Subgroups — C₄×#576×C₆₇ #30739422

Order 17152.

2826 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#576×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₄×H(5,𝔽₂)×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2951	C₂	1	3	yes	maximal, complemented
4	C₆₇×#2948	C₂	1	3	yes	maximal, complemented
5	C₆₇×#2946	C₂	1	6	yes	maximal, complemented
6	C₄×AGL(1,ℤ/8)×C₆₇	C₂	1	6	yes	maximal, complemented
7	C₄×#135×C₆₇	C₂	1	3	yes	maximal, complemented
8	C₂×C₄×D₈×C₆₇	C₂	1	3	yes	maximal, complemented
9	C₆₇×#2913	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2897	C₂	1	2	yes	maximal, complemented
11	C₄×#131×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	2	yes	—
13	#576×C₆₇	C₄	1	16	yes	direct product factor
14	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	6	yes	—
15	C₂×#135×C₆₇	C₂²	1	3	yes	—
16	C₂²×D₈×C₆₇	C₂²	1	3	yes	—
17	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
19	#520×C₆₇	—	2	2	no	—
20	#520×C₆₇	C₂²	1	2	yes	—
21	#520×C₆₇	C₂²	1	6	yes	complemented
22	#518×C₆₇	—	2	6	no	—
23	#518×C₆₇	C₂²	1	6	yes	complemented
24	C₄×#41×C₆₇	—	2	2	no	—
25	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
26	C₄×#41×C₆₇	C₂²	1	6	yes	complemented
27	C₂×C₄×D₄×C₆₇	—	2	6	no	—
28	C₂×C₄×D₄×C₆₇	C₂²	1	6	yes	complemented
29	#442×C₆₇	C₂²	1	12	yes	complemented
30	#440×C₆₇	C₂²	1	12	yes	complemented
31	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
32	C₄×Q₁₆×C₆₇	C₂²	1	3	yes	complemented
33	C₄×SD₁₆×C₆₇	C₂²	1	6	yes	complemented
34	C₄×SD₁₆×C₆₇	C₂²	1	6	yes	complemented
35	C₄×D₈×C₆₇	C₂²	1	3	yes	complemented
36	C₄×D₈×C₆₇	C₂²	1	6	yes	complemented
37	C₄×D₈×C₆₇	C₂²	1	6	yes	complemented
38	C₄×D₈×C₆₇	C₂²	1	6	yes	complemented
39	C₄×D₈×C₆₇	C₂²	1	3	yes	complemented
40	#428×C₆₇	C₂²	1	6	yes	complemented
41	#427×C₆₇	C₂²	1	3	yes	complemented
42	C₂×#107×C₆₇	C₂²	1	3	yes	complemented
43	#418×C₆₇	C₂²	1	6	yes	complemented
44	#417×C₆₇	C₂²	1	6	yes	complemented
45	#416×C₆₇	C₂²	1	6	yes	complemented
46	C₂×#102×C₆₇	C₂²	1	6	yes	complemented
47	#405×C₆₇	C₂²	1	3	yes	complemented
48	#405×C₆₇	C₂²	1	3	yes	complemented
49	C₄×M₁₆×C₆₇	C₂²	1	3	yes	complemented
50	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	complemented
51	H(5,𝔽₂)×C₆₇	—	2	8	no	—
52	H(5,𝔽₂)×C₆₇	C₂×C₄	1	16	yes	complemented
53	C₂×#41×C₆₇	—	2	2	no	—
54	C₂×#41×C₆₇	C₂³	1	6	yes	—
55	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×D₄×C₆₇	—	2	6	no	—
57	C₂²×D₄×C₆₇	C₂³	1	6	yes	—
58	C₂³×C₄×C₆₇	—	2	6	no	—
59	AGL(1,ℤ/8)×C₆₇	C₂×C₄	1	48	yes	complemented
60	#135×C₆₇	C₂×C₄	1	24	yes	complemented
61	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×SD₁₆×C₆₇	C₂³	1	6	yes	—
63	C₂×D₈×C₆₇	C₂×C₄	1	24	yes	complemented
64	C₂×D₈×C₆₇	C₂³	1	3	yes	—
65	C₂×D₈×C₆₇	C₂³	1	6	yes	—
66	#131×C₆₇	—	2	4	no	—
67	#131×C₆₇	C₂×C₄	1	8	yes	complemented
68	C₂×M₁₆×C₆₇	C₂³	1	3	yes	—
69	C₂×M₁₆×C₆₇	C₂³	1	3	yes	—
70	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
71	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
72	C₄×Q₈×C₆₇	—	2	2	no	—
73	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
74	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
75	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
76	C₄×Q₈×C₆₇	C₂³	1	6	yes	—
77	C₄×D₄×C₆₇	—	2	6	no	—
78	C₄×D₄×C₆₇	—	2	6	no	—
79	C₄×D₄×C₆₇	—	2	6	no	—
80	C₄×D₄×C₆₇	—	2	6	no	—
81	C₄×D₄×C₆₇	—	2	6	no	—
82	C₄×D₄×C₆₇	—	2	12	no	—
83	C₄×D₄×C₆₇	—	2	6	no	—
84	C₄×D₄×C₆₇	—	2	6	no	—
85	C₄×D₄×C₆₇	D₄	1	3	yes	—
86	C₄×D₄×C₆₇	D₄	1	3	yes	—
87	C₄×D₄×C₆₇	C₂³	1	6	yes	—
88	C₄×D₄×C₆₇	C₂³	1	12	yes	—
89	C₄×D₄×C₆₇	C₂³	1	6	yes	—
90	#117×C₆₇	—	2	6	no	—
91	#117×C₆₇	—	2	6	no	—
92	#117×C₆₇	C₂³	1	6	yes	—
93	#117×C₆₇	C₂³	1	3	yes	—
94	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
96	C₂×C₂²⋊C₄×C₆₇	—	2	6	no	—
97	C₂×C₂²⋊C₄×C₆₇	—	2	12	no	—
98	C₂×C₄²×C₆₇	—	2	6	no	—
99	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
100	#107×C₆₇	C₂³	1	3	yes	—
101	#107×C₆₇	C₂³	1	6	yes	—
102	#107×C₆₇	C₂³	1	1	yes	characteristic
103	#106×C₆₇	C₂³	1	6	yes	—
104	#103×C₆₇	C₂³	1	2	yes	—
105	#103×C₆₇	C₂³	1	6	yes	—
106	#102×C₆₇	C₂³	1	6	yes	—
107	#102×C₆₇	C₂³	1	12	yes	—
108	#102×C₆₇	C₂³	1	6	yes	—
109	#97×C₆₇	C₂³	1	6	yes	—
110	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
111	C₄×C₈×C₆₇	C₂³	1	3	yes	—
112	C₄×C₈×C₆₇	C₂³	1	3	yes	—
113	C₄×C₈×C₆₇	C₂³	1	3	yes	—
114	C₂⁴×C₆₇	—	2	6	no	—
115	#41×C₆₇	—	2	16	no	—
116	#41×C₆₇	—	2	12	no	—
117	#41×C₆₇	C₂²×C₄	1	4	yes	—
118	#41×C₆₇	C₂²×C₄	1	24	yes	complemented
119	#41×C₆₇	#41	1	4	yes	—
120	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
122	C₂×D₄×C₆₇	—	2	6	no	—
123	C₂×D₄×C₆₇	—	2	48	no	—
124	C₂×D₄×C₆₇	—	2	6	no	—
125	C₂×D₄×C₆₇	—	2	12	no	—
126	C₂×D₄×C₆₇	C₂²×C₄	1	24	yes	complemented
127	C₂×D₄×C₆₇	C₂×D₄	1	3	yes	—
128	C₂×D₄×C₆₇	C₂⁴	1	6	yes	—
129	C₂²×C₄×C₆₇	—	2	6	no	—
130	C₂²×C₄×C₆₇	—	2	6	no	—
131	C₂²×C₄×C₆₇	—	2	6	no	—
132	C₂²×C₄×C₆₇	—	2	6	no	—
133	C₂²×C₄×C₆₇	—	2	12	no	—
134	C₂²×C₄×C₆₇	—	8	12	no	—
135	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
136	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
137	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
138	Q₁₆×C₆₇	C₂²×C₄	1	4	yes	complemented
139	SD₁₆×C₆₇	C₂²×C₄	1	24	yes	complemented
140	D₈×C₆₇	C₂²×C₄	1	12	yes	complemented
141	D₈×C₆₇	C₂²×C₄	1	24	yes	complemented
142	M₁₆×C₆₇	—	2	6	no	—
143	M₁₆×C₆₇	C₂²×C₄	1	12	yes	complemented
144	C₂×C₈×C₆₇	—	2	6	no	—
145	C₂×C₈×C₆₇	C₂²×C₄	1	12	yes	complemented
146	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
147	C₂×C₈×C₆₇	C₂⁴	1	3	yes	—
148	C₂×C₈×C₆₇	C₂⁴	1	3	yes	—
149	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
150	C₄⋊C₄×C₆₇	—	2	6	no	—
151	C₄⋊C₄×C₆₇	—	2	6	no	—
152	C₄⋊C₄×C₆₇	—	2	2	no	—
153	C₄⋊C₄×C₆₇	C₂×D₄	1	3	yes	—
154	C₄⋊C₄×C₆₇	C₂×D₄	1	3	yes	—
155	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
156	C₄⋊C₄×C₆₇	C₂⁴	1	6	yes	—
157	C₂²⋊C₄×C₆₇	—	2	6	no	—
158	C₂²⋊C₄×C₆₇	—	2	6	no	—
159	C₂²⋊C₄×C₆₇	—	2	12	no	—
160	C₂²⋊C₄×C₆₇	—	2	6	no	—
161	C₂²⋊C₄×C₆₇	C₂×D₄	1	6	yes	—
162	C₄²×C₆₇	—	2	6	no	—
163	C₄²×C₆₇	—	2	6	no	—
164	C₄²×C₆₇	—	2	2	no	—
165	C₄²×C₆₇	C₂×D₄	1	3	yes	—
166	C₄²×C₆₇	C₂×D₄	1	3	yes	—
167	C₄²×C₆₇	C₂⁴	1	3	yes	—
168	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
169	C₂³×C₆₇	—	2	6	no	—
170	C₂³×C₆₇	—	2	24	no	—
171	C₂³×C₆₇	—	8	6	no	—
172	C₂³×C₆₇	—	8	6	no	—
173	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
174	Q₈×C₆₇	—	2	2	no	—
175	Q₈×C₆₇	C₄×D₄	1	4	yes	—
176	Q₈×C₆₇	C₂³×C₄	1	4	yes	—
177	D₄×C₆₇	—	2	24	no	—
178	D₄×C₆₇	—	2	24	no	—
179	D₄×C₆₇	—	2	6	no	—
180	D₄×C₆₇	C₄×D₄	1	12	yes	—
181	D₄×C₆₇	C₂³×C₄	1	12	yes	—
182	C₂×C₄×C₆₇	—	2	2	no	—
183	C₂×C₄×C₆₇	—	2	24	no	—
184	C₂×C₄×C₆₇	—	2	6	no	—
185	C₂×C₄×C₆₇	—	2	2	no	—
186	C₂×C₄×C₆₇	—	2	6	no	—
187	C₂×C₄×C₆₇	—	4	3	no	—
188	C₂×C₄×C₆₇	—	4	12	no	—
189	C₂×C₄×C₆₇	—	8	6	no	—
190	C₂×C₄×C₆₇	—	8	12	no	—
191	C₂×C₄×C₆₇	—	8	6	no	—
192	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
193	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
194	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
196	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
197	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₂×#41	1	6	yes	—
199	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
200	C₈×C₆₇	—	2	1	no	—
201	C₈×C₆₇	—	2	3	no	—
202	C₈×C₆₇	C₂³×C₄	1	2	yes	—
203	C₈×C₆₇	C₂³×C₄	1	6	yes	—
204	C₂²×C₆₇	—	2	3	no	—
205	C₂²×C₆₇	—	2	3	no	—
206	C₂²×C₆₇	—	2	12	no	—
207	C₂²×C₆₇	—	4	6	no	—
208	C₂²×C₆₇	—	4	6	no	—
209	C₂²×C₆₇	—	8	24	no	—
210	C₂²×C₆₇	C₂×C₄×D₄	1	6	yes	—
211	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
212	C₄×C₆₇	—	2	1	no	—
213	C₄×C₆₇	—	2	3	no	—
214	C₄×C₆₇	—	2	4	no	—
215	C₄×C₆₇	—	2	3	no	—
216	C₄×C₆₇	—	4	2	no	—
217	C₄×C₆₇	—	4	6	no	—
218	C₄×C₆₇	C₂×C₄×D₄	1	6	yes	—
219	C₄×C₆₇	#576	1	2	yes	direct product factor
220	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
221	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
222	C₄×#576	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
223	C₂×C₆₇	—	2	6	no	—
224	C₂×C₆₇	—	4	12	no	—
225	C₂×C₆₇	C₂²×C₄×D₄	1	1	yes	characteristic
226	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
227	C₂×C₆₇	#3584	1	1	yes	characteristic
228	C₂×#576	C₂×C₆₇	1	1	yes	characteristic
229	C₄×H(5,𝔽₂)	C₂×C₆₇	1	2	yes	complemented
230	#2951	C₂×C₆₇	1	3	yes	complemented
231	#2948	C₂×C₆₇	1	3	yes	complemented
232	#2946	C₂×C₆₇	1	6	yes	complemented
233	C₄×AGL(1,ℤ/8)	C₂×C₆₇	1	6	yes	complemented
234	C₄×#135	C₂×C₆₇	1	3	yes	complemented
235	C₂×C₄×D₈	C₂×C₆₇	1	3	yes	complemented
236	#2913	C₂×C₆₇	1	1	yes	characteristic, complemented
237	#2897	C₂×C₆₇	1	2	yes	complemented
238	C₄×#131	C₂×C₆₇	1	1	yes	characteristic, complemented
239	C₆₇	C₄×#576	1	1	yes	characteristic, direct product factor, Sylow
240	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	2	yes	—
241	#576	C₄×C₆₇	1	16	yes	direct product factor
242	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	6	yes	—
243	C₂×#135	C₂²×C₆₇	1	3	yes	—
244	C₂²×D₈	C₂²×C₆₇	1	3	yes	—
245	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
246	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
247	#520	—	2	2	no	—
248	#520	C₂²×C₆₇	1	2	yes	—
249	#520	C₂²×C₆₇	1	6	yes	complemented
250	#518	—	2	6	no	—
251	#518	C₂²×C₆₇	1	6	yes	complemented
252	C₄×#41	—	2	2	no	—
253	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
254	C₄×#41	C₂²×C₆₇	1	6	yes	complemented
255	C₂×C₄×D₄	—	2	6	no	—
256	C₂×C₄×D₄	C₂²×C₆₇	1	6	yes	complemented
257	#442	C₂²×C₆₇	1	12	yes	complemented
258	#440	C₂²×C₆₇	1	12	yes	complemented
259	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
260	C₄×Q₁₆	C₂²×C₆₇	1	3	yes	complemented
261	C₄×SD₁₆	C₂²×C₆₇	1	6	yes	complemented
262	C₄×SD₁₆	C₂²×C₆₇	1	6	yes	complemented
263	C₄×D₈	C₂²×C₆₇	1	3	yes	complemented
264	C₄×D₈	C₂²×C₆₇	1	6	yes	complemented
265	C₄×D₈	C₂²×C₆₇	1	6	yes	complemented
266	C₄×D₈	C₂²×C₆₇	1	6	yes	complemented
267	C₄×D₈	C₂²×C₆₇	1	3	yes	complemented
268	#428	C₂²×C₆₇	1	6	yes	complemented
269	#427	C₂²×C₆₇	1	3	yes	complemented
270	C₂×#107	C₂²×C₆₇	1	3	yes	complemented
271	#418	C₂²×C₆₇	1	6	yes	complemented
272	#417	C₂²×C₆₇	1	6	yes	complemented
273	#416	C₂²×C₆₇	1	6	yes	complemented
274	C₂×#102	C₂²×C₆₇	1	6	yes	complemented
275	#405	C₂²×C₆₇	1	3	yes	complemented
276	#405	C₂²×C₆₇	1	3	yes	complemented
277	C₄×M₁₆	C₂²×C₆₇	1	3	yes	complemented
278	C₂×C₄×C₈	C₂²×C₆₇	1	3	yes	complemented
279	H(5,𝔽₂)	—	2	8	no	—
280	H(5,𝔽₂)	C₂×C₄×C₆₇	1	16	yes	complemented
281	C₂×#41	—	2	2	no	—
282	C₂×#41	C₂³×C₆₇	1	6	yes	—
283	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
284	C₂²×D₄	—	2	6	no	—
285	C₂²×D₄	C₂³×C₆₇	1	6	yes	—
286	C₂³×C₄	—	2	6	no	—
287	AGL(1,ℤ/8)	C₂×C₄×C₆₇	1	48	yes	complemented
288	#135	C₂×C₄×C₆₇	1	24	yes	complemented
289	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
290	C₂×SD₁₆	C₂³×C₆₇	1	6	yes	—
291	C₂×D₈	C₂×C₄×C₆₇	1	24	yes	complemented
292	C₂×D₈	C₂³×C₆₇	1	3	yes	—
293	C₂×D₈	C₂³×C₆₇	1	6	yes	—
294	#131	—	2	4	no	—
295	#131	C₂×C₄×C₆₇	1	8	yes	complemented
296	C₂×M₁₆	C₂³×C₆₇	1	3	yes	—
297	C₂×M₁₆	C₂³×C₆₇	1	3	yes	—
298	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
299	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
300	C₄×Q₈	—	2	2	no	—
301	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
302	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
303	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
304	C₄×Q₈	C₂³×C₆₇	1	6	yes	—
305	C₄×D₄	—	2	6	no	—
306	C₄×D₄	—	2	6	no	—
307	C₄×D₄	—	2	6	no	—
308	C₄×D₄	—	2	6	no	—
309	C₄×D₄	—	2	6	no	—
310	C₄×D₄	—	2	12	no	—
311	C₄×D₄	—	2	6	no	—
312	C₄×D₄	—	2	6	no	—
313	C₄×D₄	D₄×C₆₇	1	3	yes	—
314	C₄×D₄	D₄×C₆₇	1	3	yes	—
315	C₄×D₄	C₂³×C₆₇	1	6	yes	—
316	C₄×D₄	C₂³×C₆₇	1	12	yes	—
317	C₄×D₄	C₂³×C₆₇	1	6	yes	—
318	#117	—	2	6	no	—
319	#117	—	2	6	no	—
320	#117	C₂³×C₆₇	1	6	yes	—
321	#117	C₂³×C₆₇	1	3	yes	—
322	C₂×C₄⋊C₄	—	2	6	no	—
323	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
324	C₂×C₂²⋊C₄	—	2	6	no	—
325	C₂×C₂²⋊C₄	—	2	12	no	—
326	C₂×C₄²	—	2	6	no	—
327	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
328	#107	C₂³×C₆₇	1	3	yes	—
329	#107	C₂³×C₆₇	1	6	yes	—
330	#107	C₂³×C₆₇	1	1	yes	characteristic
331	#106	C₂³×C₆₇	1	6	yes	—
332	#103	C₂³×C₆₇	1	2	yes	—
333	#103	C₂³×C₆₇	1	6	yes	—
334	#102	C₂³×C₆₇	1	6	yes	—
335	#102	C₂³×C₆₇	1	12	yes	—
336	#102	C₂³×C₆₇	1	6	yes	—
337	#97	C₂³×C₆₇	1	6	yes	—
338	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
339	C₄×C₈	C₂³×C₆₇	1	3	yes	—
340	C₄×C₈	C₂³×C₆₇	1	3	yes	—
341	C₄×C₈	C₂³×C₆₇	1	3	yes	—
342	C₂⁴	—	2	6	no	—
343	#41	—	2	16	no	—
344	#41	—	2	12	no	—
345	#41	C₂²×C₄×C₆₇	1	4	yes	—
346	#41	C₂²×C₄×C₆₇	1	24	yes	complemented
347	#41	#41×C₆₇	1	4	yes	—
348	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
349	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
350	C₂×D₄	—	2	6	no	—
351	C₂×D₄	—	2	48	no	—
352	C₂×D₄	—	2	6	no	—
353	C₂×D₄	—	2	12	no	—
354	C₂×D₄	C₂²×C₄×C₆₇	1	24	yes	complemented
355	C₂×D₄	C₂×D₄×C₆₇	1	3	yes	—
356	C₂×D₄	C₂⁴×C₆₇	1	6	yes	—
357	C₂²×C₄	—	2	6	no	—
358	C₂²×C₄	—	2	6	no	—
359	C₂²×C₄	—	2	6	no	—
360	C₂²×C₄	—	2	6	no	—
361	C₂²×C₄	—	2	12	no	—
362	C₂²×C₄	—	8	12	no	—
363	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
364	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
365	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
366	Q₁₆	C₂²×C₄×C₆₇	1	4	yes	complemented
367	SD₁₆	C₂²×C₄×C₆₇	1	24	yes	complemented
368	D₈	C₂²×C₄×C₆₇	1	12	yes	complemented
369	D₈	C₂²×C₄×C₆₇	1	24	yes	complemented
370	M₁₆	—	2	6	no	—
371	M₁₆	C₂²×C₄×C₆₇	1	12	yes	complemented
372	C₂×C₈	—	2	6	no	—
373	C₂×C₈	C₂²×C₄×C₆₇	1	12	yes	complemented
374	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
375	C₂×C₈	C₂⁴×C₆₇	1	3	yes	—
376	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
377	C₂×C₈	C₂⁴×C₆₇	1	3	yes	—
378	C₄⋊C₄	—	2	6	no	—
379	C₄⋊C₄	—	2	6	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	C₂×D₄×C₆₇	1	3	yes	—
382	C₄⋊C₄	C₂×D₄×C₆₇	1	3	yes	—
383	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
384	C₄⋊C₄	C₂⁴×C₆₇	1	6	yes	—
385	C₂²⋊C₄	—	2	6	no	—
386	C₂²⋊C₄	—	2	6	no	—
387	C₂²⋊C₄	—	2	12	no	—
388	C₂²⋊C₄	—	2	6	no	—
389	C₂²⋊C₄	C₂×D₄×C₆₇	1	6	yes	—
390	C₄²	—	2	2	no	—
391	C₄²	—	2	6	no	—
392	C₄²	—	2	6	no	—
393	C₄²	C₂×D₄×C₆₇	1	3	yes	—
394	C₄²	C₂×D₄×C₆₇	1	3	yes	—
395	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
396	C₄²	C₂⁴×C₆₇	1	3	yes	—
397	C₂³	—	2	6	no	—
398	C₂³	—	2	24	no	—
399	C₂³	—	8	6	no	—
400	C₂³	—	8	6	no	—
401	C₂³	C₂²×D₄×C₆₇	1	3	yes	—
402	Q₈	—	2	2	no	—
403	Q₈	C₄×D₄×C₆₇	1	4	yes	—
404	Q₈	C₂³×C₄×C₆₇	1	4	yes	—
405	D₄	—	2	24	no	—
406	D₄	—	2	24	no	—
407	D₄	—	2	6	no	—
408	D₄	C₄×D₄×C₆₇	1	12	yes	—
409	D₄	C₂³×C₄×C₆₇	1	12	yes	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	24	no	—
412	C₂×C₄	—	2	6	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	6	no	—
415	C₂×C₄	—	4	3	no	—
416	C₂×C₄	—	4	12	no	—
417	C₂×C₄	—	8	12	no	—
418	C₂×C₄	—	8	6	no	—
419	C₂×C₄	—	8	6	no	—
420	C₂×C₄	C₂³×C₄×C₆₇	1	6	yes	—
421	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
422	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
423	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
424	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
425	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
426	C₂×C₄	C₂×#41×C₆₇	1	6	yes	—
427	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
428	C₈	—	2	1	no	—
429	C₈	—	2	3	no	—
430	C₈	C₂³×C₄×C₆₇	1	2	yes	—
431	C₈	C₂³×C₄×C₆₇	1	6	yes	—
432	C₂²	—	2	3	no	—
433	C₂²	—	2	3	no	—
434	C₂²	—	2	12	no	—
435	C₂²	—	4	6	no	—
436	C₂²	—	4	6	no	—
437	C₂²	—	8	24	no	—
438	C₂²	C₂×C₄×D₄×C₆₇	1	6	yes	—
439	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
440	C₄	—	2	3	no	—
441	C₄	—	2	1	no	—
442	C₄	—	2	3	no	—
443	C₄	—	2	4	no	—
444	C₄	—	4	6	no	—
445	C₄	—	4	2	no	—
446	C₄	C₂×C₄×D₄×C₆₇	1	6	yes	—
447	C₄	#576×C₆₇	1	2	yes	direct product factor
448	C₄	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
449	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
450	C₂	—	2	6	no	—
451	C₂	—	4	12	no	—
452	C₂	C₂²×C₄×D₄×C₆₇	1	1	yes	characteristic
453	C₂	C₂×#576×C₆₇	1	1	yes	characteristic
454	C₂	C₆₇×#3584	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₄	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂²×C₆₇	C₂³×D₄
prime radical	C₂×#576×C₆₇	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₄×#41×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×H(5,𝔽₂)×C₆₇	C₂	semidirect
3. C₆₇×#2951	C₂	semidirect
4. C₆₇×#2948	C₂	semidirect
5. C₆₇×#2946	C₂	semidirect
6. C₄×AGL(1,ℤ/8)×C₆₇	C₂	semidirect
7. C₄×#135×C₆₇	C₂	semidirect
8. C₂×C₄×D₈×C₆₇	C₂	semidirect
9. C₆₇×#2913	C₂	semidirect
10. C₆₇×#2897	C₂	semidirect
11. C₄×#131×C₆₇	C₂	semidirect
13. #576×C₆₇	C₄	direct (N × H)
21. #520×C₆₇	C₂²	semidirect
23. #518×C₆₇	C₂²	semidirect
26. C₄×#41×C₆₇	C₂²	semidirect
28. C₂×C₄×D₄×C₆₇	C₂²	semidirect
29. #442×C₆₇	C₂²	semidirect
30. #440×C₆₇	C₂²	semidirect
31. C₄×Q₁₆×C₆₇	C₂²	semidirect
32. C₄×Q₁₆×C₆₇	C₂²	semidirect
33. C₄×SD₁₆×C₆₇	C₂²	semidirect
34. C₄×SD₁₆×C₆₇	C₂²	semidirect
35. C₄×D₈×C₆₇	C₂²	semidirect
36. C₄×D₈×C₆₇	C₂²	semidirect
37. C₄×D₈×C₆₇	C₂²	semidirect
38. C₄×D₈×C₆₇	C₂²	semidirect
39. C₄×D₈×C₆₇	C₂²	semidirect
40. #428×C₆₇	C₂²	semidirect
41. #427×C₆₇	C₂²	semidirect
42. C₂×#107×C₆₇	C₂²	semidirect
43. #418×C₆₇	C₂²	semidirect
44. #417×C₆₇	C₂²	semidirect
45. #416×C₆₇	C₂²	semidirect
46. C₂×#102×C₆₇	C₂²	semidirect
47. #405×C₆₇	C₂²	semidirect
48. #405×C₆₇	C₂²	semidirect
49. C₄×M₁₆×C₆₇	C₂²	semidirect
50. C₂×C₄×C₈×C₆₇	C₂²	semidirect
52. H(5,𝔽₂)×C₆₇	C₂×C₄	semidirect
59. AGL(1,ℤ/8)×C₆₇	C₂×C₄	semidirect
60. #135×C₆₇	C₂×C₄	semidirect
63. C₂×D₈×C₆₇	C₂×C₄	semidirect
67. #131×C₆₇	C₂×C₄	semidirect
118. #41×C₆₇	C₂²×C₄	semidirect
126. C₂×D₄×C₆₇	C₂²×C₄	semidirect
138. Q₁₆×C₆₇	C₂²×C₄	semidirect
139. SD₁₆×C₆₇	C₂²×C₄	semidirect
140. D₈×C₆₇	C₂²×C₄	semidirect
141. D₈×C₆₇	C₂²×C₄	semidirect
143. M₁₆×C₆₇	C₂²×C₄	semidirect
145. C₂×C₈×C₆₇	C₂²×C₄	semidirect
219. C₄×C₆₇	#576	direct (N × H)
222. C₄×#576	C₆₇	direct (N × H)
229. C₄×H(5,𝔽₂)	C₂×C₆₇	semidirect
230. #2951	C₂×C₆₇	semidirect
231. #2948	C₂×C₆₇	semidirect
232. #2946	C₂×C₆₇	semidirect
233. C₄×AGL(1,ℤ/8)	C₂×C₆₇	semidirect
234. C₄×#135	C₂×C₆₇	semidirect
235. C₂×C₄×D₈	C₂×C₆₇	semidirect
236. #2913	C₂×C₆₇	semidirect
237. #2897	C₂×C₆₇	semidirect
238. C₄×#131	C₂×C₆₇	semidirect
239. C₆₇	C₄×#576	direct (N × H)
241. #576	C₄×C₆₇	direct (N × H)
249. #520	C₂²×C₆₇	semidirect
251. #518	C₂²×C₆₇	semidirect
254. C₄×#41	C₂²×C₆₇	semidirect
256. C₂×C₄×D₄	C₂²×C₆₇	semidirect
257. #442	C₂²×C₆₇	semidirect
258. #440	C₂²×C₆₇	semidirect
259. C₄×Q₁₆	C₂²×C₆₇	semidirect
260. C₄×Q₁₆	C₂²×C₆₇	semidirect
261. C₄×SD₁₆	C₂²×C₆₇	semidirect
262. C₄×SD₁₆	C₂²×C₆₇	semidirect
263. C₄×D₈	C₂²×C₆₇	semidirect
264. C₄×D₈	C₂²×C₆₇	semidirect
265. C₄×D₈	C₂²×C₆₇	semidirect
266. C₄×D₈	C₂²×C₆₇	semidirect
267. C₄×D₈	C₂²×C₆₇	semidirect
268. #428	C₂²×C₆₇	semidirect
269. #427	C₂²×C₆₇	semidirect
270. C₂×#107	C₂²×C₆₇	semidirect
271. #418	C₂²×C₆₇	semidirect
272. #417	C₂²×C₆₇	semidirect
273. #416	C₂²×C₆₇	semidirect
274. C₂×#102	C₂²×C₆₇	semidirect
275. #405	C₂²×C₆₇	semidirect
276. #405	C₂²×C₆₇	semidirect
277. C₄×M₁₆	C₂²×C₆₇	semidirect
278. C₂×C₄×C₈	C₂²×C₆₇	semidirect
280. H(5,𝔽₂)	C₂×C₄×C₆₇	semidirect
287. AGL(1,ℤ/8)	C₂×C₄×C₆₇	semidirect
288. #135	C₂×C₄×C₆₇	semidirect
291. C₂×D₈	C₂×C₄×C₆₇	semidirect
295. #131	C₂×C₄×C₆₇	semidirect
346. #41	C₂²×C₄×C₆₇	semidirect
354. C₂×D₄	C₂²×C₄×C₆₇	semidirect
366. Q₁₆	C₂²×C₄×C₆₇	semidirect
367. SD₁₆	C₂²×C₄×C₆₇	semidirect
368. D₈	C₂²×C₄×C₆₇	semidirect
369. D₈	C₂²×C₄×C₆₇	semidirect
371. M₁₆	C₂²×C₄×C₆₇	semidirect
373. C₂×C₈	C₂²×C₄×C₆₇	semidirect
447. C₄	#576×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#576×C₆₇	2
C₄×H(5,𝔽₂)×C₆₇	2
C₆₇×#2951	2
C₆₇×#2948	2
C₆₇×#2946	2
C₄×AGL(1,ℤ/8)×C₆₇	2
C₄×#135×C₆₇	2
C₂×C₄×D₈×C₆₇	2
C₆₇×#2913	2
C₆₇×#2897	2
C₄×#131×C₆₇	2
C₄×#576	67

Subgroups — C4×#576×C67 #30739422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#576×C₆₇ #30739422