Subgroups — C₂×C₆₇×#3381 #30740605

Order 17152.

1842 conjugacy classes of subgroups.

The subgroup lattice has 506 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×#554×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3381	C₂	1	16	yes	maximal, direct product factor
4	C₂×#499×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#498×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#479×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#477×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#478×C₆₇	C₂	1	2	yes	maximal
9	C₂×#474×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×SD₁₆×C₆₇	C₂	1	2	yes	maximal
13	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	#557×C₆₇	—	2	4	no	—
15	#557×C₆₇	C₂²	1	8	yes	complemented
16	#554×C₆₇	—	2	4	no	—
17	#554×C₆₇	C₂²	1	8	yes	—
18	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
19	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#125×C₆₇	—	2	2	no	—
22	C₂×#125×C₆₇	C₂²	1	2	yes	—
23	C₂×#125×C₆₇	C₂²	1	2	yes	—
24	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
25	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
26	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
27	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
32	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
33	#499×C₆₇	C₂²	1	16	yes	complemented
34	#498×C₆₇	C₂²	1	8	yes	complemented
35	#479×C₆₇	C₂²	1	16	yes	complemented
36	#478×C₆₇	C₂²	1	16	yes	—
37	#477×C₆₇	C₂²	1	8	yes	complemented
38	#474×C₆₇	C₂²	1	8	yes	—
39	C₈×Q₈×C₆₇	C₂²	1	8	yes	complemented
40	C₄×SD₁₆×C₆₇	C₂²	1	16	yes	—
41	C₄×SD₁₆×C₆₇	C₂²	1	8	yes	—
42	C₂×#107×C₆₇	C₂²	1	2	yes	—
43	C₂×#107×C₆₇	C₂²	1	2	yes	—
44	C₂×#106×C₆₇	C₂²	1	2	yes	—
45	C₂×#106×C₆₇	C₂²	1	2	yes	—
46	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
48	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
49	C₂×#103×C₆₇	C₂²	1	2	yes	—
50	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
51	C₂×#102×C₆₇	C₂²	1	2	yes	—
52	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
53	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
54	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
55	C₂²×Q₈×C₆₇	—	2	1	no	—
56	C₂²×Q₈×C₆₇	—	2	1	no	—
57	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
58	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₆₇	—	2	1	no	—
61	C₂³×C₄×C₆₇	—	2	2	no	—
62	C₂×SD₁₆×C₆₇	Q₈	1	8	yes	complemented
63	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
64	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
66	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₆₇	—	2	8	no	—
68	C₄⋊Q₈×C₆₇	—	2	2	no	—
69	C₄⋊Q₈×C₆₇	—	2	2	no	—
70	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
71	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
72	#125×C₆₇	—	2	16	no	—
73	#125×C₆₇	—	2	4	no	—
74	#125×C₆₇	—	2	4	no	—
75	#125×C₆₇	C₂³	1	8	yes	—
76	#125×C₆₇	C₂³	1	8	yes	—
77	C₂²⋊Q₈×C₆₇	—	2	8	no	—
78	C₂²⋊Q₈×C₆₇	—	2	16	no	—
79	C₄×Q₈×C₆₇	—	2	8	no	—
80	C₄×Q₈×C₆₇	—	2	4	no	—
81	C₄×Q₈×C₆₇	—	2	2	no	—
82	C₄×Q₈×C₆₇	D₄	1	4	yes	—
83	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
84	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
85	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
86	C₄×D₄×C₆₇	—	2	4	no	—
87	C₄×D₄×C₆₇	—	2	2	no	—
88	C₄×D₄×C₆₇	C₂³	1	4	yes	—
89	C₄×D₄×C₆₇	C₂³	1	8	yes	—
90	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
91	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
92	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
93	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
94	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
95	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
96	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
97	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
99	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
105	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
106	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
107	C₂×C₄²×C₆₇	—	2	2	no	—
108	C₂×C₄²×C₆₇	—	2	1	no	—
109	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
110	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
111	#107×C₆₇	C₂³	1	8	yes	—
112	#107×C₆₇	C₂³	1	8	yes	—
113	#106×C₆₇	C₂³	1	8	yes	—
114	#106×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	8	yes	—
119	#103×C₆₇	C₂³	1	4	yes	—
120	#102×C₆₇	C₂³	1	8	yes	—
121	#102×C₆₇	C₂³	1	4	yes	—
122	C₄×C₈×C₆₇	C₂³	1	4	yes	—
123	C₄×C₈×C₆₇	C₂³	1	8	yes	—
124	C₂⁴×C₆₇	—	2	1	no	—
125	C₂×Q₈×C₆₇	—	2	4	no	—
126	C₂×Q₈×C₆₇	—	2	1	no	—
127	C₂×Q₈×C₆₇	—	2	4	no	—
128	C₂×Q₈×C₆₇	—	2	4	no	—
129	C₂×Q₈×C₆₇	—	2	2	no	—
130	C₂×Q₈×C₆₇	—	4	4	no	—
131	C₂×Q₈×C₆₇	—	8	2	no	—
132	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
133	C₂×Q₈×C₆₇	C₂×Q₈	1	4	yes	—
134	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
135	C₂×D₄×C₆₇	—	2	1	no	—
136	C₂×D₄×C₆₇	—	2	2	no	—
137	C₂×D₄×C₆₇	C₂×Q₈	1	4	yes	—
138	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
139	C₂²×C₄×C₆₇	—	2	1	no	—
140	C₂²×C₄×C₆₇	—	2	1	no	—
141	C₂²×C₄×C₆₇	—	2	2	no	—
142	C₂²×C₄×C₆₇	—	2	1	no	—
143	C₂²×C₄×C₆₇	—	2	2	no	—
144	C₂²×C₄×C₆₇	—	2	4	no	—
145	C₂²×C₄×C₆₇	—	2	8	no	—
146	C₂²×C₄×C₆₇	—	4	4	no	—
147	C₂²×C₄×C₆₇	—	8	4	no	—
148	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
149	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
150	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
153	SD₁₆×C₆₇	C₂×Q₈	1	16	yes	complemented
154	C₂×C₈×C₆₇	—	2	2	no	—
155	C₂×C₈×C₆₇	—	2	4	no	—
156	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
157	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
158	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
159	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
160	C₄⋊C₄×C₆₇	—	2	8	no	—
161	C₄⋊C₄×C₆₇	—	2	4	no	—
162	C₄⋊C₄×C₆₇	—	2	8	no	—
163	C₄⋊C₄×C₆₇	—	2	4	no	—
164	C₄⋊C₄×C₆₇	—	2	8	no	—
165	C₄⋊C₄×C₆₇	—	2	8	no	—
166	C₄⋊C₄×C₆₇	—	2	4	no	—
167	C₄⋊C₄×C₆₇	—	2	8	no	—
168	C₄⋊C₄×C₆₇	—	2	1	no	—
169	C₄⋊C₄×C₆₇	—	2	2	no	—
170	C₄⋊C₄×C₆₇	—	2	1	no	—
171	C₄⋊C₄×C₆₇	—	2	2	no	—
172	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
173	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
174	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
175	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
176	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
177	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
178	C₂²⋊C₄×C₆₇	—	2	4	no	—
179	C₂²⋊C₄×C₆₇	—	2	8	no	—
180	C₄²×C₆₇	—	2	8	no	—
181	C₄²×C₆₇	—	2	4	no	—
182	C₄²×C₆₇	C₂×D₄	1	4	yes	—
183	C₄²×C₆₇	C₂×D₄	1	2	yes	—
184	C₄²×C₆₇	C₂⁴	1	4	yes	—
185	C₄²×C₆₇	C₂⁴	1	2	yes	—
186	C₂³×C₆₇	—	2	4	no	—
187	C₂³×C₆₇	—	2	2	no	—
188	C₂³×C₆₇	—	4	2	no	—
189	C₂³×C₆₇	—	4	2	no	—
190	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
191	Q₈×C₆₇	—	2	4	no	—
192	Q₈×C₆₇	—	2	8	no	—
193	Q₈×C₆₇	—	2	2	no	—
194	Q₈×C₆₇	—	4	8	no	—
195	Q₈×C₆₇	—	8	4	no	—
196	Q₈×C₆₇	C₂²×Q₈	1	4	yes	—
197	D₄×C₆₇	—	2	4	no	—
198	D₄×C₆₇	—	2	2	no	—
199	D₄×C₆₇	C₂²×Q₈	1	4	yes	—
200	C₂×C₄×C₆₇	—	2	4	no	—
201	C₂×C₄×C₆₇	—	2	4	no	—
202	C₂×C₄×C₆₇	—	2	4	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	4	no	—
207	C₂×C₄×C₆₇	—	2	2	no	—
208	C₂×C₄×C₆₇	—	2	4	no	—
209	C₂×C₄×C₆₇	—	4	2	no	—
210	C₂×C₄×C₆₇	—	4	4	no	—
211	C₂×C₄×C₆₇	—	4	2	no	—
212	C₂×C₄×C₆₇	—	4	4	no	—
213	C₂×C₄×C₆₇	—	4	8	no	—
214	C₂×C₄×C₆₇	—	8	8	no	—
215	C₂×C₄×C₆₇	#135	1	4	yes	—
216	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
217	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
218	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
219	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
221	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
222	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
223	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
224	C₈×C₆₇	—	2	2	no	—
225	C₈×C₆₇	—	2	4	no	—
226	C₈×C₆₇	C₂²×Q₈	1	4	yes	—
227	C₂²×C₆₇	—	2	4	no	—
228	C₂²×C₆₇	—	4	8	no	—
229	C₂²×C₆₇	—	4	2	no	—
230	C₂²×C₆₇	—	4	2	no	—
231	C₂²×C₆₇	D₄×Q₈	1	2	yes	—
232	C₂²×C₆₇	C₂×#135	1	2	yes	—
233	C₂²×C₆₇	#577	1	2	yes	—
234	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
235	C₄×C₆₇	—	2	4	no	—
236	C₄×C₆₇	—	2	4	no	—
237	C₄×C₆₇	—	2	4	no	—
238	C₄×C₆₇	—	2	2	no	—
239	C₄×C₆₇	—	4	2	no	—
240	C₄×C₆₇	—	4	4	no	—
241	C₄×C₆₇	—	4	2	no	—
242	C₄×C₆₇	—	4	4	no	—
243	C₄×C₆₇	D₄×Q₈	1	2	yes	—
244	C₄×C₆₇	C₂×#135	1	4	yes	—
245	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
246	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
247	C₂×#3381	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
248	C₂×C₆₇	—	4	4	no	—
249	C₂×C₆₇	#3381	1	4	yes	direct product factor
250	C₂×C₆₇	C₂×D₄×Q₈	1	1	yes	characteristic
251	C₂×C₆₇	C₂²×#135	1	1	yes	characteristic
252	C₂×C₆₇	C₂×#577	1	1	yes	characteristic
253	C₂×#557	C₂×C₆₇	1	1	yes	characteristic, complemented
254	C₂×#554	C₂×C₆₇	1	1	yes	characteristic
255	#3381	C₂×C₆₇	1	16	yes	direct product factor
256	C₂×#499	C₂×C₆₇	1	2	yes	complemented
257	C₂×#498	C₂×C₆₇	1	1	yes	characteristic, complemented
258	C₂×#479	C₂×C₆₇	1	2	yes	complemented
259	C₂×#477	C₂×C₆₇	1	1	yes	characteristic, complemented
260	C₂×#478	C₂×C₆₇	1	2	yes	—
261	C₂×#474	C₂×C₆₇	1	1	yes	characteristic
262	C₂×C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
263	C₂×C₄×SD₁₆	C₂×C₆₇	1	2	yes	—
264	C₂×C₄×SD₁₆	C₂×C₆₇	1	1	yes	characteristic
265	C₆₇	C₂×#3381	1	1	yes	characteristic, direct product factor, Sylow
266	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
267	#557	—	2	4	no	—
268	#557	C₂²×C₆₇	1	8	yes	complemented
269	#554	—	2	4	no	—
270	#554	C₂²×C₆₇	1	8	yes	—
271	C₂×C₄⋊Q₈	—	2	1	no	—
272	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
273	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
274	C₂×#125	—	2	2	no	—
275	C₂×#125	C₂²×C₆₇	1	2	yes	—
276	C₂×#125	C₂²×C₆₇	1	2	yes	—
277	C₂×C₂²⋊Q₈	—	2	1	no	—
278	C₂×C₂²⋊Q₈	—	2	2	no	—
279	C₂×C₄×Q₈	—	2	1	no	—
280	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
281	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
282	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
283	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
284	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
285	C₂²×C₄⋊C₄	—	2	1	no	—
286	#499	C₂²×C₆₇	1	16	yes	complemented
287	#498	C₂²×C₆₇	1	8	yes	complemented
288	#479	C₂²×C₆₇	1	16	yes	complemented
289	#478	C₂²×C₆₇	1	16	yes	—
290	#477	C₂²×C₆₇	1	8	yes	complemented
291	#474	C₂²×C₆₇	1	8	yes	—
292	C₈×Q₈	C₂²×C₆₇	1	8	yes	complemented
293	C₄×SD₁₆	C₂²×C₆₇	1	8	yes	—
294	C₄×SD₁₆	C₂²×C₆₇	1	16	yes	—
295	C₂×#107	C₂²×C₆₇	1	2	yes	—
296	C₂×#107	C₂²×C₆₇	1	2	yes	—
297	C₂×#106	C₂²×C₆₇	1	2	yes	—
298	C₂×#106	C₂²×C₆₇	1	2	yes	—
299	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
300	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
301	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
302	C₂×#103	C₂²×C₆₇	1	2	yes	—
303	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
304	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
305	C₂×#102	C₂²×C₆₇	1	2	yes	—
306	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
307	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
308	C₂²×Q₈	—	2	1	no	—
309	C₂²×Q₈	—	2	1	no	—
310	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
311	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
312	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
313	C₂³×C₄	—	2	1	no	—
314	C₂³×C₄	—	2	2	no	—
315	C₂×SD₁₆	Q₈×C₆₇	1	8	yes	complemented
316	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
317	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
318	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
319	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
320	C₄⋊Q₈	—	2	8	no	—
321	C₄⋊Q₈	—	2	2	no	—
322	C₄⋊Q₈	—	2	2	no	—
323	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
324	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
325	#125	—	2	16	no	—
326	#125	—	2	4	no	—
327	#125	—	2	4	no	—
328	#125	C₂³×C₆₇	1	8	yes	—
329	#125	C₂³×C₆₇	1	8	yes	—
330	C₂²⋊Q₈	—	2	8	no	—
331	C₂²⋊Q₈	—	2	16	no	—
332	C₄×Q₈	—	2	8	no	—
333	C₄×Q₈	—	2	4	no	—
334	C₄×Q₈	—	2	2	no	—
335	C₄×Q₈	D₄×C₆₇	1	4	yes	—
336	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
337	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
338	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
339	C₄×D₄	—	2	4	no	—
340	C₄×D₄	—	2	2	no	—
341	C₄×D₄	C₂³×C₆₇	1	4	yes	—
342	C₄×D₄	C₂³×C₆₇	1	8	yes	—
343	C₂×C₄⋊C₄	—	2	2	no	—
344	C₂×C₄⋊C₄	—	2	1	no	—
345	C₂×C₄⋊C₄	—	2	2	no	—
346	C₂×C₄⋊C₄	—	2	1	no	—
347	C₂×C₄⋊C₄	—	2	2	no	—
348	C₂×C₄⋊C₄	—	2	2	no	—
349	C₂×C₄⋊C₄	—	2	1	no	—
350	C₂×C₄⋊C₄	—	2	2	no	—
351	C₂×C₄⋊C₄	—	2	8	no	—
352	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
353	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
354	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
355	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
356	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
357	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
358	C₂×C₂²⋊C₄	—	2	1	no	—
359	C₂×C₂²⋊C₄	—	2	2	no	—
360	C₂×C₄²	—	2	2	no	—
361	C₂×C₄²	—	2	1	no	—
362	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
363	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
364	#107	C₂³×C₆₇	1	8	yes	—
365	#107	C₂³×C₆₇	1	8	yes	—
366	#106	C₂³×C₆₇	1	8	yes	—
367	#106	C₂³×C₆₇	1	4	yes	—
368	#106	C₂³×C₆₇	1	8	yes	—
369	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
370	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
371	#103	C₂³×C₆₇	1	8	yes	—
372	#103	C₂³×C₆₇	1	4	yes	—
373	#102	C₂³×C₆₇	1	8	yes	—
374	#102	C₂³×C₆₇	1	4	yes	—
375	C₄×C₈	C₂³×C₆₇	1	4	yes	—
376	C₄×C₈	C₂³×C₆₇	1	8	yes	—
377	C₂⁴	—	2	1	no	—
378	C₂×Q₈	—	2	4	no	—
379	C₂×Q₈	—	2	1	no	—
380	C₂×Q₈	—	2	4	no	—
381	C₂×Q₈	—	2	4	no	—
382	C₂×Q₈	—	2	2	no	—
383	C₂×Q₈	—	4	4	no	—
384	C₂×Q₈	—	8	2	no	—
385	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
386	C₂×Q₈	C₂×Q₈×C₆₇	1	4	yes	—
387	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
388	C₂×D₄	—	2	1	no	—
389	C₂×D₄	—	2	2	no	—
390	C₂×D₄	C₂×Q₈×C₆₇	1	4	yes	—
391	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	2	4	no	—
398	C₂²×C₄	—	2	8	no	—
399	C₂²×C₄	—	4	4	no	—
400	C₂²×C₄	—	8	4	no	—
401	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
402	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
403	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
404	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
405	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
406	SD₁₆	C₂×Q₈×C₆₇	1	16	yes	complemented
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	—	2	4	no	—
409	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
410	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
411	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
412	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
413	C₄⋊C₄	—	2	8	no	—
414	C₄⋊C₄	—	2	4	no	—
415	C₄⋊C₄	—	2	8	no	—
416	C₄⋊C₄	—	2	4	no	—
417	C₄⋊C₄	—	2	8	no	—
418	C₄⋊C₄	—	2	8	no	—
419	C₄⋊C₄	—	2	4	no	—
420	C₄⋊C₄	—	2	8	no	—
421	C₄⋊C₄	—	2	1	no	—
422	C₄⋊C₄	—	2	2	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	—	2	2	no	—
425	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
426	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
427	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
428	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
429	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
430	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
431	C₂²⋊C₄	—	2	4	no	—
432	C₂²⋊C₄	—	2	8	no	—
433	C₄²	—	2	8	no	—
434	C₄²	—	2	4	no	—
435	C₄²	C₂×D₄×C₆₇	1	4	yes	—
436	C₄²	C₂×D₄×C₆₇	1	2	yes	—
437	C₄²	C₂⁴×C₆₇	1	4	yes	—
438	C₄²	C₂⁴×C₆₇	1	2	yes	—
439	C₂³	—	2	4	no	—
440	C₂³	—	2	2	no	—
441	C₂³	—	4	2	no	—
442	C₂³	—	4	2	no	—
443	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
444	Q₈	—	2	4	no	—
445	Q₈	—	2	8	no	—
446	Q₈	—	2	2	no	—
447	Q₈	—	4	8	no	—
448	Q₈	—	8	4	no	—
449	Q₈	C₂²×Q₈×C₆₇	1	4	yes	—
450	D₄	—	2	4	no	—
451	D₄	—	2	2	no	—
452	D₄	C₂²×Q₈×C₆₇	1	4	yes	—
453	C₂×C₄	—	2	4	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	4	no	—
464	C₂×C₄	—	4	4	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	4	8	no	—
467	C₂×C₄	—	8	8	no	—
468	C₂×C₄	#135×C₆₇	1	4	yes	—
469	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
470	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
471	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
472	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
473	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
474	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
475	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
476	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
477	C₈	—	2	2	no	—
478	C₈	—	2	4	no	—
479	C₈	C₂²×Q₈×C₆₇	1	4	yes	—
480	C₂²	—	2	4	no	—
481	C₂²	—	4	2	no	—
482	C₂²	—	4	2	no	—
483	C₂²	—	4	8	no	—
484	C₂²	D₄×Q₈×C₆₇	1	2	yes	—
485	C₂²	C₂×#135×C₆₇	1	2	yes	—
486	C₂²	#577×C₆₇	1	2	yes	—
487	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
488	C₄	—	2	4	no	—
489	C₄	—	2	4	no	—
490	C₄	—	2	2	no	—
491	C₄	—	2	4	no	—
492	C₄	—	4	2	no	—
493	C₄	—	4	4	no	—
494	C₄	—	4	4	no	—
495	C₄	—	4	2	no	—
496	C₄	D₄×Q₈×C₆₇	1	2	yes	—
497	C₄	C₂×#135×C₆₇	1	4	yes	—
498	C₄	C₂³×Q₈×C₆₇	1	1	yes	characteristic
499	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
500	C₂	—	4	4	no	—
501	C₂	C₆₇×#3381	1	4	yes	direct product factor
502	C₂	C₂×D₄×Q₈×C₆₇	1	1	yes	characteristic
503	C₂	C₂²×#135×C₆₇	1	1	yes	characteristic
504	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	C₂²×D₄×C₆₇	C₂³
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×#3381	C₂	direct (N × H)
4. C₂×#499×C₆₇	C₂	semidirect
5. C₂×#498×C₆₇	C₂	semidirect
6. C₂×#479×C₆₇	C₂	semidirect
7. C₂×#477×C₆₇	C₂	semidirect
10. C₂×C₈×Q₈×C₆₇	C₂	semidirect
15. #557×C₆₇	C₂²	semidirect
33. #499×C₆₇	C₂²	semidirect
34. #498×C₆₇	C₂²	semidirect
35. #479×C₆₇	C₂²	semidirect
37. #477×C₆₇	C₂²	semidirect
39. C₈×Q₈×C₆₇	C₂²	semidirect
62. C₂×SD₁₆×C₆₇	Q₈	semidirect
153. SD₁₆×C₆₇	C₂×Q₈	semidirect
247. C₂×#3381	C₆₇	direct (N × H)
249. C₂×C₆₇	#3381	direct (N × H)
253. C₂×#557	C₂×C₆₇	semidirect
255. #3381	C₂×C₆₇	direct (N × H)
256. C₂×#499	C₂×C₆₇	semidirect
257. C₂×#498	C₂×C₆₇	semidirect
258. C₂×#479	C₂×C₆₇	semidirect
259. C₂×#477	C₂×C₆₇	semidirect
262. C₂×C₈×Q₈	C₂×C₆₇	semidirect
265. C₆₇	C₂×#3381	direct (N × H)
268. #557	C₂²×C₆₇	semidirect
286. #499	C₂²×C₆₇	semidirect
287. #498	C₂²×C₆₇	semidirect
288. #479	C₂²×C₆₇	semidirect
290. #477	C₂²×C₆₇	semidirect
292. C₈×Q₈	C₂²×C₆₇	semidirect
315. C₂×SD₁₆	Q₈×C₆₇	semidirect
406. SD₁₆	C₂×Q₈×C₆₇	semidirect
501. C₂	C₆₇×#3381	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×#554×C₆₇	2
C₆₇×#3381	2
C₂×#499×C₆₇	2
C₂×#498×C₆₇	2
C₂×#479×C₆₇	2
C₂×#477×C₆₇	2
C₂×#478×C₆₇	2
C₂×#474×C₆₇	2
C₂×C₈×Q₈×C₆₇	2
C₂×C₄×SD₁₆×C₆₇	2
C₂×#3381	67

Subgroups — C2×C67×#3381 #30740605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3381 #30740605