Subgroups — C₂×C₆₇×#3368 #30740559

Order 17152.

2122 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#557×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#550×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3368	C₂	1	16	yes	maximal, direct product factor
4	C₂×#490×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#486×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#478×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#474×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#460×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#460×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#440×C₆₇	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₆₇	—	2	1	no	—
14	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
16	#557×C₆₇	—	2	4	no	—
17	#557×C₆₇	C₂²	1	8	yes	complemented
18	#550×C₆₇	—	2	4	no	—
19	#550×C₆₇	C₂²	1	8	yes	—
20	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
21	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#127×C₆₇	C₂²	1	2	yes	—
24	C₂×#125×C₆₇	—	2	2	no	—
25	C₂×#125×C₆₇	C₂²	1	2	yes	—
26	C₂×#121×C₆₇	—	2	1	no	—
27	C₂×#121×C₆₇	—	2	2	no	—
28	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
29	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
34	#490×C₆₇	C₂²	1	16	yes	complemented
35	#486×C₆₇	C₂²	1	8	yes	complemented
36	#478×C₆₇	C₂²	1	16	yes	complemented
37	#474×C₆₇	C₂²	1	8	yes	complemented
38	#460×C₆₇	C₂²	1	16	yes	complemented
39	#460×C₆₇	C₂²	1	8	yes	complemented
40	#446×C₆₇	C₂²	1	8	yes	complemented
41	#440×C₆₇	C₂²	1	16	yes	complemented
42	C₄×SD₁₆×C₆₇	C₂²	1	8	yes	complemented
43	C₂×#107×C₆₇	C₂²	1	2	yes	—
44	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
46	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
47	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
48	C₂×#102×C₆₇	C₂²	1	2	yes	—
49	C₂×#102×C₆₇	C₂²	1	2	yes	—
50	C₂×#102×C₆₇	C₂²	1	2	yes	—
51	C₂×#102×C₆₇	C₂²	1	2	yes	—
52	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
53	C₂×#97×C₆₇	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
55	C₂×#41×C₆₇	—	2	8	no	—
56	C₂×#41×C₆₇	—	4	8	no	—
57	C₂²×Q₈×C₆₇	—	2	1	no	—
58	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
59	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₂²×D₄×C₆₇	—	2	2	no	—
61	C₂²×D₄×C₆₇	—	2	1	no	—
62	C₂²×D₄×C₆₇	—	2	2	no	—
63	C₂²×D₄×C₆₇	—	2	1	no	—
64	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
66	C₂³×C₄×C₆₇	—	2	2	no	—
67	C₂³×C₄×C₆₇	—	2	1	no	—
68	C₂×SD₁₆×C₆₇	—	2	4	no	—
69	C₂×SD₁₆×C₆₇	—	2	8	no	—
70	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
71	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
72	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
74	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₄⋊Q₈×C₆₇	—	2	8	no	—
76	C₄⋊Q₈×C₆₇	—	2	2	no	—
77	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
78	#127×C₆₇	—	2	4	no	—
79	#127×C₆₇	—	2	2	no	—
80	#127×C₆₇	C₂³	1	4	yes	—
81	#127×C₆₇	C₂³	1	8	yes	—
82	#125×C₆₇	—	2	16	no	—
83	#125×C₆₇	—	2	4	no	—
84	#125×C₆₇	C₂³	1	8	yes	—
85	#121×C₆₇	—	2	8	no	—
86	#121×C₆₇	—	2	16	no	—
87	C₄×Q₈×C₆₇	—	2	8	no	—
88	C₄×Q₈×C₆₇	—	2	4	no	—
89	C₄×Q₈×C₆₇	—	2	2	no	—
90	C₄×Q₈×C₆₇	D₄	1	4	yes	—
91	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
92	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
93	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
94	C₄×D₄×C₆₇	—	2	4	no	—
95	C₄×D₄×C₆₇	—	2	2	no	—
96	C₄×D₄×C₆₇	C₂³	1	4	yes	—
97	C₄×D₄×C₆₇	C₂³	1	8	yes	—
98	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
100	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
101	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
102	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
108	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
109	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
110	C₂×C₄²×C₆₇	—	2	2	no	—
111	C₂×C₄²×C₆₇	—	2	1	no	—
112	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
113	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
114	#107×C₆₇	C₂³	1	8	yes	—
115	#106×C₆₇	C₂³	1	4	yes	—
116	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
117	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
118	#103×C₆₇	C₂³	1	4	yes	—
119	#102×C₆₇	C₂³	1	8	yes	—
120	#102×C₆₇	C₂³	1	8	yes	—
121	#102×C₆₇	C₂³	1	8	yes	—
122	#102×C₆₇	C₂³	1	8	yes	—
123	#102×C₆₇	C₂³	1	4	yes	—
124	#97×C₆₇	C₂³	1	8	yes	—
125	C₄×C₈×C₆₇	C₂³	1	4	yes	—
126	C₂⁴×C₆₇	—	2	2	no	—
127	C₂⁴×C₆₇	—	2	1	no	—
128	#41×C₆₇	—	4	32	no	—
129	C₂×Q₈×C₆₇	—	2	4	no	—
130	C₂×Q₈×C₆₇	—	2	1	no	—
131	C₂×Q₈×C₆₇	—	2	4	no	—
132	C₂×Q₈×C₆₇	—	2	2	no	—
133	C₂×Q₈×C₆₇	—	4	4	no	—
134	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
135	C₂×Q₈×C₆₇	#41	1	4	yes	complemented
136	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
137	C₂×D₄×C₆₇	—	2	2	no	—
138	C₂×D₄×C₆₇	—	2	1	no	—
139	C₂×D₄×C₆₇	—	2	8	no	—
140	C₂×D₄×C₆₇	—	2	4	no	—
141	C₂×D₄×C₆₇	—	2	8	no	—
142	C₂×D₄×C₆₇	—	2	4	no	—
143	C₂×D₄×C₆₇	—	2	2	no	—
144	C₂×D₄×C₆₇	—	2	4	no	—
145	C₂×D₄×C₆₇	—	2	2	no	—
146	C₂×D₄×C₆₇	—	2	4	no	—
147	C₂×D₄×C₆₇	—	4	8	no	—
148	C₂×D₄×C₆₇	—	4	4	no	—
149	C₂×D₄×C₆₇	—	8	4	no	—
150	C₂×D₄×C₆₇	—	8	2	no	—
151	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
152	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₆₇	—	2	1	no	—
154	C₂²×C₄×C₆₇	—	2	1	no	—
155	C₂²×C₄×C₆₇	—	2	2	no	—
156	C₂²×C₄×C₆₇	—	2	1	no	—
157	C₂²×C₄×C₆₇	—	2	8	no	—
158	C₂²×C₄×C₆₇	—	2	4	no	—
159	C₂²×C₄×C₆₇	—	4	8	no	—
160	C₂²×C₄×C₆₇	—	4	4	no	—
161	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
164	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
166	SD₁₆×C₆₇	—	2	8	no	—
167	SD₁₆×C₆₇	—	2	16	no	—
168	C₂×C₈×C₆₇	—	2	2	no	—
169	C₂×C₈×C₆₇	—	2	4	no	—
170	C₂×C₈×C₆₇	—	2	2	no	—
171	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
172	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
173	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
174	C₄⋊C₄×C₆₇	—	2	8	no	—
175	C₄⋊C₄×C₆₇	—	2	4	no	—
176	C₄⋊C₄×C₆₇	—	2	8	no	—
177	C₄⋊C₄×C₆₇	—	2	4	no	—
178	C₄⋊C₄×C₆₇	—	2	8	no	—
179	C₄⋊C₄×C₆₇	—	2	1	no	—
180	C₄⋊C₄×C₆₇	—	2	1	no	—
181	C₄⋊C₄×C₆₇	—	2	2	no	—
182	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
183	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
184	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
185	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
186	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
187	C₂²⋊C₄×C₆₇	—	2	8	no	—
188	C₂²⋊C₄×C₆₇	—	2	4	no	—
189	C₄²×C₆₇	—	2	8	no	—
190	C₄²×C₆₇	—	2	4	no	—
191	C₄²×C₆₇	C₂×D₄	1	4	yes	—
192	C₄²×C₆₇	C₂×D₄	1	2	yes	—
193	C₄²×C₆₇	C₂⁴	1	4	yes	—
194	C₄²×C₆₇	C₂⁴	1	2	yes	—
195	C₂³×C₆₇	—	2	4	no	—
196	C₂³×C₆₇	—	2	2	no	—
197	C₂³×C₆₇	—	4	4	no	—
198	C₂³×C₆₇	—	4	2	no	—
199	C₂³×C₆₇	—	4	4	no	—
200	C₂³×C₆₇	—	4	2	no	—
201	C₂³×C₆₇	—	4	4	no	—
202	C₂³×C₆₇	—	4	2	no	—
203	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
204	Q₈×C₆₇	—	2	4	no	—
205	Q₈×C₆₇	—	2	8	no	—
206	Q₈×C₆₇	—	2	2	no	—
207	Q₈×C₆₇	—	4	8	no	—
208	Q₈×C₆₇	C₂×#41	1	4	yes	complemented
209	D₄×C₆₇	—	2	2	no	—
210	D₄×C₆₇	—	2	4	no	—
211	D₄×C₆₇	—	2	2	no	—
212	D₄×C₆₇	—	2	4	no	—
213	D₄×C₆₇	—	4	16	no	—
214	D₄×C₆₇	—	4	4	no	—
215	D₄×C₆₇	—	4	2	no	—
216	D₄×C₆₇	—	4	8	no	—
217	D₄×C₆₇	—	8	8	no	—
218	D₄×C₆₇	—	8	4	no	—
219	C₂×C₄×C₆₇	—	2	4	no	—
220	C₂×C₄×C₆₇	—	2	4	no	—
221	C₂×C₄×C₆₇	—	2	4	no	—
222	C₂×C₄×C₆₇	—	2	2	no	—
223	C₂×C₄×C₆₇	—	2	2	no	—
224	C₂×C₄×C₆₇	—	2	2	no	—
225	C₂×C₄×C₆₇	—	2	4	no	—
226	C₂×C₄×C₆₇	—	2	2	no	—
227	C₂×C₄×C₆₇	—	4	16	no	—
228	C₂×C₄×C₆₇	—	4	2	no	—
229	C₂×C₄×C₆₇	—	4	4	no	—
230	C₂×C₄×C₆₇	—	4	2	no	—
231	C₂×C₄×C₆₇	—	4	8	no	—
232	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
233	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
234	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
235	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
236	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
238	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
239	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
240	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
241	C₈×C₆₇	—	2	2	no	—
242	C₈×C₆₇	—	2	4	no	—
243	C₈×C₆₇	—	2	2	no	—
244	C₂²×C₆₇	—	4	4	no	—
245	C₂²×C₆₇	—	4	2	no	—
246	C₂²×C₆₇	—	4	4	no	—
247	C₂²×C₆₇	—	4	2	no	—
248	C₂²×C₆₇	—	4	4	no	—
249	C₂²×C₆₇	—	4	2	no	—
250	C₂²×C₆₇	—	8	8	no	—
251	C₂²×C₆₇	—	8	4	no	—
252	C₂²×C₆₇	#548	1	2	yes	—
253	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
254	C₂²×C₆₇	#577	1	2	yes	—
255	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
256	C₄×C₆₇	—	2	4	no	—
257	C₄×C₆₇	—	2	4	no	—
258	C₄×C₆₇	—	2	4	no	—
259	C₄×C₆₇	—	2	2	no	—
260	C₄×C₆₇	—	4	2	no	—
261	C₄×C₆₇	—	4	4	no	—
262	C₄×C₆₇	—	4	2	no	—
263	C₄×C₆₇	#548	1	2	yes	—
264	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
265	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
266	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
267	C₂×#3368	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
268	C₂×C₆₇	—	8	4	no	—
269	C₂×C₆₇	—	8	2	no	—
270	C₂×C₆₇	#3368	1	4	yes	direct product factor
271	C₂×C₆₇	C₂×#548	1	1	yes	characteristic
272	C₂×C₆₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
273	C₂×C₆₇	C₂×#577	1	1	yes	characteristic
274	C₂×#557	C₂×C₆₇	1	1	yes	characteristic, complemented
275	C₂×#550	C₂×C₆₇	1	1	yes	characteristic
276	#3368	C₂×C₆₇	1	16	yes	direct product factor
277	C₂×#490	C₂×C₆₇	1	2	yes	complemented
278	C₂×#486	C₂×C₆₇	1	1	yes	characteristic, complemented
279	C₂×#478	C₂×C₆₇	1	2	yes	complemented
280	C₂×#474	C₂×C₆₇	1	1	yes	characteristic, complemented
281	C₂×#460	C₂×C₆₇	1	1	yes	characteristic, complemented
282	C₂×#460	C₂×C₆₇	1	2	yes	complemented
283	C₂×#446	C₂×C₆₇	1	1	yes	characteristic, complemented
284	C₂×#440	C₂×C₆₇	1	2	yes	complemented
285	C₂×C₄×SD₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
286	C₆₇	C₂×#3368	1	1	yes	characteristic, direct product factor, Sylow
287	C₂²×#41	—	2	1	no	—
288	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
289	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
290	#557	—	2	4	no	—
291	#557	C₂²×C₆₇	1	8	yes	complemented
292	#550	—	2	4	no	—
293	#550	C₂²×C₆₇	1	8	yes	—
294	C₂×C₄⋊Q₈	—	2	1	no	—
295	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
296	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
297	C₂×#127	C₂²×C₆₇	1	2	yes	—
298	C₂×#125	—	2	2	no	—
299	C₂×#125	C₂²×C₆₇	1	2	yes	—
300	C₂×#121	—	2	1	no	—
301	C₂×#121	—	2	2	no	—
302	C₂×C₄×Q₈	—	2	1	no	—
303	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
304	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
305	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
306	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
307	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
308	#490	C₂²×C₆₇	1	16	yes	complemented
309	#486	C₂²×C₆₇	1	8	yes	complemented
310	#478	C₂²×C₆₇	1	16	yes	complemented
311	#474	C₂²×C₆₇	1	8	yes	complemented
312	#460	C₂²×C₆₇	1	16	yes	complemented
313	#460	C₂²×C₆₇	1	8	yes	complemented
314	#446	C₂²×C₆₇	1	8	yes	complemented
315	#440	C₂²×C₆₇	1	16	yes	complemented
316	C₄×SD₁₆	C₂²×C₆₇	1	8	yes	complemented
317	C₂×#107	C₂²×C₆₇	1	2	yes	—
318	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
319	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
320	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
321	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
322	C₂×#102	C₂²×C₆₇	1	2	yes	—
323	C₂×#102	C₂²×C₆₇	1	2	yes	—
324	C₂×#102	C₂²×C₆₇	1	2	yes	—
325	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
326	C₂×#102	C₂²×C₆₇	1	2	yes	—
327	C₂×#97	C₂²×C₆₇	1	2	yes	—
328	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
329	C₂×#41	—	2	8	no	—
330	C₂×#41	—	4	8	no	—
331	C₂²×Q₈	—	2	1	no	—
332	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
333	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
334	C₂²×D₄	—	2	2	no	—
335	C₂²×D₄	—	2	1	no	—
336	C₂²×D₄	—	2	2	no	—
337	C₂²×D₄	—	2	1	no	—
338	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
339	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
340	C₂³×C₄	—	2	2	no	—
341	C₂³×C₄	—	2	1	no	—
342	C₂×SD₁₆	—	2	4	no	—
343	C₂×SD₁₆	—	2	8	no	—
344	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
345	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
346	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
347	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
348	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
349	C₄⋊Q₈	—	2	8	no	—
350	C₄⋊Q₈	—	2	2	no	—
351	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
352	#127	—	2	4	no	—
353	#127	—	2	2	no	—
354	#127	C₂³×C₆₇	1	4	yes	—
355	#127	C₂³×C₆₇	1	8	yes	—
356	#125	—	2	16	no	—
357	#125	—	2	4	no	—
358	#125	C₂³×C₆₇	1	8	yes	—
359	#121	—	2	8	no	—
360	#121	—	2	16	no	—
361	C₄×Q₈	—	2	8	no	—
362	C₄×Q₈	—	2	4	no	—
363	C₄×Q₈	—	2	2	no	—
364	C₄×Q₈	D₄×C₆₇	1	4	yes	—
365	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
366	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
367	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
368	C₄×D₄	—	2	4	no	—
369	C₄×D₄	—	2	2	no	—
370	C₄×D₄	C₂³×C₆₇	1	4	yes	—
371	C₄×D₄	C₂³×C₆₇	1	8	yes	—
372	C₂×C₄⋊C₄	—	2	2	no	—
373	C₂×C₄⋊C₄	—	2	1	no	—
374	C₂×C₄⋊C₄	—	2	2	no	—
375	C₂×C₄⋊C₄	—	2	1	no	—
376	C₂×C₄⋊C₄	—	2	2	no	—
377	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
378	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
379	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
380	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
381	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
382	C₂×C₂²⋊C₄	—	2	2	no	—
383	C₂×C₂²⋊C₄	—	2	1	no	—
384	C₂×C₄²	—	2	2	no	—
385	C₂×C₄²	—	2	1	no	—
386	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
387	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
388	#107	C₂³×C₆₇	1	8	yes	—
389	#106	C₂³×C₆₇	1	4	yes	—
390	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
391	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
392	#103	C₂³×C₆₇	1	4	yes	—
393	#102	C₂³×C₆₇	1	4	yes	—
394	#102	C₂³×C₆₇	1	8	yes	—
395	#102	C₂³×C₆₇	1	8	yes	—
396	#102	C₂³×C₆₇	1	8	yes	—
397	#102	C₂³×C₆₇	1	8	yes	—
398	#97	C₂³×C₆₇	1	8	yes	—
399	C₄×C₈	C₂³×C₆₇	1	4	yes	—
400	C₂⁴	—	2	2	no	—
401	C₂⁴	—	2	1	no	—
402	#41	—	4	32	no	—
403	C₂×Q₈	—	2	4	no	—
404	C₂×Q₈	—	2	1	no	—
405	C₂×Q₈	—	2	4	no	—
406	C₂×Q₈	—	2	2	no	—
407	C₂×Q₈	—	4	4	no	—
408	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
409	C₂×Q₈	#41×C₆₇	1	4	yes	complemented
410	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
411	C₂×D₄	—	2	2	no	—
412	C₂×D₄	—	2	1	no	—
413	C₂×D₄	—	2	8	no	—
414	C₂×D₄	—	2	4	no	—
415	C₂×D₄	—	2	8	no	—
416	C₂×D₄	—	2	4	no	—
417	C₂×D₄	—	2	2	no	—
418	C₂×D₄	—	2	4	no	—
419	C₂×D₄	—	2	4	no	—
420	C₂×D₄	—	2	2	no	—
421	C₂×D₄	—	4	8	no	—
422	C₂×D₄	—	4	4	no	—
423	C₂×D₄	—	8	4	no	—
424	C₂×D₄	—	8	2	no	—
425	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
426	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	1	no	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	2	8	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	4	8	no	—
434	C₂²×C₄	—	4	4	no	—
435	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
436	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
437	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
438	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
439	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
440	SD₁₆	—	2	8	no	—
441	SD₁₆	—	2	16	no	—
442	C₂×C₈	—	2	2	no	—
443	C₂×C₈	—	2	4	no	—
444	C₂×C₈	—	2	2	no	—
445	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
446	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
447	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
448	C₄⋊C₄	—	2	8	no	—
449	C₄⋊C₄	—	2	4	no	—
450	C₄⋊C₄	—	2	8	no	—
451	C₄⋊C₄	—	2	4	no	—
452	C₄⋊C₄	—	2	8	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	2	2	no	—
456	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
457	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
458	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
459	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
460	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
461	C₂²⋊C₄	—	2	8	no	—
462	C₂²⋊C₄	—	2	4	no	—
463	C₄²	—	2	8	no	—
464	C₄²	—	2	4	no	—
465	C₄²	C₂×D₄×C₆₇	1	4	yes	—
466	C₄²	C₂×D₄×C₆₇	1	2	yes	—
467	C₄²	C₂⁴×C₆₇	1	2	yes	—
468	C₄²	C₂⁴×C₆₇	1	4	yes	—
469	C₂³	—	2	4	no	—
470	C₂³	—	2	2	no	—
471	C₂³	—	4	4	no	—
472	C₂³	—	4	2	no	—
473	C₂³	—	4	4	no	—
474	C₂³	—	4	2	no	—
475	C₂³	—	4	4	no	—
476	C₂³	—	4	2	no	—
477	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
478	Q₈	—	2	4	no	—
479	Q₈	—	2	8	no	—
480	Q₈	—	2	2	no	—
481	Q₈	—	4	8	no	—
482	Q₈	C₂×#41×C₆₇	1	4	yes	complemented
483	D₄	—	2	2	no	—
484	D₄	—	2	4	no	—
485	D₄	—	2	2	no	—
486	D₄	—	2	4	no	—
487	D₄	—	4	16	no	—
488	D₄	—	4	4	no	—
489	D₄	—	4	2	no	—
490	D₄	—	4	8	no	—
491	D₄	—	8	8	no	—
492	D₄	—	8	4	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	4	16	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	—	4	4	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	—	4	8	no	—
506	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
507	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
508	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
509	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
510	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
511	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
512	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
513	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
514	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
515	C₈	—	2	2	no	—
516	C₈	—	2	4	no	—
517	C₈	—	2	2	no	—
518	C₂²	—	4	4	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	4	no	—
521	C₂²	—	4	2	no	—
522	C₂²	—	4	4	no	—
523	C₂²	—	4	2	no	—
524	C₂²	—	8	8	no	—
525	C₂²	—	8	4	no	—
526	C₂²	#548×C₆₇	1	2	yes	—
527	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
528	C₂²	#577×C₆₇	1	2	yes	—
529	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
530	C₄	—	2	4	no	—
531	C₄	—	2	4	no	—
532	C₄	—	2	2	no	—
533	C₄	—	2	4	no	—
534	C₄	—	4	2	no	—
535	C₄	—	4	4	no	—
536	C₄	—	4	2	no	—
537	C₄	#548×C₆₇	1	2	yes	—
538	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	4	yes	—
539	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
540	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
541	C₂	—	8	4	no	—
542	C₂	—	8	2	no	—
543	C₂	C₆₇×#3368	1	4	yes	direct product factor
544	C₂	C₂×#548×C₆₇	1	1	yes	characteristic
545	C₂	C₂²×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
546	C₂	C₂×#577×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂×#550×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#557×C₆₇	C₂	semidirect
3. C₆₇×#3368	C₂	direct (N × H)
4. C₂×#490×C₆₇	C₂	semidirect
5. C₂×#486×C₆₇	C₂	semidirect
6. C₂×#478×C₆₇	C₂	semidirect
7. C₂×#474×C₆₇	C₂	semidirect
8. C₂×#460×C₆₇	C₂	semidirect
9. C₂×#460×C₆₇	C₂	semidirect
10. C₂×#446×C₆₇	C₂	semidirect
11. C₂×#440×C₆₇	C₂	semidirect
12. C₂×C₄×SD₁₆×C₆₇	C₂	semidirect
17. #557×C₆₇	C₂²	semidirect
34. #490×C₆₇	C₂²	semidirect
35. #486×C₆₇	C₂²	semidirect
36. #478×C₆₇	C₂²	semidirect
37. #474×C₆₇	C₂²	semidirect
38. #460×C₆₇	C₂²	semidirect
39. #460×C₆₇	C₂²	semidirect
40. #446×C₆₇	C₂²	semidirect
41. #440×C₆₇	C₂²	semidirect
42. C₄×SD₁₆×C₆₇	C₂²	semidirect
135. C₂×Q₈×C₆₇	#41	semidirect
208. Q₈×C₆₇	C₂×#41	semidirect
267. C₂×#3368	C₆₇	direct (N × H)
270. C₂×C₆₇	#3368	direct (N × H)
274. C₂×#557	C₂×C₆₇	semidirect
276. #3368	C₂×C₆₇	direct (N × H)
277. C₂×#490	C₂×C₆₇	semidirect
278. C₂×#486	C₂×C₆₇	semidirect
279. C₂×#478	C₂×C₆₇	semidirect
280. C₂×#474	C₂×C₆₇	semidirect
281. C₂×#460	C₂×C₆₇	semidirect
282. C₂×#460	C₂×C₆₇	semidirect
283. C₂×#446	C₂×C₆₇	semidirect
284. C₂×#440	C₂×C₆₇	semidirect
285. C₂×C₄×SD₁₆	C₂×C₆₇	semidirect
286. C₆₇	C₂×#3368	direct (N × H)
291. #557	C₂²×C₆₇	semidirect
308. #490	C₂²×C₆₇	semidirect
309. #486	C₂²×C₆₇	semidirect
310. #478	C₂²×C₆₇	semidirect
311. #474	C₂²×C₆₇	semidirect
312. #460	C₂²×C₆₇	semidirect
313. #460	C₂²×C₆₇	semidirect
314. #446	C₂²×C₆₇	semidirect
315. #440	C₂²×C₆₇	semidirect
316. C₄×SD₁₆	C₂²×C₆₇	semidirect
409. C₂×Q₈	#41×C₆₇	semidirect
482. Q₈	C₂×#41×C₆₇	semidirect
543. C₂	C₆₇×#3368	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×#557×C₆₇	2
C₂×#550×C₆₇	2
C₆₇×#3368	2
C₂×#490×C₆₇	2
C₂×#486×C₆₇	2
C₂×#478×C₆₇	2
C₂×#474×C₆₇	2
C₂×#460×C₆₇	2
C₂×#446×C₆₇	2
C₂×#440×C₆₇	2
C₂×C₄×SD₁₆×C₆₇	2
C₂×#3368	67

Subgroups — C2×C67×#3368 #30740559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3368 #30740559