Subgroups — C₆₇×#22641 #30701892

Order 17152.

780 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1940	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1924	C₂	1	2	yes	maximal
4	C₆₇×#1923	C₂	1	2	yes	maximal
5	C₆₇×#1839	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1839	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1771	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1749	C₂	1	1	yes	maximal, characteristic
12	C₂×#117×C₆₇	C₄	1	1	yes	characteristic
13	C₂×#117×C₆₇	C₄	1	1	yes	characteristic
14	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×#97×C₆₇	C₂²	1	2	yes	—
21	C₂×#97×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
28	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
29	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
31	#376×C₆₇	C₄	1	1	yes	characteristic
32	#376×C₆₇	C₄	1	1	yes	characteristic
33	C₄³×C₆₇	C₂²	1	1	yes	characteristic
34	C₄³×C₆₇	C₂²	1	1	yes	characteristic
35	#336×C₆₇	C₂²	1	1	yes	characteristic
36	#336×C₆₇	C₂²	1	1	yes	characteristic
37	#336×C₆₇	C₂²	1	1	yes	characteristic
38	#336×C₆₇	C₂²	1	1	yes	characteristic
39	#336×C₆₇	C₂²	1	2	yes	—
40	#336×C₆₇	C₂²	1	2	yes	—
41	#336×C₆₇	C₂²	1	2	yes	—
42	#336×C₆₇	C₂²	1	2	yes	—
43	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	#117×C₆₇	—	2	4	no	—
51	#117×C₆₇	—	2	4	no	—
52	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₆₇	—	2	4	no	—
61	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₆₇	—	2	4	no	—
78	C₄⋊C₈×C₆₇	—	2	4	no	—
79	C₄⋊C₈×C₆₇	—	2	4	no	—
80	C₂²⋊C₈×C₆₇	—	2	4	no	—
81	C₂²⋊C₈×C₆₇	—	4	2	no	—
82	#97×C₆₇	—	2	4	no	—
83	#97×C₆₇	—	4	2	no	—
84	C₄×C₈×C₆₇	—	4	2	no	—
85	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
86	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
87	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
88	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
89	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
90	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
91	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
92	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
93	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	—	2	2	no	—
96	C₂²×C₄×C₆₇	—	2	2	no	—
97	C₂²×C₄×C₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	2	1	no	—
102	C₂²×C₄×C₆₇	—	2	1	no	—
103	C₂²×C₄×C₆₇	—	4	1	no	—
104	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
110	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
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	—	2	2	no	—
129	C₂×C₈×C₆₇	#41	1	4	yes	—
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	2	2	no	—
133	C₄⋊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	M₁₆	1	4	yes	—
138	C₄²×C₆₇	—	2	2	no	—
139	C₄²×C₆₇	—	2	2	no	—
140	C₄²×C₆₇	—	2	2	no	—
141	C₄²×C₆₇	—	2	2	no	—
142	C₄²×C₆₇	—	2	2	no	—
143	C₄²×C₆₇	—	2	2	no	—
144	C₄²×C₆₇	—	2	2	no	—
145	C₄²×C₆₇	—	2	2	no	—
146	C₄²×C₆₇	—	2	2	no	—
147	C₄²×C₆₇	—	2	2	no	—
148	C₄²×C₆₇	—	2	1	no	—
149	C₄²×C₆₇	—	2	1	no	—
150	C₄²×C₆₇	—	2	1	no	—
151	C₄²×C₆₇	—	2	1	no	—
152	C₄²×C₆₇	—	4	1	no	—
153	C₄²×C₆₇	—	4	1	no	—
154	C₄²×C₆₇	M₁₆	1	4	yes	—
155	C₄²×C₆₇	#41	1	4	yes	—
156	C₂³×C₆₇	—	2	1	no	—
157	C₂³×C₆₇	—	2	1	no	—
158	C₂³×C₆₇	—	2	1	no	—
159	C₂³×C₆₇	—	2	1	no	—
160	C₂³×C₆₇	—	2	1	no	—
161	C₂³×C₆₇	—	2	1	no	—
162	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
163	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
164	C₂×C₄×C₆₇	—	2	8	no	—
165	C₂×C₄×C₆₇	—	2	1	no	—
166	C₂×C₄×C₆₇	—	2	1	no	—
167	C₂×C₄×C₆₇	—	2	1	no	—
168	C₂×C₄×C₆₇	—	2	1	no	—
169	C₂×C₄×C₆₇	—	2	1	no	—
170	C₂×C₄×C₆₇	—	2	1	no	—
171	C₂×C₄×C₆₇	—	2	1	no	—
172	C₂×C₄×C₆₇	—	2	1	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	1	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	1	no	—
180	C₂×C₄×C₆₇	—	2	1	no	—
181	C₂×C₄×C₆₇	—	2	1	no	—
182	C₂×C₄×C₆₇	—	2	1	no	—
183	C₂×C₄×C₆₇	—	2	1	no	—
184	C₂×C₄×C₆₇	—	2	1	no	—
185	C₂×C₄×C₆₇	—	2	1	no	—
186	C₂×C₄×C₆₇	—	2	1	no	—
187	C₂×C₄×C₆₇	—	2	1	no	—
188	C₂×C₄×C₆₇	—	2	1	no	—
189	C₂×C₄×C₆₇	—	2	1	no	—
190	C₂×C₄×C₆₇	—	2	1	no	—
191	C₂×C₄×C₆₇	—	2	1	no	—
192	C₂×C₄×C₆₇	—	2	1	no	—
193	C₂×C₄×C₆₇	—	2	1	no	—
194	C₂×C₄×C₆₇	—	2	1	no	—
195	C₂×C₄×C₆₇	—	2	1	no	—
196	C₂×C₄×C₆₇	—	4	4	no	—
197	C₂×C₄×C₆₇	—	4	2	no	—
198	C₂×C₄×C₆₇	—	4	2	no	—
199	C₂×C₄×C₆₇	#117	1	2	yes	—
200	C₂×C₄×C₆₇	#117	1	2	yes	—
201	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
202	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
203	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
204	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
205	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
206	C₂×C₄×C₆₇	#131	1	2	yes	—
207	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
208	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
209	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
210	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
211	C₈×C₆₇	—	2	4	no	—
212	C₈×C₆₇	—	4	2	no	—
213	C₈×C₆₇	—	4	2	no	—
214	C₈×C₆₇	—	4	2	no	—
215	C₂²×C₆₇	—	2	2	no	—
216	C₂²×C₆₇	—	2	2	no	—
217	C₂²×C₆₇	—	2	2	no	—
218	C₂²×C₆₇	—	4	1	no	—
219	C₂²×C₆₇	—	4	1	no	—
220	C₂²×C₆₇	—	4	1	no	—
221	C₂²×C₆₇	—	4	1	no	—
222	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
223	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
224	C₂²×C₆₇	#520	1	1	yes	characteristic
225	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
226	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
227	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
228	C₂²×C₆₇	#568	1	1	yes	characteristic
229	C₄×C₆₇	—	2	2	no	—
230	C₄×C₆₇	—	2	2	no	—
231	C₄×C₆₇	—	2	2	no	—
232	C₄×C₆₇	—	2	2	no	—
233	C₄×C₆₇	—	2	2	no	—
234	C₄×C₆₇	—	2	2	no	—
235	C₄×C₆₇	—	2	2	no	—
236	C₄×C₆₇	—	2	2	no	—
237	C₄×C₆₇	—	4	1	no	—
238	C₄×C₆₇	—	4	1	no	—
239	C₄×C₆₇	—	4	1	no	—
240	C₄×C₆₇	—	4	1	no	—
241	C₄×C₆₇	—	4	1	no	—
242	C₄×C₆₇	#553	1	2	yes	—
243	C₄×C₆₇	#556	1	2	yes	—
244	#22641	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
245	C₂×C₆₇	—	4	2	no	—
246	C₂×C₆₇	#2347	1	1	yes	characteristic
247	C₂×C₆₇	#2920	1	1	yes	characteristic
248	C₂×C₆₇	#2924	1	1	yes	characteristic
249	C₂×C₆₇	#2965	1	1	yes	characteristic
250	C₂×C₆₇	#2970	1	1	yes	characteristic
251	C₂×C₆₇	#2989	1	1	yes	characteristic
252	C₂×C₆₇	#2990	1	1	yes	characteristic
253	C₄×#117	C₂×C₆₇	1	1	yes	characteristic
254	#1940	C₂×C₆₇	1	2	yes	complemented
255	#1924	C₂×C₆₇	1	2	yes	—
256	#1923	C₂×C₆₇	1	2	yes	—
257	#1839	C₂×C₆₇	1	1	yes	characteristic, complemented
258	#1839	C₂×C₆₇	1	1	yes	characteristic, complemented
259	#1797	C₂×C₆₇	1	1	yes	characteristic
260	#1771	C₂×C₆₇	1	2	yes	complemented
261	#1767	C₂×C₆₇	1	1	yes	characteristic, complemented
262	C₄×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
263	#1749	C₂×C₆₇	1	1	yes	characteristic
264	C₆₇	#22641	1	1	yes	characteristic, direct product factor, Sylow
265	C₂×#117	C₄×C₆₇	1	1	yes	characteristic
266	C₂×#117	C₄×C₆₇	1	1	yes	characteristic
267	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
268	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
269	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
270	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
271	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
272	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
273	C₂×#97	C₂²×C₆₇	1	2	yes	—
274	C₂×#97	C₂²×C₆₇	1	2	yes	—
275	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
276	C₄×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
277	C₄×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
278	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
279	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
280	C₄×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
281	C₄×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
282	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
283	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
284	#376	C₄×C₆₇	1	1	yes	characteristic
285	#376	C₄×C₆₇	1	1	yes	characteristic
286	C₄³	C₂²×C₆₇	1	1	yes	characteristic
287	C₄³	C₂²×C₆₇	1	1	yes	characteristic
288	#336	C₂²×C₆₇	1	1	yes	characteristic
289	#336	C₂²×C₆₇	1	1	yes	characteristic
290	#336	C₂²×C₆₇	1	1	yes	characteristic
291	#336	C₂²×C₆₇	1	2	yes	—
292	#336	C₂²×C₆₇	1	2	yes	—
293	#336	C₂²×C₆₇	1	2	yes	—
294	#336	C₂²×C₆₇	1	2	yes	—
295	#336	C₂²×C₆₇	1	1	yes	characteristic
296	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
297	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
298	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
299	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
300	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
301	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
302	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
303	#117	—	2	4	no	—
304	#117	—	2	4	no	—
305	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
306	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
307	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
308	C₂×C₄⋊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	1	yes	characteristic
311	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
312	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
313	C₂×C₄²	—	2	4	no	—
314	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
315	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
316	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
317	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
318	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
319	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
320	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
321	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
322	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
323	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
324	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
325	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
326	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
327	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
328	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
329	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
330	C₄⋊C₈	—	2	4	no	—
331	C₄⋊C₈	—	2	4	no	—
332	C₄⋊C₈	—	2	4	no	—
333	C₂²⋊C₈	—	2	4	no	—
334	C₂²⋊C₈	—	4	2	no	—
335	#97	—	2	4	no	—
336	#97	—	4	2	no	—
337	C₄×C₈	—	4	2	no	—
338	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
339	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
340	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
341	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
342	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
343	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
344	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
345	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
346	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	4	1	no	—
357	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
358	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
359	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
360	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
361	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
362	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
363	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
364	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
365	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
366	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
367	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
368	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
369	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
370	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
371	C₂×C₈	—	2	2	no	—
372	C₂×C₈	—	2	2	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	#41×C₆₇	1	4	yes	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₂²⋊C₄	—	2	2	no	—
388	C₂²⋊C₄	—	2	2	no	—
389	C₂²⋊C₄	—	2	2	no	—
390	C₂²⋊C₄	M₁₆×C₆₇	1	4	yes	—
391	C₄²	—	2	2	no	—
392	C₄²	—	2	2	no	—
393	C₄²	—	2	2	no	—
394	C₄²	—	2	2	no	—
395	C₄²	—	2	2	no	—
396	C₄²	—	2	2	no	—
397	C₄²	—	2	2	no	—
398	C₄²	—	2	2	no	—
399	C₄²	—	2	2	no	—
400	C₄²	—	2	2	no	—
401	C₄²	—	2	1	no	—
402	C₄²	—	2	1	no	—
403	C₄²	—	2	1	no	—
404	C₄²	—	2	1	no	—
405	C₄²	—	4	1	no	—
406	C₄²	—	4	1	no	—
407	C₄²	M₁₆×C₆₇	1	4	yes	—
408	C₄²	#41×C₆₇	1	4	yes	—
409	C₂³	—	2	1	no	—
410	C₂³	—	2	1	no	—
411	C₂³	—	2	1	no	—
412	C₂³	—	2	1	no	—
413	C₂³	—	2	1	no	—
414	C₂³	—	2	1	no	—
415	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
416	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
417	C₂×C₄	—	2	8	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	4	4	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	#117×C₆₇	1	2	yes	—
453	C₂×C₄	#117×C₆₇	1	2	yes	—
454	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
455	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
456	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
457	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
458	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
459	C₂×C₄	#131×C₆₇	1	2	yes	—
460	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
461	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
462	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
463	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
464	C₈	—	2	4	no	—
465	C₈	—	4	2	no	—
466	C₈	—	4	2	no	—
467	C₈	—	4	2	no	—
468	C₂²	—	2	2	no	—
469	C₂²	—	2	2	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	4	1	no	—
472	C₂²	—	4	1	no	—
473	C₂²	—	4	1	no	—
474	C₂²	—	4	1	no	—
475	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
476	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
477	C₂²	#520×C₆₇	1	1	yes	characteristic
478	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
479	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
480	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
481	C₂²	#568×C₆₇	1	1	yes	characteristic
482	C₄	—	2	2	no	—
483	C₄	—	2	2	no	—
484	C₄	—	2	2	no	—
485	C₄	—	2	2	no	—
486	C₄	—	2	2	no	—
487	C₄	—	2	2	no	—
488	C₄	—	2	2	no	—
489	C₄	—	2	2	no	—
490	C₄	—	4	1	no	—
491	C₄	—	4	1	no	—
492	C₄	—	4	1	no	—
493	C₄	—	4	1	no	—
494	C₄	—	4	1	no	—
495	C₄	#553×C₆₇	1	2	yes	—
496	C₄	#556×C₆₇	1	2	yes	—
497	C₂	—	4	2	no	—
498	C₂	C₆₇×#2347	1	1	yes	characteristic
499	C₂	C₆₇×#2920	1	1	yes	characteristic
500	C₂	C₆₇×#2924	1	1	yes	characteristic
501	C₂	C₆₇×#2965	1	1	yes	characteristic
502	C₂	C₆₇×#2970	1	1	yes	characteristic
503	C₂	C₆₇×#2989	1	1	yes	characteristic
504	C₂	C₆₇×#2990	1	1	yes	characteristic

	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	C₂⁴×C₆₇	C₂²×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
2. C₆₇×#1940	C₂	semidirect
5. C₆₇×#1839	C₂	semidirect
6. C₆₇×#1839	C₂	semidirect
8. C₆₇×#1771	C₂	semidirect
9. C₆₇×#1767	C₂	semidirect
10. C₄×C₄⋊C₈×C₆₇	C₂	semidirect
244. #22641	C₆₇	direct (N × H)
254. #1940	C₂×C₆₇	semidirect
257. #1839	C₂×C₆₇	semidirect
258. #1839	C₂×C₆₇	semidirect
260. #1771	C₂×C₆₇	semidirect
261. #1767	C₂×C₆₇	semidirect
262. C₄×C₄⋊C₈	C₂×C₆₇	semidirect
264. C₆₇	#22641	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#117×C₆₇	2
C₆₇×#1940	2
C₆₇×#1924	2
C₆₇×#1923	2
C₆₇×#1839	2
C₆₇×#1797	2
C₆₇×#1771	2
C₆₇×#1767	2
C₄×C₄⋊C₈×C₆₇	2
C₆₇×#1749	2
#22641	67

Subgroups — C67×#22641 #30701892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22641 #30701892