Subgroups — C₆₇×#21924 #30701175

Order 17152.

770 conjugacy classes of subgroups.

The subgroup lattice has 536 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×#1924	C₂	1	2	yes	maximal
3	C₆₇×#1923	C₂	1	2	yes	maximal
4	C₆₇×#1917	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1916	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1842	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1831	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic
10	C₆₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
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	1	yes	characteristic
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄⋊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×#97×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
30	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
31	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
33	#376×C₆₇	C₄	1	1	yes	characteristic
34	#376×C₆₇	C₄	1	1	yes	characteristic
35	C₄³×C₆₇	C₂²	1	1	yes	characteristic
36	C₄³×C₆₇	C₂²	1	1	yes	characteristic
37	#336×C₆₇	C₂²	1	2	yes	—
38	#336×C₆₇	C₂²	1	2	yes	—
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	2	no	—
61	C₂×C₄²×C₆₇	—	2	2	no	—
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	C₂×C₄	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	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₆₇	—	2	2	no	—
79	C₄⋊C₈×C₆₇	—	2	2	no	—
80	C₄⋊C₈×C₆₇	—	2	2	no	—
81	C₄⋊C₈×C₆₇	—	2	2	no	—
82	C₄⋊C₈×C₆₇	—	4	2	no	—
83	C₄⋊C₈×C₆₇	—	4	2	no	—
84	C₂²⋊C₈×C₆₇	—	4	2	no	—
85	C₂²⋊C₈×C₆₇	—	4	2	no	—
86	#97×C₆₇	—	4	2	no	—
87	C₄×C₈×C₆₇	—	4	2	no	—
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	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
94	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
95	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
96	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	2	no	—
102	C₂²×C₄×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	—	2	1	no	—
104	C₂²×C₄×C₆₇	—	2	1	no	—
105	C₂²×C₄×C₆₇	—	2	1	no	—
106	C₂²×C₄×C₆₇	—	2	1	no	—
107	C₂²×C₄×C₆₇	—	2	1	no	—
108	C₂²×C₄×C₆₇	—	2	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
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	—	2	2	no	—
130	C₂×C₈×C₆₇	—	2	2	no	—
131	C₂×C₈×C₆₇	—	2	2	no	—
132	C₂×C₈×C₆₇	—	2	2	no	—
133	C₂×C₈×C₆₇	—	2	2	no	—
134	C₂×C₈×C₆₇	—	2	2	no	—
135	C₄⋊C₄×C₆₇	—	2	2	no	—
136	C₄⋊C₄×C₆₇	—	2	2	no	—
137	C₄⋊C₄×C₆₇	—	2	2	no	—
138	C₄⋊C₄×C₆₇	M₁₆	1	4	yes	—
139	C₂²⋊C₄×C₆₇	—	2	2	no	—
140	C₂²⋊C₄×C₆₇	—	2	2	no	—
141	C₂²⋊C₄×C₆₇	—	2	2	no	—
142	C₂²⋊C₄×C₆₇	M₁₆	1	4	yes	—
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	2	no	—
149	C₄²×C₆₇	—	2	2	no	—
150	C₄²×C₆₇	—	2	2	no	—
151	C₄²×C₆₇	—	2	2	no	—
152	C₄²×C₆₇	—	2	2	no	—
153	C₄²×C₆₇	—	2	1	no	—
154	C₄²×C₆₇	—	2	1	no	—
155	C₄²×C₆₇	—	2	1	no	—
156	C₄²×C₆₇	—	2	1	no	—
157	C₄²×C₆₇	—	2	1	no	—
158	C₄²×C₆₇	—	2	1	no	—
159	C₄²×C₆₇	—	2	1	no	—
160	C₄²×C₆₇	—	2	1	no	—
161	C₄²×C₆₇	—	2	1	no	—
162	C₄²×C₆₇	—	2	1	no	—
163	C₄²×C₆₇	#41	1	2	yes	—
164	C₄²×C₆₇	#41	1	2	yes	—
165	C₂³×C₆₇	—	2	1	no	—
166	C₂³×C₆₇	—	2	1	no	—
167	C₂³×C₆₇	—	2	1	no	—
168	C₂³×C₆₇	—	2	1	no	—
169	C₂³×C₆₇	—	2	1	no	—
170	C₂³×C₆₇	—	2	1	no	—
171	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
172	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
173	C₂×C₄×C₆₇	—	2	8	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	—	2	1	no	—
197	C₂×C₄×C₆₇	—	2	1	no	—
198	C₂×C₄×C₆₇	—	2	1	no	—
199	C₂×C₄×C₆₇	—	2	1	no	—
200	C₂×C₄×C₆₇	—	2	1	no	—
201	C₂×C₄×C₆₇	—	2	1	no	—
202	C₂×C₄×C₆₇	—	2	1	no	—
203	C₂×C₄×C₆₇	—	2	1	no	—
204	C₂×C₄×C₆₇	—	4	1	no	—
205	C₂×C₄×C₆₇	—	4	1	no	—
206	C₂×C₄×C₆₇	—	4	1	no	—
207	C₂×C₄×C₆₇	—	4	1	no	—
208	C₂×C₄×C₆₇	—	4	4	no	—
209	C₂×C₄×C₆₇	#117	1	2	yes	—
210	C₂×C₄×C₆₇	#117	1	2	yes	—
211	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
212	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
213	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
214	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
215	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
216	C₂×C₄×C₆₇	#131	1	2	yes	—
217	C₂×C₄×C₆₇	#131	1	2	yes	—
218	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
219	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
220	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
221	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
222	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
223	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
224	C₈×C₆₇	—	4	2	no	—
225	C₈×C₆₇	—	4	2	no	—
226	C₈×C₆₇	—	4	2	no	—
227	C₈×C₆₇	—	4	2	no	—
228	C₂²×C₆₇	—	2	2	no	—
229	C₂²×C₆₇	—	2	2	no	—
230	C₂²×C₆₇	—	2	2	no	—
231	C₂²×C₆₇	—	4	1	no	—
232	C₂²×C₆₇	—	4	1	no	—
233	C₂²×C₆₇	—	4	1	no	—
234	C₂²×C₆₇	—	4	1	no	—
235	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
236	C₂²×C₆₇	#520	1	1	yes	characteristic
237	C₂²×C₆₇	#520	1	1	yes	characteristic
238	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
239	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
240	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
241	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
242	C₄×C₆₇	—	2	2	no	—
243	C₄×C₆₇	—	2	2	no	—
244	C₄×C₆₇	—	2	2	no	—
245	C₄×C₆₇	—	2	2	no	—
246	C₄×C₆₇	—	2	2	no	—
247	C₄×C₆₇	—	2	2	no	—
248	C₄×C₆₇	—	4	1	no	—
249	C₄×C₆₇	—	4	1	no	—
250	C₄×C₆₇	—	4	1	no	—
251	C₄×C₆₇	—	4	1	no	—
252	C₄×C₆₇	—	4	1	no	—
253	C₄×C₆₇	—	4	1	no	—
254	C₄×C₆₇	#433	1	4	yes	—
255	C₄×C₆₇	#540	1	1	yes	characteristic
256	C₄×C₆₇	#541	1	1	yes	characteristic
257	C₄×C₆₇	#542	1	1	yes	characteristic
258	C₄×C₆₇	#542	1	1	yes	characteristic
259	#21924	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
260	C₂×C₆₇	—	4	2	no	—
261	C₂×C₆₇	#2314	1	1	yes	characteristic
262	C₂×C₆₇	C₂×#433	1	1	yes	characteristic
263	C₂×C₆₇	#2920	1	1	yes	characteristic
264	C₂×C₆₇	#2990	1	1	yes	characteristic
265	C₂×C₆₇	#2990	1	1	yes	characteristic
266	C₂×C₆₇	#2990	1	1	yes	characteristic
267	C₂×C₆₇	#2991	1	1	yes	characteristic
268	C₄×#117	C₂×C₆₇	1	1	yes	characteristic
269	#1924	C₂×C₆₇	1	2	yes	—
270	#1923	C₂×C₆₇	1	2	yes	—
271	#1917	C₂×C₆₇	1	2	yes	complemented
272	#1916	C₂×C₆₇	1	2	yes	complemented
273	#1842	C₂×C₆₇	1	1	yes	characteristic, complemented
274	#1831	C₂×C₆₇	1	1	yes	characteristic, complemented
275	#1797	C₂×C₆₇	1	1	yes	characteristic
276	#1797	C₂×C₆₇	1	1	yes	characteristic
277	#1767	C₂×C₆₇	1	1	yes	characteristic, complemented
278	C₄×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
279	C₆₇	#21924	1	1	yes	characteristic, direct product factor, Sylow
280	C₂×#117	C₄×C₆₇	1	1	yes	characteristic
281	C₂×#117	C₄×C₆₇	1	1	yes	characteristic
282	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
283	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
284	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
285	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
286	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
287	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
288	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
289	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
290	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
291	C₂×#97	C₂²×C₆₇	1	2	yes	—
292	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
293	C₄×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
294	C₄×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
295	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
296	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
297	C₄×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
298	C₄×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
299	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
300	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
301	#376	C₄×C₆₇	1	1	yes	characteristic
302	#376	C₄×C₆₇	1	1	yes	characteristic
303	C₄³	C₂²×C₆₇	1	1	yes	characteristic
304	C₄³	C₂²×C₆₇	1	1	yes	characteristic
305	#336	C₂²×C₆₇	1	2	yes	—
306	#336	C₂²×C₆₇	1	2	yes	—
307	#336	C₂²×C₆₇	1	2	yes	—
308	#336	C₂²×C₆₇	1	2	yes	—
309	#336	C₂²×C₆₇	1	2	yes	—
310	#336	C₂²×C₆₇	1	2	yes	—
311	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
312	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
313	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
314	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
315	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
316	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
317	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
318	#117	—	2	4	no	—
319	#117	—	2	4	no	—
320	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
321	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
322	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
323	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
324	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
325	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
326	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
327	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
328	C₂×C₄²	—	2	2	no	—
329	C₂×C₄²	—	2	2	no	—
330	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
331	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
332	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
333	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
334	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
335	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
336	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
337	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
338	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
339	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
340	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
341	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
342	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
343	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
344	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
345	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
346	C₄⋊C₈	—	2	2	no	—
347	C₄⋊C₈	—	2	2	no	—
348	C₄⋊C₈	—	2	2	no	—
349	C₄⋊C₈	—	2	2	no	—
350	C₄⋊C₈	—	4	2	no	—
351	C₄⋊C₈	—	4	2	no	—
352	C₂²⋊C₈	—	4	2	no	—
353	C₂²⋊C₈	—	4	2	no	—
354	#97	—	4	2	no	—
355	C₄×C₈	—	4	2	no	—
356	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
357	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
358	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
359	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
360	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
361	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
362	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
363	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
364	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	—	2	2	no	—
368	C₂²×C₄	—	2	2	no	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	1	no	—
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	—	2	1	no	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
378	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
379	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
380	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
381	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
382	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
383	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
384	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
385	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
386	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
387	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
388	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
389	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
390	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	2	no	—
403	C₄⋊C₄	—	2	2	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	M₁₆×C₆₇	1	4	yes	—
407	C₂²⋊C₄	—	2	2	no	—
408	C₂²⋊C₄	—	2	2	no	—
409	C₂²⋊C₄	—	2	2	no	—
410	C₂²⋊C₄	M₁₆×C₆₇	1	4	yes	—
411	C₄²	—	2	2	no	—
412	C₄²	—	2	2	no	—
413	C₄²	—	2	2	no	—
414	C₄²	—	2	2	no	—
415	C₄²	—	2	2	no	—
416	C₄²	—	2	2	no	—
417	C₄²	—	2	2	no	—
418	C₄²	—	2	2	no	—
419	C₄²	—	2	2	no	—
420	C₄²	—	2	2	no	—
421	C₄²	—	2	1	no	—
422	C₄²	—	2	1	no	—
423	C₄²	—	2	1	no	—
424	C₄²	—	2	1	no	—
425	C₄²	—	2	1	no	—
426	C₄²	—	2	1	no	—
427	C₄²	—	2	1	no	—
428	C₄²	—	2	1	no	—
429	C₄²	—	2	1	no	—
430	C₄²	—	2	1	no	—
431	C₄²	#41×C₆₇	1	2	yes	—
432	C₄²	#41×C₆₇	1	2	yes	—
433	C₂³	—	2	1	no	—
434	C₂³	—	2	1	no	—
435	C₂³	—	2	1	no	—
436	C₂³	—	2	1	no	—
437	C₂³	—	2	1	no	—
438	C₂³	—	2	1	no	—
439	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
440	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
441	C₂×C₄	—	2	8	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	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	4	no	—
477	C₂×C₄	#117×C₆₇	1	2	yes	—
478	C₂×C₄	#117×C₆₇	1	2	yes	—
479	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
480	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
481	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
482	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
483	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
484	C₂×C₄	#131×C₆₇	1	2	yes	—
485	C₂×C₄	#131×C₆₇	1	2	yes	—
486	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
487	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
488	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
489	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
490	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
491	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
492	C₈	—	4	2	no	—
493	C₈	—	4	2	no	—
494	C₈	—	4	2	no	—
495	C₈	—	4	2	no	—
496	C₂²	—	2	2	no	—
497	C₂²	—	2	2	no	—
498	C₂²	—	2	2	no	—
499	C₂²	—	4	1	no	—
500	C₂²	—	4	1	no	—
501	C₂²	—	4	1	no	—
502	C₂²	—	4	1	no	—
503	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
504	C₂²	#520×C₆₇	1	1	yes	characteristic
505	C₂²	#520×C₆₇	1	1	yes	characteristic
506	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
507	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
508	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
509	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
510	C₄	—	2	2	no	—
511	C₄	—	2	2	no	—
512	C₄	—	2	2	no	—
513	C₄	—	2	2	no	—
514	C₄	—	2	2	no	—
515	C₄	—	2	2	no	—
516	C₄	—	4	1	no	—
517	C₄	—	4	1	no	—
518	C₄	—	4	1	no	—
519	C₄	—	4	1	no	—
520	C₄	—	4	1	no	—
521	C₄	—	4	1	no	—
522	C₄	#433×C₆₇	1	4	yes	—
523	C₄	#540×C₆₇	1	1	yes	characteristic
524	C₄	#541×C₆₇	1	1	yes	characteristic
525	C₄	#542×C₆₇	1	1	yes	characteristic
526	C₄	#542×C₆₇	1	1	yes	characteristic
527	C₂	—	4	2	no	—
528	C₂	C₆₇×#2314	1	1	yes	characteristic
529	C₂	C₂×#433×C₆₇	1	1	yes	characteristic
530	C₂	C₆₇×#2920	1	1	yes	characteristic
531	C₂	C₆₇×#2990	1	1	yes	characteristic
532	C₂	C₆₇×#2990	1	1	yes	characteristic
533	C₂	C₆₇×#2990	1	1	yes	characteristic
534	C₂	C₆₇×#2991	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
4. C₆₇×#1917	C₂	semidirect
5. C₆₇×#1916	C₂	semidirect
6. C₆₇×#1842	C₂	semidirect
7. C₆₇×#1831	C₂	semidirect
10. C₆₇×#1767	C₂	semidirect
11. C₄×C₄⋊C₈×C₆₇	C₂	semidirect
259. #21924	C₆₇	direct (N × H)
271. #1917	C₂×C₆₇	semidirect
272. #1916	C₂×C₆₇	semidirect
273. #1842	C₂×C₆₇	semidirect
274. #1831	C₂×C₆₇	semidirect
277. #1767	C₂×C₆₇	semidirect
278. C₄×C₄⋊C₈	C₂×C₆₇	semidirect
279. C₆₇	#21924	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×#1924	2
C₆₇×#1923	2
C₆₇×#1917	2
C₆₇×#1916	2
C₆₇×#1842	2
C₆₇×#1831	2
C₆₇×#1797	2
C₆₇×#1767	2
C₄×C₄⋊C₈×C₆₇	2
#21924	67

Subgroups — C67×#21924 #30701175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21924 #30701175