Subgroups — C₆₇×#60369 #30739620

Order 17152.

4938 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#585×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×AGL(1,ℤ/8)×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#535×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3010	C₂	1	8	yes	maximal, complemented
5	C₆₇×#3008	C₂	1	8	yes	maximal, complemented
6	C₆₇×#3006	C₂	1	8	yes	maximal, complemented
7	C₆₇×#2896	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2879	C₂	1	1	yes	maximal, characteristic, complemented
9	#585×C₆₇	—	2	16	no	—
10	C₂×2₋¹⁺⁴×C₆₇	—	2	1	no	—
11	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	4	yes	complemented
12	C₂×H(5,𝔽₂)×C₆₇	—	2	2	no	—
13	C₂×H(5,𝔽₂)×C₆₇	—	2	1	no	—
14	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	4	yes	complemented
15	C₂²×#41×C₆₇	—	2	4	no	—
16	C₂²×#41×C₆₇	C₂²	1	2	yes	—
17	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×#41×C₆₇	C₂²	1	4	yes	complemented
19	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
20	C₂×AGL(1,ℤ/8)×C₆₇	—	2	16	no	—
21	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	8	yes	complemented
22	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	8	yes	complemented
23	C₂²×SD₁₆×C₆₇	C₂²	1	4	yes	complemented
24	C₂²×D₈×C₆₇	C₂²	1	4	yes	complemented
25	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	complemented
26	#535×C₆₇	—	2	4	no	—
27	#535×C₆₇	C₂²	1	8	yes	complemented
28	C₂×#127×C₆₇	—	2	1	no	—
29	C₂×#124×C₆₇	—	2	1	no	—
30	C₂×#121×C₆₇	C₂²	1	4	yes	complemented
31	C₂×#120×C₆₇	—	2	2	no	—
32	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic, complemented
33	#449×C₆₇	C₂²	1	32	yes	complemented
34	#448×C₆₇	C₂²	1	16	yes	complemented
35	#447×C₆₇	C₂²	1	16	yes	complemented
36	#419×C₆₇	C₂²	1	4	yes	complemented
37	#418×C₆₇	C₂²	1	4	yes	complemented
38	#417×C₆₇	C₂²	1	8	yes	complemented
39	#416×C₆₇	C₂²	1	8	yes	complemented
40	#408×C₆₇	C₂²	1	8	yes	complemented
41	#407×C₆₇	C₂²	1	4	yes	complemented
42	C₂⁵×C₆₇	—	2	1	no	—
43	2₋¹⁺⁴×C₆₇	—	2	16	no	—
44	2₋¹⁺⁴×C₆₇	—	2	16	no	—
45	2₋¹⁺⁴×C₆₇	—	2	16	no	—
46	H(5,𝔽₂)×C₆₇	—	2	16	no	—
47	H(5,𝔽₂)×C₆₇	—	2	32	no	—
48	H(5,𝔽₂)×C₆₇	—	2	16	no	—
49	H(5,𝔽₂)×C₆₇	—	2	16	no	—
50	C₂×#41×C₆₇	—	2	64	no	—
51	C₂×#41×C₆₇	—	2	32	no	—
52	C₂×#41×C₆₇	—	2	4	no	—
53	C₂×#41×C₆₇	—	2	8	no	—
54	C₂×#41×C₆₇	—	2	4	no	—
55	C₂×#41×C₆₇	—	2	2	no	—
56	C₂×#41×C₆₇	—	2	4	no	—
57	C₂×#41×C₆₇	—	2	2	no	—
58	C₂×#41×C₆₇	—	2	8	no	—
59	C₂×#41×C₆₇	—	2	8	no	—
60	C₂×#41×C₆₇	—	2	4	no	—
61	C₂×#41×C₆₇	D₄	1	4	yes	—
62	C₂×#41×C₆₇	D₄	1	16	yes	complemented
63	C₂×#41×C₆₇	C₂³	1	4	yes	—
64	C₂×#41×C₆₇	C₂³	1	4	yes	—
65	C₂×#41×C₆₇	C₂³	1	4	yes	—
66	C₂×#41×C₆₇	C₂³	1	8	yes	complemented
67	C₂²×Q₈×C₆₇	—	2	4	no	—
68	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
69	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
70	C₂²×Q₈×C₆₇	C₂³	1	4	yes	complemented
71	C₂²×D₄×C₆₇	—	2	2	no	—
72	C₂²×D₄×C₆₇	—	2	4	no	—
73	C₂²×D₄×C₆₇	—	2	4	no	—
74	C₂²×D₄×C₆₇	—	2	8	no	—
75	C₂²×D₄×C₆₇	—	2	4	no	—
76	C₂²×D₄×C₆₇	—	2	8	no	—
77	C₂²×D₄×C₆₇	—	2	1	no	—
78	C₂²×D₄×C₆₇	—	2	2	no	—
79	C₂²×D₄×C₆₇	D₄	1	2	yes	—
80	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
81	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
82	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
83	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
84	C₂²×D₄×C₆₇	C₂³	1	4	yes	complemented
85	C₂³×C₄×C₆₇	—	2	2	no	—
86	C₂³×C₄×C₆₇	—	2	4	no	—
87	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
88	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
89	AGL(1,ℤ/8)×C₆₇	—	2	32	no	—
90	AGL(1,ℤ/8)×C₆₇	—	2	32	no	—
91	C₂×SD₁₆×C₆₇	—	2	16	no	—
92	C₂×SD₁₆×C₆₇	C₂³	1	16	yes	—
93	C₂×D₈×C₆₇	—	2	16	no	—
94	C₂×D₈×C₆₇	C₂³	1	16	yes	—
95	C₂×M₁₆×C₆₇	—	2	8	no	—
96	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
97	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
98	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
99	#127×C₆₇	—	2	8	no	—
100	#124×C₆₇	—	2	8	no	—
101	#121×C₆₇	—	2	8	no	—
102	#121×C₆₇	C₂³	1	16	yes	—
103	#120×C₆₇	—	2	16	no	—
104	#117×C₆₇	—	2	2	no	—
105	#117×C₆₇	C₂³	1	4	yes	—
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
107	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
108	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
109	C₂×C₄²×C₆₇	—	2	1	no	—
110	#103×C₆₇	C₂³	1	16	yes	complemented
111	#102×C₆₇	C₂³	1	16	yes	complemented
112	C₂²⋊C₈×C₆₇	C₂³	1	16	yes	complemented
113	C₂⁴×C₆₇	—	2	2	no	—
114	C₂⁴×C₆₇	—	2	4	no	—
115	C₂⁴×C₆₇	—	2	2	no	—
116	C₂⁴×C₆₇	—	2	4	no	—
117	C₂⁴×C₆₇	—	4	4	no	—
118	C₂⁴×C₆₇	—	8	4	no	—
119	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
120	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
121	#41×C₆₇	—	2	32	no	—
122	#41×C₆₇	—	2	64	no	—
123	#41×C₆₇	—	2	32	no	—
124	#41×C₆₇	—	2	32	no	—
125	#41×C₆₇	—	2	32	no	—
126	#41×C₆₇	—	2	16	no	—
127	#41×C₆₇	—	2	16	no	—
128	#41×C₆₇	—	2	16	no	—
129	#41×C₆₇	—	2	16	no	—
130	#41×C₆₇	—	2	16	no	—
131	#41×C₆₇	—	2	8	no	—
132	#41×C₆₇	—	2	8	no	—
133	#41×C₆₇	—	2	8	no	—
134	#41×C₆₇	—	2	8	no	—
135	#41×C₆₇	C₂×D₄	1	16	yes	—
136	#41×C₆₇	C₂×D₄	1	16	yes	complemented
137	C₂×Q₈×C₆₇	—	2	32	no	—
138	C₂×Q₈×C₆₇	—	2	16	no	—
139	C₂×Q₈×C₆₇	—	2	4	no	—
140	C₂×Q₈×C₆₇	—	2	2	no	—
141	C₂×Q₈×C₆₇	—	2	4	no	—
142	C₂×Q₈×C₆₇	—	2	4	no	—
143	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
144	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	complemented
145	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
146	C₂×D₄×C₆₇	—	2	32	no	—
147	C₂×D₄×C₆₇	—	2	64	no	—
148	C₂×D₄×C₆₇	—	2	32	no	—
149	C₂×D₄×C₆₇	—	2	16	no	—
150	C₂×D₄×C₆₇	—	2	8	no	—
151	C₂×D₄×C₆₇	—	2	4	no	—
152	C₂×D₄×C₆₇	—	2	4	no	—
153	C₂×D₄×C₆₇	—	2	8	no	—
154	C₂×D₄×C₆₇	—	2	16	no	—
155	C₂×D₄×C₆₇	—	2	16	no	—
156	C₂×D₄×C₆₇	—	2	4	no	—
157	C₂×D₄×C₆₇	—	2	2	no	—
158	C₂×D₄×C₆₇	—	2	4	no	—
159	C₂×D₄×C₆₇	—	2	4	no	—
160	C₂×D₄×C₆₇	—	2	8	no	—
161	C₂×D₄×C₆₇	—	2	8	no	—
162	C₂×D₄×C₆₇	—	2	8	no	—
163	C₂×D₄×C₆₇	—	2	2	no	—
164	C₂×D₄×C₆₇	—	2	4	no	—
165	C₂×D₄×C₆₇	—	4	8	no	—
166	C₂×D₄×C₆₇	—	4	4	no	—
167	C₂×D₄×C₆₇	—	8	8	no	—
168	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
169	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
170	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	complemented
171	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
172	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₆₇	—	2	32	no	—
174	C₂²×C₄×C₆₇	—	2	2	no	—
175	C₂²×C₄×C₆₇	—	2	4	no	—
176	C₂²×C₄×C₆₇	—	2	16	no	—
177	C₂²×C₄×C₆₇	—	2	4	no	—
178	C₂²×C₄×C₆₇	—	2	4	no	—
179	C₂²×C₄×C₆₇	—	2	8	no	—
180	C₂²×C₄×C₆₇	—	2	8	no	—
181	C₂²×C₄×C₆₇	—	2	2	no	—
182	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
183	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
184	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
185	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
186	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
187	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
188	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
189	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
190	SD₁₆×C₆₇	—	2	32	no	—
191	D₈×C₆₇	—	2	32	no	—
192	M₁₆×C₆₇	—	2	8	no	—
193	M₁₆×C₆₇	—	2	8	no	—
194	C₂×C₈×C₆₇	—	2	8	no	—
195	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
196	C₄⋊C₄×C₆₇	—	2	2	no	—
197	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
198	C₂²⋊C₄×C₆₇	—	2	4	no	—
199	C₂²⋊C₄×C₆₇	—	2	16	no	—
200	C₄²×C₆₇	—	2	4	no	—
201	C₂³×C₆₇	—	2	16	no	—
202	C₂³×C₆₇	—	2	4	no	—
203	C₂³×C₆₇	—	2	4	no	—
204	C₂³×C₆₇	—	2	8	no	—
205	C₂³×C₆₇	—	2	4	no	—
206	C₂³×C₆₇	—	4	2	no	—
207	C₂³×C₆₇	—	4	4	no	—
208	C₂³×C₆₇	—	4	8	no	—
209	C₂³×C₆₇	—	4	4	no	—
210	C₂³×C₆₇	—	4	8	no	—
211	C₂³×C₆₇	—	8	4	no	—
212	C₂³×C₆₇	—	8	8	no	—
213	C₂³×C₆₇	—	8	8	no	—
214	C₂³×C₆₇	—	8	16	no	—
215	C₂³×C₆₇	#120	1	4	yes	—
216	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
217	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
218	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
219	Q₈×C₆₇	—	2	16	no	—
220	Q₈×C₆₇	—	2	8	no	—
221	Q₈×C₆₇	—	2	8	no	—
222	Q₈×C₆₇	—	2	4	no	—
223	Q₈×C₆₇	C₂²×D₄	1	8	yes	—
224	D₄×C₆₇	—	2	32	no	—
225	D₄×C₆₇	—	2	16	no	—
226	D₄×C₆₇	—	2	16	no	—
227	D₄×C₆₇	—	2	16	no	—
228	D₄×C₆₇	—	2	16	no	—
229	D₄×C₆₇	—	2	8	no	—
230	D₄×C₆₇	—	2	8	no	—
231	D₄×C₆₇	—	2	4	no	—
232	D₄×C₆₇	—	2	8	no	—
233	D₄×C₆₇	—	4	4	no	—
234	D₄×C₆₇	—	4	8	no	—
235	D₄×C₆₇	—	8	16	no	—
236	D₄×C₆₇	C₂²×D₄	1	8	yes	—
237	C₂×C₄×C₆₇	—	2	16	no	—
238	C₂×C₄×C₆₇	—	2	32	no	—
239	C₂×C₄×C₆₇	—	2	8	no	—
240	C₂×C₄×C₆₇	—	2	4	no	—
241	C₂×C₄×C₆₇	—	2	4	no	—
242	C₂×C₄×C₆₇	—	2	16	no	—
243	C₂×C₄×C₆₇	—	2	8	no	—
244	C₂×C₄×C₆₇	—	2	16	no	—
245	C₂×C₄×C₆₇	—	2	8	no	—
246	C₂×C₄×C₆₇	—	2	4	no	—
247	C₂×C₄×C₆₇	—	4	4	no	—
248	C₂×C₄×C₆₇	#120	1	8	yes	—
249	C₂×C₄×C₆₇	#120	1	4	yes	—
250	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
251	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
252	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
253	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
254	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
255	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
256	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
257	C₈×C₆₇	—	2	8	no	—
258	C₂²×C₆₇	—	2	2	no	—
259	C₂²×C₆₇	—	2	4	no	—
260	C₂²×C₆₇	—	2	8	no	—
261	C₂²×C₆₇	—	2	4	no	—
262	C₂²×C₆₇	—	2	8	no	—
263	C₂²×C₆₇	—	4	4	no	—
264	C₂²×C₆₇	—	4	8	no	—
265	C₂²×C₆₇	—	4	8	no	—
266	C₂²×C₆₇	—	4	4	no	—
267	C₂²×C₆₇	—	8	4	no	—
268	C₂²×C₆₇	—	8	8	no	—
269	C₂²×C₆₇	—	8	8	no	—
270	C₂²×C₆₇	—	8	16	no	—
271	C₂²×C₆₇	—	8	16	no	—
272	C₂²×C₆₇	C₂×#120	1	4	yes	—
273	C₂²×C₆₇	C₂×#120	1	2	yes	—
274	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
275	C₄×C₆₇	—	2	8	no	—
276	C₄×C₆₇	—	2	4	no	—
277	C₄×C₆₇	—	4	4	no	—
278	C₄×C₆₇	C₂×#120	1	2	yes	—
279	C₄×C₆₇	C₂×#120	1	4	yes	—
280	C₄×C₆₇	C₂³×D₄	1	2	yes	—
281	#60369	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
282	C₂×C₆₇	—	2	2	no	—
283	C₂×C₆₇	—	2	4	no	—
284	C₂×C₆₇	—	4	4	no	—
285	C₂×C₆₇	—	4	8	no	—
286	C₂×C₆₇	—	8	4	no	—
287	C₂×C₆₇	C₂²×#120	1	1	yes	characteristic
288	C₂×C₆₇	#3585	1	2	yes	—
289	C₂×#585	C₂×C₆₇	1	1	yes	characteristic, complemented
290	C₂²×AGL(1,ℤ/8)	C₂×C₆₇	1	2	yes	complemented
291	C₂×#535	C₂×C₆₇	1	1	yes	characteristic, complemented
292	#3010	C₂×C₆₇	1	8	yes	complemented
293	#3008	C₂×C₆₇	1	8	yes	complemented
294	#3006	C₂×C₆₇	1	8	yes	complemented
295	#2896	C₂×C₆₇	1	2	yes	complemented
296	#2879	C₂×C₆₇	1	1	yes	characteristic, complemented
297	C₆₇	#60369	1	1	yes	characteristic, direct product factor, Sylow
298	#585	—	2	16	no	—
299	C₂×2₋¹⁺⁴	—	2	1	no	—
300	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	4	yes	complemented
301	C₂×H(5,𝔽₂)	—	2	2	no	—
302	C₂×H(5,𝔽₂)	—	2	1	no	—
303	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	4	yes	complemented
304	C₂²×#41	—	2	4	no	—
305	C₂²×#41	C₂²×C₆₇	1	2	yes	—
306	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
307	C₂²×#41	C₂²×C₆₇	1	4	yes	complemented
308	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
309	C₂×AGL(1,ℤ/8)	—	2	16	no	—
310	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	8	yes	complemented
311	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	8	yes	complemented
312	C₂²×SD₁₆	C₂²×C₆₇	1	4	yes	complemented
313	C₂²×D₈	C₂²×C₆₇	1	4	yes	complemented
314	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	complemented
315	#535	—	2	4	no	—
316	#535	C₂²×C₆₇	1	8	yes	complemented
317	C₂×#127	—	2	1	no	—
318	C₂×#124	—	2	1	no	—
319	C₂×#121	C₂²×C₆₇	1	4	yes	complemented
320	C₂×#120	—	2	2	no	—
321	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic, complemented
322	#449	C₂²×C₆₇	1	32	yes	complemented
323	#448	C₂²×C₆₇	1	16	yes	complemented
324	#447	C₂²×C₆₇	1	16	yes	complemented
325	#419	C₂²×C₆₇	1	4	yes	complemented
326	#418	C₂²×C₆₇	1	4	yes	complemented
327	#417	C₂²×C₆₇	1	8	yes	complemented
328	#416	C₂²×C₆₇	1	8	yes	complemented
329	#408	C₂²×C₆₇	1	8	yes	complemented
330	#407	C₂²×C₆₇	1	4	yes	complemented
331	C₂⁵	—	2	1	no	—
332	2₋¹⁺⁴	—	2	16	no	—
333	2₋¹⁺⁴	—	2	16	no	—
334	2₋¹⁺⁴	—	2	16	no	—
335	H(5,𝔽₂)	—	2	16	no	—
336	H(5,𝔽₂)	—	2	32	no	—
337	H(5,𝔽₂)	—	2	16	no	—
338	H(5,𝔽₂)	—	2	16	no	—
339	C₂×#41	—	2	64	no	—
340	C₂×#41	—	2	32	no	—
341	C₂×#41	—	2	4	no	—
342	C₂×#41	—	2	8	no	—
343	C₂×#41	—	2	4	no	—
344	C₂×#41	—	2	2	no	—
345	C₂×#41	—	2	4	no	—
346	C₂×#41	—	2	2	no	—
347	C₂×#41	—	2	8	no	—
348	C₂×#41	—	2	8	no	—
349	C₂×#41	—	2	4	no	—
350	C₂×#41	D₄×C₆₇	1	4	yes	—
351	C₂×#41	D₄×C₆₇	1	16	yes	complemented
352	C₂×#41	C₂³×C₆₇	1	4	yes	—
353	C₂×#41	C₂³×C₆₇	1	4	yes	—
354	C₂×#41	C₂³×C₆₇	1	4	yes	—
355	C₂×#41	C₂³×C₆₇	1	8	yes	complemented
356	C₂²×Q₈	—	2	4	no	—
357	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
358	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
359	C₂²×Q₈	C₂³×C₆₇	1	4	yes	complemented
360	C₂²×D₄	—	2	2	no	—
361	C₂²×D₄	—	2	4	no	—
362	C₂²×D₄	—	2	4	no	—
363	C₂²×D₄	—	2	8	no	—
364	C₂²×D₄	—	2	4	no	—
365	C₂²×D₄	—	2	8	no	—
366	C₂²×D₄	—	2	1	no	—
367	C₂²×D₄	—	2	2	no	—
368	C₂²×D₄	D₄×C₆₇	1	2	yes	—
369	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
370	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
371	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
372	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
373	C₂²×D₄	C₂³×C₆₇	1	4	yes	complemented
374	C₂³×C₄	—	2	2	no	—
375	C₂³×C₄	—	2	4	no	—
376	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
377	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
378	AGL(1,ℤ/8)	—	2	32	no	—
379	AGL(1,ℤ/8)	—	2	32	no	—
380	C₂×SD₁₆	—	2	16	no	—
381	C₂×SD₁₆	C₂³×C₆₇	1	16	yes	—
382	C₂×D₈	—	2	16	no	—
383	C₂×D₈	C₂³×C₆₇	1	16	yes	—
384	C₂×M₁₆	—	2	8	no	—
385	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
386	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
387	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
388	#127	—	2	8	no	—
389	#124	—	2	8	no	—
390	#121	—	2	8	no	—
391	#121	C₂³×C₆₇	1	16	yes	—
392	#120	—	2	16	no	—
393	#117	—	2	2	no	—
394	#117	C₂³×C₆₇	1	4	yes	—
395	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
396	C₂×C₂²⋊C₄	—	2	1	no	—
397	C₂×C₂²⋊C₄	—	2	4	no	—
398	C₂×C₄²	—	2	1	no	—
399	#103	C₂³×C₆₇	1	16	yes	complemented
400	#102	C₂³×C₆₇	1	16	yes	complemented
401	C₂²⋊C₈	C₂³×C₆₇	1	16	yes	complemented
402	C₂⁴	—	2	2	no	—
403	C₂⁴	—	2	4	no	—
404	C₂⁴	—	2	2	no	—
405	C₂⁴	—	2	4	no	—
406	C₂⁴	—	4	4	no	—
407	C₂⁴	—	8	4	no	—
408	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
409	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
410	#41	—	2	32	no	—
411	#41	—	2	64	no	—
412	#41	—	2	32	no	—
413	#41	—	2	32	no	—
414	#41	—	2	32	no	—
415	#41	—	2	16	no	—
416	#41	—	2	16	no	—
417	#41	—	2	16	no	—
418	#41	—	2	16	no	—
419	#41	—	2	16	no	—
420	#41	—	2	8	no	—
421	#41	—	2	8	no	—
422	#41	—	2	8	no	—
423	#41	—	2	8	no	—
424	#41	C₂×D₄×C₆₇	1	16	yes	—
425	#41	C₂×D₄×C₆₇	1	16	yes	complemented
426	C₂×Q₈	—	2	32	no	—
427	C₂×Q₈	—	2	16	no	—
428	C₂×Q₈	—	2	4	no	—
429	C₂×Q₈	—	2	2	no	—
430	C₂×Q₈	—	2	4	no	—
431	C₂×Q₈	—	2	4	no	—
432	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
433	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	complemented
434	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
435	C₂×D₄	—	2	32	no	—
436	C₂×D₄	—	2	64	no	—
437	C₂×D₄	—	2	32	no	—
438	C₂×D₄	—	2	16	no	—
439	C₂×D₄	—	2	8	no	—
440	C₂×D₄	—	2	4	no	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	—	2	8	no	—
443	C₂×D₄	—	2	16	no	—
444	C₂×D₄	—	2	16	no	—
445	C₂×D₄	—	2	4	no	—
446	C₂×D₄	—	2	2	no	—
447	C₂×D₄	—	2	4	no	—
448	C₂×D₄	—	2	4	no	—
449	C₂×D₄	—	2	8	no	—
450	C₂×D₄	—	2	8	no	—
451	C₂×D₄	—	2	8	no	—
452	C₂×D₄	—	2	2	no	—
453	C₂×D₄	—	2	4	no	—
454	C₂×D₄	—	4	8	no	—
455	C₂×D₄	—	4	4	no	—
456	C₂×D₄	—	8	8	no	—
457	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
458	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
459	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	complemented
460	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
461	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
462	C₂²×C₄	—	2	32	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	4	no	—
465	C₂²×C₄	—	2	16	no	—
466	C₂²×C₄	—	2	4	no	—
467	C₂²×C₄	—	2	4	no	—
468	C₂²×C₄	—	2	8	no	—
469	C₂²×C₄	—	2	8	no	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
472	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
473	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
474	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
475	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
476	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
477	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
478	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
479	SD₁₆	—	2	32	no	—
480	D₈	—	2	32	no	—
481	M₁₆	—	2	8	no	—
482	M₁₆	—	2	8	no	—
483	C₂×C₈	—	2	8	no	—
484	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
487	C₂²⋊C₄	—	2	4	no	—
488	C₂²⋊C₄	—	2	16	no	—
489	C₄²	—	2	4	no	—
490	C₂³	—	2	16	no	—
491	C₂³	—	2	4	no	—
492	C₂³	—	2	4	no	—
493	C₂³	—	2	8	no	—
494	C₂³	—	2	4	no	—
495	C₂³	—	4	2	no	—
496	C₂³	—	4	4	no	—
497	C₂³	—	4	8	no	—
498	C₂³	—	4	4	no	—
499	C₂³	—	4	8	no	—
500	C₂³	—	8	8	no	—
501	C₂³	—	8	4	no	—
502	C₂³	—	8	8	no	—
503	C₂³	—	8	16	no	—
504	C₂³	#120×C₆₇	1	4	yes	—
505	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
506	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
507	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
508	Q₈	—	2	16	no	—
509	Q₈	—	2	8	no	—
510	Q₈	—	2	8	no	—
511	Q₈	—	2	4	no	—
512	Q₈	C₂²×D₄×C₆₇	1	8	yes	—
513	D₄	—	2	32	no	—
514	D₄	—	2	16	no	—
515	D₄	—	2	16	no	—
516	D₄	—	2	16	no	—
517	D₄	—	2	16	no	—
518	D₄	—	2	8	no	—
519	D₄	—	2	8	no	—
520	D₄	—	2	4	no	—
521	D₄	—	2	8	no	—
522	D₄	—	4	4	no	—
523	D₄	—	4	8	no	—
524	D₄	—	8	16	no	—
525	D₄	C₂²×D₄×C₆₇	1	8	yes	—
526	C₂×C₄	—	2	16	no	—
527	C₂×C₄	—	2	32	no	—
528	C₂×C₄	—	2	8	no	—
529	C₂×C₄	—	2	4	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	16	no	—
532	C₂×C₄	—	2	8	no	—
533	C₂×C₄	—	2	16	no	—
534	C₂×C₄	—	2	8	no	—
535	C₂×C₄	—	2	4	no	—
536	C₂×C₄	—	4	4	no	—
537	C₂×C₄	#120×C₆₇	1	8	yes	—
538	C₂×C₄	#120×C₆₇	1	4	yes	—
539	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
540	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
541	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
542	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
543	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
544	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
545	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
546	C₈	—	2	8	no	—
547	C₂²	—	2	2	no	—
548	C₂²	—	2	4	no	—
549	C₂²	—	2	8	no	—
550	C₂²	—	2	4	no	—
551	C₂²	—	2	8	no	—
552	C₂²	—	4	4	no	—
553	C₂²	—	4	8	no	—
554	C₂²	—	4	8	no	—
555	C₂²	—	4	4	no	—
556	C₂²	—	8	4	no	—
557	C₂²	—	8	8	no	—
558	C₂²	—	8	8	no	—
559	C₂²	—	8	16	no	—
560	C₂²	—	8	16	no	—
561	C₂²	C₂×#120×C₆₇	1	4	yes	—
562	C₂²	C₂×#120×C₆₇	1	2	yes	—
563	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
564	C₄	—	2	8	no	—
565	C₄	—	2	4	no	—
566	C₄	—	4	4	no	—
567	C₄	C₂×#120×C₆₇	1	2	yes	—
568	C₄	C₂×#120×C₆₇	1	4	yes	—
569	C₄	C₂³×D₄×C₆₇	1	2	yes	—
570	C₂	—	2	2	no	—
571	C₂	—	2	4	no	—
572	C₂	—	4	4	no	—
573	C₂	—	4	8	no	—
574	C₂	—	8	4	no	—
575	C₂	C₂²×#120×C₆₇	1	1	yes	characteristic
576	C₂	C₆₇×#3585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂²×#41×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#585×C₆₇	C₂	semidirect
2. C₂²×AGL(1,ℤ/8)×C₆₇	C₂	semidirect
3. C₂×#535×C₆₇	C₂	semidirect
4. C₆₇×#3010	C₂	semidirect
5. C₆₇×#3008	C₂	semidirect
6. C₆₇×#3006	C₂	semidirect
7. C₆₇×#2896	C₂	semidirect
8. C₆₇×#2879	C₂	semidirect
11. C₂×2₋¹⁺⁴×C₆₇	C₂²	semidirect
14. C₂×H(5,𝔽₂)×C₆₇	C₂²	semidirect
18. C₂²×#41×C₆₇	C₂²	semidirect
19. C₂³×D₄×C₆₇	C₂²	semidirect
21. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
22. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
23. C₂²×SD₁₆×C₆₇	C₂²	semidirect
24. C₂²×D₈×C₆₇	C₂²	semidirect
25. C₂²×M₁₆×C₆₇	C₂²	semidirect
27. #535×C₆₇	C₂²	semidirect
30. C₂×#121×C₆₇	C₂²	semidirect
32. C₂×#117×C₆₇	C₂²	semidirect
33. #449×C₆₇	C₂²	semidirect
34. #448×C₆₇	C₂²	semidirect
35. #447×C₆₇	C₂²	semidirect
36. #419×C₆₇	C₂²	semidirect
37. #418×C₆₇	C₂²	semidirect
38. #417×C₆₇	C₂²	semidirect
39. #416×C₆₇	C₂²	semidirect
40. #408×C₆₇	C₂²	semidirect
41. #407×C₆₇	C₂²	semidirect
62. C₂×#41×C₆₇	D₄	semidirect
66. C₂×#41×C₆₇	C₂³	semidirect
70. C₂²×Q₈×C₆₇	C₂³	semidirect
84. C₂²×D₄×C₆₇	C₂³	semidirect
110. #103×C₆₇	C₂³	semidirect
111. #102×C₆₇	C₂³	semidirect
112. C₂²⋊C₈×C₆₇	C₂³	semidirect
136. #41×C₆₇	C₂×D₄	semidirect
144. C₂×Q₈×C₆₇	C₂×D₄	semidirect
170. C₂×D₄×C₆₇	C₂×D₄	semidirect
281. #60369	C₆₇	direct (N × H)
289. C₂×#585	C₂×C₆₇	semidirect
290. C₂²×AGL(1,ℤ/8)	C₂×C₆₇	semidirect
291. C₂×#535	C₂×C₆₇	semidirect
292. #3010	C₂×C₆₇	semidirect
293. #3008	C₂×C₆₇	semidirect
294. #3006	C₂×C₆₇	semidirect
295. #2896	C₂×C₆₇	semidirect
296. #2879	C₂×C₆₇	semidirect
297. C₆₇	#60369	direct (N × H)
300. C₂×2₋¹⁺⁴	C₂²×C₆₇	semidirect
303. C₂×H(5,𝔽₂)	C₂²×C₆₇	semidirect
307. C₂²×#41	C₂²×C₆₇	semidirect
308. C₂³×D₄	C₂²×C₆₇	semidirect
310. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
311. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
312. C₂²×SD₁₆	C₂²×C₆₇	semidirect
313. C₂²×D₈	C₂²×C₆₇	semidirect
314. C₂²×M₁₆	C₂²×C₆₇	semidirect
316. #535	C₂²×C₆₇	semidirect
319. C₂×#121	C₂²×C₆₇	semidirect
321. C₂×#117	C₂²×C₆₇	semidirect
322. #449	C₂²×C₆₇	semidirect
323. #448	C₂²×C₆₇	semidirect
324. #447	C₂²×C₆₇	semidirect
325. #419	C₂²×C₆₇	semidirect
326. #418	C₂²×C₆₇	semidirect
327. #417	C₂²×C₆₇	semidirect
328. #416	C₂²×C₆₇	semidirect
329. #408	C₂²×C₆₇	semidirect
330. #407	C₂²×C₆₇	semidirect
351. C₂×#41	D₄×C₆₇	semidirect
355. C₂×#41	C₂³×C₆₇	semidirect
359. C₂²×Q₈	C₂³×C₆₇	semidirect
373. C₂²×D₄	C₂³×C₆₇	semidirect
399. #103	C₂³×C₆₇	semidirect
400. #102	C₂³×C₆₇	semidirect
401. C₂²⋊C₈	C₂³×C₆₇	semidirect
425. #41	C₂×D₄×C₆₇	semidirect
433. C₂×Q₈	C₂×D₄×C₆₇	semidirect
459. C₂×D₄	C₂×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×#585×C₆₇	2
C₂²×AGL(1,ℤ/8)×C₆₇	2
C₂×#535×C₆₇	2
C₆₇×#3010	2
C₆₇×#3008	2
C₆₇×#3006	2
C₆₇×#2896	2
C₆₇×#2879	2
#60369	67

Subgroups — C67×#60369 #30739620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60369 #30739620