Subgroups — C₆₇×#60388 #30739639

Order 17152.

4874 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×H(7,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3585	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3498	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3021	C₂	1	6	yes	maximal, complemented
5	C₆₇×#3019	C₂	1	6	yes	maximal, complemented
6	C₆₇×#3018	C₂	1	6	yes	maximal, complemented
7	C₂×#453×C₆₇	C₂	1	6	yes	maximal, complemented
8	C₆₇×#2901	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2888	C₂	1	1	yes	maximal, characteristic, complemented
10	#585×C₆₇	—	2	8	no	—
11	#585×C₆₇	—	2	2	no	—
12	#585×C₆₇	C₂²	1	2	yes	—
13	#585×C₆₇	C₂²	1	6	yes	complemented
14	C₂×H(5,𝔽₂)×C₆₇	—	2	8	no	—
15	C₂×H(5,𝔽₂)×C₆₇	—	2	6	no	—
16	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	6	yes	complemented
18	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	1	yes	characteristic, complemented
19	#577×C₆₇	—	2	8	no	—
20	#576×C₆₇	—	2	8	no	—
21	#575×C₆₇	C₂²	1	12	yes	complemented
22	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	12	yes	complemented
23	#568×C₆₇	C₂²	1	2	yes	complemented
24	#550×C₆₇	—	2	2	no	—
25	D₄²×C₆₇	—	2	6	no	—
26	#535×C₆₇	C₂²	1	3	yes	complemented
27	#535×C₆₇	C₂²	1	3	yes	complemented
28	#532×C₆₇	C₂²	1	3	yes	complemented
29	C₂×#127×C₆₇	C₂²	1	3	yes	complemented
30	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic, complemented
31	#456×C₆₇	C₂²	1	4	yes	complemented
32	#455×C₆₇	C₂²	1	12	yes	complemented
33	#454×C₆₇	C₂²	1	12	yes	complemented
34	#453×C₆₇	C₂²	1	12	yes	complemented
35	#453×C₆₇	C₂²	1	24	yes	complemented
36	#421×C₆₇	C₂²	1	12	yes	complemented
37	C₂×#104×C₆₇	C₂²	1	12	yes	complemented
38	#413×C₆₇	C₂²	1	6	yes	complemented
39	C₂×#100×C₆₇	C₂²	1	6	yes	complemented
40	2₋¹⁺⁴×C₆₇	—	2	8	no	—
41	2₋¹⁺⁴×C₆₇	—	2	2	no	—
42	2₋¹⁺⁴×C₆₇	—	2	12	no	—
43	2₋¹⁺⁴×C₆₇	D₄	1	8	yes	complemented
44	2₋¹⁺⁴×C₆₇	C₂³	1	4	yes	—
45	H(5,𝔽₂)×C₆₇	—	2	8	no	—
46	H(5,𝔽₂)×C₆₇	—	2	48	no	—
47	H(5,𝔽₂)×C₆₇	—	2	8	no	—
48	H(5,𝔽₂)×C₆₇	—	2	24	no	—
49	H(5,𝔽₂)×C₆₇	—	2	6	no	—
50	H(5,𝔽₂)×C₆₇	—	2	12	no	—
51	H(5,𝔽₂)×C₆₇	—	2	12	no	—
52	H(5,𝔽₂)×C₆₇	—	2	12	no	—
53	H(5,𝔽₂)×C₆₇	—	2	8	no	—
54	H(5,𝔽₂)×C₆₇	D₄	1	8	yes	complemented
55	H(5,𝔽₂)×C₆₇	C₂³	1	12	yes	—
56	C₂×#41×C₆₇	—	2	16	no	—
57	C₂×#41×C₆₇	—	2	24	no	—
58	C₂×#41×C₆₇	—	2	8	no	—
59	C₂×#41×C₆₇	—	2	24	no	—
60	C₂×#41×C₆₇	—	2	1	no	—
61	C₂×#41×C₆₇	—	2	12	no	—
62	C₂×#41×C₆₇	—	2	12	no	—
63	C₂×#41×C₆₇	D₄	1	2	yes	—
64	C₂×#41×C₆₇	C₂³	1	12	yes	—
65	C₂×#41×C₆₇	C₂³	1	3	yes	—
66	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×#41×C₆₇	C₂³	1	3	yes	complemented
68	C₂²×D₄×C₆₇	—	2	24	no	—
69	C₂²×D₄×C₆₇	—	2	12	no	—
70	C₂²×D₄×C₆₇	—	2	12	no	—
71	C₂²×D₄×C₆₇	—	2	6	no	—
72	C₂²×D₄×C₆₇	—	2	3	no	—
73	C₂²×D₄×C₆₇	D₄	1	6	yes	—
74	C₂²×D₄×C₆₇	C₂³	1	3	yes	—
75	C₂²×D₄×C₆₇	C₂³	1	3	yes	complemented
76	#137×C₆₇	—	2	12	no	—
77	#137×C₆₇	C₂³	1	8	yes	complemented
78	AGL(1,ℤ/8)×C₆₇	—	2	24	no	—
79	AGL(1,ℤ/8)×C₆₇	—	2	12	no	—
80	AGL(1,ℤ/8)×C₆₇	C₂³	1	24	yes	complemented
81	#135×C₆₇	—	2	24	no	—
82	C₂×SD₁₆×C₆₇	—	2	12	no	—
83	C₂×D₈×C₆₇	—	2	12	no	—
84	#131×C₆₇	—	2	8	no	—
85	C₂×M₁₆×C₆₇	C₂³	1	12	yes	complemented
86	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic, complemented
87	#127×C₆₇	—	2	6	no	—
88	#127×C₆₇	C₂³	1	3	yes	complemented
89	#127×C₆₇	C₂³	1	6	yes	complemented
90	#124×C₆₇	C₂³	1	3	yes	complemented
91	#124×C₆₇	C₂³	1	3	yes	complemented
92	#121×C₆₇	—	2	6	no	—
93	#121×C₆₇	—	2	6	no	—
94	#121×C₆₇	—	2	6	no	—
95	#120×C₆₇	—	2	6	no	—
96	#120×C₆₇	—	2	6	no	—
97	C₄×Q₈×C₆₇	—	2	1	no	—
98	C₄×D₄×C₆₇	—	2	3	no	—
99	C₄×D₄×C₆₇	—	2	6	no	—
100	#117×C₆₇	C₂³	1	3	yes	complemented
101	C₂×C₄²×C₆₇	C₂³	1	3	yes	complemented
102	#104×C₆₇	C₂³	1	8	yes	complemented
103	#104×C₆₇	C₂³	1	24	yes	complemented
104	#101×C₆₇	C₂³	1	4	yes	complemented
105	#100×C₆₇	C₂³	1	12	yes	complemented
106	C₂⁴×C₆₇	—	2	12	no	—
107	C₂⁴×C₆₇	—	2	3	no	—
108	C₂⁴×C₆₇	C₂×D₄	1	6	yes	—
109	#41×C₆₇	—	2	48	no	—
110	#41×C₆₇	—	2	48	no	—
111	#41×C₆₇	—	2	24	no	—
112	#41×C₆₇	—	2	24	no	—
113	#41×C₆₇	—	2	8	no	—
114	#41×C₆₇	—	2	8	no	—
115	#41×C₆₇	—	2	24	no	—
116	#41×C₆₇	—	2	24	no	—
117	#41×C₆₇	—	2	8	no	—
118	#41×C₆₇	—	2	8	no	—
119	#41×C₆₇	—	2	12	no	—
120	#41×C₆₇	—	2	12	no	—
121	#41×C₆₇	—	2	12	no	—
122	#41×C₆₇	—	2	12	no	—
123	#41×C₆₇	—	4	4	no	—
124	#41×C₆₇	C₂×D₄	1	8	yes	—
125	#41×C₆₇	C₂×D₄	1	24	yes	complemented
126	#41×C₆₇	C₂⁴	1	8	yes	—
127	C₂×Q₈×C₆₇	—	2	12	no	—
128	C₂×Q₈×C₆₇	—	2	8	no	—
129	C₂×Q₈×C₆₇	—	2	8	no	—
130	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₂×Q₈×C₆₇	C₂×D₄	1	12	yes	complemented
132	C₂×Q₈×C₆₇	C₂⁴	1	1	yes	characteristic
133	C₂×Q₈×C₆₇	C₂⁴	1	1	yes	characteristic
134	C₂×D₄×C₆₇	—	2	48	no	—
135	C₂×D₄×C₆₇	—	2	48	no	—
136	C₂×D₄×C₆₇	—	2	24	no	—
137	C₂×D₄×C₆₇	—	2	48	no	—
138	C₂×D₄×C₆₇	—	2	12	no	—
139	C₂×D₄×C₆₇	—	2	24	no	—
140	C₂×D₄×C₆₇	—	2	12	no	—
141	C₂×D₄×C₆₇	—	2	24	no	—
142	C₂×D₄×C₆₇	—	2	24	no	—
143	C₂×D₄×C₆₇	—	2	24	no	—
144	C₂×D₄×C₆₇	—	2	6	no	—
145	C₂×D₄×C₆₇	—	2	3	no	—
146	C₂×D₄×C₆₇	—	2	12	no	—
147	C₂×D₄×C₆₇	—	2	12	no	—
148	C₂×D₄×C₆₇	—	2	12	no	—
149	C₂×D₄×C₆₇	—	2	3	no	—
150	C₂×D₄×C₆₇	—	4	12	no	—
151	C₂×D₄×C₆₇	—	8	12	no	—
152	C₂×D₄×C₆₇	C₂×D₄	1	6	yes	—
153	C₂×D₄×C₆₇	C₂×D₄	1	6	yes	—
154	C₂×D₄×C₆₇	C₂×D₄	1	3	yes	—
155	C₂×D₄×C₆₇	C₂×D₄	1	12	yes	complemented
156	C₂×D₄×C₆₇	C₂⁴	1	3	yes	—
157	C₂×D₄×C₆₇	C₂⁴	1	3	yes	—
158	C₂²×C₄×C₆₇	—	2	12	no	—
159	C₂²×C₄×C₆₇	—	2	24	no	—
160	C₂²×C₄×C₆₇	—	2	12	no	—
161	C₂²×C₄×C₆₇	—	2	3	no	—
162	C₂²×C₄×C₆₇	—	2	3	no	—
163	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
164	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
165	Q₁₆×C₆₇	—	2	8	no	—
166	SD₁₆×C₆₇	—	2	24	no	—
167	SD₁₆×C₆₇	—	2	8	no	—
168	D₈×C₆₇	—	2	24	no	—
169	M₁₆×C₆₇	—	2	12	no	—
170	M₁₆×C₆₇	C₂⁴	1	8	yes	—
171	C₂×C₈×C₆₇	—	2	12	no	—
172	C₄⋊C₄×C₆₇	—	2	3	no	—
173	C₄⋊C₄×C₆₇	—	2	2	no	—
174	C₂²⋊C₄×C₆₇	—	2	3	no	—
175	C₂²⋊C₄×C₆₇	—	2	6	no	—
176	C₂²⋊C₄×C₆₇	—	2	6	no	—
177	C₄²×C₆₇	—	2	3	no	—
178	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
179	C₄²×C₆₇	C₂⁴	1	3	yes	—
180	C₂³×C₆₇	—	2	24	no	—
181	C₂³×C₆₇	—	2	12	no	—
182	C₂³×C₆₇	—	2	12	no	—
183	C₂³×C₆₇	—	2	3	no	—
184	C₂³×C₆₇	—	4	6	no	—
185	C₂³×C₆₇	—	8	6	no	—
186	C₂³×C₆₇	—	8	6	no	—
187	C₂³×C₆₇	—	16	12	no	—
188	C₂³×C₆₇	C₂²×D₄	1	6	yes	—
189	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
190	Q₈×C₆₇	—	2	8	no	—
191	Q₈×C₆₇	—	2	12	no	—
192	Q₈×C₆₇	—	2	2	no	—
193	Q₈×C₆₇	—	2	2	no	—
194	Q₈×C₆₇	#120	1	4	yes	—
195	Q₈×C₆₇	C₂²×D₄	1	8	yes	—
196	D₄×C₆₇	—	2	48	no	—
197	D₄×C₆₇	—	2	8	no	—
198	D₄×C₆₇	—	2	24	no	—
199	D₄×C₆₇	—	2	12	no	—
200	D₄×C₆₇	—	2	8	no	—
201	D₄×C₆₇	—	2	12	no	—
202	D₄×C₆₇	—	2	12	no	—
203	D₄×C₆₇	—	2	6	no	—
204	D₄×C₆₇	—	2	6	no	—
205	D₄×C₆₇	—	4	12	no	—
206	D₄×C₆₇	—	4	6	no	—
207	D₄×C₆₇	—	4	6	no	—
208	D₄×C₆₇	—	8	12	no	—
209	D₄×C₆₇	#120	1	12	yes	—
210	D₄×C₆₇	C₂²×D₄	1	8	yes	—
211	C₂×C₄×C₆₇	—	2	24	no	—
212	C₂×C₄×C₆₇	—	2	16	no	—
213	C₂×C₄×C₆₇	—	2	12	no	—
214	C₂×C₄×C₆₇	—	2	1	no	—
215	C₂×C₄×C₆₇	—	2	24	no	—
216	C₂×C₄×C₆₇	—	2	12	no	—
217	C₂×C₄×C₆₇	—	2	8	no	—
218	C₂×C₄×C₆₇	—	2	3	no	—
219	C₂×C₄×C₆₇	—	2	1	no	—
220	C₂×C₄×C₆₇	—	4	6	no	—
221	C₂×C₄×C₆₇	—	4	6	no	—
222	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
223	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
224	C₂×C₄×C₆₇	C₂²×D₄	1	12	yes	—
225	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
226	C₈×C₆₇	—	2	8	no	—
227	C₂²×C₆₇	—	2	6	no	—
228	C₂²×C₆₇	—	2	8	no	—
229	C₂²×C₆₇	—	4	3	no	—
230	C₂²×C₆₇	—	4	6	no	—
231	C₂²×C₆₇	—	4	6	no	—
232	C₂²×C₆₇	—	4	3	no	—
233	C₂²×C₆₇	—	8	12	no	—
234	C₂²×C₆₇	—	8	12	no	—
235	C₂²×C₆₇	—	8	24	no	—
236	C₂²×C₆₇	—	16	6	no	—
237	C₂²×C₆₇	C₂×#120	1	6	yes	—
238	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
239	C₄×C₆₇	—	2	2	no	—
240	C₄×C₆₇	—	2	8	no	—
241	C₄×C₆₇	—	2	2	no	—
242	C₄×C₆₇	—	4	3	no	—
243	C₄×C₆₇	—	4	1	no	—
244	C₄×C₆₇	C₂×#120	1	6	yes	—
245	C₄×C₆₇	C₂³×D₄	1	2	yes	—
246	#60388	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
247	C₂×C₆₇	—	2	1	no	—
248	C₂×C₆₇	—	2	6	no	—
249	C₂×C₆₇	—	4	6	no	—
250	C₂×C₆₇	—	4	8	no	—
251	C₂×C₆₇	—	8	3	no	—
252	C₂×C₆₇	C₂²×#120	1	1	yes	characteristic
253	H(7,𝔽₂)	C₂×C₆₇	1	1	yes	characteristic, complemented
254	#3585	C₂×C₆₇	1	2	yes	complemented
255	#3498	C₂×C₆₇	1	1	yes	characteristic, complemented
256	#3021	C₂×C₆₇	1	6	yes	complemented
257	#3019	C₂×C₆₇	1	6	yes	complemented
258	#3018	C₂×C₆₇	1	6	yes	complemented
259	C₂×#453	C₂×C₆₇	1	6	yes	complemented
260	#2901	C₂×C₆₇	1	2	yes	complemented
261	#2888	C₂×C₆₇	1	1	yes	characteristic, complemented
262	C₆₇	#60388	1	1	yes	characteristic, direct product factor, Sylow
263	#585	—	2	8	no	—
264	#585	—	2	2	no	—
265	#585	C₂²×C₆₇	1	2	yes	—
266	#585	C₂²×C₆₇	1	6	yes	complemented
267	C₂×H(5,𝔽₂)	—	2	8	no	—
268	C₂×H(5,𝔽₂)	—	2	6	no	—
269	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	1	yes	characteristic
270	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	6	yes	complemented
271	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	1	yes	characteristic, complemented
272	#577	—	2	8	no	—
273	#576	—	2	8	no	—
274	#575	C₂²×C₆₇	1	12	yes	complemented
275	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	12	yes	complemented
276	#568	C₂²×C₆₇	1	2	yes	complemented
277	#550	—	2	2	no	—
278	D₄²	—	2	6	no	—
279	#535	C₂²×C₆₇	1	3	yes	complemented
280	#535	C₂²×C₆₇	1	3	yes	complemented
281	#532	C₂²×C₆₇	1	3	yes	complemented
282	C₂×#127	C₂²×C₆₇	1	3	yes	complemented
283	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic, complemented
284	#456	C₂²×C₆₇	1	4	yes	complemented
285	#455	C₂²×C₆₇	1	12	yes	complemented
286	#454	C₂²×C₆₇	1	12	yes	complemented
287	#453	C₂²×C₆₇	1	12	yes	complemented
288	#453	C₂²×C₆₇	1	24	yes	complemented
289	#421	C₂²×C₆₇	1	12	yes	complemented
290	C₂×#104	C₂²×C₆₇	1	12	yes	complemented
291	#413	C₂²×C₆₇	1	6	yes	complemented
292	C₂×#100	C₂²×C₆₇	1	6	yes	complemented
293	2₋¹⁺⁴	—	2	8	no	—
294	2₋¹⁺⁴	—	2	2	no	—
295	2₋¹⁺⁴	—	2	12	no	—
296	2₋¹⁺⁴	D₄×C₆₇	1	8	yes	complemented
297	2₋¹⁺⁴	C₂³×C₆₇	1	4	yes	—
298	H(5,𝔽₂)	—	2	8	no	—
299	H(5,𝔽₂)	—	2	48	no	—
300	H(5,𝔽₂)	—	2	8	no	—
301	H(5,𝔽₂)	—	2	24	no	—
302	H(5,𝔽₂)	—	2	6	no	—
303	H(5,𝔽₂)	—	2	12	no	—
304	H(5,𝔽₂)	—	2	12	no	—
305	H(5,𝔽₂)	—	2	12	no	—
306	H(5,𝔽₂)	—	2	8	no	—
307	H(5,𝔽₂)	D₄×C₆₇	1	8	yes	complemented
308	H(5,𝔽₂)	C₂³×C₆₇	1	12	yes	—
309	C₂×#41	—	2	16	no	—
310	C₂×#41	—	2	24	no	—
311	C₂×#41	—	2	8	no	—
312	C₂×#41	—	2	24	no	—
313	C₂×#41	—	2	1	no	—
314	C₂×#41	—	2	12	no	—
315	C₂×#41	—	2	12	no	—
316	C₂×#41	D₄×C₆₇	1	2	yes	—
317	C₂×#41	C₂³×C₆₇	1	12	yes	—
318	C₂×#41	C₂³×C₆₇	1	3	yes	—
319	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
320	C₂×#41	C₂³×C₆₇	1	3	yes	complemented
321	C₂²×D₄	—	2	24	no	—
322	C₂²×D₄	—	2	12	no	—
323	C₂²×D₄	—	2	12	no	—
324	C₂²×D₄	—	2	6	no	—
325	C₂²×D₄	—	2	3	no	—
326	C₂²×D₄	D₄×C₆₇	1	6	yes	—
327	C₂²×D₄	C₂³×C₆₇	1	3	yes	—
328	C₂²×D₄	C₂³×C₆₇	1	3	yes	complemented
329	#137	—	2	12	no	—
330	#137	C₂³×C₆₇	1	8	yes	complemented
331	AGL(1,ℤ/8)	—	2	24	no	—
332	AGL(1,ℤ/8)	—	2	12	no	—
333	AGL(1,ℤ/8)	C₂³×C₆₇	1	24	yes	complemented
334	#135	—	2	24	no	—
335	C₂×SD₁₆	—	2	12	no	—
336	C₂×D₈	—	2	12	no	—
337	#131	—	2	8	no	—
338	C₂×M₁₆	C₂³×C₆₇	1	12	yes	complemented
339	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic, complemented
340	#127	—	2	6	no	—
341	#127	C₂³×C₆₇	1	3	yes	complemented
342	#127	C₂³×C₆₇	1	6	yes	complemented
343	#124	C₂³×C₆₇	1	3	yes	complemented
344	#124	C₂³×C₆₇	1	3	yes	complemented
345	#121	—	2	6	no	—
346	#121	—	2	6	no	—
347	#121	—	2	6	no	—
348	#120	—	2	6	no	—
349	#120	—	2	6	no	—
350	C₄×Q₈	—	2	1	no	—
351	C₄×D₄	—	2	3	no	—
352	C₄×D₄	—	2	6	no	—
353	#117	C₂³×C₆₇	1	3	yes	complemented
354	C₂×C₄²	C₂³×C₆₇	1	3	yes	complemented
355	#104	C₂³×C₆₇	1	8	yes	complemented
356	#104	C₂³×C₆₇	1	24	yes	complemented
357	#101	C₂³×C₆₇	1	4	yes	complemented
358	#100	C₂³×C₆₇	1	12	yes	complemented
359	C₂⁴	—	2	12	no	—
360	C₂⁴	—	2	3	no	—
361	C₂⁴	C₂×D₄×C₆₇	1	6	yes	—
362	#41	—	2	48	no	—
363	#41	—	2	48	no	—
364	#41	—	2	24	no	—
365	#41	—	2	24	no	—
366	#41	—	2	8	no	—
367	#41	—	2	8	no	—
368	#41	—	2	24	no	—
369	#41	—	2	24	no	—
370	#41	—	2	8	no	—
371	#41	—	2	8	no	—
372	#41	—	2	12	no	—
373	#41	—	2	12	no	—
374	#41	—	2	12	no	—
375	#41	—	2	12	no	—
376	#41	—	4	4	no	—
377	#41	C₂×D₄×C₆₇	1	8	yes	—
378	#41	C₂×D₄×C₆₇	1	24	yes	complemented
379	#41	C₂⁴×C₆₇	1	8	yes	—
380	C₂×Q₈	—	2	12	no	—
381	C₂×Q₈	—	2	8	no	—
382	C₂×Q₈	—	2	8	no	—
383	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
384	C₂×Q₈	C₂×D₄×C₆₇	1	12	yes	complemented
385	C₂×Q₈	C₂⁴×C₆₇	1	1	yes	characteristic
386	C₂×Q₈	C₂⁴×C₆₇	1	1	yes	characteristic
387	C₂×D₄	—	2	48	no	—
388	C₂×D₄	—	2	48	no	—
389	C₂×D₄	—	2	24	no	—
390	C₂×D₄	—	2	48	no	—
391	C₂×D₄	—	2	12	no	—
392	C₂×D₄	—	2	24	no	—
393	C₂×D₄	—	2	12	no	—
394	C₂×D₄	—	2	24	no	—
395	C₂×D₄	—	2	24	no	—
396	C₂×D₄	—	2	24	no	—
397	C₂×D₄	—	2	6	no	—
398	C₂×D₄	—	2	3	no	—
399	C₂×D₄	—	2	12	no	—
400	C₂×D₄	—	2	12	no	—
401	C₂×D₄	—	2	12	no	—
402	C₂×D₄	—	2	3	no	—
403	C₂×D₄	—	4	12	no	—
404	C₂×D₄	—	8	12	no	—
405	C₂×D₄	C₂×D₄×C₆₇	1	6	yes	—
406	C₂×D₄	C₂×D₄×C₆₇	1	6	yes	—
407	C₂×D₄	C₂×D₄×C₆₇	1	3	yes	—
408	C₂×D₄	C₂×D₄×C₆₇	1	12	yes	complemented
409	C₂×D₄	C₂⁴×C₆₇	1	3	yes	—
410	C₂×D₄	C₂⁴×C₆₇	1	3	yes	—
411	C₂²×C₄	—	2	12	no	—
412	C₂²×C₄	—	2	24	no	—
413	C₂²×C₄	—	2	12	no	—
414	C₂²×C₄	—	2	3	no	—
415	C₂²×C₄	—	2	3	no	—
416	C₂²×C₄	C₂×D₄×C₆₇	1	6	yes	—
417	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
418	Q₁₆	—	2	8	no	—
419	SD₁₆	—	2	24	no	—
420	SD₁₆	—	2	8	no	—
421	D₈	—	2	24	no	—
422	M₁₆	—	2	12	no	—
423	M₁₆	C₂⁴×C₆₇	1	8	yes	—
424	C₂×C₈	—	2	12	no	—
425	C₄⋊C₄	—	2	3	no	—
426	C₄⋊C₄	—	2	2	no	—
427	C₂²⋊C₄	—	2	3	no	—
428	C₂²⋊C₄	—	2	6	no	—
429	C₂²⋊C₄	—	2	6	no	—
430	C₄²	—	2	3	no	—
431	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
432	C₄²	C₂⁴×C₆₇	1	3	yes	—
433	C₂³	—	2	24	no	—
434	C₂³	—	2	12	no	—
435	C₂³	—	2	12	no	—
436	C₂³	—	2	3	no	—
437	C₂³	—	4	6	no	—
438	C₂³	—	8	6	no	—
439	C₂³	—	8	6	no	—
440	C₂³	—	16	12	no	—
441	C₂³	C₂²×D₄×C₆₇	1	6	yes	—
442	C₂³	C₂²×D₄×C₆₇	1	3	yes	—
443	Q₈	—	2	8	no	—
444	Q₈	—	2	12	no	—
445	Q₈	—	2	2	no	—
446	Q₈	—	2	2	no	—
447	Q₈	#120×C₆₇	1	4	yes	—
448	Q₈	C₂²×D₄×C₆₇	1	8	yes	—
449	D₄	—	2	48	no	—
450	D₄	—	2	8	no	—
451	D₄	—	2	24	no	—
452	D₄	—	2	12	no	—
453	D₄	—	2	8	no	—
454	D₄	—	2	12	no	—
455	D₄	—	2	12	no	—
456	D₄	—	2	6	no	—
457	D₄	—	2	6	no	—
458	D₄	—	4	12	no	—
459	D₄	—	4	6	no	—
460	D₄	—	4	6	no	—
461	D₄	—	8	12	no	—
462	D₄	#120×C₆₇	1	12	yes	—
463	D₄	C₂²×D₄×C₆₇	1	8	yes	—
464	C₂×C₄	—	2	24	no	—
465	C₂×C₄	—	2	16	no	—
466	C₂×C₄	—	2	12	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	24	no	—
469	C₂×C₄	—	2	12	no	—
470	C₂×C₄	—	2	8	no	—
471	C₂×C₄	—	2	3	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	4	6	no	—
474	C₂×C₄	—	4	6	no	—
475	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
476	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
477	C₂×C₄	C₂²×D₄×C₆₇	1	12	yes	—
478	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
479	C₈	—	2	8	no	—
480	C₂²	—	2	6	no	—
481	C₂²	—	2	8	no	—
482	C₂²	—	4	3	no	—
483	C₂²	—	4	6	no	—
484	C₂²	—	4	6	no	—
485	C₂²	—	4	3	no	—
486	C₂²	—	8	12	no	—
487	C₂²	—	8	12	no	—
488	C₂²	—	8	24	no	—
489	C₂²	—	16	6	no	—
490	C₂²	C₂×#120×C₆₇	1	6	yes	—
491	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
492	C₄	—	2	2	no	—
493	C₄	—	2	8	no	—
494	C₄	—	2	2	no	—
495	C₄	—	4	3	no	—
496	C₄	—	4	1	no	—
497	C₄	C₂×#120×C₆₇	1	6	yes	—
498	C₄	C₂³×D₄×C₆₇	1	2	yes	—
499	C₂	—	2	1	no	—
500	C₂	—	2	6	no	—
501	C₂	—	4	6	no	—
502	C₂	—	4	8	no	—
503	C₂	—	8	3	no	—
504	C₂	C₂²×#120×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	C₂²×#120
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂×#41×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×H(7,𝔽₂)	C₂	semidirect
2. C₆₇×#3585	C₂	semidirect
3. C₆₇×#3498	C₂	semidirect
4. C₆₇×#3021	C₂	semidirect
5. C₆₇×#3019	C₂	semidirect
6. C₆₇×#3018	C₂	semidirect
7. C₂×#453×C₆₇	C₂	semidirect
8. C₆₇×#2901	C₂	semidirect
9. C₆₇×#2888	C₂	semidirect
13. #585×C₆₇	C₂²	semidirect
17. C₂×H(5,𝔽₂)×C₆₇	C₂²	semidirect
18. C₂×H(5,𝔽₂)×C₆₇	C₂²	semidirect
21. #575×C₆₇	C₂²	semidirect
22. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
23. #568×C₆₇	C₂²	semidirect
26. #535×C₆₇	C₂²	semidirect
27. #535×C₆₇	C₂²	semidirect
28. #532×C₆₇	C₂²	semidirect
29. C₂×#127×C₆₇	C₂²	semidirect
30. C₄×#41×C₆₇	C₂²	semidirect
31. #456×C₆₇	C₂²	semidirect
32. #455×C₆₇	C₂²	semidirect
33. #454×C₆₇	C₂²	semidirect
34. #453×C₆₇	C₂²	semidirect
35. #453×C₆₇	C₂²	semidirect
36. #421×C₆₇	C₂²	semidirect
37. C₂×#104×C₆₇	C₂²	semidirect
38. #413×C₆₇	C₂²	semidirect
39. C₂×#100×C₆₇	C₂²	semidirect
43. 2₋¹⁺⁴×C₆₇	D₄	semidirect
54. H(5,𝔽₂)×C₆₇	D₄	semidirect
67. C₂×#41×C₆₇	C₂³	semidirect
75. C₂²×D₄×C₆₇	C₂³	semidirect
77. #137×C₆₇	C₂³	semidirect
80. AGL(1,ℤ/8)×C₆₇	C₂³	semidirect
85. C₂×M₁₆×C₆₇	C₂³	semidirect
86. C₄⋊Q₈×C₆₇	C₂³	semidirect
88. #127×C₆₇	C₂³	semidirect
89. #127×C₆₇	C₂³	semidirect
90. #124×C₆₇	C₂³	semidirect
91. #124×C₆₇	C₂³	semidirect
100. #117×C₆₇	C₂³	semidirect
101. C₂×C₄²×C₆₇	C₂³	semidirect
102. #104×C₆₇	C₂³	semidirect
103. #104×C₆₇	C₂³	semidirect
104. #101×C₆₇	C₂³	semidirect
105. #100×C₆₇	C₂³	semidirect
125. #41×C₆₇	C₂×D₄	semidirect
131. C₂×Q₈×C₆₇	C₂×D₄	semidirect
155. C₂×D₄×C₆₇	C₂×D₄	semidirect
246. #60388	C₆₇	direct (N × H)
253. H(7,𝔽₂)	C₂×C₆₇	semidirect
254. #3585	C₂×C₆₇	semidirect
255. #3498	C₂×C₆₇	semidirect
256. #3021	C₂×C₆₇	semidirect
257. #3019	C₂×C₆₇	semidirect
258. #3018	C₂×C₆₇	semidirect
259. C₂×#453	C₂×C₆₇	semidirect
260. #2901	C₂×C₆₇	semidirect
261. #2888	C₂×C₆₇	semidirect
262. C₆₇	#60388	direct (N × H)
266. #585	C₂²×C₆₇	semidirect
270. C₂×H(5,𝔽₂)	C₂²×C₆₇	semidirect
271. C₂×H(5,𝔽₂)	C₂²×C₆₇	semidirect
274. #575	C₂²×C₆₇	semidirect
275. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
276. #568	C₂²×C₆₇	semidirect
279. #535	C₂²×C₆₇	semidirect
280. #535	C₂²×C₆₇	semidirect
281. #532	C₂²×C₆₇	semidirect
282. C₂×#127	C₂²×C₆₇	semidirect
283. C₄×#41	C₂²×C₆₇	semidirect
284. #456	C₂²×C₆₇	semidirect
285. #455	C₂²×C₆₇	semidirect
286. #454	C₂²×C₆₇	semidirect
287. #453	C₂²×C₆₇	semidirect
288. #453	C₂²×C₆₇	semidirect
289. #421	C₂²×C₆₇	semidirect
290. C₂×#104	C₂²×C₆₇	semidirect
291. #413	C₂²×C₆₇	semidirect
292. C₂×#100	C₂²×C₆₇	semidirect
296. 2₋¹⁺⁴	D₄×C₆₇	semidirect
307. H(5,𝔽₂)	D₄×C₆₇	semidirect
320. C₂×#41	C₂³×C₆₇	semidirect
328. C₂²×D₄	C₂³×C₆₇	semidirect
330. #137	C₂³×C₆₇	semidirect
333. AGL(1,ℤ/8)	C₂³×C₆₇	semidirect
338. C₂×M₁₆	C₂³×C₆₇	semidirect
339. C₄⋊Q₈	C₂³×C₆₇	semidirect
341. #127	C₂³×C₆₇	semidirect
342. #127	C₂³×C₆₇	semidirect
343. #124	C₂³×C₆₇	semidirect
344. #124	C₂³×C₆₇	semidirect
353. #117	C₂³×C₆₇	semidirect
354. C₂×C₄²	C₂³×C₆₇	semidirect
355. #104	C₂³×C₆₇	semidirect
356. #104	C₂³×C₆₇	semidirect
357. #101	C₂³×C₆₇	semidirect
358. #100	C₂³×C₆₇	semidirect
378. #41	C₂×D₄×C₆₇	semidirect
384. C₂×Q₈	C₂×D₄×C₆₇	semidirect
408. 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
C₆₇×H(7,𝔽₂)	2
C₆₇×#3585	2
C₆₇×#3498	2
C₆₇×#3021	2
C₆₇×#3019	2
C₆₇×#3018	2
C₂×#453×C₆₇	2
C₆₇×#2901	2
C₆₇×#2888	2
#60388	67

Subgroups — C67×#60388 #30739639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60388 #30739639