	no	—
576	C₄	—	4	2	no	—
577	C₄	—	4	2	no	—
578	C₄	—	8	2	no	—
579	C₄	#385×C₆₇	1	2	yes	—
580	C₄	#385×C₆₇	1	2	yes	—
581	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
582	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
583	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
584	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
585	C₂	—	2	1	no	—
586	C₂	—	2	1	no	—
587	C₂	—	2	2	no	—
588	C₂	—	4	4	no	—
589	C₂	C₆₇×#1860	1	4	yes	direct product factor
590	C₂	C₂×#385×C₆₇	1	1	yes	characteristic
591	C₂	C₂×#455×C₆₇	1	1	yes	characteristic
592	C₂	C₂×#454×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	#120
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#428×C₆₇	C₂	semidirect
5. C₂×#382×C₆₇	C₂	semidirect
6. C₆₇×#1860	C₂	direct (N × H)
7. C₂×#343×C₆₇	C₂	semidirect
8. C₂×#339×C₆₇	C₂	semidirect
26. #428×C₆₇	C₂²	semidirect
29. C₂×#104×C₆₇	C₄	semidirect
36. #382×C₆₇	C₂²	semidirect
38. #343×C₆₇	C₂²	semidirect
39. #339×C₆₇	C₂²	semidirect
62. C₂×M₁₆×C₆₇	C₂×C₄	semidirect
102. C₂×C₄²×C₆₇	C₂×C₄	semidirect
106. #104×C₆₇	C₂×C₄	semidirect
170. M₁₆×C₆₇	C₂²×C₄	semidirect
196. C₄²×C₆₇	C₂²×C₄	semidirect
288. C₂×#1860	C₆₇	direct (N × H)
293. C₂×C₆₇	#1860	direct (N × H)
298. C₂×#428	C₂×C₆₇	semidirect
301. C₂×#382	C₂×C₆₇	semidirect
302. #1860	C₂×C₆₇	direct (N × H)
303. C₂×#343	C₂×C₆₇	semidirect
304. C₂×#339	C₂×C₆₇	semidirect
305. C₆₇	C₂×#1860	direct (N × H)
323. #428	C₂²×C₆₇	semidirect
326. C₂×#104	C₄×C₆₇	semidirect
333. #382	C₂²×C₆₇	semidirect
335. #343	C₂²×C₆₇	semidirect
336. #339	C₂²×C₆₇	semidirect
359. C₂×M₁₆	C₂×C₄×C₆₇	semidirect
399. C₂×C₄²	C₂×C₄×C₆₇	semidirect
403. #104	C₂×C₄×C₆₇	semidirect
467. M₁₆	C₂²×C₄×C₆₇	semidirect
493. C₄²	C₂²×C₄×C₆₇	semidirect
589. C₂	C₆₇×#1860	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×#520×C₆₇	2
C₂×#428×C₆₇	2
C₂²×#104×C₆₇	2
C₂×#417×C₆₇	2
C₂×#382×C₆₇	2
C₆₇×#1860	2
C₂×#343×C₆₇	2
C₂×#339×C₆₇	2
C₂×#1860	67

Subgroups — C2×C67×#1860 #30700113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1860 #30700113