Subgroups — C₂×C₆₇×#1850 #30700083

Order 17152.

1854 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂²×C₂²⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1850	C₂	1	8	yes	maximal, direct product factor
6	C₂×#336×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₆₇	—	2	4	no	—
9	C₂³×C₈×C₆₇	—	2	8	no	—
10	C₂³×C₈×C₆₇	C₂²	1	4	yes	—
11	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
14	C₂²×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
15	C₂²×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
16	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₆₇	—	2	8	no	—
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	8	yes	complemented
19	C₂×C₂²⋊C₈×C₆₇	—	2	4	no	—
20	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
21	C₄×C₂²⋊C₄×C₆₇	C₄	1	4	yes	—
22	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
23	C₂×#95×C₆₇	C₄	1	2	yes	—
24	#336×C₆₇	C₂²	1	8	yes	complemented
25	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₆₇	—	2	4	no	—
27	C₂³×C₄×C₆₇	—	2	4	no	—
28	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₆₇	D₄	1	4	yes	—
33	C₂³×C₄×C₆₇	Q₈	1	4	yes	—
34	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
35	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	—	2	32	no	—
37	C₂²×C₈×C₆₇	—	2	8	no	—
38	C₂²×C₈×C₆₇	—	2	4	no	—
39	C₂²×C₈×C₆₇	—	2	4	no	—
40	C₂²×C₈×C₆₇	—	2	16	no	—
41	C₂²×C₈×C₆₇	—	2	2	no	—
42	C₂²×C₈×C₆₇	—	2	2	no	—
43	C₂²×C₈×C₆₇	—	2	4	no	—
44	C₂²×C₈×C₆₇	—	2	4	no	—
45	C₂²×C₈×C₆₇	D₄	1	8	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
49	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
50	C₂×C₂²⋊C₄×C₆₇	C₈	1	8	yes	complemented
51	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
52	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₆₇	—	2	4	no	—
54	C₂×C₄²×C₆₇	—	2	4	no	—
55	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
56	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
57	C₄⋊C₈×C₆₇	—	2	16	no	—
58	C₂²⋊C₈×C₆₇	—	2	8	no	—
59	#95×C₆₇	C₂×C₄	1	8	yes	—
60	C₂⁴×C₆₇	—	2	4	no	—
61	C₂⁴×C₆₇	—	2	4	no	—
62	C₂⁴×C₆₇	C₄⋊C₄	1	4	yes	—
63	C₂⁴×C₆₇	C₂×C₈	1	4	yes	—
64	C₂⁴×C₆₇	M₁₆	1	4	yes	—
65	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
66	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	—	2	4	no	—
68	C₂²×C₄×C₆₇	—	2	4	no	—
69	C₂²×C₄×C₆₇	—	2	4	no	—
70	C₂²×C₄×C₆₇	—	2	8	no	—
71	C₂²×C₄×C₆₇	—	2	8	no	—
72	C₂²×C₄×C₆₇	—	2	16	no	—
73	C₂²×C₄×C₆₇	—	2	8	no	—
74	C₂²×C₄×C₆₇	—	2	4	no	—
75	C₂²×C₄×C₆₇	—	2	4	no	—
76	C₂²×C₄×C₆₇	—	2	4	no	—
77	C₂²×C₄×C₆₇	—	2	4	no	—
78	C₂²×C₄×C₆₇	—	2	4	no	—
79	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
80	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
81	C₂²×C₄×C₆₇	C₂×C₈	1	8	yes	complemented
82	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
83	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
85	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
87	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
89	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
90	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
91	C₂²×C₄×C₆₇	#41	1	4	yes	—
92	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
93	C₂×C₈×C₆₇	—	2	32	no	—
94	C₂×C₈×C₆₇	—	2	16	no	—
95	C₂×C₈×C₆₇	—	2	4	no	—
96	C₂×C₈×C₆₇	—	2	4	no	—
97	C₂×C₈×C₆₇	—	2	4	no	—
98	C₂×C₈×C₆₇	—	2	2	no	—
99	C₂×C₈×C₆₇	—	2	2	no	—
100	C₂×C₈×C₆₇	—	2	4	no	—
101	C₂×C₈×C₆₇	—	4	4	no	—
102	C₂×C₈×C₆₇	—	4	4	no	—
103	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
104	C₂²⋊C₄×C₆₇	—	2	8	no	—
105	C₂²⋊C₄×C₆₇	C₂×C₈	1	16	yes	complemented
106	C₄²×C₆₇	—	2	8	no	—
107	C₄²×C₆₇	—	2	4	no	—
108	C₄²×C₆₇	—	2	4	no	—
109	C₂³×C₆₇	—	2	4	no	—
110	C₂³×C₆₇	—	2	4	no	—
111	C₂³×C₆₇	—	2	8	no	—
112	C₂³×C₆₇	—	2	8	no	—
113	C₂³×C₆₇	—	2	16	no	—
114	C₂³×C₆₇	—	2	16	no	—
115	C₂³×C₆₇	C₄⋊C₈	1	16	yes	complemented
116	C₂³×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
117	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂³×C₆₇	C₄×D₄	1	4	yes	—
119	C₂³×C₆₇	C₂²×C₈	1	4	yes	—
120	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
121	C₂³×C₆₇	C₂×M₁₆	1	4	yes	—
122	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
123	C₂³×C₆₇	#131	1	4	yes	—
124	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	—	2	4	no	—
126	C₂×C₄×C₆₇	—	2	4	no	—
127	C₂×C₄×C₆₇	—	2	4	no	—
128	C₂×C₄×C₆₇	—	2	8	no	—
129	C₂×C₄×C₆₇	—	2	16	no	—
130	C₂×C₄×C₆₇	—	2	4	no	—
131	C₂×C₄×C₆₇	—	2	8	no	—
132	C₂×C₄×C₆₇	—	2	4	no	—
133	C₂×C₄×C₆₇	—	2	16	no	—
134	C₂×C₄×C₆₇	—	2	4	no	—
135	C₂×C₄×C₆₇	—	2	4	no	—
136	C₂×C₄×C₆₇	—	2	4	no	—
137	C₂×C₄×C₆₇	—	4	8	no	—
138	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
139	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
141	C₂×C₄×C₆₇	#120	1	4	yes	—
142	C₂×C₄×C₆₇	C₂²⋊Q₈	1	8	yes	—
143	C₂×C₄×C₆₇	#123	1	4	yes	—
144	C₂×C₄×C₆₇	C₂²×C₈	1	8	yes	complemented
145	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
146	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
149	C₈×C₆₇	—	2	8	no	—
150	C₈×C₆₇	—	4	2	no	—
151	C₈×C₆₇	—	4	2	no	—
152	C₂²×C₆₇	—	2	32	no	—
153	C₂²×C₆₇	—	2	4	no	—
154	C₂²×C₆₇	—	2	8	no	—
155	C₂²×C₆₇	—	2	8	no	—
156	C₂²×C₆₇	—	2	4	no	—
157	C₂²×C₆₇	#385	1	4	yes	—
158	C₂²×C₆₇	C₂×C₄⋊C₈	1	4	yes	—
159	C₂²×C₆₇	C₂×C₄⋊C₈	1	8	yes	complemented
160	C₂²×C₆₇	#424	1	4	yes	—
161	C₂²×C₆₇	C₈×D₄	1	8	yes	—
162	C₂²×C₆₇	#435	1	8	yes	—
163	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
164	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
165	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
166	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
167	C₂²×C₆₇	C₂×#131	1	2	yes	—
168	C₄×C₆₇	—	2	8	no	—
169	C₄×C₆₇	—	2	4	no	—
170	C₄×C₆₇	—	4	4	no	—
171	C₄×C₆₇	#385	1	4	yes	—
172	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
173	C₄×C₆₇	C₂×C₂²⋊Q₈	1	2	yes	—
174	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
175	C₂×#1850	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂×C₆₇	—	2	8	no	—
177	C₂×C₆₇	#1850	1	8	yes	direct product factor
178	C₂×C₆₇	C₂×#385	1	1	yes	characteristic
179	C₂×C₆₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
180	C₂×C₆₇	C₂×#424	1	1	yes	characteristic
181	C₂×C₆₇	C₂×C₈×D₄	1	2	yes	—
182	C₂×C₆₇	C₂×#435	1	2	yes	—
183	C₂⁴×C₈	C₂×C₆₇	1	1	yes	characteristic
184	C₂²×C₄⋊C₈	C₂×C₆₇	1	2	yes	complemented
185	C₂²×C₂²⋊C₈	C₂×C₆₇	1	1	yes	characteristic
186	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic
187	#1850	C₂×C₆₇	1	8	yes	direct product factor
188	C₂×#336	C₂×C₆₇	1	2	yes	complemented
189	C₆₇	C₂×#1850	1	1	yes	characteristic, direct product factor, Sylow
190	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
191	C₂³×C₈	—	2	4	no	—
192	C₂³×C₈	—	2	8	no	—
193	C₂³×C₈	C₂²×C₆₇	1	4	yes	—
194	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
195	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
196	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
197	C₂²×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
198	C₂²×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
199	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
200	C₂×C₄⋊C₈	—	2	8	no	—
201	C₂×C₄⋊C₈	C₂²×C₆₇	1	8	yes	complemented
202	C₂×C₂²⋊C₈	—	2	4	no	—
203	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
204	C₄×C₂²⋊C₄	C₄×C₆₇	1	4	yes	—
205	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
206	C₂×#95	C₄×C₆₇	1	2	yes	—
207	#336	C₂²×C₆₇	1	8	yes	complemented
208	C₂⁵	C₂×C₄×C₆₇	1	1	yes	characteristic
209	C₂³×C₄	—	2	4	no	—
210	C₂³×C₄	—	2	4	no	—
211	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
212	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
213	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
214	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
215	C₂³×C₄	D₄×C₆₇	1	4	yes	—
216	C₂³×C₄	Q₈×C₆₇	1	4	yes	—
217	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
218	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
219	C₂²×C₈	—	2	32	no	—
220	C₂²×C₈	—	2	8	no	—
221	C₂²×C₈	—	2	4	no	—
222	C₂²×C₈	—	2	4	no	—
223	C₂²×C₈	—	2	16	no	—
224	C₂²×C₈	—	2	2	no	—
225	C₂²×C₈	—	2	2	no	—
226	C₂²×C₈	—	2	4	no	—
227	C₂²×C₈	—	2	4	no	—
228	C₂²×C₈	D₄×C₆₇	1	8	yes	—
229	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
230	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
231	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
232	C₂×C₂²⋊C₄	—	2	4	no	—
233	C₂×C₂²⋊C₄	C₈×C₆₇	1	8	yes	complemented
234	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
235	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
236	C₂×C₄²	—	2	4	no	—
237	C₂×C₄²	—	2	4	no	—
238	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
239	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
240	C₄⋊C₈	—	2	16	no	—
241	C₂²⋊C₈	—	2	8	no	—
242	#95	C₂×C₄×C₆₇	1	8	yes	—
243	C₂⁴	—	2	4	no	—
244	C₂⁴	—	2	4	no	—
245	C₂⁴	C₄⋊C₄×C₆₇	1	4	yes	—
246	C₂⁴	C₂×C₈×C₆₇	1	4	yes	—
247	C₂⁴	M₁₆×C₆₇	1	4	yes	—
248	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
249	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
250	C₂²×C₄	—	2	4	no	—
251	C₂²×C₄	—	2	4	no	—
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	—	2	8	no	—
254	C₂²×C₄	—	2	8	no	—
255	C₂²×C₄	—	2	16	no	—
256	C₂²×C₄	—	2	8	no	—
257	C₂²×C₄	—	2	4	no	—
258	C₂²×C₄	—	2	4	no	—
259	C₂²×C₄	—	2	4	no	—
260	C₂²×C₄	—	2	4	no	—
261	C₂²×C₄	—	2	4	no	—
262	C₂²×C₄	C₄⋊C₄×C₆₇	1	4	yes	—
263	C₂²×C₄	C₄⋊C₄×C₆₇	1	8	yes	—
264	C₂²×C₄	C₂×C₈×C₆₇	1	8	yes	complemented
265	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
266	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
267	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
268	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
270	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
271	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
272	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
273	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
274	C₂²×C₄	#41×C₆₇	1	4	yes	—
275	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
276	C₂×C₈	—	2	32	no	—
277	C₂×C₈	—	2	16	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	4	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	—	4	4	no	—
285	C₂×C₈	—	4	4	no	—
286	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
287	C₂²⋊C₄	—	2	8	no	—
288	C₂²⋊C₄	C₂×C₈×C₆₇	1	16	yes	complemented
289	C₄²	—	2	8	no	—
290	C₄²	—	2	4	no	—
291	C₄²	—	2	4	no	—
292	C₂³	—	2	4	no	—
293	C₂³	—	2	4	no	—
294	C₂³	—	2	8	no	—
295	C₂³	—	2	8	no	—
296	C₂³	—	2	16	no	—
297	C₂³	—	2	16	no	—
298	C₂³	C₄⋊C₈×C₆₇	1	16	yes	complemented
299	C₂³	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
300	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
301	C₂³	C₄×D₄×C₆₇	1	4	yes	—
302	C₂³	C₂²×C₈×C₆₇	1	4	yes	—
303	C₂³	C₂²×C₈×C₆₇	1	2	yes	—
304	C₂³	C₂×M₁₆×C₆₇	1	4	yes	—
305	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
306	C₂³	#131×C₆₇	1	4	yes	—
307	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	2	8	no	—
312	C₂×C₄	—	2	16	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	8	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	16	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	4	8	no	—
321	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
322	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
323	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
324	C₂×C₄	#120×C₆₇	1	4	yes	—
325	C₂×C₄	C₂²⋊Q₈×C₆₇	1	8	yes	—
326	C₂×C₄	#123×C₆₇	1	4	yes	—
327	C₂×C₄	C₂²×C₈×C₆₇	1	8	yes	complemented
328	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
329	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
330	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
331	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
332	C₈	—	2	8	no	—
333	C₈	—	4	2	no	—
334	C₈	—	4	2	no	—
335	C₂²	—	2	4	no	—
336	C₂²	—	2	32	no	—
337	C₂²	—	2	4	no	—
338	C₂²	—	2	8	no	—
339	C₂²	—	2	8	no	—
340	C₂²	#385×C₆₇	1	4	yes	—
341	C₂²	C₂×C₄⋊C₈×C₆₇	1	4	yes	—
342	C₂²	C₂×C₄⋊C₈×C₆₇	1	8	yes	complemented
343	C₂²	#424×C₆₇	1	4	yes	—
344	C₂²	C₈×D₄×C₆₇	1	8	yes	—
345	C₂²	#435×C₆₇	1	8	yes	—
346	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
347	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
348	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
349	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
350	C₂²	C₂×#131×C₆₇	1	2	yes	—
351	C₄	—	2	4	no	—
352	C₄	—	2	8	no	—
353	C₄	—	4	4	no	—
354	C₄	#385×C₆₇	1	4	yes	—
355	C₄	C₂×#120×C₆₇	1	1	yes	characteristic
356	C₄	C₂×C₂²⋊Q₈×C₆₇	1	2	yes	—
357	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
358	C₂	—	2	8	no	—
359	C₂	C₆₇×#1850	1	8	yes	direct product factor
360	C₂	C₂×#385×C₆₇	1	1	yes	characteristic
361	C₂	C₂²×C₄⋊C₈×C₆₇	1	1	yes	characteristic
362	C₂	C₂×#424×C₆₇	1	1	yes	characteristic
363	C₂	C₂×C₈×D₄×C₆₇	1	2	yes	—
364	C₂	C₂×#435×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₆₇	C₂³
commutator	C₂²	C₂³×C₈×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂⁴×C₆₇	C₂²×C₄
prime radical	C₂⁵×C₆₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊C₈×C₆₇	C₂	semidirect
5. C₆₇×#1850	C₂	direct (N × H)
6. C₂×#336×C₆₇	C₂	semidirect
18. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
24. #336×C₆₇	C₂²	semidirect
50. C₂×C₂²⋊C₄×C₆₇	C₈	semidirect
81. C₂²×C₄×C₆₇	C₂×C₈	semidirect
105. C₂²⋊C₄×C₆₇	C₂×C₈	semidirect
115. C₂³×C₆₇	C₄⋊C₈	semidirect
144. C₂×C₄×C₆₇	C₂²×C₈	semidirect
159. C₂²×C₆₇	C₂×C₄⋊C₈	semidirect
175. C₂×#1850	C₆₇	direct (N × H)
177. C₂×C₆₇	#1850	direct (N × H)
184. C₂²×C₄⋊C₈	C₂×C₆₇	semidirect
187. #1850	C₂×C₆₇	direct (N × H)
188. C₂×#336	C₂×C₆₇	semidirect
189. C₆₇	C₂×#1850	direct (N × H)
201. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
207. #336	C₂²×C₆₇	semidirect
233. C₂×C₂²⋊C₄	C₈×C₆₇	semidirect
264. C₂²×C₄	C₂×C₈×C₆₇	semidirect
288. C₂²⋊C₄	C₂×C₈×C₆₇	semidirect
298. C₂³	C₄⋊C₈×C₆₇	semidirect
327. C₂×C₄	C₂²×C₈×C₆₇	semidirect
342. C₂²	C₂×C₄⋊C₈×C₆₇	semidirect
359. C₂	C₆₇×#1850	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×C₈×C₆₇	2
C₂²×C₄⋊C₈×C₆₇	2
C₂²×C₂²⋊C₈×C₆₇	2
C₂×C₄×C₂²⋊C₄×C₆₇	2
C₆₇×#1850	2
C₂×#336×C₆₇	2
C₂×#1850	67

Subgroups — C2×C67×#1850 #30700083

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1850 #30700083