Subgroups — C₂×C₆₇×#2544 #30714137

Order 17152.

4174 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#127×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#124×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#124×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2544	C₂	1	16	yes	maximal, direct product factor
6	C₂×#394×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₂×#391×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#390×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#386×C₆₇	C₂	1	4	yes	maximal, complemented
10	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₆₇	C₂²	1	4	yes	—
14	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
16	C₂×#127×C₆₇	—	2	8	no	—
17	C₂×#127×C₆₇	C₂²	1	8	yes	—
18	C₂×#124×C₆₇	—	2	8	no	—
19	C₂×#124×C₆₇	—	2	8	no	—
20	C₂×#124×C₆₇	C₂²	1	8	yes	complemented
21	C₂×#124×C₆₇	C₂²	1	8	yes	complemented
22	C₂×C₄×D₄×C₆₇	—	2	8	no	—
23	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	complemented
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
25	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
27	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
28	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
29	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
30	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
31	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
32	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
33	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
34	#394×C₆₇	C₂²	1	32	yes	complemented
35	#391×C₆₇	C₂²	1	8	yes	complemented
36	#390×C₆₇	C₂²	1	8	yes	complemented
37	#386×C₆₇	C₂²	1	32	yes	complemented
38	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	complemented
39	C₂×#95×C₆₇	C₂²	1	2	yes	—
40	C₂×#95×C₆₇	C₂²	1	4	yes	—
41	C₂⁵×C₆₇	C₂³	1	2	yes	—
42	C₂⁵×C₆₇	C₂³	1	2	yes	—
43	C₂²×Q₈×C₆₇	—	2	4	no	—
44	C₂²×Q₈×C₆₇	—	2	4	no	—
45	C₂²×Q₈×C₆₇	—	2	4	no	—
46	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
47	C₂²×D₄×C₆₇	—	2	16	no	—
48	C₂²×D₄×C₆₇	—	2	4	no	—
49	C₂²×D₄×C₆₇	—	2	4	no	—
50	C₂²×D₄×C₆₇	—	2	4	no	—
51	C₂²×D₄×C₆₇	—	2	4	no	—
52	C₂²×D₄×C₆₇	—	2	16	no	—
53	C₂²×D₄×C₆₇	—	2	4	no	—
54	C₂²×D₄×C₆₇	—	4	8	no	—
55	C₂²×D₄×C₆₇	D₄	1	8	yes	—
56	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
57	C₂²×D₄×C₆₇	C₂³	1	16	yes	—
58	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
59	C₂³×C₄×C₆₇	—	2	8	no	—
60	C₂³×C₄×C₆₇	—	4	8	no	—
61	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
62	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
63	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
64	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
67	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
68	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
69	#127×C₆₇	—	2	32	no	—
70	#124×C₆₇	—	2	32	no	—
71	#124×C₆₇	—	2	32	no	—
72	C₄×D₄×C₆₇	—	2	32	no	—
73	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
75	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
76	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
77	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
78	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
79	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
80	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
81	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
82	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
83	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
84	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
85	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
86	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
87	C₂×C₄²×C₆₇	—	2	4	no	—
88	C₂×C₄²×C₆₇	—	2	4	no	—
89	C₂×C₄²×C₆₇	D₄	1	8	yes	complemented
90	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
91	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
92	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
93	#95×C₆₇	C₂³	1	8	yes	—
94	#95×C₆₇	C₂³	1	16	yes	—
95	C₂⁴×C₆₇	—	2	4	no	—
96	C₂⁴×C₆₇	—	2	4	no	—
97	C₂⁴×C₆₇	—	2	8	no	—
98	C₂⁴×C₆₇	—	2	8	no	—
99	C₂⁴×C₆₇	—	2	4	no	—
100	C₂⁴×C₆₇	—	2	4	no	—
101	C₂⁴×C₆₇	—	2	4	no	—
102	C₂⁴×C₆₇	—	2	8	no	—
103	C₂⁴×C₆₇	—	2	8	no	—
104	C₂⁴×C₆₇	C₂×D₄	1	8	yes	—
105	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
106	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
107	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
108	C₂×Q₈×C₆₇	—	2	8	no	—
109	C₂×Q₈×C₆₇	—	2	8	no	—
110	C₂×Q₈×C₆₇	—	2	8	no	—
111	C₂×Q₈×C₆₇	—	4	16	no	—
112	C₂×D₄×C₆₇	—	2	32	no	—
113	C₂×D₄×C₆₇	—	2	32	no	—
114	C₂×D₄×C₆₇	—	2	8	no	—
115	C₂×D₄×C₆₇	—	2	8	no	—
116	C₂×D₄×C₆₇	—	2	8	no	—
117	C₂×D₄×C₆₇	—	2	8	no	—
118	C₂×D₄×C₆₇	—	2	32	no	—
119	C₂×D₄×C₆₇	—	2	8	no	—
120	C₂×D₄×C₆₇	—	4	16	no	—
121	C₂×D₄×C₆₇	—	4	64	no	—
122	C₂×D₄×C₆₇	—	4	16	no	—
123	C₂×D₄×C₆₇	C₂×D₄	1	16	yes	—
124	C₂²×C₄×C₆₇	—	2	4	no	—
125	C₂²×C₄×C₆₇	—	2	4	no	—
126	C₂²×C₄×C₆₇	—	2	4	no	—
127	C₂²×C₄×C₆₇	—	2	4	no	—
128	C₂²×C₄×C₆₇	—	2	4	no	—
129	C₂²×C₄×C₆₇	—	2	4	no	—
130	C₂²×C₄×C₆₇	—	2	4	no	—
131	C₂²×C₄×C₆₇	—	2	4	no	—
132	C₂²×C₄×C₆₇	—	2	4	no	—
133	C₂²×C₄×C₆₇	—	2	4	no	—
134	C₂²×C₄×C₆₇	—	2	4	no	—
135	C₂²×C₄×C₆₇	—	2	8	no	—
136	C₂²×C₄×C₆₇	—	2	16	no	—
137	C₂²×C₄×C₆₇	—	4	16	no	—
138	C₂²×C₄×C₆₇	—	4	32	no	—
139	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
140	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
141	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
142	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
143	C₂²×C₄×C₆₇	#41	1	8	yes	—
144	C₂²×C₄×C₆₇	#41	1	4	yes	—
145	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
146	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
147	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
149	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
150	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
151	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
152	C₄⋊C₄×C₆₇	—	2	8	no	—
153	C₄⋊C₄×C₆₇	—	2	16	no	—
154	C₂²⋊C₄×C₆₇	—	2	32	no	—
155	C₂²⋊C₄×C₆₇	—	2	16	no	—
156	C₂²⋊C₄×C₆₇	—	2	16	no	—
157	C₂²⋊C₄×C₆₇	—	2	16	no	—
158	C₂²⋊C₄×C₆₇	—	4	16	no	—
159	C₄²×C₆₇	—	2	8	no	—
160	C₄²×C₆₇	—	2	8	no	—
161	C₄²×C₆₇	C₂×D₄	1	16	yes	complemented
162	C₂³×C₆₇	—	2	16	no	—
163	C₂³×C₆₇	—	2	32	no	—
164	C₂³×C₆₇	—	2	4	no	—
165	C₂³×C₆₇	—	2	4	no	—
166	C₂³×C₆₇	—	2	8	no	—
167	C₂³×C₆₇	—	2	8	no	—
168	C₂³×C₆₇	—	2	4	no	—
169	C₂³×C₆₇	—	2	4	no	—
170	C₂³×C₆₇	—	2	4	no	—
171	C₂³×C₆₇	—	2	8	no	—
172	C₂³×C₆₇	—	2	8	no	—
173	C₂³×C₆₇	—	4	8	no	—
174	C₂³×C₆₇	—	4	8	no	—
175	C₂³×C₆₇	—	4	16	no	—
176	C₂³×C₆₇	—	4	8	no	—
177	C₂³×C₆₇	—	4	8	no	—
178	C₂³×C₆₇	—	4	8	no	—
179	C₂³×C₆₇	—	4	16	no	—
180	C₂³×C₆₇	—	4	16	no	—
181	C₂³×C₆₇	C₂²×D₄	1	8	yes	—
182	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
183	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
184	C₂³×C₆₇	C₂×#41	1	2	yes	—
185	C₂³×C₆₇	C₂×#41	1	4	yes	—
186	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
187	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
188	Q₈×C₆₇	—	4	16	no	—
189	D₄×C₆₇	—	2	32	no	—
190	D₄×C₆₇	—	4	64	no	—
191	D₄×C₆₇	—	4	16	no	—
192	C₂×C₄×C₆₇	—	2	8	no	—
193	C₂×C₄×C₆₇	—	2	8	no	—
194	C₂×C₄×C₆₇	—	2	16	no	—
195	C₂×C₄×C₆₇	—	2	4	no	—
196	C₂×C₄×C₆₇	—	2	4	no	—
197	C₂×C₄×C₆₇	—	2	4	no	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	2	4	no	—
200	C₂×C₄×C₆₇	—	2	4	no	—
201	C₂×C₄×C₆₇	—	2	4	no	—
202	C₂×C₄×C₆₇	—	2	4	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	4	no	—
205	C₂×C₄×C₆₇	—	2	4	no	—
206	C₂×C₄×C₆₇	—	2	8	no	—
207	C₂×C₄×C₆₇	—	4	32	no	—
208	C₂×C₄×C₆₇	—	4	8	no	—
209	C₂×C₄×C₆₇	—	4	8	no	—
210	C₂×C₄×C₆₇	—	4	8	no	—
211	C₂×C₄×C₆₇	#121	1	16	yes	—
212	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
213	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
214	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
215	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
216	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
217	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
218	C₂²×C₆₇	—	2	8	no	—
219	C₂²×C₆₇	—	2	16	no	—
220	C₂²×C₆₇	—	4	4	no	—
221	C₂²×C₆₇	—	4	4	no	—
222	C₂²×C₆₇	—	4	8	no	—
223	C₂²×C₆₇	—	4	32	no	—
224	C₂²×C₆₇	—	4	4	no	—
225	C₂²×C₆₇	—	4	4	no	—
226	C₂²×C₆₇	—	4	4	no	—
227	C₂²×C₆₇	—	4	8	no	—
228	C₂²×C₆₇	—	4	8	no	—
229	C₂²×C₆₇	—	8	16	no	—
230	C₂²×C₆₇	C₂×#121	1	4	yes	—
231	C₂²×C₆₇	#532	1	4	yes	—
232	C₂²×C₆₇	#535	1	4	yes	—
233	C₂²×C₆₇	#546	1	8	yes	—
234	C₂²×C₆₇	#559	1	8	yes	—
235	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
236	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
237	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
238	C₂²×C₆₇	C₂²×#41	1	2	yes	—
239	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
240	C₄×C₆₇	—	2	4	no	—
241	C₄×C₆₇	—	2	4	no	—
242	C₄×C₆₇	—	2	8	no	—
243	C₄×C₆₇	—	4	4	no	—
244	C₄×C₆₇	—	4	4	no	—
245	C₄×C₆₇	—	4	4	no	—
246	C₄×C₆₇	C₂×#121	1	8	yes	—
247	C₂×#2544	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
248	C₂×C₆₇	—	4	8	no	—
249	C₂×C₆₇	—	8	4	no	—
250	C₂×C₆₇	#2544	1	8	yes	direct product factor
251	C₂×C₆₇	C₂²×#121	1	1	yes	characteristic
252	C₂×C₆₇	C₂×#532	1	1	yes	characteristic
253	C₂×C₆₇	C₂×#535	1	1	yes	characteristic
254	C₂×C₆₇	C₂×#546	1	2	yes	—
255	C₂×C₆₇	C₂×#559	1	2	yes	—
256	C₂²×#127	C₂×C₆₇	1	1	yes	characteristic
257	C₂²×#124	C₂×C₆₇	1	1	yes	characteristic, complemented
258	C₂²×#124	C₂×C₆₇	1	1	yes	characteristic, complemented
259	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
260	#2544	C₂×C₆₇	1	16	yes	direct product factor
261	C₂×#394	C₂×C₆₇	1	4	yes	complemented
262	C₂×#391	C₂×C₆₇	1	1	yes	characteristic, complemented
263	C₂×#390	C₂×C₆₇	1	1	yes	characteristic, complemented
264	C₂×#386	C₂×C₆₇	1	4	yes	complemented
265	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
266	C₆₇	C₂×#2544	1	1	yes	characteristic, direct product factor, Sylow
267	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
268	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
269	C₂³×D₄	C₂²×C₆₇	1	4	yes	—
270	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
271	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
272	C₂×#127	—	2	8	no	—
273	C₂×#127	C₂²×C₆₇	1	8	yes	—
274	C₂×#124	—	2	8	no	—
275	C₂×#124	—	2	8	no	—
276	C₂×#124	C₂²×C₆₇	1	8	yes	complemented
277	C₂×#124	C₂²×C₆₇	1	8	yes	complemented
278	C₂×C₄×D₄	—	2	8	no	—
279	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	complemented
280	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
281	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
282	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
283	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
284	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
285	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
286	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
287	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
288	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
289	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
290	#394	C₂²×C₆₇	1	32	yes	complemented
291	#391	C₂²×C₆₇	1	8	yes	complemented
292	#390	C₂²×C₆₇	1	8	yes	complemented
293	#386	C₂²×C₆₇	1	32	yes	complemented
294	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	complemented
295	C₂×#95	C₂²×C₆₇	1	4	yes	—
296	C₂×#95	C₂²×C₆₇	1	2	yes	—
297	C₂⁵	C₂³×C₆₇	1	2	yes	—
298	C₂⁵	C₂³×C₆₇	1	2	yes	—
299	C₂²×Q₈	—	2	4	no	—
300	C₂²×Q₈	—	2	4	no	—
301	C₂²×Q₈	—	2	4	no	—
302	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
303	C₂²×D₄	—	2	16	no	—
304	C₂²×D₄	—	2	4	no	—
305	C₂²×D₄	—	2	4	no	—
306	C₂²×D₄	—	2	4	no	—
307	C₂²×D₄	—	2	4	no	—
308	C₂²×D₄	—	2	16	no	—
309	C₂²×D₄	—	2	4	no	—
310	C₂²×D₄	—	4	8	no	—
311	C₂²×D₄	D₄×C₆₇	1	8	yes	—
312	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
313	C₂²×D₄	C₂³×C₆₇	1	16	yes	—
314	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
315	C₂³×C₄	—	2	8	no	—
316	C₂³×C₄	—	4	8	no	—
317	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
318	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
319	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
320	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
321	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
322	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
323	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
324	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
325	#127	—	2	32	no	—
326	#124	—	2	32	no	—
327	#124	—	2	32	no	—
328	C₄×D₄	—	2	32	no	—
329	C₂×C₄⋊C₄	—	2	4	no	—
330	C₂×C₄⋊C₄	—	2	8	no	—
331	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
332	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
333	C₂×C₂²⋊C₄	—	2	16	no	—
334	C₂×C₂²⋊C₄	—	2	8	no	—
335	C₂×C₂²⋊C₄	—	2	8	no	—
336	C₂×C₂²⋊C₄	—	2	8	no	—
337	C₂×C₂²⋊C₄	—	4	8	no	—
338	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
339	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
340	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
341	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
342	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
343	C₂×C₄²	—	2	4	no	—
344	C₂×C₄²	—	2	4	no	—
345	C₂×C₄²	D₄×C₆₇	1	8	yes	complemented
346	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
347	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
348	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
349	#95	C₂³×C₆₇	1	8	yes	—
350	#95	C₂³×C₆₇	1	16	yes	—
351	C₂⁴	—	2	4	no	—
352	C₂⁴	—	2	4	no	—
353	C₂⁴	—	2	8	no	—
354	C₂⁴	—	2	8	no	—
355	C₂⁴	—	2	4	no	—
356	C₂⁴	—	2	4	no	—
357	C₂⁴	—	2	4	no	—
358	C₂⁴	—	2	8	no	—
359	C₂⁴	—	2	8	no	—
360	C₂⁴	C₂×D₄×C₆₇	1	8	yes	—
361	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
362	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
363	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
364	C₂×Q₈	—	2	8	no	—
365	C₂×Q₈	—	2	8	no	—
366	C₂×Q₈	—	2	8	no	—
367	C₂×Q₈	—	4	16	no	—
368	C₂×D₄	—	2	32	no	—
369	C₂×D₄	—	2	32	no	—
370	C₂×D₄	—	2	8	no	—
371	C₂×D₄	—	2	8	no	—
372	C₂×D₄	—	2	8	no	—
373	C₂×D₄	—	2	8	no	—
374	C₂×D₄	—	2	32	no	—
375	C₂×D₄	—	2	8	no	—
376	C₂×D₄	—	4	16	no	—
377	C₂×D₄	—	4	64	no	—
378	C₂×D₄	—	4	16	no	—
379	C₂×D₄	C₂×D₄×C₆₇	1	16	yes	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	4	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	4	no	—
385	C₂²×C₄	—	2	4	no	—
386	C₂²×C₄	—	2	4	no	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	2	4	no	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	2	8	no	—
392	C₂²×C₄	—	2	16	no	—
393	C₂²×C₄	—	4	16	no	—
394	C₂²×C₄	—	4	32	no	—
395	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
396	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
397	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
398	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
399	C₂²×C₄	#41×C₆₇	1	8	yes	—
400	C₂²×C₄	#41×C₆₇	1	4	yes	—
401	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
402	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
403	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
404	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
405	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
406	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
407	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
408	C₄⋊C₄	—	2	8	no	—
409	C₄⋊C₄	—	2	16	no	—
410	C₂²⋊C₄	—	2	32	no	—
411	C₂²⋊C₄	—	2	16	no	—
412	C₂²⋊C₄	—	2	16	no	—
413	C₂²⋊C₄	—	2	16	no	—
414	C₂²⋊C₄	—	4	16	no	—
415	C₄²	—	2	8	no	—
416	C₄²	—	2	8	no	—
417	C₄²	C₂×D₄×C₆₇	1	16	yes	complemented
418	C₂³	—	2	16	no	—
419	C₂³	—	2	32	no	—
420	C₂³	—	2	4	no	—
421	C₂³	—	2	4	no	—
422	C₂³	—	2	8	no	—
423	C₂³	—	2	8	no	—
424	C₂³	—	2	4	no	—
425	C₂³	—	2	4	no	—
426	C₂³	—	2	4	no	—
427	C₂³	—	2	8	no	—
428	C₂³	—	2	8	no	—
429	C₂³	—	4	8	no	—
430	C₂³	—	4	8	no	—
431	C₂³	—	4	16	no	—
432	C₂³	—	4	8	no	—
433	C₂³	—	4	8	no	—
434	C₂³	—	4	8	no	—
435	C₂³	—	4	16	no	—
436	C₂³	—	4	16	no	—
437	C₂³	C₂²×D₄×C₆₇	1	8	yes	—
438	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
439	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
440	C₂³	C₂×#41×C₆₇	1	2	yes	—
441	C₂³	C₂×#41×C₆₇	1	4	yes	—
442	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
443	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
444	Q₈	—	4	16	no	—
445	D₄	—	2	32	no	—
446	D₄	—	4	64	no	—
447	D₄	—	4	16	no	—
448	C₂×C₄	—	2	8	no	—
449	C₂×C₄	—	2	8	no	—
450	C₂×C₄	—	2	16	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	2	4	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	4	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	8	no	—
463	C₂×C₄	—	4	32	no	—
464	C₂×C₄	—	4	8	no	—
465	C₂×C₄	—	4	8	no	—
466	C₂×C₄	—	4	8	no	—
467	C₂×C₄	#121×C₆₇	1	16	yes	—
468	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
469	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
470	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
471	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
472	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
473	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
474	C₂²	—	2	8	no	—
475	C₂²	—	2	16	no	—
476	C₂²	—	4	32	no	—
477	C₂²	—	4	4	no	—
478	C₂²	—	4	4	no	—
479	C₂²	—	4	8	no	—
480	C₂²	—	4	4	no	—
481	C₂²	—	4	4	no	—
482	C₂²	—	4	4	no	—
483	C₂²	—	4	8	no	—
484	C₂²	—	4	8	no	—
485	C₂²	—	8	16	no	—
486	C₂²	C₂×#121×C₆₇	1	4	yes	—
487	C₂²	#532×C₆₇	1	4	yes	—
488	C₂²	#535×C₆₇	1	4	yes	—
489	C₂²	#546×C₆₇	1	8	yes	—
490	C₂²	#559×C₆₇	1	8	yes	—
491	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
492	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
493	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
494	C₂²	C₂²×#41×C₆₇	1	2	yes	—
495	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
496	C₄	—	2	4	no	—
497	C₄	—	2	4	no	—
498	C₄	—	2	8	no	—
499	C₄	—	4	4	no	—
500	C₄	—	4	4	no	—
501	C₄	—	4	4	no	—
502	C₄	C₂×#121×C₆₇	1	8	yes	—
503	C₂	—	4	8	no	—
504	C₂	—	8	4	no	—
505	C₂	C₆₇×#2544	1	8	yes	direct product factor
506	C₂	C₂²×#121×C₆₇	1	1	yes	characteristic
507	C₂	C₂×#532×C₆₇	1	1	yes	characteristic
508	C₂	C₂×#535×C₆₇	1	1	yes	characteristic
509	C₂	C₂×#546×C₆₇	1	2	yes	—
510	C₂	C₂×#559×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂²×#127×C₆₇	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#124×C₆₇	C₂	semidirect
3. C₂²×#124×C₆₇	C₂	semidirect
4. C₂²×C₄×D₄×C₆₇	C₂	semidirect
5. C₆₇×#2544	C₂	direct (N × H)
6. C₂×#394×C₆₇	C₂	semidirect
7. C₂×#391×C₆₇	C₂	semidirect
8. C₂×#390×C₆₇	C₂	semidirect
9. C₂×#386×C₆₇	C₂	semidirect
10. C₂×C₄×C₄⋊C₄×C₆₇	C₂	semidirect
20. C₂×#124×C₆₇	C₂²	semidirect
21. C₂×#124×C₆₇	C₂²	semidirect
23. C₂×C₄×D₄×C₆₇	C₂²	semidirect
34. #394×C₆₇	C₂²	semidirect
35. #391×C₆₇	C₂²	semidirect
36. #390×C₆₇	C₂²	semidirect
37. #386×C₆₇	C₂²	semidirect
38. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
89. C₂×C₄²×C₆₇	D₄	semidirect
161. C₄²×C₆₇	C₂×D₄	semidirect
247. C₂×#2544	C₆₇	direct (N × H)
250. C₂×C₆₇	#2544	direct (N × H)
257. C₂²×#124	C₂×C₆₇	semidirect
258. C₂²×#124	C₂×C₆₇	semidirect
259. C₂²×C₄×D₄	C₂×C₆₇	semidirect
260. #2544	C₂×C₆₇	direct (N × H)
261. C₂×#394	C₂×C₆₇	semidirect
262. C₂×#391	C₂×C₆₇	semidirect
263. C₂×#390	C₂×C₆₇	semidirect
264. C₂×#386	C₂×C₆₇	semidirect
265. C₂×C₄×C₄⋊C₄	C₂×C₆₇	semidirect
266. C₆₇	C₂×#2544	direct (N × H)
276. C₂×#124	C₂²×C₆₇	semidirect
277. C₂×#124	C₂²×C₆₇	semidirect
279. C₂×C₄×D₄	C₂²×C₆₇	semidirect
290. #394	C₂²×C₆₇	semidirect
291. #391	C₂²×C₆₇	semidirect
292. #390	C₂²×C₆₇	semidirect
293. #386	C₂²×C₆₇	semidirect
294. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
345. C₂×C₄²	D₄×C₆₇	semidirect
417. C₄²	C₂×D₄×C₆₇	semidirect
505. C₂	C₆₇×#2544	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#127×C₆₇	2
C₂²×#124×C₆₇	2
C₂²×C₄×D₄×C₆₇	2
C₆₇×#2544	2
C₂×#394×C₆₇	2
C₂×#391×C₆₇	2
C₂×#390×C₆₇	2
C₂×#386×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₂×#2544	67

Subgroups — C2×C67×#2544 #30714137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2544 #30714137