Subgroups — C₆₇×#22171 #30701422

Order 17152.

918 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#435×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#2392	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1924	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1917	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1869	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1869	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1850	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1798	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1729	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
16	#435×C₆₇	—	2	8	no	—
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×#97×C₆₇	C₂²	1	2	yes	—
25	#398×C₆₇	C₄	1	1	yes	characteristic
26	#397×C₆₇	C₄	1	1	yes	characteristic
27	#396×C₆₇	C₄	1	2	yes	—
28	#394×C₆₇	C₄	1	1	yes	characteristic
29	#394×C₆₇	C₄	1	2	yes	—
30	#392×C₆₇	C₄	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
33	#336×C₆₇	C₂²	1	2	yes	—
34	#336×C₆₇	C₂²	1	2	yes	—
35	#336×C₆₇	C₂²	1	2	yes	—
36	#336×C₆₇	C₂²	1	2	yes	—
37	#336×C₆₇	C₂²	1	2	yes	—
38	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
39	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
41	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₆₇	—	4	1	no	—
43	C₂×M₁₆×C₆₇	—	4	1	no	—
44	C₂×M₁₆×C₆₇	D₄	1	4	yes	complemented
45	C₂²×C₈×C₆₇	—	2	2	no	—
46	C₂²×C₈×C₆₇	—	2	4	no	—
47	C₂²×C₈×C₆₇	—	4	2	no	—
48	C₂²×C₈×C₆₇	—	4	2	no	—
49	C₂²×C₈×C₆₇	—	4	1	no	—
50	C₂²×C₈×C₆₇	—	4	1	no	—
51	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
52	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
54	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
55	C₄×D₄×C₆₇	—	2	4	no	—
56	C₄×D₄×C₆₇	—	2	4	no	—
57	C₄×D₄×C₆₇	—	4	1	no	—
58	C₄×D₄×C₆₇	—	4	1	no	—
59	#117×C₆₇	—	4	1	no	—
60	#117×C₆₇	—	4	1	no	—
61	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
66	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
67	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
68	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
69	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
72	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
76	C₄⋊C₈×C₆₇	—	2	4	no	—
77	C₄⋊C₈×C₆₇	—	4	1	no	—
78	C₄⋊C₈×C₆₇	—	4	1	no	—
79	C₂²⋊C₈×C₆₇	—	2	4	no	—
80	C₂²⋊C₈×C₆₇	—	2	4	no	—
81	C₂²⋊C₈×C₆₇	—	4	2	no	—
82	C₂²⋊C₈×C₆₇	—	4	2	no	—
83	#97×C₆₇	—	2	4	no	—
84	#95×C₆₇	C₂×C₄	1	2	yes	—
85	#95×C₆₇	C₂×C₄	1	2	yes	—
86	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
87	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
88	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
90	C₂×D₄×C₆₇	—	2	4	no	—
91	C₂×D₄×C₆₇	—	2	4	no	—
92	C₂×D₄×C₆₇	—	2	4	no	—
93	C₂×D₄×C₆₇	—	4	1	no	—
94	C₂×D₄×C₆₇	—	4	2	no	—
95	C₂²×C₄×C₆₇	—	2	1	no	—
96	C₂²×C₄×C₆₇	—	2	1	no	—
97	C₂²×C₄×C₆₇	—	2	1	no	—
98	C₂²×C₄×C₆₇	—	2	1	no	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
107	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₆₇	#41	1	2	yes	—
115	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
116	M₁₆×C₆₇	—	4	4	no	—
117	C₂×C₈×C₆₇	—	2	2	no	—
118	C₂×C₈×C₆₇	—	2	2	no	—
119	C₂×C₈×C₆₇	—	2	2	no	—
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	2	2	no	—
127	C₂×C₈×C₆₇	—	2	2	no	—
128	C₂×C₈×C₆₇	—	4	8	no	—
129	C₂×C₈×C₆₇	—	4	4	no	—
130	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
131	C₄⋊C₄×C₆₇	—	2	4	no	—
132	C₄⋊C₄×C₆₇	—	4	2	no	—
133	C₄⋊C₄×C₆₇	—	4	1	no	—
134	C₄⋊C₄×C₆₇	—	4	1	no	—
135	C₄⋊C₄×C₆₇	—	4	1	no	—
136	C₂²⋊C₄×C₆₇	—	2	4	no	—
137	C₂²⋊C₄×C₆₇	—	2	4	no	—
138	C₂²⋊C₄×C₆₇	—	2	4	no	—
139	C₂²⋊C₄×C₆₇	—	4	2	no	—
140	C₂²⋊C₄×C₆₇	—	4	1	no	—
141	C₂²⋊C₄×C₆₇	—	4	1	no	—
142	C₂²⋊C₄×C₆₇	—	4	2	no	—
143	C₄²×C₆₇	—	2	2	no	—
144	C₄²×C₆₇	—	2	2	no	—
145	C₄²×C₆₇	—	2	1	no	—
146	C₄²×C₆₇	—	2	1	no	—
147	C₄²×C₆₇	—	2	1	no	—
148	C₄²×C₆₇	—	2	1	no	—
149	C₂³×C₆₇	—	2	2	no	—
150	C₂³×C₆₇	—	2	1	no	—
151	C₂³×C₆₇	—	2	2	no	—
152	C₂³×C₆₇	—	2	1	no	—
153	C₂³×C₆₇	—	2	2	no	—
154	C₂³×C₆₇	—	2	2	no	—
155	C₂³×C₆₇	—	2	2	no	—
156	C₂³×C₆₇	—	2	2	no	—
157	C₂³×C₆₇	—	2	2	no	—
158	C₂³×C₆₇	—	2	2	no	—
159	C₂³×C₆₇	C₄×D₄	1	2	yes	—
160	C₂³×C₆₇	#131	1	4	yes	—
161	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
162	D₄×C₆₇	—	4	8	no	—
163	D₄×C₆₇	—	4	4	no	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	2	1	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	2	2	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	2	2	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	2	1	no	—
174	C₂×C₄×C₆₇	—	2	1	no	—
175	C₂×C₄×C₆₇	—	2	1	no	—
176	C₂×C₄×C₆₇	—	2	1	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	2	2	no	—
181	C₂×C₄×C₆₇	—	2	2	no	—
182	C₂×C₄×C₆₇	—	2	2	no	—
183	C₂×C₄×C₆₇	—	2	2	no	—
184	C₂×C₄×C₆₇	—	2	2	no	—
185	C₂×C₄×C₆₇	—	2	2	no	—
186	C₂×C₄×C₆₇	—	2	2	no	—
187	C₂×C₄×C₆₇	—	4	2	no	—
188	C₂×C₄×C₆₇	—	4	2	no	—
189	C₂×C₄×C₆₇	—	4	2	no	—
190	C₂×C₄×C₆₇	—	4	2	no	—
191	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
192	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
195	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
196	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
197	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
198	C₈×C₆₇	—	2	4	no	—
199	C₈×C₆₇	—	4	2	no	—
200	C₈×C₆₇	—	4	2	no	—
201	C₈×C₆₇	—	4	2	no	—
202	C₂²×C₆₇	—	2	2	no	—
203	C₂²×C₆₇	—	2	4	no	—
204	C₂²×C₆₇	—	2	4	no	—
205	C₂²×C₆₇	—	2	4	no	—
206	C₂²×C₆₇	—	2	4	no	—
207	C₂²×C₆₇	—	4	2	no	—
208	C₂²×C₆₇	—	4	2	no	—
209	C₂²×C₆₇	—	4	1	no	—
210	C₂²×C₆₇	—	4	1	no	—
211	C₂²×C₆₇	—	4	2	no	—
212	C₂²×C₆₇	—	4	2	no	—
213	C₂²×C₆₇	—	4	2	no	—
214	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
215	C₂²×C₆₇	#518	1	1	yes	characteristic
216	C₂²×C₆₇	#520	1	1	yes	characteristic
217	C₂²×C₆₇	C₂×#131	1	2	yes	—
218	C₂²×C₆₇	#568	1	2	yes	—
219	C₄×C₆₇	—	2	2	no	—
220	C₄×C₆₇	—	4	2	no	—
221	C₄×C₆₇	—	4	2	no	—
222	C₄×C₆₇	—	4	2	no	—
223	C₄×C₆₇	—	4	1	no	—
224	C₄×C₆₇	—	4	1	no	—
225	C₄×C₆₇	—	4	1	no	—
226	C₄×C₆₇	—	4	1	no	—
227	C₄×C₆₇	#534	1	1	yes	characteristic
228	C₄×C₆₇	#537	1	1	yes	characteristic
229	C₄×C₆₇	#538	1	1	yes	characteristic
230	C₄×C₆₇	#539	1	1	yes	characteristic
231	#22171	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
232	C₂×C₆₇	—	4	2	no	—
233	C₂×C₆₇	—	4	4	no	—
234	C₂×C₆₇	#2325	1	1	yes	characteristic
235	C₂×C₆₇	#2933	1	2	yes	—
236	C₂×C₆₇	#2977	1	1	yes	characteristic
237	C₂×C₆₇	#2980	1	1	yes	characteristic
238	C₂×C₆₇	#2992	1	2	yes	—
239	C₂×#435	C₂×C₆₇	1	2	yes	complemented
240	#2392	C₂×C₆₇	1	1	yes	characteristic
241	#1924	C₂×C₆₇	1	2	yes	complemented
242	#1917	C₂×C₆₇	1	2	yes	complemented
243	#1869	C₂×C₆₇	1	2	yes	complemented
244	#1869	C₂×C₆₇	1	2	yes	complemented
245	#1850	C₂×C₆₇	1	2	yes	complemented
246	#1798	C₂×C₆₇	1	1	yes	characteristic, complemented
247	#1729	C₂×C₆₇	1	1	yes	characteristic, complemented
248	C₆₇	#22171	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
250	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
251	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
252	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
253	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
254	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
255	#435	—	2	8	no	—
256	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
257	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
258	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
259	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
260	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
261	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
262	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
263	C₂×#97	C₂²×C₆₇	1	2	yes	—
264	#398	C₄×C₆₇	1	1	yes	characteristic
265	#397	C₄×C₆₇	1	1	yes	characteristic
266	#396	C₄×C₆₇	1	2	yes	—
267	#394	C₄×C₆₇	1	2	yes	—
268	#394	C₄×C₆₇	1	1	yes	characteristic
269	#392	C₄×C₆₇	1	1	yes	characteristic
270	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
271	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
272	#336	C₂²×C₆₇	1	2	yes	—
273	#336	C₂²×C₆₇	1	2	yes	—
274	#336	C₂²×C₆₇	1	2	yes	—
275	#336	C₂²×C₆₇	1	2	yes	—
276	#336	C₂²×C₆₇	1	2	yes	—
277	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
278	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
279	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
280	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
281	C₂×M₁₆	—	4	1	no	—
282	C₂×M₁₆	—	4	1	no	—
283	C₂×M₁₆	D₄×C₆₇	1	4	yes	complemented
284	C₂²×C₈	—	2	2	no	—
285	C₂²×C₈	—	2	4	no	—
286	C₂²×C₈	—	4	2	no	—
287	C₂²×C₈	—	4	2	no	—
288	C₂²×C₈	—	4	1	no	—
289	C₂²×C₈	—	4	1	no	—
290	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
291	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
292	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
293	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
294	C₄×D₄	—	2	4	no	—
295	C₄×D₄	—	2	4	no	—
296	C₄×D₄	—	4	1	no	—
297	C₄×D₄	—	4	1	no	—
298	#117	—	4	1	no	—
299	#117	—	4	1	no	—
300	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
301	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
302	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
304	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
305	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
306	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
307	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
308	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
309	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
310	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
311	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
312	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
313	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
314	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
315	C₄⋊C₈	—	2	4	no	—
316	C₄⋊C₈	—	4	1	no	—
317	C₄⋊C₈	—	4	1	no	—
318	C₂²⋊C₈	—	2	4	no	—
319	C₂²⋊C₈	—	2	4	no	—
320	C₂²⋊C₈	—	4	2	no	—
321	C₂²⋊C₈	—	4	2	no	—
322	#97	—	2	4	no	—
323	#95	C₂×C₄×C₆₇	1	2	yes	—
324	#95	C₂×C₄×C₆₇	1	2	yes	—
325	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
326	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
327	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
328	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
329	C₂×D₄	—	2	4	no	—
330	C₂×D₄	—	2	4	no	—
331	C₂×D₄	—	2	4	no	—
332	C₂×D₄	—	4	1	no	—
333	C₂×D₄	—	4	2	no	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
345	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
346	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
347	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
348	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
349	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
350	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
351	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
353	C₂²×C₄	#41×C₆₇	1	2	yes	—
354	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
355	M₁₆	—	4	4	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	4	8	no	—
368	C₂×C₈	—	4	4	no	—
369	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
370	C₄⋊C₄	—	2	4	no	—
371	C₄⋊C₄	—	4	1	no	—
372	C₄⋊C₄	—	4	1	no	—
373	C₄⋊C₄	—	4	1	no	—
374	C₄⋊C₄	—	4	2	no	—
375	C₂²⋊C₄	—	2	4	no	—
376	C₂²⋊C₄	—	2	4	no	—
377	C₂²⋊C₄	—	2	4	no	—
378	C₂²⋊C₄	—	4	1	no	—
379	C₂²⋊C₄	—	4	1	no	—
380	C₂²⋊C₄	—	4	2	no	—
381	C₂²⋊C₄	—	4	2	no	—
382	C₄²	—	2	2	no	—
383	C₄²	—	2	2	no	—
384	C₄²	—	2	1	no	—
385	C₄²	—	2	1	no	—
386	C₄²	—	2	1	no	—
387	C₄²	—	2	1	no	—
388	C₂³	—	2	2	no	—
389	C₂³	—	2	1	no	—
390	C₂³	—	2	2	no	—
391	C₂³	—	2	1	no	—
392	C₂³	—	2	2	no	—
393	C₂³	—	2	2	no	—
394	C₂³	—	2	2	no	—
395	C₂³	—	2	2	no	—
396	C₂³	—	2	2	no	—
397	C₂³	—	2	2	no	—
398	C₂³	C₄×D₄×C₆₇	1	2	yes	—
399	C₂³	#131×C₆₇	1	4	yes	—
400	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
401	D₄	—	4	8	no	—
402	D₄	—	4	4	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
431	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
432	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
433	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
434	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
435	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
436	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
437	C₈	—	2	4	no	—
438	C₈	—	4	2	no	—
439	C₈	—	4	2	no	—
440	C₈	—	4	2	no	—
441	C₂²	—	2	2	no	—
442	C₂²	—	2	4	no	—
443	C₂²	—	2	4	no	—
444	C₂²	—	2	4	no	—
445	C₂²	—	2	4	no	—
446	C₂²	—	4	2	no	—
447	C₂²	—	4	2	no	—
448	C₂²	—	4	1	no	—
449	C₂²	—	4	1	no	—
450	C₂²	—	4	2	no	—
451	C₂²	—	4	2	no	—
452	C₂²	—	4	2	no	—
453	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
454	C₂²	#518×C₆₇	1	1	yes	characteristic
455	C₂²	#520×C₆₇	1	1	yes	characteristic
456	C₂²	C₂×#131×C₆₇	1	2	yes	—
457	C₂²	#568×C₆₇	1	2	yes	—
458	C₄	—	2	2	no	—
459	C₄	—	4	2	no	—
460	C₄	—	4	2	no	—
461	C₄	—	4	2	no	—
462	C₄	—	4	1	no	—
463	C₄	—	4	1	no	—
464	C₄	—	4	1	no	—
465	C₄	—	4	1	no	—
466	C₄	#534×C₆₇	1	1	yes	characteristic
467	C₄	#537×C₆₇	1	1	yes	characteristic
468	C₄	#538×C₆₇	1	1	yes	characteristic
469	C₄	#539×C₆₇	1	1	yes	characteristic
470	C₂	—	4	2	no	—
471	C₂	—	4	4	no	—
472	C₂	C₆₇×#2325	1	1	yes	characteristic
473	C₂	C₆₇×#2933	1	2	yes	—
474	C₂	C₆₇×#2977	1	1	yes	characteristic
475	C₂	C₆₇×#2980	1	1	yes	characteristic
476	C₂	C₆₇×#2992	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	#392×C₆₇	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#435×C₆₇	C₂	semidirect
3. C₆₇×#1924	C₂	semidirect
4. C₆₇×#1917	C₂	semidirect
5. C₆₇×#1869	C₂	semidirect
6. C₆₇×#1869	C₂	semidirect
7. C₆₇×#1850	C₂	semidirect
8. C₆₇×#1798	C₂	semidirect
9. C₆₇×#1729	C₂	semidirect
44. C₂×M₁₆×C₆₇	D₄	semidirect
231. #22171	C₆₇	direct (N × H)
239. C₂×#435	C₂×C₆₇	semidirect
241. #1924	C₂×C₆₇	semidirect
242. #1917	C₂×C₆₇	semidirect
243. #1869	C₂×C₆₇	semidirect
244. #1869	C₂×C₆₇	semidirect
245. #1850	C₂×C₆₇	semidirect
246. #1798	C₂×C₆₇	semidirect
247. #1729	C₂×C₆₇	semidirect
248. C₆₇	#22171	direct (N × H)
283. C₂×M₁₆	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#435×C₆₇	2
C₆₇×#2392	2
C₆₇×#1924	2
C₆₇×#1917	2
C₆₇×#1869	2
C₆₇×#1850	2
C₆₇×#1798	2
C₆₇×#1729	2
#22171	67

Subgroups — C67×#22171 #30701422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22171 #30701422