Subgroups — C₆₇×#22147 #30701398

Order 17152.

988 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#460×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₄×#127×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2295	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1982	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1968	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1948	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1934	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1877	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1864	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1803	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
13	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₆₇	—	2	1	no	—
16	C₂×C₄×D₄×C₆₇	—	2	1	no	—
17	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
18	#460×C₆₇	C₄	1	8	yes	complemented
19	C₂×#106×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#103×C₆₇	C₂²	1	2	yes	—
23	C₂×#102×C₆₇	C₂²	1	2	yes	—
24	C₂×#102×C₆₇	C₂²	1	2	yes	—
25	C₂×#102×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
27	#391×C₆₇	C₂²	1	1	yes	characteristic
28	#391×C₆₇	C₂²	1	1	yes	characteristic
29	#390×C₆₇	—	2	1	no	—
30	#390×C₆₇	—	2	1	no	—
31	#390×C₆₇	C₂²	1	2	yes	—
32	#389×C₆₇	C₂²	1	1	yes	characteristic
33	#389×C₆₇	C₂²	1	1	yes	characteristic
34	#387×C₆₇	—	2	1	no	—
35	#387×C₆₇	—	2	1	no	—
36	#384×C₆₇	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
38	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
39	#376×C₆₇	—	2	1	no	—
40	C₄³×C₆₇	C₂²	1	1	yes	characteristic
41	#340×C₆₇	C₂²	1	2	yes	—
42	#340×C₆₇	C₂²	1	2	yes	—
43	#336×C₆₇	C₂²	1	2	yes	—
44	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₆₇	—	2	1	no	—
46	C₂²×D₄×C₆₇	—	2	1	no	—
47	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
48	C₂³×C₄×C₆₇	—	2	2	no	—
49	C₂×SD₁₆×C₆₇	—	4	2	no	—
50	C₂×SD₁₆×C₆₇	—	4	2	no	—
51	C₂×SD₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
52	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
54	#127×C₆₇	—	2	2	no	—
55	#127×C₆₇	C₂×C₄	1	4	yes	—
56	C₄×Q₈×C₆₇	—	2	2	no	—
57	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
58	C₄×D₄×C₆₇	—	4	2	no	—
59	C₄×D₄×C₆₇	—	4	2	no	—
60	C₄×D₄×C₆₇	—	4	4	no	—
61	C₄×D₄×C₆₇	—	4	4	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
74	C₂×C₄²×C₆₇	—	2	1	no	—
75	C₂×C₄²×C₆₇	—	2	1	no	—
76	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
78	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
80	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
83	#106×C₆₇	—	4	2	no	—
84	C₄⋊C₈×C₆₇	—	2	2	no	—
85	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	complemented
86	#103×C₆₇	—	4	2	no	—
87	#102×C₆₇	—	4	2	no	—
88	#102×C₆₇	—	4	2	no	—
89	#102×C₆₇	C₂×C₄	1	8	yes	complemented
90	C₄×C₈×C₆₇	—	4	2	no	—
91	#95×C₆₇	—	2	1	no	—
92	#95×C₆₇	—	2	1	no	—
93	#95×C₆₇	—	2	1	no	—
94	#95×C₆₇	—	2	1	no	—
95	#95×C₆₇	—	2	1	no	—
96	#95×C₆₇	—	2	1	no	—
97	#95×C₆₇	—	2	1	no	—
98	C₂⁴×C₆₇	—	2	2	no	—
99	C₂×Q₈×C₆₇	—	2	1	no	—
100	C₂×Q₈×C₆₇	—	2	1	no	—
101	C₂×Q₈×C₆₇	—	2	1	no	—
102	C₂×Q₈×C₆₇	—	2	1	no	—
103	C₂×Q₈×C₆₇	—	2	1	no	—
104	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
105	C₂×D₄×C₆₇	—	2	4	no	—
106	C₂×D₄×C₆₇	—	2	4	no	—
107	C₂×D₄×C₆₇	—	2	2	no	—
108	C₂×D₄×C₆₇	—	2	2	no	—
109	C₂×D₄×C₆₇	—	2	2	no	—
110	C₂×D₄×C₆₇	—	2	2	no	—
111	C₂×D₄×C₆₇	—	2	2	no	—
112	C₂×D₄×C₆₇	—	4	2	no	—
113	C₂×D₄×C₆₇	—	4	2	no	—
114	C₂×D₄×C₆₇	—	4	2	no	—
115	C₂×D₄×C₆₇	—	4	2	no	—
116	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
117	C₂²×C₄×C₆₇	—	2	1	no	—
118	C₂²×C₄×C₆₇	—	2	1	no	—
119	C₂²×C₄×C₆₇	—	2	1	no	—
120	C₂²×C₄×C₆₇	—	2	1	no	—
121	C₂²×C₄×C₆₇	—	2	2	no	—
122	C₂²×C₄×C₆₇	—	4	4	no	—
123	C₂²×C₄×C₆₇	—	4	4	no	—
124	C₂²×C₄×C₆₇	—	4	4	no	—
125	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
132	SD₁₆×C₆₇	—	4	8	no	—
133	C₂×C₈×C₆₇	—	2	2	no	—
134	C₂×C₈×C₆₇	—	2	2	no	—
135	C₂×C₈×C₆₇	—	2	2	no	—
136	C₂×C₈×C₆₇	—	2	2	no	—
137	C₂×C₈×C₆₇	—	2	2	no	—
138	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
139	C₄⋊C₄×C₆₇	—	2	4	no	—
140	C₄⋊C₄×C₆₇	—	2	1	no	—
141	C₄⋊C₄×C₆₇	—	2	1	no	—
142	C₄⋊C₄×C₆₇	—	2	1	no	—
143	C₄⋊C₄×C₆₇	—	2	2	no	—
144	C₄⋊C₄×C₆₇	—	2	1	no	—
145	C₄⋊C₄×C₆₇	—	2	2	no	—
146	C₄⋊C₄×C₆₇	—	2	1	no	—
147	C₄⋊C₄×C₆₇	—	4	2	no	—
148	C₄⋊C₄×C₆₇	—	4	2	no	—
149	C₄⋊C₄×C₆₇	—	4	2	no	—
150	C₄⋊C₄×C₆₇	—	4	2	no	—
151	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
152	C₂²⋊C₄×C₆₇	—	4	4	no	—
153	C₂²⋊C₄×C₆₇	—	4	4	no	—
154	C₂²⋊C₄×C₆₇	—	4	4	no	—
155	C₄²×C₆₇	—	2	4	no	—
156	C₄²×C₆₇	—	2	2	no	—
157	C₄²×C₆₇	—	2	2	no	—
158	C₄²×C₆₇	—	2	1	no	—
159	C₄²×C₆₇	—	2	1	no	—
160	C₄²×C₆₇	—	2	1	no	—
161	C₄²×C₆₇	—	2	1	no	—
162	C₄²×C₆₇	—	4	1	no	—
163	C₄²×C₆₇	—	4	2	no	—
164	C₄²×C₆₇	SD₁₆	1	4	yes	complemented
165	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
166	C₄²×C₆₇	#41	1	2	yes	—
167	C₂³×C₆₇	—	2	4	no	—
168	C₂³×C₆₇	—	4	2	no	—
169	C₂³×C₆₇	—	4	2	no	—
170	C₂³×C₆₇	—	4	2	no	—
171	C₂³×C₆₇	—	4	2	no	—
172	C₂³×C₆₇	—	4	2	no	—
173	C₂³×C₆₇	—	4	2	no	—
174	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
175	Q₈×C₆₇	—	2	2	no	—
176	Q₈×C₆₇	—	2	2	no	—
177	Q₈×C₆₇	—	4	1	no	—
178	Q₈×C₆₇	—	4	1	no	—
179	D₄×C₆₇	—	2	4	no	—
180	D₄×C₆₇	—	2	4	no	—
181	D₄×C₆₇	—	4	8	no	—
182	D₄×C₆₇	—	4	2	no	—
183	D₄×C₆₇	—	4	2	no	—
184	D₄×C₆₇	—	4	8	no	—
185	C₂×C₄×C₆₇	—	2	1	no	—
186	C₂×C₄×C₆₇	—	2	1	no	—
187	C₂×C₄×C₆₇	—	2	1	no	—
188	C₂×C₄×C₆₇	—	2	1	no	—
189	C₂×C₄×C₆₇	—	2	1	no	—
190	C₂×C₄×C₆₇	—	2	1	no	—
191	C₂×C₄×C₆₇	—	2	1	no	—
192	C₂×C₄×C₆₇	—	2	1	no	—
193	C₂×C₄×C₆₇	—	2	2	no	—
194	C₂×C₄×C₆₇	—	2	2	no	—
195	C₂×C₄×C₆₇	—	2	1	no	—
196	C₂×C₄×C₆₇	—	2	1	no	—
197	C₂×C₄×C₆₇	—	2	1	no	—
198	C₂×C₄×C₆₇	—	4	1	no	—
199	C₂×C₄×C₆₇	—	4	1	no	—
200	C₂×C₄×C₆₇	—	4	1	no	—
201	C₂×C₄×C₆₇	—	4	1	no	—
202	C₂×C₄×C₆₇	—	4	2	no	—
203	C₂×C₄×C₆₇	—	4	1	no	—
204	C₂×C₄×C₆₇	—	4	1	no	—
205	C₂×C₄×C₆₇	—	4	2	no	—
206	C₂×C₄×C₆₇	—	4	1	no	—
207	C₂×C₄×C₆₇	—	4	1	no	—
208	C₂×C₄×C₆₇	—	4	2	no	—
209	C₂×C₄×C₆₇	—	4	1	no	—
210	C₂×C₄×C₆₇	—	4	1	no	—
211	C₂×C₄×C₆₇	—	8	8	no	—
212	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
213	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
214	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
215	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
216	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
217	C₂×C₄×C₆₇	#135	1	2	yes	—
218	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
219	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
220	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
221	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
222	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
223	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
224	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
225	C₈×C₆₇	—	2	4	no	—
226	C₈×C₆₇	—	4	2	no	—
227	C₂²×C₆₇	—	4	4	no	—
228	C₂²×C₆₇	—	4	4	no	—
229	C₂²×C₆₇	—	4	4	no	—
230	C₂²×C₆₇	—	8	2	no	—
231	C₂²×C₆₇	—	8	2	no	—
232	C₂²×C₆₇	—	8	2	no	—
233	C₂²×C₆₇	—	8	2	no	—
234	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
235	C₂²×C₆₇	#535	1	1	yes	characteristic
236	C₂²×C₆₇	#537	1	1	yes	characteristic
237	C₂²×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
238	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
239	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
240	C₂²×C₆₇	#575	1	1	yes	characteristic
241	C₄×C₆₇	—	2	2	no	—
242	C₄×C₆₇	—	2	2	no	—
243	C₄×C₆₇	—	2	2	no	—
244	C₄×C₆₇	—	2	2	no	—
245	C₄×C₆₇	—	4	2	no	—
246	C₄×C₆₇	—	4	2	no	—
247	C₄×C₆₇	—	4	1	no	—
248	C₄×C₆₇	—	8	2	no	—
249	C₄×C₆₇	—	8	1	no	—
250	C₄×C₆₇	—	8	1	no	—
251	C₄×C₆₇	C₄×SD₁₆	1	4	yes	—
252	C₄×C₆₇	#518	1	1	yes	characteristic
253	C₄×C₆₇	#520	1	1	yes	characteristic
254	C₄×C₆₇	#540	1	1	yes	characteristic
255	C₄×C₆₇	#542	1	1	yes	characteristic
256	#22147	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
257	C₂×C₆₇	—	8	4	no	—
258	C₂×C₆₇	#2324	1	1	yes	characteristic
259	C₂×C₆₇	C₂×C₄×SD₁₆	1	1	yes	characteristic
260	C₂×C₆₇	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
261	C₂×C₆₇	#3208	1	1	yes	characteristic
262	C₂×C₆₇	#3212	1	1	yes	characteristic
263	C₂×C₆₇	#3246	1	1	yes	characteristic
264	C₂×C₆₇	#3247	1	1	yes	characteristic
265	C₂×#460	C₂×C₆₇	1	1	yes	characteristic
266	C₄×#127	C₂×C₆₇	1	1	yes	characteristic
267	#2295	C₂×C₆₇	1	1	yes	characteristic, complemented
268	#1982	C₂×C₆₇	1	1	yes	characteristic
269	#1968	C₂×C₆₇	1	2	yes	complemented
270	#1948	C₂×C₆₇	1	1	yes	characteristic, complemented
271	#1934	C₂×C₆₇	1	2	yes	complemented
272	#1877	C₂×C₆₇	1	2	yes	complemented
273	#1864	C₂×C₆₇	1	2	yes	complemented
274	#1803	C₂×C₆₇	1	1	yes	characteristic, complemented
275	C₄×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
276	C₆₇	#22147	1	1	yes	characteristic, direct product factor, Sylow
277	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
278	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
279	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
280	C₂×C₄×D₄	—	2	1	no	—
281	C₂×C₄×D₄	—	2	1	no	—
282	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
283	#460	C₄×C₆₇	1	8	yes	complemented
284	C₂×#106	C₂²×C₆₇	1	2	yes	—
285	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
286	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
287	C₂×#103	C₂²×C₆₇	1	2	yes	—
288	C₂×#102	C₂²×C₆₇	1	2	yes	—
289	C₂×#102	C₂²×C₆₇	1	2	yes	—
290	C₂×#102	C₂²×C₆₇	1	2	yes	—
291	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
292	#391	C₂²×C₆₇	1	1	yes	characteristic
293	#391	C₂²×C₆₇	1	1	yes	characteristic
294	#390	—	2	1	no	—
295	#390	—	2	1	no	—
296	#390	C₂²×C₆₇	1	2	yes	—
297	#389	C₂²×C₆₇	1	1	yes	characteristic
298	#389	C₂²×C₆₇	1	1	yes	characteristic
299	#387	—	2	1	no	—
300	#387	—	2	1	no	—
301	#384	C₂²×C₆₇	1	1	yes	characteristic
302	C₄×C₄⋊C₄	—	2	1	no	—
303	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
304	#376	—	2	1	no	—
305	C₄³	C₂²×C₆₇	1	1	yes	characteristic
306	#340	C₂²×C₆₇	1	2	yes	—
307	#340	C₂²×C₆₇	1	2	yes	—
308	#336	C₂²×C₆₇	1	2	yes	—
309	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
310	C₂²×D₄	—	2	1	no	—
311	C₂²×D₄	—	2	1	no	—
312	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
313	C₂³×C₄	—	2	2	no	—
314	C₂×SD₁₆	—	4	2	no	—
315	C₂×SD₁₆	—	4	2	no	—
316	C₂×SD₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
317	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
318	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
319	#127	—	2	2	no	—
320	#127	C₂×C₄×C₆₇	1	4	yes	—
321	C₄×Q₈	—	2	2	no	—
322	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
323	C₄×D₄	—	4	2	no	—
324	C₄×D₄	—	4	2	no	—
325	C₄×D₄	—	4	4	no	—
326	C₄×D₄	—	4	4	no	—
327	C₂×C₄⋊C₄	—	2	1	no	—
328	C₂×C₄⋊C₄	—	2	1	no	—
329	C₂×C₄⋊C₄	—	2	1	no	—
330	C₂×C₄⋊C₄	—	2	1	no	—
331	C₂×C₄⋊C₄	—	2	1	no	—
332	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
333	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
334	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
335	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
336	C₂×C₂²⋊C₄	—	2	2	no	—
337	C₂×C₂²⋊C₄	—	2	2	no	—
338	C₂×C₂²⋊C₄	—	2	2	no	—
339	C₂×C₄²	—	2	1	no	—
340	C₂×C₄²	—	2	1	no	—
341	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
342	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
343	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
344	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
345	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
346	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
347	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
348	#106	—	4	2	no	—
349	C₄⋊C₈	—	2	2	no	—
350	C₄⋊C₈	C₂×C₄×C₆₇	1	4	yes	complemented
351	#103	—	4	2	no	—
352	#102	—	4	2	no	—
353	#102	—	4	2	no	—
354	#102	C₂×C₄×C₆₇	1	8	yes	complemented
355	C₄×C₈	—	4	2	no	—
356	#95	—	2	1	no	—
357	#95	—	2	1	no	—
358	#95	—	2	1	no	—
359	#95	—	2	1	no	—
360	#95	—	2	1	no	—
361	#95	—	2	1	no	—
362	#95	—	2	1	no	—
363	C₂⁴	—	2	2	no	—
364	C₂×Q₈	—	2	1	no	—
365	C₂×Q₈	—	2	1	no	—
366	C₂×Q₈	—	2	1	no	—
367	C₂×Q₈	—	2	1	no	—
368	C₂×Q₈	—	2	1	no	—
369	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
370	C₂×D₄	—	2	4	no	—
371	C₂×D₄	—	2	4	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	2	no	—
377	C₂×D₄	—	4	2	no	—
378	C₂×D₄	—	4	2	no	—
379	C₂×D₄	—	4	2	no	—
380	C₂×D₄	—	4	2	no	—
381	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	4	4	no	—
388	C₂²×C₄	—	4	4	no	—
389	C₂²×C₄	—	4	4	no	—
390	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
391	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
392	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
393	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
394	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
395	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
396	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
397	SD₁₆	—	4	8	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
404	C₄⋊C₄	—	2	4	no	—
405	C₄⋊C₄	—	2	1	no	—
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	1	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	4	2	no	—
413	C₄⋊C₄	—	4	2	no	—
414	C₄⋊C₄	—	4	2	no	—
415	C₄⋊C₄	—	4	2	no	—
416	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
417	C₂²⋊C₄	—	4	4	no	—
418	C₂²⋊C₄	—	4	4	no	—
419	C₂²⋊C₄	—	4	4	no	—
420	C₄²	—	2	4	no	—
421	C₄²	—	2	2	no	—
422	C₄²	—	2	2	no	—
423	C₄²	—	2	1	no	—
424	C₄²	—	2	1	no	—
425	C₄²	—	2	1	no	—
426	C₄²	—	2	1	no	—
427	C₄²	—	4	1	no	—
428	C₄²	—	4	2	no	—
429	C₄²	SD₁₆×C₆₇	1	4	yes	complemented
430	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
431	C₄²	#41×C₆₇	1	2	yes	—
432	C₂³	—	2	4	no	—
433	C₂³	—	4	2	no	—
434	C₂³	—	4	2	no	—
435	C₂³	—	4	2	no	—
436	C₂³	—	4	2	no	—
437	C₂³	—	4	2	no	—
438	C₂³	—	4	2	no	—
439	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
440	Q₈	—	2	2	no	—
441	Q₈	—	2	2	no	—
442	Q₈	—	4	1	no	—
443	Q₈	—	4	1	no	—
444	D₄	—	2	4	no	—
445	D₄	—	2	4	no	—
446	D₄	—	4	8	no	—
447	D₄	—	4	2	no	—
448	D₄	—	4	2	no	—
449	D₄	—	4	8	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	4	1	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	8	8	no	—
477	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
478	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
479	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
480	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
481	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
482	C₂×C₄	#135×C₆₇	1	2	yes	—
483	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
484	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
485	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
486	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
487	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
488	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
489	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
490	C₈	—	2	4	no	—
491	C₈	—	4	2	no	—
492	C₂²	—	4	4	no	—
493	C₂²	—	4	4	no	—
494	C₂²	—	4	4	no	—
495	C₂²	—	8	2	no	—
496	C₂²	—	8	2	no	—
497	C₂²	—	8	2	no	—
498	C₂²	—	8	2	no	—
499	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
500	C₂²	#535×C₆₇	1	1	yes	characteristic
501	C₂²	#537×C₆₇	1	1	yes	characteristic
502	C₂²	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
503	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
504	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
505	C₂²	#575×C₆₇	1	1	yes	characteristic
506	C₄	—	2	2	no	—
507	C₄	—	2	2	no	—
508	C₄	—	2	2	no	—
509	C₄	—	2	2	no	—
510	C₄	—	4	2	no	—
511	C₄	—	4	2	no	—
512	C₄	—	4	1	no	—
513	C₄	—	8	2	no	—
514	C₄	—	8	1	no	—
515	C₄	—	8	1	no	—
516	C₄	C₄×SD₁₆×C₆₇	1	4	yes	—
517	C₄	#518×C₆₇	1	1	yes	characteristic
518	C₄	#520×C₆₇	1	1	yes	characteristic
519	C₄	#540×C₆₇	1	1	yes	characteristic
520	C₄	#542×C₆₇	1	1	yes	characteristic
521	C₂	—	8	4	no	—
522	C₂	C₆₇×#2324	1	1	yes	characteristic
523	C₂	C₂×C₄×SD₁₆×C₆₇	1	1	yes	characteristic
524	C₂	C₄×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
525	C₂	C₆₇×#3208	1	1	yes	characteristic
526	C₂	C₆₇×#3212	1	1	yes	characteristic
527	C₂	C₆₇×#3246	1	1	yes	characteristic
528	C₂	C₆₇×#3247	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×#127×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2295	C₂	semidirect
5. C₆₇×#1968	C₂	semidirect
6. C₆₇×#1948	C₂	semidirect
7. C₆₇×#1934	C₂	semidirect
8. C₆₇×#1877	C₂	semidirect
9. C₆₇×#1864	C₂	semidirect
10. C₆₇×#1803	C₂	semidirect
11. C₄×C₄⋊C₈×C₆₇	C₂	semidirect
18. #460×C₆₇	C₄	semidirect
51. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
57. C₄×Q₈×C₆₇	C₂×C₄	semidirect
85. C₄⋊C₈×C₆₇	C₂×C₄	semidirect
89. #102×C₆₇	C₂×C₄	semidirect
164. C₄²×C₆₇	SD₁₆	semidirect
256. #22147	C₆₇	direct (N × H)
267. #2295	C₂×C₆₇	semidirect
269. #1968	C₂×C₆₇	semidirect
270. #1948	C₂×C₆₇	semidirect
271. #1934	C₂×C₆₇	semidirect
272. #1877	C₂×C₆₇	semidirect
273. #1864	C₂×C₆₇	semidirect
274. #1803	C₂×C₆₇	semidirect
275. C₄×C₄⋊C₈	C₂×C₆₇	semidirect
276. C₆₇	#22147	direct (N × H)
283. #460	C₄×C₆₇	semidirect
316. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
322. C₄×Q₈	C₂×C₄×C₆₇	semidirect
350. C₄⋊C₈	C₂×C₄×C₆₇	semidirect
354. #102	C₂×C₄×C₆₇	semidirect
429. C₄²	SD₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#460×C₆₇	2
C₄×#127×C₆₇	2
C₆₇×#2295	2
C₆₇×#1982	2
C₆₇×#1968	2
C₆₇×#1948	2
C₆₇×#1934	2
C₆₇×#1877	2
C₆₇×#1864	2
C₆₇×#1803	2
C₄×C₄⋊C₈×C₆₇	2
#22147	67

Subgroups — C67×#22147 #30701398

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22147 #30701398