Subgroups — C₆₇×#22500 #30701751

Order 17152.

1370 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#547×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2354	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1990	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1988	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1895	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1875	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1824	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1802	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#102×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1736	C₂	1	2	yes	maximal, complemented
12	C₂²×#41×C₆₇	—	2	1	no	—
13	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
14	#547×C₆₇	—	2	4	no	—
15	#547×C₆₇	C₄	1	8	yes	complemented
16	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#123×C₆₇	—	2	2	no	—
18	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
19	C₂×#121×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
24	#436×C₆₇	—	2	4	no	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#102×C₆₇	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
31	#390×C₆₇	—	2	1	no	—
32	#388×C₆₇	—	2	2	no	—
33	#384×C₆₇	C₂²	1	1	yes	characteristic
34	#380×C₆₇	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
36	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
37	#340×C₆₇	C₂²	1	2	yes	—
38	#340×C₆₇	C₂²	1	2	yes	—
39	#340×C₆₇	C₂²	1	2	yes	—
40	#340×C₆₇	C₂²	1	2	yes	—
41	#340×C₆₇	C₂²	1	1	yes	characteristic
42	#340×C₆₇	C₂²	1	2	yes	—
43	#340×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×#41×C₆₇	—	2	8	no	—
45	C₂×#41×C₆₇	—	4	4	no	—
46	C₂×#41×C₆₇	—	4	4	no	—
47	C₂²×Q₈×C₆₇	—	2	1	no	—
48	C₂²×D₄×C₆₇	—	2	2	no	—
49	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
50	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₆₇	—	2	2	no	—
52	C₂³×C₄×C₆₇	—	2	1	no	—
53	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
54	C₂×M₁₆×C₆₇	—	2	4	no	—
55	C₂×M₁₆×C₆₇	—	2	4	no	—
56	C₂×M₁₆×C₆₇	—	4	2	no	—
57	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
58	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈×C₆₇	—	2	2	no	—
61	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
62	#123×C₆₇	—	2	16	no	—
63	C₂²⋊Q₈×C₆₇	—	2	8	no	—
64	#121×C₆₇	—	2	4	no	—
65	#121×C₆₇	C₂×C₄	1	8	yes	—
66	C₄×D₄×C₆₇	—	2	2	no	—
67	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
68	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
69	C₄×D₄×C₆₇	D₄	1	4	yes	—
70	#117×C₆₇	—	2	4	no	—
71	#117×C₆₇	—	4	2	no	—
72	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
73	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
76	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
77	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
78	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
84	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
85	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
89	C₂×C₄²×C₆₇	—	2	2	no	—
90	C₂×C₄²×C₆₇	—	2	1	no	—
91	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
92	C₄⋊C₈×C₆₇	—	2	2	no	—
93	#102×C₆₇	—	2	2	no	—
94	#102×C₆₇	—	2	2	no	—
95	#102×C₆₇	—	4	2	no	—
96	C₂²⋊C₈×C₆₇	—	2	4	no	—
97	C₄×C₈×C₆₇	—	2	2	no	—
98	#95×C₆₇	—	2	2	no	—
99	#95×C₆₇	—	2	1	no	—
100	C₂⁴×C₆₇	—	2	1	no	—
101	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
102	#41×C₆₇	—	4	32	no	—
103	C₂×Q₈×C₆₇	—	2	4	no	—
104	C₂×Q₈×C₆₇	—	4	2	no	—
105	C₂×Q₈×C₆₇	—	4	2	no	—
106	C₂×D₄×C₆₇	—	2	8	no	—
107	C₂×D₄×C₆₇	—	2	2	no	—
108	C₂×D₄×C₆₇	—	2	2	no	—
109	C₂×D₄×C₆₇	—	2	1	no	—
110	C₂×D₄×C₆₇	—	2	1	no	—
111	C₂×D₄×C₆₇	—	2	1	no	—
112	C₂×D₄×C₆₇	—	2	1	no	—
113	C₂×D₄×C₆₇	—	2	1	no	—
114	C₂×D₄×C₆₇	—	4	4	no	—
115	C₂×D₄×C₆₇	—	4	4	no	—
116	C₂×D₄×C₆₇	C₂²⋊C₄	1	4	yes	—
117	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
118	C₂²×C₄×C₆₇	—	2	2	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	—	2	2	no	—
121	C₂²×C₄×C₆₇	—	2	2	no	—
122	C₂²×C₄×C₆₇	—	2	8	no	—
123	C₂²×C₄×C₆₇	—	2	1	no	—
124	C₂²×C₄×C₆₇	—	2	1	no	—
125	C₂²×C₄×C₆₇	—	2	2	no	—
126	C₂²×C₄×C₆₇	—	2	4	no	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	—	2	1	no	—
129	C₂²×C₄×C₆₇	—	4	4	no	—
130	C₂²×C₄×C₆₇	—	4	4	no	—
131	C₂²×C₄×C₆₇	—	4	2	no	—
132	C₂²×C₄×C₆₇	—	4	2	no	—
133	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
134	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
137	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
139	M₁₆×C₆₇	—	4	8	no	—
140	C₂×C₈×C₆₇	—	2	1	no	—
141	C₂×C₈×C₆₇	—	2	1	no	—
142	C₂×C₈×C₆₇	—	2	2	no	—
143	C₂×C₈×C₆₇	—	2	1	no	—
144	C₂×C₈×C₆₇	—	2	1	no	—
145	C₂×C₈×C₆₇	—	2	2	no	—
146	C₂×C₈×C₆₇	—	2	2	no	—
147	C₂×C₈×C₆₇	—	2	1	no	—
148	C₂×C₈×C₆₇	#41	1	2	yes	complemented
149	C₄⋊C₄×C₆₇	—	2	8	no	—
150	C₄⋊C₄×C₆₇	—	2	4	no	—
151	C₄⋊C₄×C₆₇	—	2	1	no	—
152	C₄⋊C₄×C₆₇	—	2	2	no	—
153	C₄⋊C₄×C₆₇	—	2	1	no	—
154	C₄⋊C₄×C₆₇	—	2	1	no	—
155	C₄⋊C₄×C₆₇	—	2	2	no	—
156	C₄⋊C₄×C₆₇	—	2	2	no	—
157	C₄⋊C₄×C₆₇	—	2	1	no	—
158	C₄⋊C₄×C₆₇	—	4	2	no	—
159	C₄⋊C₄×C₆₇	—	8	1	no	—
160	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
161	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
162	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
163	C₄⋊C₄×C₆₇	#41	1	2	yes	complemented
164	C₂²⋊C₄×C₆₇	—	2	4	no	—
165	C₂²⋊C₄×C₆₇	—	2	8	no	—
166	C₂²⋊C₄×C₆₇	—	4	4	no	—
167	C₂²⋊C₄×C₆₇	—	8	2	no	—
168	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
169	C₄²×C₆₇	—	4	4	no	—
170	C₄²×C₆₇	—	4	2	no	—
171	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
172	C₄²×C₆₇	C₂×D₄	1	2	yes	—
173	C₂³×C₆₇	—	2	2	no	—
174	C₂³×C₆₇	—	2	2	no	—
175	C₂³×C₆₇	—	2	2	no	—
176	C₂³×C₆₇	—	2	2	no	—
177	C₂³×C₆₇	—	2	2	no	—
178	C₂³×C₆₇	—	2	2	no	—
179	C₂³×C₆₇	—	2	2	no	—
180	C₂³×C₆₇	—	4	1	no	—
181	C₂³×C₆₇	—	4	1	no	—
182	C₂³×C₆₇	—	4	1	no	—
183	C₂³×C₆₇	—	4	1	no	—
184	C₂³×C₆₇	—	4	1	no	—
185	C₂³×C₆₇	—	4	1	no	—
186	C₂³×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
187	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
188	Q₈×C₆₇	—	4	8	no	—
189	D₄×C₆₇	—	2	2	no	—
190	D₄×C₆₇	—	2	2	no	—
191	D₄×C₆₇	—	2	8	no	—
192	D₄×C₆₇	—	4	16	no	—
193	D₄×C₆₇	—	4	1	no	—
194	D₄×C₆₇	—	4	1	no	—
195	C₂×C₄×C₆₇	—	2	2	no	—
196	C₂×C₄×C₆₇	—	2	2	no	—
197	C₂×C₄×C₆₇	—	2	1	no	—
198	C₂×C₄×C₆₇	—	2	1	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	2	no	—
204	C₂×C₄×C₆₇	—	2	4	no	—
205	C₂×C₄×C₆₇	—	2	2	no	—
206	C₂×C₄×C₆₇	—	4	16	no	—
207	C₂×C₄×C₆₇	—	4	1	no	—
208	C₂×C₄×C₆₇	—	4	1	no	—
209	C₂×C₄×C₆₇	—	4	1	no	—
210	C₂×C₄×C₆₇	—	4	2	no	—
211	C₂×C₄×C₆₇	—	4	2	no	—
212	C₂×C₄×C₆₇	—	4	1	no	—
213	C₂×C₄×C₆₇	—	4	8	no	—
214	C₂×C₄×C₆₇	—	4	2	no	—
215	C₂×C₄×C₆₇	—	4	1	no	—
216	C₂×C₄×C₆₇	—	4	1	no	—
217	C₂×C₄×C₆₇	—	4	2	no	—
218	C₂×C₄×C₆₇	—	4	1	no	—
219	C₂×C₄×C₆₇	—	4	2	no	—
220	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
221	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
222	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
223	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
224	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
225	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
226	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
228	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
229	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
230	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
231	C₈×C₆₇	—	2	2	no	—
232	C₈×C₆₇	—	4	2	no	—
233	C₈×C₆₇	—	4	1	no	—
234	C₂²×C₆₇	—	2	4	no	—
235	C₂²×C₆₇	—	2	4	no	—
236	C₂²×C₆₇	—	2	4	no	—
237	C₂²×C₆₇	—	4	2	no	—
238	C₂²×C₆₇	—	4	2	no	—
239	C₂²×C₆₇	—	4	2	no	—
240	C₂²×C₆₇	—	4	2	no	—
241	C₂²×C₆₇	—	4	2	no	—
242	C₂²×C₆₇	—	4	2	no	—
243	C₂²×C₆₇	—	4	2	no	—
244	C₂²×C₆₇	—	8	1	no	—
245	C₂²×C₆₇	—	8	1	no	—
246	C₂²×C₆₇	—	8	1	no	—
247	C₂²×C₆₇	—	8	1	no	—
248	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
249	C₂²×C₆₇	#546	1	1	yes	characteristic
250	C₂²×C₆₇	#547	1	1	yes	characteristic
251	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
252	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
253	C₂²×C₆₇	#577	1	1	yes	characteristic
254	C₂²×C₆₇	#578	1	1	yes	characteristic
255	C₄×C₆₇	—	2	4	no	—
256	C₄×C₆₇	—	2	2	no	—
257	C₄×C₆₇	—	4	2	no	—
258	C₄×C₆₇	—	4	2	no	—
259	C₄×C₆₇	—	4	4	no	—
260	C₄×C₆₇	—	8	2	no	—
261	C₄×C₆₇	—	8	1	no	—
262	C₄×C₆₇	#418	1	4	yes	—
263	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
264	C₄×C₆₇	#520	1	1	yes	characteristic
265	C₄×C₆₇	#546	1	1	yes	characteristic
266	C₄×C₆₇	#547	1	1	yes	characteristic
267	#22500	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
268	C₂×C₆₇	—	4	4	no	—
269	C₂×C₆₇	—	8	2	no	—
270	C₂×C₆₇	#2339	1	1	yes	characteristic
271	C₂×C₆₇	C₂×#418	1	1	yes	characteristic
272	C₂×C₆₇	#2898	1	1	yes	characteristic
273	C₂×C₆₇	#3330	1	1	yes	characteristic
274	C₂×C₆₇	#3332	1	1	yes	characteristic
275	C₂×C₆₇	#3353	1	1	yes	characteristic
276	C₂×C₆₇	#3354	1	1	yes	characteristic
277	C₂×#547	C₂×C₆₇	1	1	yes	characteristic
278	C₂×#436	C₂×C₆₇	1	1	yes	characteristic, complemented
279	#2354	C₂×C₆₇	1	1	yes	characteristic, complemented
280	#1990	C₂×C₆₇	1	1	yes	characteristic, complemented
281	#1988	C₂×C₆₇	1	1	yes	characteristic, complemented
282	#1895	C₂×C₆₇	1	2	yes	complemented
283	#1875	C₂×C₆₇	1	2	yes	complemented
284	#1824	C₂×C₆₇	1	2	yes	complemented
285	#1802	C₂×C₆₇	1	1	yes	characteristic, complemented
286	C₄×#102	C₂×C₆₇	1	1	yes	characteristic, complemented
287	#1736	C₂×C₆₇	1	2	yes	complemented
288	C₆₇	#22500	1	1	yes	characteristic, direct product factor, Sylow
289	C₂²×#41	—	2	1	no	—
290	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
291	#547	—	2	4	no	—
292	#547	C₄×C₆₇	1	8	yes	complemented
293	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
294	C₂×#123	—	2	2	no	—
295	C₂×C₂²⋊Q₈	—	2	1	no	—
296	C₂×#121	C₂²×C₆₇	1	2	yes	—
297	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
298	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
299	C₂×#117	C₂²×C₆₇	1	2	yes	—
300	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
301	#436	—	2	4	no	—
302	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
303	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
304	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
305	C₂×#102	C₂²×C₆₇	1	2	yes	—
306	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
307	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
308	#390	—	2	1	no	—
309	#388	—	2	2	no	—
310	#384	C₂²×C₆₇	1	1	yes	characteristic
311	#380	C₂²×C₆₇	1	2	yes	—
312	C₄×C₄⋊C₄	—	2	1	no	—
313	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
314	#340	C₂²×C₆₇	1	2	yes	—
315	#340	C₂²×C₆₇	1	1	yes	characteristic
316	#340	C₂²×C₆₇	1	2	yes	—
317	#340	C₂²×C₆₇	1	2	yes	—
318	#340	C₂²×C₆₇	1	2	yes	—
319	#340	C₂²×C₆₇	1	2	yes	—
320	#340	C₂²×C₆₇	1	1	yes	characteristic
321	C₂×#41	—	2	8	no	—
322	C₂×#41	—	4	4	no	—
323	C₂×#41	—	4	4	no	—
324	C₂²×Q₈	—	2	1	no	—
325	C₂²×D₄	—	2	2	no	—
326	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
327	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
328	C₂³×C₄	—	2	2	no	—
329	C₂³×C₄	—	2	1	no	—
330	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
331	C₂×M₁₆	—	2	4	no	—
332	C₂×M₁₆	—	2	4	no	—
333	C₂×M₁₆	—	4	2	no	—
334	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
335	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
336	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
337	C₄⋊Q₈	—	2	2	no	—
338	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
339	#123	—	2	16	no	—
340	C₂²⋊Q₈	—	2	8	no	—
341	#121	—	2	4	no	—
342	#121	C₂×C₄×C₆₇	1	8	yes	—
343	C₄×D₄	—	2	2	no	—
344	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
345	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
346	C₄×D₄	D₄×C₆₇	1	4	yes	—
347	#117	—	2	4	no	—
348	#117	—	4	2	no	—
349	C₂×C₄⋊C₄	—	2	2	no	—
350	C₂×C₄⋊C₄	—	2	4	no	—
351	C₂×C₄⋊C₄	—	2	1	no	—
352	C₂×C₄⋊C₄	—	2	1	no	—
353	C₂×C₄⋊C₄	—	2	1	no	—
354	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
355	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
356	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
357	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
358	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
359	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
360	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
361	C₂×C₂²⋊C₄	—	2	1	no	—
362	C₂×C₂²⋊C₄	—	2	2	no	—
363	C₂×C₂²⋊C₄	—	2	2	no	—
364	C₂×C₂²⋊C₄	—	2	2	no	—
365	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
366	C₂×C₄²	—	2	2	no	—
367	C₂×C₄²	—	2	1	no	—
368	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
369	C₄⋊C₈	—	2	2	no	—
370	#102	—	2	2	no	—
371	#102	—	2	2	no	—
372	#102	—	4	2	no	—
373	C₂²⋊C₈	—	2	4	no	—
374	C₄×C₈	—	2	2	no	—
375	#95	—	2	2	no	—
376	#95	—	2	1	no	—
377	C₂⁴	—	2	1	no	—
378	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
379	#41	—	4	32	no	—
380	C₂×Q₈	—	2	4	no	—
381	C₂×Q₈	—	4	2	no	—
382	C₂×Q₈	—	4	2	no	—
383	C₂×D₄	—	2	8	no	—
384	C₂×D₄	—	2	2	no	—
385	C₂×D₄	—	2	2	no	—
386	C₂×D₄	—	2	1	no	—
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	—	2	1	no	—
389	C₂×D₄	—	2	1	no	—
390	C₂×D₄	—	2	1	no	—
391	C₂×D₄	—	4	4	no	—
392	C₂×D₄	—	4	4	no	—
393	C₂×D₄	C₂²⋊C₄×C₆₇	1	4	yes	—
394	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
395	C₂²×C₄	—	2	2	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	—	2	8	no	—
400	C₂²×C₄	—	2	1	no	—
401	C₂²×C₄	—	2	1	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	4	no	—
404	C₂²×C₄	—	2	1	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	4	4	no	—
407	C₂²×C₄	—	4	4	no	—
408	C₂²×C₄	—	4	2	no	—
409	C₂²×C₄	—	4	2	no	—
410	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
411	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
412	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
413	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
414	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
415	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
416	M₁₆	—	4	8	no	—
417	C₂×C₈	—	2	1	no	—
418	C₂×C₈	—	2	1	no	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	2	2	no	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	2	1	no	—
425	C₂×C₈	#41×C₆₇	1	2	yes	complemented
426	C₄⋊C₄	—	2	8	no	—
427	C₄⋊C₄	—	2	4	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	2	2	no	—
430	C₄⋊C₄	—	2	1	no	—
431	C₄⋊C₄	—	2	1	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	2	no	—
434	C₄⋊C₄	—	2	1	no	—
435	C₄⋊C₄	—	4	2	no	—
436	C₄⋊C₄	—	8	1	no	—
437	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
438	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
439	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
440	C₄⋊C₄	#41×C₆₇	1	2	yes	complemented
441	C₂²⋊C₄	—	2	4	no	—
442	C₂²⋊C₄	—	2	8	no	—
443	C₂²⋊C₄	—	4	4	no	—
444	C₂²⋊C₄	—	8	2	no	—
445	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
446	C₄²	—	4	4	no	—
447	C₄²	—	4	2	no	—
448	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
449	C₄²	C₂×D₄×C₆₇	1	2	yes	—
450	C₂³	—	2	2	no	—
451	C₂³	—	2	2	no	—
452	C₂³	—	2	2	no	—
453	C₂³	—	2	2	no	—
454	C₂³	—	2	2	no	—
455	C₂³	—	2	2	no	—
456	C₂³	—	2	2	no	—
457	C₂³	—	4	1	no	—
458	C₂³	—	4	1	no	—
459	C₂³	—	4	1	no	—
460	C₂³	—	4	1	no	—
461	C₂³	—	4	1	no	—
462	C₂³	—	4	1	no	—
463	C₂³	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
464	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
465	Q₈	—	4	8	no	—
466	D₄	—	2	2	no	—
467	D₄	—	2	2	no	—
468	D₄	—	2	8	no	—
469	D₄	—	4	16	no	—
470	D₄	—	4	1	no	—
471	D₄	—	4	1	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	2	4	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	4	16	no	—
484	C₂×C₄	—	4	1	no	—
485	C₂×C₄	—	4	1	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	1	no	—
490	C₂×C₄	—	4	8	no	—
491	C₂×C₄	—	4	2	no	—
492	C₂×C₄	—	4	1	no	—
493	C₂×C₄	—	4	1	no	—
494	C₂×C₄	—	4	2	no	—
495	C₂×C₄	—	4	1	no	—
496	C₂×C₄	—	4	2	no	—
497	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
498	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
499	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
500	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
501	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
502	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
503	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
504	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
505	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
506	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
507	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
508	C₈	—	2	2	no	—
509	C₈	—	4	2	no	—
510	C₈	—	4	1	no	—
511	C₂²	—	2	4	no	—
512	C₂²	—	2	4	no	—
513	C₂²	—	2	4	no	—
514	C₂²	—	4	2	no	—
515	C₂²	—	4	2	no	—
516	C₂²	—	4	2	no	—
517	C₂²	—	4	2	no	—
518	C₂²	—	4	2	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	2	no	—
521	C₂²	—	8	1	no	—
522	C₂²	—	8	1	no	—
523	C₂²	—	8	1	no	—
524	C₂²	—	8	1	no	—
525	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
526	C₂²	#546×C₆₇	1	1	yes	characteristic
527	C₂²	#547×C₆₇	1	1	yes	characteristic
528	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
529	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
530	C₂²	#577×C₆₇	1	1	yes	characteristic
531	C₂²	#578×C₆₇	1	1	yes	characteristic
532	C₄	—	2	4	no	—
533	C₄	—	2	2	no	—
534	C₄	—	4	2	no	—
535	C₄	—	4	2	no	—
536	C₄	—	4	4	no	—
537	C₄	—	8	2	no	—
538	C₄	—	8	1	no	—
539	C₄	#418×C₆₇	1	4	yes	—
540	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
541	C₄	#520×C₆₇	1	1	yes	characteristic
542	C₄	#546×C₆₇	1	1	yes	characteristic
543	C₄	#547×C₆₇	1	1	yes	characteristic
544	C₂	—	4	4	no	—
545	C₂	—	8	2	no	—
546	C₂	C₆₇×#2339	1	1	yes	characteristic
547	C₂	C₂×#418×C₆₇	1	1	yes	characteristic
548	C₂	C₆₇×#2898	1	1	yes	characteristic
549	C₂	C₆₇×#3330	1	1	yes	characteristic
550	C₂	C₆₇×#3332	1	1	yes	characteristic
551	C₂	C₆₇×#3353	1	1	yes	characteristic
552	C₂	C₆₇×#3354	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#547×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#436×C₆₇	C₂	semidirect
3. C₆₇×#2354	C₂	semidirect
4. C₆₇×#1990	C₂	semidirect
5. C₆₇×#1988	C₂	semidirect
6. C₆₇×#1895	C₂	semidirect
7. C₆₇×#1875	C₂	semidirect
8. C₆₇×#1824	C₂	semidirect
9. C₆₇×#1802	C₂	semidirect
10. C₄×#102×C₆₇	C₂	semidirect
11. C₆₇×#1736	C₂	semidirect
15. #547×C₆₇	C₄	semidirect
148. C₂×C₈×C₆₇	#41	semidirect
163. C₄⋊C₄×C₆₇	#41	semidirect
267. #22500	C₆₇	direct (N × H)
278. C₂×#436	C₂×C₆₇	semidirect
279. #2354	C₂×C₆₇	semidirect
280. #1990	C₂×C₆₇	semidirect
281. #1988	C₂×C₆₇	semidirect
282. #1895	C₂×C₆₇	semidirect
283. #1875	C₂×C₆₇	semidirect
284. #1824	C₂×C₆₇	semidirect
285. #1802	C₂×C₆₇	semidirect
286. C₄×#102	C₂×C₆₇	semidirect
287. #1736	C₂×C₆₇	semidirect
288. C₆₇	#22500	direct (N × H)
292. #547	C₄×C₆₇	semidirect
425. C₂×C₈	#41×C₆₇	semidirect
440. C₄⋊C₄	#41×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#547×C₆₇	2
C₂×#436×C₆₇	2
C₆₇×#2354	2
C₆₇×#1990	2
C₆₇×#1988	2
C₆₇×#1895	2
C₆₇×#1875	2
C₆₇×#1824	2
C₆₇×#1802	2
C₄×#102×C₆₇	2
C₆₇×#1736	2
#22500	67

Subgroups — C67×#22500 #30701751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22500 #30701751