Subgroups — C₆₇×#22286 #30701537

Order 17152.

820 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3155	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2957	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2948	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1978	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1976	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1911	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1910	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1904	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1903	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1833	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1747	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	2	yes	—
13	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
15	#568×C₆₇	C₂²	1	2	yes	—
16	#535×C₆₇	C₂²	1	2	yes	—
17	#518×C₆₇	C₂²	1	2	yes	—
18	#496×C₆₇	C₄	1	4	yes	complemented
19	#495×C₆₇	C₄	1	2	yes	complemented
20	#493×C₆₇	C₄	1	2	yes	complemented
21	#444×C₆₇	—	2	2	no	—
22	#443×C₆₇	—	2	2	no	—
23	#442×C₆₇	—	2	2	no	—
24	C₄×D₈×C₆₇	—	2	2	no	—
25	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
26	#427×C₆₇	C₂²	1	1	yes	characteristic
27	#421×C₆₇	C₂²	1	2	yes	—
28	#421×C₆₇	C₂²	1	2	yes	—
29	#418×C₆₇	C₂²	1	2	yes	—
30	#418×C₆₇	C₂²	1	2	yes	—
31	C₂×#100×C₆₇	C₂²	1	2	yes	—
32	#410×C₆₇	C₂²	1	2	yes	—
33	#405×C₆₇	C₂²	1	1	yes	characteristic
34	#405×C₆₇	C₂²	1	1	yes	characteristic
35	#405×C₆₇	C₂²	1	1	yes	characteristic
36	#344×C₆₇	C₂²	1	2	yes	—
37	#338×C₆₇	C₂²	1	1	yes	characteristic
38	H(3,ℤ/4)×C₆₇	C₂²	1	1	yes	characteristic
39	H(3,ℤ/4)×C₆₇	C₂²	1	1	yes	characteristic
40	H(3,ℤ/4)×C₆₇	C₂²	1	1	yes	characteristic
41	H(3,ℤ/4)×C₆₇	C₂²	1	2	yes	—
42	C₂×#41×C₆₇	C₂³	1	2	yes	—
43	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
44	AGL(1,ℤ/8)×C₆₇	—	4	2	no	—
45	AGL(1,ℤ/8)×C₆₇	—	4	2	no	—
46	#135×C₆₇	—	2	1	no	—
47	#135×C₆₇	—	2	1	no	—
48	#135×C₆₇	—	2	1	no	—
49	#135×C₆₇	—	2	1	no	—
50	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
51	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	complemented
52	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
53	C₂×D₈×C₆₇	—	2	1	no	—
54	C₂×D₈×C₆₇	—	2	1	no	—
55	C₂×D₈×C₆₇	—	2	1	no	—
56	C₂×D₈×C₆₇	—	2	1	no	—
57	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
58	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
59	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
60	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	complemented
61	#131×C₆₇	—	2	4	no	—
62	#131×C₆₇	—	4	2	no	—
63	C₂×M₁₆×C₆₇	—	4	2	no	—
64	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
65	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	#127×C₆₇	C₂×C₄	1	4	yes	complemented
70	#124×C₆₇	C₂×C₄	1	4	yes	complemented
71	#121×C₆₇	—	4	2	no	—
72	#120×C₆₇	—	4	2	no	—
73	C₄×D₄×C₆₇	—	2	4	no	—
74	C₄×D₄×C₆₇	—	4	2	no	—
75	#117×C₆₇	C₂³	1	2	yes	—
76	#117×C₆₇	C₂³	1	1	yes	characteristic
77	#117×C₆₇	C₂³	1	1	yes	characteristic
78	#117×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₆₇	—	4	2	no	—
80	#108×C₆₇	—	2	1	no	—
81	#108×C₆₇	—	2	1	no	—
82	#107×C₆₇	—	2	1	no	—
83	#106×C₆₇	—	2	1	no	—
84	#104×C₆₇	—	2	4	no	—
85	#104×C₆₇	—	4	2	no	—
86	#102×C₆₇	—	2	4	no	—
87	#102×C₆₇	—	4	2	no	—
88	#100×C₆₇	—	4	2	no	—
89	#99×C₆₇	—	4	2	no	—
90	#97×C₆₇	—	2	1	no	—
91	#97×C₆₇	C₂×C₄	1	2	yes	complemented
92	#97×C₆₇	D₄	1	2	yes	complemented
93	C₄×C₈×C₆₇	—	2	1	no	—
94	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	complemented
95	C₄×C₈×C₆₇	D₄	1	2	yes	complemented
96	C₂⁴×C₆₇	—	2	2	no	—
97	#41×C₆₇	—	2	2	no	—
98	#41×C₆₇	—	2	2	no	—
99	#41×C₆₇	—	2	2	no	—
100	#41×C₆₇	—	2	2	no	—
101	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
102	C₂×D₄×C₆₇	—	2	2	no	—
103	C₂×D₄×C₆₇	—	2	2	no	—
104	C₂×D₄×C₆₇	—	2	2	no	—
105	C₂×D₄×C₆₇	—	2	2	no	—
106	C₂×D₄×C₆₇	—	2	2	no	—
107	C₂×D₄×C₆₇	—	2	2	no	—
108	C₂×D₄×C₆₇	—	4	4	no	—
109	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
110	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₆₇	—	2	2	no	—
112	C₂²×C₄×C₆₇	—	4	4	no	—
113	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
114	Q₁₆×C₆₇	—	2	1	no	—
115	Q₁₆×C₆₇	—	2	1	no	—
116	SD₁₆×C₆₇	—	2	2	no	—
117	SD₁₆×C₆₇	—	2	2	no	—
118	SD₁₆×C₆₇	—	4	4	no	—
119	D₈×C₆₇	—	2	2	no	—
120	D₈×C₆₇	—	2	2	no	—
121	D₈×C₆₇	—	2	2	no	—
122	D₈×C₆₇	—	2	2	no	—
123	D₈×C₆₇	—	2	1	no	—
124	D₈×C₆₇	—	2	1	no	—
125	D₈×C₆₇	—	4	4	no	—
126	M₁₆×C₆₇	—	2	1	no	—
127	M₁₆×C₆₇	—	2	1	no	—
128	M₁₆×C₆₇	—	2	1	no	—
129	M₁₆×C₆₇	—	2	1	no	—
130	M₁₆×C₆₇	—	2	2	no	—
131	M₁₆×C₆₇	—	2	1	no	—
132	M₁₆×C₆₇	—	2	2	no	—
133	M₁₆×C₆₇	—	2	1	no	—
134	M₁₆×C₆₇	—	4	4	no	—
135	C₂×C₈×C₆₇	—	2	1	no	—
136	C₂×C₈×C₆₇	—	2	2	no	—
137	C₂×C₈×C₆₇	—	4	4	no	—
138	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
139	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
142	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
145	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
146	C₄⋊C₄×C₆₇	—	2	1	no	—
147	C₄⋊C₄×C₆₇	—	2	1	no	—
148	C₄⋊C₄×C₆₇	—	2	2	no	—
149	C₄⋊C₄×C₆₇	—	2	1	no	—
150	C₂²⋊C₄×C₆₇	—	2	1	no	—
151	C₂²⋊C₄×C₆₇	—	2	1	no	—
152	C₂²⋊C₄×C₆₇	—	2	2	no	—
153	C₂²⋊C₄×C₆₇	—	2	1	no	—
154	C₂²⋊C₄×C₆₇	—	4	4	no	—
155	C₂²⋊C₄×C₆₇	—	4	4	no	—
156	C₄²×C₆₇	—	2	1	no	—
157	C₄²×C₆₇	—	2	2	no	—
158	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
159	C₄²×C₆₇	C₂×D₄	1	2	yes	—
160	C₂³×C₆₇	—	2	4	no	—
161	C₂³×C₆₇	—	2	2	no	—
162	C₂³×C₆₇	—	4	2	no	—
163	C₂³×C₆₇	—	4	2	no	—
164	C₂³×C₆₇	—	16	2	no	—
165	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
166	Q₈×C₆₇	—	2	2	no	—
167	Q₈×C₆₇	—	2	2	no	—
168	D₄×C₆₇	—	2	4	no	—
169	D₄×C₆₇	—	2	2	no	—
170	D₄×C₆₇	—	2	2	no	—
171	D₄×C₆₇	—	2	4	no	—
172	D₄×C₆₇	—	4	2	no	—
173	D₄×C₆₇	—	4	2	no	—
174	D₄×C₆₇	—	4	2	no	—
175	D₄×C₆₇	—	4	2	no	—
176	D₄×C₆₇	—	16	4	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	4	2	no	—
181	C₂×C₄×C₆₇	—	4	2	no	—
182	C₂×C₄×C₆₇	—	4	2	no	—
183	C₂×C₄×C₆₇	—	4	1	no	—
184	C₂×C₄×C₆₇	—	16	4	no	—
185	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
188	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
189	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
190	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
191	C₈×C₆₇	—	2	2	no	—
192	C₈×C₆₇	—	2	2	no	—
193	C₈×C₆₇	—	4	2	no	—
194	C₈×C₆₇	—	4	1	no	—
195	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
196	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
197	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
198	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
199	C₂²×C₆₇	—	4	1	no	—
200	C₂²×C₆₇	—	4	2	no	—
201	C₂²×C₆₇	—	4	4	no	—
202	C₂²×C₆₇	—	8	2	no	—
203	C₂²×C₆₇	—	8	4	no	—
204	C₂²×C₆₇	—	8	4	no	—
205	C₂²×C₆₇	—	16	2	no	—
206	C₂²×C₆₇	#518	1	1	yes	characteristic
207	C₂²×C₆₇	#535	1	1	yes	characteristic
208	C₂²×C₆₇	#540	1	1	yes	characteristic
209	C₄×C₆₇	—	4	2	no	—
210	C₄×C₆₇	—	4	2	no	—
211	C₄×C₆₇	—	4	2	no	—
212	C₄×C₆₇	—	8	1	no	—
213	C₄×C₆₇	—	8	2	no	—
214	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
215	C₄×C₆₇	#518	1	1	yes	characteristic
216	C₄×C₆₇	#535	1	1	yes	characteristic
217	C₄×C₆₇	#540	1	1	yes	characteristic
218	#22286	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
219	C₂×C₆₇	—	2	1	no	—
220	C₂×C₆₇	—	2	1	no	—
221	C₂×C₆₇	—	2	1	no	—
222	C₂×C₆₇	—	8	2	no	—
223	C₂×C₆₇	—	8	4	no	—
224	C₂×C₆₇	#2328	1	1	yes	characteristic
225	#3155	C₂×C₆₇	1	1	yes	characteristic
226	#2957	C₂×C₆₇	1	1	yes	characteristic, complemented
227	#2948	C₂×C₆₇	1	1	yes	characteristic, complemented
228	#1978	C₂×C₆₇	1	1	yes	characteristic, complemented
229	#1976	C₂×C₆₇	1	1	yes	characteristic, complemented
230	#1911	C₂×C₆₇	1	2	yes	complemented
231	#1910	C₂×C₆₇	1	2	yes	complemented
232	#1904	C₂×C₆₇	1	2	yes	complemented
233	#1903	C₂×C₆₇	1	2	yes	complemented
234	#1833	C₂×C₆₇	1	1	yes	characteristic, complemented
235	#1747	C₂×C₆₇	1	1	yes	characteristic, complemented
236	C₆₇	#22286	1	1	yes	characteristic, direct product factor, Sylow
237	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	2	yes	—
238	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
239	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
240	#568	C₂²×C₆₇	1	2	yes	—
241	#535	C₂²×C₆₇	1	2	yes	—
242	#518	C₂²×C₆₇	1	2	yes	—
243	#496	C₄×C₆₇	1	4	yes	complemented
244	#495	C₄×C₆₇	1	2	yes	complemented
245	#493	C₄×C₆₇	1	2	yes	complemented
246	#444	—	2	2	no	—
247	#443	—	2	2	no	—
248	#442	—	2	2	no	—
249	C₄×D₈	—	2	2	no	—
250	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
251	#427	C₂²×C₆₇	1	1	yes	characteristic
252	#421	C₂²×C₆₇	1	2	yes	—
253	#421	C₂²×C₆₇	1	2	yes	—
254	#418	C₂²×C₆₇	1	2	yes	—
255	#418	C₂²×C₆₇	1	2	yes	—
256	C₂×#100	C₂²×C₆₇	1	2	yes	—
257	#410	C₂²×C₆₇	1	2	yes	—
258	#405	C₂²×C₆₇	1	1	yes	characteristic
259	#405	C₂²×C₆₇	1	1	yes	characteristic
260	#405	C₂²×C₆₇	1	1	yes	characteristic
261	#344	C₂²×C₆₇	1	2	yes	—
262	#338	C₂²×C₆₇	1	1	yes	characteristic
263	H(3,ℤ/4)	C₂²×C₆₇	1	2	yes	—
264	H(3,ℤ/4)	C₂²×C₆₇	1	1	yes	characteristic
265	H(3,ℤ/4)	C₂²×C₆₇	1	1	yes	characteristic
266	H(3,ℤ/4)	C₂²×C₆₇	1	1	yes	characteristic
267	C₂×#41	C₂³×C₆₇	1	2	yes	—
268	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
269	AGL(1,ℤ/8)	—	4	2	no	—
270	AGL(1,ℤ/8)	—	4	2	no	—
271	#135	—	2	1	no	—
272	#135	—	2	1	no	—
273	#135	—	2	1	no	—
274	#135	—	2	1	no	—
275	C₂×Q₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
276	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	complemented
277	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
278	C₂×D₈	—	2	1	no	—
279	C₂×D₈	—	2	1	no	—
280	C₂×D₈	—	2	1	no	—
281	C₂×D₈	—	2	1	no	—
282	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
283	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
284	C₂×D₈	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
285	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	complemented
286	#131	—	2	4	no	—
287	#131	—	4	2	no	—
288	C₂×M₁₆	—	4	2	no	—
289	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
290	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
291	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
292	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
293	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
294	#127	C₂×C₄×C₆₇	1	4	yes	complemented
295	#124	C₂×C₄×C₆₇	1	4	yes	complemented
296	#121	—	4	2	no	—
297	#120	—	4	2	no	—
298	C₄×D₄	—	2	4	no	—
299	C₄×D₄	—	4	2	no	—
300	#117	C₂³×C₆₇	1	2	yes	—
301	#117	C₂³×C₆₇	1	1	yes	characteristic
302	#117	C₂³×C₆₇	1	1	yes	characteristic
303	#117	C₂³×C₆₇	1	1	yes	characteristic
304	C₂×C₂²⋊C₄	—	4	2	no	—
305	#108	—	2	1	no	—
306	#108	—	2	1	no	—
307	#107	—	2	1	no	—
308	#106	—	2	1	no	—
309	#104	—	2	4	no	—
310	#104	—	4	2	no	—
311	#102	—	2	4	no	—
312	#102	—	4	2	no	—
313	#100	—	4	2	no	—
314	#99	—	4	2	no	—
315	#97	—	2	1	no	—
316	#97	C₂×C₄×C₆₇	1	2	yes	complemented
317	#97	D₄×C₆₇	1	2	yes	complemented
318	C₄×C₈	—	2	1	no	—
319	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	complemented
320	C₄×C₈	D₄×C₆₇	1	2	yes	complemented
321	C₂⁴	—	2	2	no	—
322	#41	—	2	2	no	—
323	#41	—	2	2	no	—
324	#41	—	2	2	no	—
325	#41	—	2	2	no	—
326	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	2	2	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	—	4	4	no	—
334	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
335	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	4	4	no	—
338	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
339	Q₁₆	—	2	1	no	—
340	Q₁₆	—	2	1	no	—
341	SD₁₆	—	2	2	no	—
342	SD₁₆	—	2	2	no	—
343	SD₁₆	—	4	4	no	—
344	D₈	—	2	2	no	—
345	D₈	—	2	2	no	—
346	D₈	—	2	2	no	—
347	D₈	—	2	2	no	—
348	D₈	—	2	1	no	—
349	D₈	—	2	1	no	—
350	D₈	—	4	4	no	—
351	M₁₆	—	2	1	no	—
352	M₁₆	—	2	1	no	—
353	M₁₆	—	2	1	no	—
354	M₁₆	—	2	1	no	—
355	M₁₆	—	2	2	no	—
356	M₁₆	—	2	1	no	—
357	M₁₆	—	2	2	no	—
358	M₁₆	—	2	1	no	—
359	M₁₆	—	4	4	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	4	4	no	—
363	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
364	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
365	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
366	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
367	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
368	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
369	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
370	C₂×C₈	#41×C₆₇	1	1	yes	characteristic
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	1	no	—
375	C₂²⋊C₄	—	2	1	no	—
376	C₂²⋊C₄	—	2	1	no	—
377	C₂²⋊C₄	—	2	1	no	—
378	C₂²⋊C₄	—	2	2	no	—
379	C₂²⋊C₄	—	4	4	no	—
380	C₂²⋊C₄	—	4	4	no	—
381	C₄²	—	2	1	no	—
382	C₄²	—	2	2	no	—
383	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
384	C₄²	C₂×D₄×C₆₇	1	2	yes	—
385	C₂³	—	2	4	no	—
386	C₂³	—	2	2	no	—
387	C₂³	—	4	2	no	—
388	C₂³	—	4	2	no	—
389	C₂³	—	16	2	no	—
390	C₂³	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
391	Q₈	—	2	2	no	—
392	Q₈	—	2	2	no	—
393	D₄	—	2	4	no	—
394	D₄	—	2	2	no	—
395	D₄	—	2	2	no	—
396	D₄	—	2	4	no	—
397	D₄	—	4	2	no	—
398	D₄	—	4	2	no	—
399	D₄	—	4	2	no	—
400	D₄	—	4	2	no	—
401	D₄	—	16	4	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	16	4	no	—
410	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
411	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
412	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
413	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
414	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
415	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
416	C₈	—	2	2	no	—
417	C₈	—	2	2	no	—
418	C₈	—	4	2	no	—
419	C₈	—	4	1	no	—
420	C₈	C₄×D₄×C₆₇	1	1	yes	characteristic
421	C₈	C₄×D₄×C₆₇	1	1	yes	characteristic
422	C₈	C₄×D₄×C₆₇	1	1	yes	characteristic
423	C₈	C₄×D₄×C₆₇	1	1	yes	characteristic
424	C₂²	—	4	1	no	—
425	C₂²	—	4	2	no	—
426	C₂²	—	4	4	no	—
427	C₂²	—	8	2	no	—
428	C₂²	—	8	4	no	—
429	C₂²	—	8	4	no	—
430	C₂²	—	16	2	no	—
431	C₂²	#518×C₆₇	1	1	yes	characteristic
432	C₂²	#535×C₆₇	1	1	yes	characteristic
433	C₂²	#540×C₆₇	1	1	yes	characteristic
434	C₄	—	4	2	no	—
435	C₄	—	4	2	no	—
436	C₄	—	4	2	no	—
437	C₄	—	8	2	no	—
438	C₄	—	8	1	no	—
439	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
440	C₄	#518×C₆₇	1	1	yes	characteristic
441	C₄	#535×C₆₇	1	1	yes	characteristic
442	C₄	#540×C₆₇	1	1	yes	characteristic
443	C₂	—	2	1	no	—
444	C₂	—	2	1	no	—
445	C₂	—	2	1	no	—
446	C₂	—	8	4	no	—
447	C₂	—	8	2	no	—
448	C₂	C₆₇×#2328	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	#2328
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#3155	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×#2948	C₂	semidirect
4. C₆₇×#1978	C₂	semidirect
5. C₆₇×#1976	C₂	semidirect
6. C₆₇×#1911	C₂	semidirect
7. C₆₇×#1910	C₂	semidirect
8. C₆₇×#1904	C₂	semidirect
9. C₆₇×#1903	C₂	semidirect
10. C₆₇×#1833	C₂	semidirect
11. C₆₇×#1747	C₂	semidirect
18. #496×C₆₇	C₄	semidirect
19. #495×C₆₇	C₄	semidirect
20. #493×C₆₇	C₄	semidirect
50. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
51. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
52. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
59. C₂×D₈×C₆₇	C₂×C₄	semidirect
60. C₂×D₈×C₆₇	C₂×C₄	semidirect
69. #127×C₆₇	C₂×C₄	semidirect
70. #124×C₆₇	C₂×C₄	semidirect
91. #97×C₆₇	C₂×C₄	semidirect
92. #97×C₆₇	D₄	semidirect
94. C₄×C₈×C₆₇	C₂×C₄	semidirect
95. C₄×C₈×C₆₇	D₄	semidirect
218. #22286	C₆₇	direct (N × H)
226. #2957	C₂×C₆₇	semidirect
227. #2948	C₂×C₆₇	semidirect
228. #1978	C₂×C₆₇	semidirect
229. #1976	C₂×C₆₇	semidirect
230. #1911	C₂×C₆₇	semidirect
231. #1910	C₂×C₆₇	semidirect
232. #1904	C₂×C₆₇	semidirect
233. #1903	C₂×C₆₇	semidirect
234. #1833	C₂×C₆₇	semidirect
235. #1747	C₂×C₆₇	semidirect
236. C₆₇	#22286	direct (N × H)
243. #496	C₄×C₆₇	semidirect
244. #495	C₄×C₆₇	semidirect
245. #493	C₄×C₆₇	semidirect
275. C₂×Q₁₆	C₂×C₄×C₆₇	semidirect
276. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
277. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
284. C₂×D₈	C₂×C₄×C₆₇	semidirect
285. C₂×D₈	C₂×C₄×C₆₇	semidirect
294. #127	C₂×C₄×C₆₇	semidirect
295. #124	C₂×C₄×C₆₇	semidirect
316. #97	C₂×C₄×C₆₇	semidirect
317. #97	D₄×C₆₇	semidirect
319. C₄×C₈	C₂×C₄×C₆₇	semidirect
320. C₄×C₈	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×#3155	2
C₆₇×#2957	2
C₆₇×#2948	2
C₆₇×#1978	2
C₆₇×#1976	2
C₆₇×#1911	2
C₆₇×#1910	2
C₆₇×#1904	2
C₆₇×#1903	2
C₆₇×#1833	2
C₆₇×#1747	2
#22286	67

Subgroups — C67×#22286 #30701537

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22286 #30701537