Subgroups — C₆₇×#22459 #30701710

Order 17152.

1312 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#577×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2957	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2889	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2888	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1977	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1903	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1902	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1813	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1796	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1795	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1757	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	1	yes	characteristic
14	#577×C₆₇	—	2	4	no	—
15	#577×C₆₇	—	2	4	no	—
16	C₂×#137×C₆₇	C₄	1	2	yes	complemented
17	C₂×#137×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×AGL(1,ℤ/8)×C₆₇	C₄	1	2	yes	complemented
19	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#135×C₆₇	C₄	1	2	yes	complemented
21	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×SD₁₆×C₆₇	C₄	1	2	yes	complemented
23	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
24	#568×C₆₇	C₂²	1	2	yes	—
25	#568×C₆₇	C₂²	1	2	yes	complemented
26	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
27	#444×C₆₇	—	2	4	no	—
28	#443×C₆₇	—	2	4	no	—
29	C₂×#108×C₆₇	C₂²	1	2	yes	complemented
30	#421×C₆₇	C₂²	1	2	yes	—
31	#421×C₆₇	C₂²	1	2	yes	complemented
32	#419×C₆₇	C₂²	1	2	yes	complemented
33	#418×C₆₇	C₂²	1	2	yes	—
34	#413×C₆₇	—	2	2	no	—
35	#413×C₆₇	—	2	2	no	—
36	#413×C₆₇	C₂²	1	1	yes	characteristic
37	#413×C₆₇	C₂²	1	1	yes	characteristic
38	#413×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#101×C₆₇	—	2	2	no	—
40	C₂×#101×C₆₇	C₂²	1	2	yes	complemented
41	C₂×#100×C₆₇	—	2	2	no	—
42	C₂×#100×C₆₇	C₂²	1	2	yes	—
43	#410×C₆₇	C₂²	1	1	yes	characteristic
44	#405×C₆₇	C₂²	1	2	yes	complemented
45	#344×C₆₇	C₂²	1	2	yes	complemented
46	#338×C₆₇	C₂²	1	2	yes	complemented
47	2₋¹⁺⁴×C₆₇	—	2	4	no	—
48	2₋¹⁺⁴×C₆₇	—	4	2	no	—
49	H(5,𝔽₂)×C₆₇	—	2	4	no	—
50	H(5,𝔽₂)×C₆₇	—	4	2	no	—
51	C₂×#41×C₆₇	—	2	1	no	—
52	C₂×#41×C₆₇	—	2	1	no	—
53	C₂×#41×C₆₇	—	2	2	no	—
54	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
55	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
57	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
58	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×Q₈×C₆₇	—	2	1	no	—
60	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
61	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₆₇	—	2	1	no	—
63	C₂²×D₄×C₆₇	—	2	2	no	—
64	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
65	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
66	#137×C₆₇	—	2	4	no	—
67	#137×C₆₇	—	2	8	no	—
68	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
69	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
70	#135×C₆₇	—	2	4	no	—
71	#135×C₆₇	—	2	4	no	—
72	#135×C₆₇	D₄	1	4	yes	—
73	#135×C₆₇	D₄	1	4	yes	—
74	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
75	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₂×SD₁₆×C₆₇	—	2	4	no	—
77	C₂×SD₁₆×C₆₇	—	2	4	no	—
78	C₂×SD₁₆×C₆₇	—	2	2	no	—
79	C₂×SD₁₆×C₆₇	—	2	2	no	—
80	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
81	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
82	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
83	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
84	C₂×SD₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
85	C₂×SD₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
86	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
87	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
88	#131×C₆₇	—	2	2	no	—
89	#131×C₆₇	—	2	2	no	—
90	#131×C₆₇	—	2	4	no	—
91	#131×C₆₇	—	2	4	no	—
92	#131×C₆₇	—	4	2	no	—
93	#131×C₆₇	—	4	2	no	—
94	C₂×M₁₆×C₆₇	—	2	4	no	—
95	C₂×M₁₆×C₆₇	—	2	4	no	—
96	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
97	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
98	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
99	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
100	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
101	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
102	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
103	#117×C₆₇	C₂³	1	2	yes	—
104	#108×C₆₇	—	2	2	no	—
105	#108×C₆₇	—	2	2	no	—
106	#104×C₆₇	—	2	4	no	—
107	#104×C₆₇	—	2	4	no	—
108	#103×C₆₇	—	4	2	no	—
109	#102×C₆₇	—	4	2	no	—
110	#101×C₆₇	—	2	1	no	—
111	#101×C₆₇	—	2	1	no	—
112	#101×C₆₇	—	2	4	no	—
113	#101×C₆₇	D₄	1	2	yes	—
114	#100×C₆₇	—	2	1	no	—
115	#100×C₆₇	—	2	1	no	—
116	#100×C₆₇	—	2	4	no	—
117	#100×C₆₇	D₄	1	2	yes	—
118	#99×C₆₇	—	2	2	no	—
119	#97×C₆₇	—	2	2	no	—
120	C₄×C₈×C₆₇	—	2	2	no	—
121	C₂⁴×C₆₇	—	2	2	no	—
122	C₂⁴×C₆₇	—	2	1	no	—
123	#41×C₆₇	—	2	2	no	—
124	#41×C₆₇	—	2	2	no	—
125	#41×C₆₇	—	2	2	no	—
126	#41×C₆₇	—	2	2	no	—
127	#41×C₆₇	—	2	4	no	—
128	#41×C₆₇	—	2	4	no	—
129	#41×C₆₇	—	4	2	no	—
130	#41×C₆₇	—	4	2	no	—
131	#41×C₆₇	—	4	4	no	—
132	#41×C₆₇	—	4	4	no	—
133	#41×C₆₇	—	4	8	no	—
134	#41×C₆₇	C₂×D₄	1	4	yes	—
135	#41×C₆₇	C₂×D₄	1	4	yes	—
136	C₂×Q₈×C₆₇	—	2	2	no	—
137	C₂×Q₈×C₆₇	—	2	1	no	—
138	C₂×Q₈×C₆₇	—	2	4	no	—
139	C₂×Q₈×C₆₇	—	2	2	no	—
140	C₂×Q₈×C₆₇	—	2	2	no	—
141	C₂×Q₈×C₆₇	—	4	2	no	—
142	C₂×Q₈×C₆₇	—	4	4	no	—
143	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
144	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
145	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
146	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
147	C₂×D₄×C₆₇	—	2	1	no	—
148	C₂×D₄×C₆₇	—	2	1	no	—
149	C₂×D₄×C₆₇	—	2	4	no	—
150	C₂×D₄×C₆₇	—	2	2	no	—
151	C₂×D₄×C₆₇	—	2	2	no	—
152	C₂×D₄×C₆₇	—	2	2	no	—
153	C₂×D₄×C₆₇	—	2	2	no	—
154	C₂×D₄×C₆₇	—	2	1	no	—
155	C₂×D₄×C₆₇	—	2	2	no	—
156	C₂×D₄×C₆₇	—	4	2	no	—
157	C₂×D₄×C₆₇	—	4	4	no	—
158	C₂×D₄×C₆₇	—	4	8	no	—
159	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
160	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
161	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
162	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
163	C₂×D₄×C₆₇	#41	1	2	yes	—
164	C₂²×C₄×C₆₇	—	2	1	no	—
165	C₂²×C₄×C₆₇	—	2	1	no	—
166	C₂²×C₄×C₆₇	—	2	2	no	—
167	C₂²×C₄×C₆₇	—	2	1	no	—
168	C₂²×C₄×C₆₇	—	2	2	no	—
169	C₂²×C₄×C₆₇	—	2	2	no	—
170	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
171	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
172	Q₁₆×C₆₇	—	2	4	no	—
173	Q₁₆×C₆₇	C₂²⋊C₄	1	4	yes	—
174	SD₁₆×C₆₇	—	2	4	no	—
175	SD₁₆×C₆₇	—	2	4	no	—
176	SD₁₆×C₆₇	—	2	2	no	—
177	SD₁₆×C₆₇	—	2	2	no	—
178	SD₁₆×C₆₇	—	2	4	no	—
179	SD₁₆×C₆₇	C₂²⋊C₄	1	2	yes	—
180	SD₁₆×C₆₇	C₂²⋊C₄	1	2	yes	—
181	SD₁₆×C₆₇	C₂²⋊C₄	1	2	yes	—
182	SD₁₆×C₆₇	C₂²⋊C₄	1	2	yes	—
183	D₈×C₆₇	—	2	4	no	—
184	D₈×C₆₇	C₂²⋊C₄	1	4	yes	—
185	M₁₆×C₆₇	—	2	2	no	—
186	M₁₆×C₆₇	—	2	2	no	—
187	M₁₆×C₆₇	—	2	4	no	—
188	M₁₆×C₆₇	—	2	2	no	—
189	M₁₆×C₆₇	—	2	2	no	—
190	M₁₆×C₆₇	—	2	2	no	—
191	M₁₆×C₆₇	—	2	2	no	—
192	M₁₆×C₆₇	—	4	4	no	—
193	M₁₆×C₆₇	—	4	4	no	—
194	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
195	C₂×C₈×C₆₇	—	2	2	no	—
196	C₂×C₈×C₆₇	—	2	2	no	—
197	C₂×C₈×C₆₇	—	2	2	no	—
198	C₂×C₈×C₆₇	—	2	2	no	—
199	C₂×C₈×C₆₇	—	2	2	no	—
200	C₂×C₈×C₆₇	—	4	4	no	—
201	C₂×C₈×C₆₇	—	4	4	no	—
202	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
203	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
204	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
205	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
206	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
207	C₄⋊C₄×C₆₇	—	2	2	no	—
208	C₂²⋊C₄×C₆₇	—	2	2	no	—
209	C₄²×C₆₇	—	2	2	no	—
210	C₂³×C₆₇	—	2	1	no	—
211	C₂³×C₆₇	—	2	1	no	—
212	C₂³×C₆₇	—	2	2	no	—
213	C₂³×C₆₇	—	4	4	no	—
214	C₂³×C₆₇	—	4	2	no	—
215	C₂³×C₆₇	—	16	1	no	—
216	C₂³×C₆₇	—	16	2	no	—
217	C₂³×C₆₇	C₄×D₄	1	2	yes	—
218	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
219	Q₈×C₆₇	—	2	1	no	—
220	Q₈×C₆₇	—	2	1	no	—
221	Q₈×C₆₇	—	2	2	no	—
222	Q₈×C₆₇	—	2	2	no	—
223	Q₈×C₆₇	—	2	2	no	—
224	Q₈×C₆₇	—	4	4	no	—
225	Q₈×C₆₇	—	4	2	no	—
226	Q₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
227	Q₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
228	D₄×C₆₇	—	2	4	no	—
229	D₄×C₆₇	—	2	4	no	—
230	D₄×C₆₇	—	2	4	no	—
231	D₄×C₆₇	—	2	2	no	—
232	D₄×C₆₇	—	2	1	no	—
233	D₄×C₆₇	—	2	1	no	—
234	D₄×C₆₇	—	2	2	no	—
235	D₄×C₆₇	—	2	2	no	—
236	D₄×C₆₇	—	4	4	no	—
237	D₄×C₆₇	—	4	4	no	—
238	D₄×C₆₇	—	4	4	no	—
239	D₄×C₆₇	—	4	2	no	—
240	D₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
241	D₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
242	C₂×C₄×C₆₇	—	2	1	no	—
243	C₂×C₄×C₆₇	—	2	1	no	—
244	C₂×C₄×C₆₇	—	2	2	no	—
245	C₂×C₄×C₆₇	—	2	2	no	—
246	C₂×C₄×C₆₇	—	2	4	no	—
247	C₂×C₄×C₆₇	—	2	4	no	—
248	C₂×C₄×C₆₇	—	2	2	no	—
249	C₂×C₄×C₆₇	—	4	4	no	—
250	C₂×C₄×C₆₇	—	4	4	no	—
251	C₂×C₄×C₆₇	—	4	4	no	—
252	C₂×C₄×C₆₇	—	4	2	no	—
253	C₂×C₄×C₆₇	—	4	2	no	—
254	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
255	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
256	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
257	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
259	C₈×C₆₇	—	2	2	no	—
260	C₈×C₆₇	—	2	4	no	—
261	C₈×C₆₇	—	4	2	no	—
262	C₈×C₆₇	—	4	2	no	—
263	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
264	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
265	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
266	C₂²×C₆₇	—	2	2	no	—
267	C₂²×C₆₇	—	2	2	no	—
268	C₂²×C₆₇	—	2	2	no	—
269	C₂²×C₆₇	—	4	1	no	—
270	C₂²×C₆₇	—	4	2	no	—
271	C₂²×C₆₇	—	4	2	no	—
272	C₂²×C₆₇	—	8	1	no	—
273	C₂²×C₆₇	—	8	1	no	—
274	C₂²×C₆₇	—	8	2	no	—
275	C₂²×C₆₇	—	16	2	no	—
276	C₂²×C₆₇	—	16	4	no	—
277	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
278	C₂²×C₆₇	#546	1	2	yes	—
279	C₄×C₆₇	—	2	2	no	—
280	C₄×C₆₇	—	2	2	no	—
281	C₄×C₆₇	—	2	2	no	—
282	C₄×C₆₇	—	4	2	no	—
283	C₄×C₆₇	—	8	2	no	—
284	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
285	C₄×C₆₇	#518	1	1	yes	characteristic
286	C₄×C₆₇	D₄²	1	2	yes	—
287	#22459	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
288	C₂×C₆₇	—	2	2	no	—
289	C₂×C₆₇	—	2	1	no	—
290	C₂×C₆₇	—	4	2	no	—
291	C₂×C₆₇	—	4	2	no	—
292	C₂×C₆₇	—	4	2	no	—
293	C₂×C₆₇	—	8	2	no	—
294	C₂×C₆₇	D₄×C₂²⋊C₄	1	1	yes	characteristic
295	C₂×#577	C₂×C₆₇	1	1	yes	characteristic
296	#2957	C₂×C₆₇	1	2	yes	complemented
297	#2889	C₂×C₆₇	1	1	yes	characteristic, complemented
298	#2888	C₂×C₆₇	1	1	yes	characteristic, complemented
299	#1977	C₂×C₆₇	1	2	yes	complemented
300	#1903	C₂×C₆₇	1	2	yes	complemented
301	#1902	C₂×C₆₇	1	2	yes	complemented
302	#1813	C₂×C₆₇	1	1	yes	characteristic, complemented
303	#1796	C₂×C₆₇	1	1	yes	characteristic, complemented
304	#1795	C₂×C₆₇	1	1	yes	characteristic, complemented
305	#1757	C₂×C₆₇	1	1	yes	characteristic, complemented
306	C₆₇	#22459	1	1	yes	characteristic, direct product factor, Sylow
307	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	1	yes	characteristic
308	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	1	yes	characteristic
309	#577	—	2	4	no	—
310	#577	—	2	4	no	—
311	C₂×#137	C₄×C₆₇	1	2	yes	complemented
312	C₂×#137	C₂²×C₆₇	1	1	yes	characteristic
313	C₂×AGL(1,ℤ/8)	C₄×C₆₇	1	2	yes	complemented
314	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	1	yes	characteristic
315	C₂×#135	C₄×C₆₇	1	2	yes	complemented
316	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
317	C₂²×SD₁₆	C₄×C₆₇	1	2	yes	complemented
318	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
319	#568	C₂²×C₆₇	1	2	yes	—
320	#568	C₂²×C₆₇	1	2	yes	complemented
321	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
322	#444	—	2	4	no	—
323	#443	—	2	4	no	—
324	C₂×#108	C₂²×C₆₇	1	2	yes	complemented
325	#421	C₂²×C₆₇	1	2	yes	—
326	#421	C₂²×C₆₇	1	2	yes	complemented
327	#419	C₂²×C₆₇	1	2	yes	complemented
328	#418	C₂²×C₆₇	1	2	yes	—
329	#413	—	2	2	no	—
330	#413	—	2	2	no	—
331	#413	C₂²×C₆₇	1	1	yes	characteristic
332	#413	C₂²×C₆₇	1	1	yes	characteristic
333	#413	C₂²×C₆₇	1	1	yes	characteristic
334	C₂×#101	—	2	2	no	—
335	C₂×#101	C₂²×C₆₇	1	2	yes	complemented
336	C₂×#100	—	2	2	no	—
337	C₂×#100	C₂²×C₆₇	1	2	yes	—
338	#410	C₂²×C₆₇	1	1	yes	characteristic
339	#405	C₂²×C₆₇	1	2	yes	complemented
340	#344	C₂²×C₆₇	1	2	yes	complemented
341	#338	C₂²×C₆₇	1	2	yes	complemented
342	2₋¹⁺⁴	—	2	4	no	—
343	2₋¹⁺⁴	—	4	2	no	—
344	H(5,𝔽₂)	—	2	4	no	—
345	H(5,𝔽₂)	—	4	2	no	—
346	C₂×#41	—	2	1	no	—
347	C₂×#41	—	2	1	no	—
348	C₂×#41	—	2	2	no	—
349	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
350	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
351	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
352	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
353	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
354	C₂²×Q₈	—	2	1	no	—
355	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
356	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
357	C₂²×D₄	—	2	1	no	—
358	C₂²×D₄	—	2	2	no	—
359	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
360	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
361	#137	—	2	4	no	—
362	#137	—	2	8	no	—
363	AGL(1,ℤ/8)	—	2	4	no	—
364	AGL(1,ℤ/8)	—	2	8	no	—
365	#135	—	2	4	no	—
366	#135	—	2	4	no	—
367	#135	D₄×C₆₇	1	4	yes	—
368	#135	D₄×C₆₇	1	4	yes	—
369	C₂×Q₁₆	C₂×C₄×C₆₇	1	2	yes	—
370	C₂×Q₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
371	C₂×SD₁₆	—	2	4	no	—
372	C₂×SD₁₆	—	2	4	no	—
373	C₂×SD₁₆	—	2	2	no	—
374	C₂×SD₁₆	—	2	2	no	—
375	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
376	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
377	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
378	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
379	C₂×SD₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
380	C₂×SD₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
381	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
382	C₂×D₈	C₂×C₄×C₆₇	1	1	yes	characteristic
383	#131	—	2	2	no	—
384	#131	—	2	2	no	—
385	#131	—	2	4	no	—
386	#131	—	2	4	no	—
387	#131	—	4	2	no	—
388	#131	—	4	2	no	—
389	C₂×M₁₆	—	2	4	no	—
390	C₂×M₁₆	—	2	4	no	—
391	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
392	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
393	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
394	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
395	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
396	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
397	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
398	#117	C₂³×C₆₇	1	2	yes	—
399	#108	—	2	2	no	—
400	#108	—	2	2	no	—
401	#104	—	2	4	no	—
402	#104	—	2	4	no	—
403	#103	—	4	2	no	—
404	#102	—	4	2	no	—
405	#101	—	2	1	no	—
406	#101	—	2	1	no	—
407	#101	—	2	4	no	—
408	#101	D₄×C₆₇	1	2	yes	—
409	#100	—	2	1	no	—
410	#100	—	2	1	no	—
411	#100	—	2	4	no	—
412	#100	D₄×C₆₇	1	2	yes	—
413	#99	—	2	2	no	—
414	#97	—	2	2	no	—
415	C₄×C₈	—	2	2	no	—
416	C₂⁴	—	2	2	no	—
417	C₂⁴	—	2	1	no	—
418	#41	—	2	2	no	—
419	#41	—	2	2	no	—
420	#41	—	2	2	no	—
421	#41	—	2	2	no	—
422	#41	—	2	4	no	—
423	#41	—	2	4	no	—
424	#41	—	4	2	no	—
425	#41	—	4	2	no	—
426	#41	—	4	4	no	—
427	#41	—	4	4	no	—
428	#41	—	4	8	no	—
429	#41	C₂×D₄×C₆₇	1	4	yes	—
430	#41	C₂×D₄×C₆₇	1	4	yes	—
431	C₂×Q₈	—	2	2	no	—
432	C₂×Q₈	—	2	1	no	—
433	C₂×Q₈	—	2	4	no	—
434	C₂×Q₈	—	2	2	no	—
435	C₂×Q₈	—	2	2	no	—
436	C₂×Q₈	—	4	2	no	—
437	C₂×Q₈	—	4	4	no	—
438	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
439	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
440	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
441	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
442	C₂×D₄	—	2	1	no	—
443	C₂×D₄	—	2	1	no	—
444	C₂×D₄	—	2	4	no	—
445	C₂×D₄	—	2	2	no	—
446	C₂×D₄	—	2	2	no	—
447	C₂×D₄	—	2	2	no	—
448	C₂×D₄	—	2	2	no	—
449	C₂×D₄	—	2	1	no	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	4	2	no	—
452	C₂×D₄	—	4	4	no	—
453	C₂×D₄	—	4	8	no	—
454	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
455	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
456	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
457	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
458	C₂×D₄	#41×C₆₇	1	2	yes	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
466	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
467	Q₁₆	—	2	4	no	—
468	Q₁₆	C₂²⋊C₄×C₆₇	1	4	yes	—
469	SD₁₆	—	2	4	no	—
470	SD₁₆	—	2	4	no	—
471	SD₁₆	—	2	2	no	—
472	SD₁₆	—	2	2	no	—
473	SD₁₆	—	2	4	no	—
474	SD₁₆	C₂²⋊C₄×C₆₇	1	2	yes	—
475	SD₁₆	C₂²⋊C₄×C₆₇	1	2	yes	—
476	SD₁₆	C₂²⋊C₄×C₆₇	1	2	yes	—
477	SD₁₆	C₂²⋊C₄×C₆₇	1	2	yes	—
478	D₈	—	2	4	no	—
479	D₈	C₂²⋊C₄×C₆₇	1	4	yes	—
480	M₁₆	—	2	2	no	—
481	M₁₆	—	2	2	no	—
482	M₁₆	—	2	4	no	—
483	M₁₆	—	2	2	no	—
484	M₁₆	—	2	2	no	—
485	M₁₆	—	2	2	no	—
486	M₁₆	—	2	2	no	—
487	M₁₆	—	4	4	no	—
488	M₁₆	—	4	4	no	—
489	M₁₆	C₂×D₄×C₆₇	1	4	yes	—
490	C₂×C₈	—	2	2	no	—
491	C₂×C₈	—	2	2	no	—
492	C₂×C₈	—	2	2	no	—
493	C₂×C₈	—	2	2	no	—
494	C₂×C₈	—	2	2	no	—
495	C₂×C₈	—	4	4	no	—
496	C₂×C₈	—	4	4	no	—
497	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
498	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
499	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
500	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
501	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
502	C₄⋊C₄	—	2	2	no	—
503	C₂²⋊C₄	—	2	2	no	—
504	C₄²	—	2	2	no	—
505	C₂³	—	2	1	no	—
506	C₂³	—	2	1	no	—
507	C₂³	—	2	2	no	—
508	C₂³	—	4	4	no	—
509	C₂³	—	4	2	no	—
510	C₂³	—	16	1	no	—
511	C₂³	—	16	2	no	—
512	C₂³	C₄×D₄×C₆₇	1	2	yes	—
513	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
514	Q₈	—	2	1	no	—
515	Q₈	—	2	1	no	—
516	Q₈	—	2	2	no	—
517	Q₈	—	2	2	no	—
518	Q₈	—	2	2	no	—
519	Q₈	—	4	4	no	—
520	Q₈	—	4	2	no	—
521	Q₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
522	Q₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
523	D₄	—	2	4	no	—
524	D₄	—	2	4	no	—
525	D₄	—	2	4	no	—
526	D₄	—	2	2	no	—
527	D₄	—	2	1	no	—
528	D₄	—	2	1	no	—
529	D₄	—	2	2	no	—
530	D₄	—	2	2	no	—
531	D₄	—	4	4	no	—
532	D₄	—	4	4	no	—
533	D₄	—	4	4	no	—
534	D₄	—	4	2	no	—
535	D₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
536	D₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	4	4	no	—
545	C₂×C₄	—	4	4	no	—
546	C₂×C₄	—	4	4	no	—
547	C₂×C₄	—	4	2	no	—
548	C₂×C₄	—	4	2	no	—
549	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
550	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
551	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
552	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
553	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
554	C₈	—	2	2	no	—
555	C₈	—	2	4	no	—
556	C₈	—	4	2	no	—
557	C₈	—	4	2	no	—
558	C₈	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
559	C₈	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
560	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
561	C₂²	—	2	2	no	—
562	C₂²	—	2	2	no	—
563	C₂²	—	2	2	no	—
564	C₂²	—	4	1	no	—
565	C₂²	—	4	2	no	—
566	C₂²	—	4	2	no	—
567	C₂²	—	8	1	no	—
568	C₂²	—	8	1	no	—
569	C₂²	—	8	2	no	—
570	C₂²	—	16	2	no	—
571	C₂²	—	16	4	no	—
572	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
573	C₂²	#546×C₆₇	1	2	yes	—
574	C₄	—	2	2	no	—
575	C₄	—	2	2	no	—
576	C₄	—	2	2	no	—
577	C₄	—	4	2	no	—
578	C₄	—	8	2	no	—
579	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
580	C₄	#518×C₆₇	1	1	yes	characteristic
581	C₄	D₄²×C₆₇	1	2	yes	—
582	C₂	—	2	2	no	—
583	C₂	—	2	1	no	—
584	C₂	—	4	2	no	—
585	C₂	—	4	2	no	—
586	C₂	—	4	2	no	—
587	C₂	—	8	2	no	—
588	C₂	D₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	D₄×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#577×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2957	C₂	semidirect
3. C₆₇×#2889	C₂	semidirect
4. C₆₇×#2888	C₂	semidirect
5. C₆₇×#1977	C₂	semidirect
6. C₆₇×#1903	C₂	semidirect
7. C₆₇×#1902	C₂	semidirect
8. C₆₇×#1813	C₂	semidirect
9. C₆₇×#1796	C₂	semidirect
10. C₆₇×#1795	C₂	semidirect
11. C₆₇×#1757	C₂	semidirect
16. C₂×#137×C₆₇	C₄	semidirect
18. C₂×AGL(1,ℤ/8)×C₆₇	C₄	semidirect
20. C₂×#135×C₆₇	C₄	semidirect
22. C₂²×SD₁₆×C₆₇	C₄	semidirect
25. #568×C₆₇	C₂²	semidirect
29. C₂×#108×C₆₇	C₂²	semidirect
31. #421×C₆₇	C₂²	semidirect
32. #419×C₆₇	C₂²	semidirect
40. C₂×#101×C₆₇	C₂²	semidirect
44. #405×C₆₇	C₂²	semidirect
45. #344×C₆₇	C₂²	semidirect
46. #338×C₆₇	C₂²	semidirect
287. #22459	C₆₇	direct (N × H)
296. #2957	C₂×C₆₇	semidirect
297. #2889	C₂×C₆₇	semidirect
298. #2888	C₂×C₆₇	semidirect
299. #1977	C₂×C₆₇	semidirect
300. #1903	C₂×C₆₇	semidirect
301. #1902	C₂×C₆₇	semidirect
302. #1813	C₂×C₆₇	semidirect
303. #1796	C₂×C₆₇	semidirect
304. #1795	C₂×C₆₇	semidirect
305. #1757	C₂×C₆₇	semidirect
306. C₆₇	#22459	direct (N × H)
311. C₂×#137	C₄×C₆₇	semidirect
313. C₂×AGL(1,ℤ/8)	C₄×C₆₇	semidirect
315. C₂×#135	C₄×C₆₇	semidirect
317. C₂²×SD₁₆	C₄×C₆₇	semidirect
320. #568	C₂²×C₆₇	semidirect
324. C₂×#108	C₂²×C₆₇	semidirect
326. #421	C₂²×C₆₇	semidirect
327. #419	C₂²×C₆₇	semidirect
335. C₂×#101	C₂²×C₆₇	semidirect
339. #405	C₂²×C₆₇	semidirect
340. #344	C₂²×C₆₇	semidirect
341. #338	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#577×C₆₇	2
C₆₇×#2957	2
C₆₇×#2889	2
C₆₇×#2888	2
C₆₇×#1977	2
C₆₇×#1903	2
C₆₇×#1902	2
C₆₇×#1813	2
C₆₇×#1796	2
C₆₇×#1795	2
C₆₇×#1757	2
#22459	67

Subgroups — C67×#22459 #30701710

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22459 #30701710