Subgroups — C₄×#472×C₆₇ #30700804

Order 17152.

946 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#472×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₄×AGL(1,ℤ/8)×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#131×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1978	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1952	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1931	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1907	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1816	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#108×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#100×C₆₇	C₂	1	2	yes	maximal, complemented
12	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	2	yes	—
13	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
16	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
18	#472×C₆₇	C₄	1	8	yes	direct product factor
19	#442×C₆₇	—	2	2	no	—
20	#440×C₆₇	—	2	2	no	—
21	C₄×SD₁₆×C₆₇	—	2	2	no	—
22	C₄×D₈×C₆₇	—	2	1	no	—
23	C₄×D₈×C₆₇	—	2	1	no	—
24	C₄×D₈×C₆₇	—	2	1	no	—
25	C₄×D₈×C₆₇	—	2	1	no	—
26	C₄×D₈×C₆₇	—	2	2	no	—
27	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
29	#428×C₆₇	C₂²	1	2	yes	—
30	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
31	#418×C₆₇	C₂²	1	2	yes	—
32	#417×C₆₇	C₂²	1	2	yes	—
33	C₂×#102×C₆₇	C₂²	1	2	yes	—
34	C₂×#100×C₆₇	C₂²	1	2	yes	—
35	C₂×#100×C₆₇	C₂²	1	2	yes	—
36	#405×C₆₇	—	2	1	no	—
37	#405×C₆₇	—	2	1	no	—
38	#405×C₆₇	C₂²	1	1	yes	characteristic
39	#405×C₆₇	C₂²	1	1	yes	characteristic
40	C₄×M₁₆×C₆₇	—	2	1	no	—
41	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
42	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
43	C₂×C₄×C₈×C₆₇	—	2	1	no	—
44	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
45	#343×C₆₇	C₂²	1	2	yes	—
46	#343×C₆₇	C₂²	1	2	yes	—
47	#341×C₆₇	C₂²	1	2	yes	—
48	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
50	C₂³×C₄×C₆₇	—	2	2	no	—
51	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
52	AGL(1,ℤ/8)×C₆₇	C₂×C₄	1	8	yes	complemented
53	C₂×SD₁₆×C₆₇	—	2	2	no	—
54	C₂×D₈×C₆₇	—	2	2	no	—
55	C₂×D₈×C₆₇	—	2	1	no	—
56	C₂×D₈×C₆₇	—	2	1	no	—
57	C₂×D₈×C₆₇	—	2	2	no	—
58	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	complemented
59	#131×C₆₇	—	2	2	no	—
60	#131×C₆₇	—	2	2	no	—
61	#131×C₆₇	—	2	2	no	—
62	#131×C₆₇	C₂×C₄	1	4	yes	complemented
63	C₂×M₁₆×C₆₇	—	2	1	no	—
64	C₂×M₁₆×C₆₇	—	2	1	no	—
65	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
66	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
68	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₆₇	—	2	1	no	—
70	C₂²×C₈×C₆₇	—	2	1	no	—
71	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
74	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
75	C₄×D₄×C₆₇	—	2	1	no	—
76	C₄×D₄×C₆₇	—	2	1	no	—
77	C₄×D₄×C₆₇	—	2	2	no	—
78	C₄×D₄×C₆₇	—	2	2	no	—
79	C₄×D₄×C₆₇	—	2	2	no	—
80	C₄×D₄×C₆₇	—	2	2	no	—
81	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
82	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
83	#117×C₆₇	—	2	1	no	—
84	#117×C₆₇	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
86	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
87	C₂×C₄²×C₆₇	—	2	1	no	—
88	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
89	#108×C₆₇	—	2	2	no	—
90	#108×C₆₇	C₂×C₄	1	4	yes	complemented
91	#107×C₆₇	—	2	1	no	—
92	#107×C₆₇	—	2	1	no	—
93	#107×C₆₇	—	2	2	no	—
94	#106×C₆₇	—	2	2	no	—
95	#103×C₆₇	—	2	2	no	—
96	#102×C₆₇	—	2	2	no	—
97	#102×C₆₇	—	2	2	no	—
98	#102×C₆₇	—	2	2	no	—
99	#102×C₆₇	—	2	2	no	—
100	#102×C₆₇	—	2	2	no	—
101	#100×C₆₇	—	2	4	no	—
102	#100×C₆₇	C₂×C₄	1	8	yes	complemented
103	#97×C₆₇	—	2	1	no	—
104	#97×C₆₇	—	2	1	no	—
105	#97×C₆₇	—	2	1	no	—
106	#97×C₆₇	—	2	2	no	—
107	C₄×C₈×C₆₇	—	2	1	no	—
108	C₄×C₈×C₆₇	—	2	1	no	—
109	C₄×C₈×C₆₇	—	2	1	no	—
110	C₄×C₈×C₆₇	—	2	2	no	—
111	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
112	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
113	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
114	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
115	C₂⁴×C₆₇	—	2	2	no	—
116	#41×C₆₇	—	2	1	no	—
117	#41×C₆₇	—	2	1	no	—
118	#41×C₆₇	C₂²×C₄	1	2	yes	—
119	#41×C₆₇	#41	1	2	yes	—
120	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂×D₄×C₆₇	—	2	1	no	—
122	C₂×D₄×C₆₇	—	2	2	no	—
123	C₂×D₄×C₆₇	—	2	2	no	—
124	C₂×D₄×C₆₇	—	2	2	no	—
125	C₂×D₄×C₆₇	—	2	2	no	—
126	C₂×D₄×C₆₇	—	2	2	no	—
127	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
128	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	—	2	1	no	—
130	C₂²×C₄×C₆₇	—	2	1	no	—
131	C₂²×C₄×C₆₇	—	2	1	no	—
132	C₂²×C₄×C₆₇	—	2	2	no	—
133	C₂²×C₄×C₆₇	—	4	2	no	—
134	C₂²×C₄×C₆₇	—	4	2	no	—
135	C₂²×C₄×C₆₇	—	8	4	no	—
136	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
137	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
139	SD₁₆×C₆₇	—	2	8	no	—
140	D₈×C₆₇	—	2	8	no	—
141	D₈×C₆₇	—	2	4	no	—
142	D₈×C₆₇	—	2	4	no	—
143	M₁₆×C₆₇	—	2	4	no	—
144	M₁₆×C₆₇	—	2	1	no	—
145	M₁₆×C₆₇	—	2	1	no	—
146	M₁₆×C₆₇	—	2	2	no	—
147	M₁₆×C₆₇	—	2	4	no	—
148	M₁₆×C₆₇	—	2	2	no	—
149	M₁₆×C₆₇	—	2	2	no	—
150	M₁₆×C₆₇	C₂²×C₄	1	2	yes	—
151	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
152	M₁₆×C₆₇	#41	1	2	yes	—
153	C₂×C₈×C₆₇	—	2	4	no	—
154	C₂×C₈×C₆₇	—	2	2	no	—
155	C₂×C₈×C₆₇	—	2	1	no	—
156	C₂×C₈×C₆₇	—	2	1	no	—
157	C₂×C₈×C₆₇	—	2	1	no	—
158	C₂×C₈×C₆₇	—	2	1	no	—
159	C₂×C₈×C₆₇	—	2	4	no	—
160	C₂×C₈×C₆₇	—	2	2	no	—
161	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
162	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
163	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
164	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
165	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
166	C₂×C₈×C₆₇	#41	1	2	yes	—
167	C₄⋊C₄×C₆₇	—	2	1	no	—
168	C₄⋊C₄×C₆₇	—	2	1	no	—
169	C₄⋊C₄×C₆₇	—	2	2	no	—
170	C₄⋊C₄×C₆₇	—	2	2	no	—
171	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
172	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
173	C₂²⋊C₄×C₆₇	—	2	1	no	—
174	C₂²⋊C₄×C₆₇	—	2	1	no	—
175	C₂²⋊C₄×C₆₇	—	2	1	no	—
176	C₂²⋊C₄×C₆₇	—	4	2	no	—
177	C₂²⋊C₄×C₆₇	—	4	2	no	—
178	C₄²×C₆₇	—	2	1	no	—
179	C₄²×C₆₇	—	2	1	no	—
180	C₄²×C₆₇	—	2	1	no	—
181	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
182	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
183	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
184	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
185	C₂³×C₆₇	—	2	1	no	—
186	C₂³×C₆₇	—	2	4	no	—
187	C₂³×C₆₇	—	4	2	no	—
188	C₂³×C₆₇	—	4	2	no	—
189	C₂³×C₆₇	—	8	2	no	—
190	C₂³×C₆₇	—	8	2	no	—
191	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
192	Q₈×C₆₇	—	2	1	no	—
193	Q₈×C₆₇	C₄×D₄	1	2	yes	—
194	D₄×C₆₇	—	2	2	no	—
195	D₄×C₆₇	—	2	2	no	—
196	D₄×C₆₇	—	2	4	no	—
197	D₄×C₆₇	—	2	4	no	—
198	D₄×C₆₇	—	2	1	no	—
199	D₄×C₆₇	—	4	2	no	—
200	D₄×C₆₇	—	4	2	no	—
201	D₄×C₆₇	C₄×D₄	1	2	yes	—
202	C₂×C₄×C₆₇	—	2	1	no	—
203	C₂×C₄×C₆₇	—	2	2	no	—
204	C₂×C₄×C₆₇	—	2	2	no	—
205	C₂×C₄×C₆₇	—	2	2	no	—
206	C₂×C₄×C₆₇	—	2	1	no	—
207	C₂×C₄×C₆₇	—	2	1	no	—
208	C₂×C₄×C₆₇	—	4	1	no	—
209	C₂×C₄×C₆₇	—	4	2	no	—
210	C₂×C₄×C₆₇	—	4	2	no	—
211	C₂×C₄×C₆₇	—	4	2	no	—
212	C₂×C₄×C₆₇	—	8	4	no	—
213	C₂×C₄×C₆₇	—	8	4	no	—
214	C₂×C₄×C₆₇	—	8	2	no	—
215	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
216	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
217	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
218	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
219	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
220	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
223	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
224	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
225	C₈×C₆₇	—	2	2	no	—
226	C₈×C₆₇	—	2	4	no	—
227	C₈×C₆₇	—	2	1	no	—
228	C₈×C₆₇	—	2	2	no	—
229	C₈×C₆₇	—	2	1	no	—
230	C₈×C₆₇	—	4	2	no	—
231	C₈×C₆₇	C₄×D₄	1	2	yes	—
232	C₈×C₆₇	C₄×D₄	1	2	yes	—
233	C₂²×C₆₇	—	2	1	no	—
234	C₂²×C₆₇	—	2	1	no	—
235	C₂²×C₆₇	—	2	2	no	—
236	C₂²×C₆₇	—	4	1	no	—
237	C₂²×C₆₇	—	4	1	no	—
238	C₂²×C₆₇	—	4	4	no	—
239	C₂²×C₆₇	—	8	4	no	—
240	C₂²×C₆₇	—	8	4	no	—
241	C₂²×C₆₇	—	8	2	no	—
242	C₂²×C₆₇	—	8	2	no	—
243	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
244	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
245	C₂²×C₆₇	#529	1	1	yes	characteristic
246	C₄×C₆₇	—	2	1	no	—
247	C₄×C₆₇	—	2	1	no	—
248	C₄×C₆₇	—	2	1	no	—
249	C₄×C₆₇	—	2	2	no	—
250	C₄×C₆₇	—	4	1	no	—
251	C₄×C₆₇	—	4	1	no	—
252	C₄×C₆₇	—	8	2	no	—
253	C₄×C₆₇	#472	1	2	yes	direct product factor
254	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
255	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
256	C₄×C₆₇	#525	1	1	yes	characteristic
257	C₄×C₆₇	#532	1	1	yes	characteristic
258	C₄×#472	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
259	C₂×C₆₇	—	2	1	no	—
260	C₂×C₆₇	—	2	1	no	—
261	C₂×C₆₇	—	4	2	no	—
262	C₂×C₆₇	—	8	4	no	—
263	C₂×C₆₇	C₄×#121	1	1	yes	characteristic
264	C₂×C₆₇	C₂×#472	1	1	yes	characteristic
265	C₂×C₆₇	#3067	1	1	yes	characteristic
266	C₂×#472	C₂×C₆₇	1	1	yes	characteristic
267	C₄×AGL(1,ℤ/8)	C₂×C₆₇	1	2	yes	complemented
268	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
269	C₄×#131	C₂×C₆₇	1	1	yes	characteristic, complemented
270	#1978	C₂×C₆₇	1	1	yes	characteristic, complemented
271	#1952	C₂×C₆₇	1	1	yes	characteristic, complemented
272	#1931	C₂×C₆₇	1	2	yes	complemented
273	#1907	C₂×C₆₇	1	2	yes	complemented
274	#1816	C₂×C₆₇	1	1	yes	characteristic, complemented
275	C₄×#108	C₂×C₆₇	1	1	yes	characteristic, complemented
276	C₄×#100	C₂×C₆₇	1	2	yes	complemented
277	C₆₇	C₄×#472	1	1	yes	characteristic, direct product factor, Sylow
278	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	2	yes	—
279	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
280	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
281	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
282	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
283	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
284	#472	C₄×C₆₇	1	8	yes	direct product factor
285	#442	—	2	2	no	—
286	#440	—	2	2	no	—
287	C₄×SD₁₆	—	2	2	no	—
288	C₄×D₈	—	2	2	no	—
289	C₄×D₈	—	2	1	no	—
290	C₄×D₈	—	2	1	no	—
291	C₄×D₈	—	2	1	no	—
292	C₄×D₈	—	2	1	no	—
293	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
294	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
295	#428	C₂²×C₆₇	1	2	yes	—
296	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
297	#418	C₂²×C₆₇	1	2	yes	—
298	#417	C₂²×C₆₇	1	2	yes	—
299	C₂×#102	C₂²×C₆₇	1	2	yes	—
300	C₂×#100	C₂²×C₆₇	1	2	yes	—
301	C₂×#100	C₂²×C₆₇	1	2	yes	—
302	#405	—	2	1	no	—
303	#405	—	2	1	no	—
304	#405	C₂²×C₆₇	1	1	yes	characteristic
305	#405	C₂²×C₆₇	1	1	yes	characteristic
306	C₄×M₁₆	—	2	1	no	—
307	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
308	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
309	C₂×C₄×C₈	—	2	1	no	—
310	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
311	#343	C₂²×C₆₇	1	2	yes	—
312	#343	C₂²×C₆₇	1	2	yes	—
313	#341	C₂²×C₆₇	1	2	yes	—
314	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
315	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
316	C₂³×C₄	—	2	2	no	—
317	AGL(1,ℤ/8)	—	2	4	no	—
318	AGL(1,ℤ/8)	C₂×C₄×C₆₇	1	8	yes	complemented
319	C₂×SD₁₆	—	2	2	no	—
320	C₂×D₈	—	2	2	no	—
321	C₂×D₈	—	2	1	no	—
322	C₂×D₈	—	2	1	no	—
323	C₂×D₈	—	2	2	no	—
324	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	complemented
325	#131	—	2	2	no	—
326	#131	—	2	2	no	—
327	#131	—	2	2	no	—
328	#131	C₂×C₄×C₆₇	1	4	yes	complemented
329	C₂×M₁₆	—	2	1	no	—
330	C₂×M₁₆	—	2	1	no	—
331	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
332	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
333	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
334	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
335	C₂²×C₈	—	2	1	no	—
336	C₂²×C₈	—	2	1	no	—
337	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
338	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
339	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
340	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
341	C₄×D₄	—	2	1	no	—
342	C₄×D₄	—	2	1	no	—
343	C₄×D₄	—	2	2	no	—
344	C₄×D₄	—	2	2	no	—
345	C₄×D₄	—	2	2	no	—
346	C₄×D₄	—	2	2	no	—
347	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
348	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
349	#117	—	2	1	no	—
350	#117	C₂³×C₆₇	1	1	yes	characteristic
351	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
352	C₂×C₂²⋊C₄	—	2	2	no	—
353	C₂×C₄²	—	2	1	no	—
354	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
355	#108	—	2	2	no	—
356	#108	C₂×C₄×C₆₇	1	4	yes	complemented
357	#107	—	2	1	no	—
358	#107	—	2	1	no	—
359	#107	—	2	2	no	—
360	#106	—	2	2	no	—
361	#103	—	2	2	no	—
362	#102	—	2	2	no	—
363	#102	—	2	2	no	—
364	#102	—	2	2	no	—
365	#102	—	2	2	no	—
366	#102	—	2	2	no	—
367	#100	—	2	4	no	—
368	#100	C₂×C₄×C₆₇	1	8	yes	complemented
369	#97	—	2	1	no	—
370	#97	—	2	1	no	—
371	#97	—	2	1	no	—
372	#97	—	2	2	no	—
373	C₄×C₈	—	2	1	no	—
374	C₄×C₈	—	2	1	no	—
375	C₄×C₈	—	2	1	no	—
376	C₄×C₈	—	2	2	no	—
377	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
378	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
379	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
380	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
381	C₂⁴	—	2	2	no	—
382	#41	—	2	1	no	—
383	#41	—	2	1	no	—
384	#41	C₂²×C₄×C₆₇	1	2	yes	—
385	#41	#41×C₆₇	1	2	yes	—
386	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	—	2	2	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	—	2	2	no	—
393	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
394	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	—	4	2	no	—
400	C₂²×C₄	—	4	2	no	—
401	C₂²×C₄	—	8	4	no	—
402	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
403	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
404	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
405	SD₁₆	—	2	8	no	—
406	D₈	—	2	8	no	—
407	D₈	—	2	4	no	—
408	D₈	—	2	4	no	—
409	M₁₆	—	2	4	no	—
410	M₁₆	—	2	1	no	—
411	M₁₆	—	2	1	no	—
412	M₁₆	—	2	2	no	—
413	M₁₆	—	2	4	no	—
414	M₁₆	—	2	2	no	—
415	M₁₆	—	2	2	no	—
416	M₁₆	C₂²×C₄×C₆₇	1	2	yes	—
417	M₁₆	C₂²×C₄×C₆₇	1	4	yes	—
418	M₁₆	#41×C₆₇	1	2	yes	—
419	C₂×C₈	—	2	4	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	2	1	no	—
424	C₂×C₈	—	2	1	no	—
425	C₂×C₈	—	2	4	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
428	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
429	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
430	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
431	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
432	C₂×C₈	#41×C₆₇	1	2	yes	—
433	C₄⋊C₄	—	2	1	no	—
434	C₄⋊C₄	—	2	1	no	—
435	C₄⋊C₄	—	2	2	no	—
436	C₄⋊C₄	—	2	2	no	—
437	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
438	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
439	C₂²⋊C₄	—	2	1	no	—
440	C₂²⋊C₄	—	2	1	no	—
441	C₂²⋊C₄	—	2	1	no	—
442	C₂²⋊C₄	—	4	2	no	—
443	C₂²⋊C₄	—	4	2	no	—
444	C₄²	—	2	1	no	—
445	C₄²	—	2	1	no	—
446	C₄²	—	2	1	no	—
447	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
448	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
449	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
450	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
451	C₂³	—	2	1	no	—
452	C₂³	—	2	4	no	—
453	C₂³	—	4	2	no	—
454	C₂³	—	4	2	no	—
455	C₂³	—	8	2	no	—
456	C₂³	—	8	2	no	—
457	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
458	Q₈	—	2	1	no	—
459	Q₈	C₄×D₄×C₆₇	1	2	yes	—
460	D₄	—	2	2	no	—
461	D₄	—	2	2	no	—
462	D₄	—	2	4	no	—
463	D₄	—	2	4	no	—
464	D₄	—	2	1	no	—
465	D₄	—	4	2	no	—
466	D₄	—	4	2	no	—
467	D₄	C₄×D₄×C₆₇	1	2	yes	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	2	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	8	4	no	—
479	C₂×C₄	—	8	4	no	—
480	C₂×C₄	—	8	2	no	—
481	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
482	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
483	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
484	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
485	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
486	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
487	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
488	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
489	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
490	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
491	C₈	—	2	2	no	—
492	C₈	—	2	4	no	—
493	C₈	—	2	1	no	—
494	C₈	—	2	2	no	—
495	C₈	—	2	1	no	—
496	C₈	—	4	2	no	—
497	C₈	C₄×D₄×C₆₇	1	2	yes	—
498	C₈	C₄×D₄×C₆₇	1	2	yes	—
499	C₂²	—	2	1	no	—
500	C₂²	—	2	1	no	—
501	C₂²	—	2	2	no	—
502	C₂²	—	4	1	no	—
503	C₂²	—	4	1	no	—
504	C₂²	—	4	4	no	—
505	C₂²	—	8	4	no	—
506	C₂²	—	8	4	no	—
507	C₂²	—	8	2	no	—
508	C₂²	—	8	2	no	—
509	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
510	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
511	C₂²	#529×C₆₇	1	1	yes	characteristic
512	C₄	—	2	1	no	—
513	C₄	—	2	1	no	—
514	C₄	—	2	1	no	—
515	C₄	—	2	2	no	—
516	C₄	—	4	1	no	—
517	C₄	—	4	1	no	—
518	C₄	—	8	2	no	—
519	C₄	#472×C₆₇	1	2	yes	direct product factor
520	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
521	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
522	C₄	#525×C₆₇	1	1	yes	characteristic
523	C₄	#532×C₆₇	1	1	yes	characteristic
524	C₂	—	2	1	no	—
525	C₂	—	2	1	no	—
526	C₂	—	4	2	no	—
527	C₂	—	8	4	no	—
528	C₂	C₄×#121×C₆₇	1	1	yes	characteristic
529	C₂	C₂×#472×C₆₇	1	1	yes	characteristic
530	C₂	C₆₇×#3067	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	#121
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×#121
prime radical	C₂×#472×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×AGL(1,ℤ/8)×C₆₇	C₂	semidirect
3. C₂×C₄×D₈×C₆₇	C₂	semidirect
4. C₄×#131×C₆₇	C₂	semidirect
5. C₆₇×#1978	C₂	semidirect
6. C₆₇×#1952	C₂	semidirect
7. C₆₇×#1931	C₂	semidirect
8. C₆₇×#1907	C₂	semidirect
9. C₆₇×#1816	C₂	semidirect
10. C₄×#108×C₆₇	C₂	semidirect
11. C₄×#100×C₆₇	C₂	semidirect
18. #472×C₆₇	C₄	direct (N × H)
52. AGL(1,ℤ/8)×C₆₇	C₂×C₄	semidirect
58. C₂×D₈×C₆₇	C₂×C₄	semidirect
62. #131×C₆₇	C₂×C₄	semidirect
90. #108×C₆₇	C₂×C₄	semidirect
102. #100×C₆₇	C₂×C₄	semidirect
253. C₄×C₆₇	#472	direct (N × H)
258. C₄×#472	C₆₇	direct (N × H)
267. C₄×AGL(1,ℤ/8)	C₂×C₆₇	semidirect
268. C₂×C₄×D₈	C₂×C₆₇	semidirect
269. C₄×#131	C₂×C₆₇	semidirect
270. #1978	C₂×C₆₇	semidirect
271. #1952	C₂×C₆₇	semidirect
272. #1931	C₂×C₆₇	semidirect
273. #1907	C₂×C₆₇	semidirect
274. #1816	C₂×C₆₇	semidirect
275. C₄×#108	C₂×C₆₇	semidirect
276. C₄×#100	C₂×C₆₇	semidirect
277. C₆₇	C₄×#472	direct (N × H)
284. #472	C₄×C₆₇	direct (N × H)
318. AGL(1,ℤ/8)	C₂×C₄×C₆₇	semidirect
324. C₂×D₈	C₂×C₄×C₆₇	semidirect
328. #131	C₂×C₄×C₆₇	semidirect
356. #108	C₂×C₄×C₆₇	semidirect
368. #100	C₂×C₄×C₆₇	semidirect
519. C₄	#472×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×#472×C₆₇	2
C₄×AGL(1,ℤ/8)×C₆₇	2
C₂×C₄×D₈×C₆₇	2
C₄×#131×C₆₇	2
C₆₇×#1978	2
C₆₇×#1952	2
C₆₇×#1931	2
C₆₇×#1907	2
C₆₇×#1816	2
C₄×#108×C₆₇	2
C₄×#100×C₆₇	2
C₄×#472	67

Subgroups — C4×#472×C67 #30700804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#472×C₆₇ #30700804