Subgroups — C₂×C₆₇×#3335 #30740477

Order 17152.

2158 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#554×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#547×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3335	C₂	1	16	yes	maximal, direct product factor
4	C₂×#498×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#481×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#483×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#475×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#476×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#465×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#106×C₆₇	C₂	1	2	yes	maximal, complemented
13	C₂²×#41×C₆₇	—	2	1	no	—
14	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
16	#554×C₆₇	—	2	4	no	—
17	#554×C₆₇	C₂²	1	8	yes	complemented
18	#547×C₆₇	—	2	4	no	—
19	#547×C₆₇	C₂²	1	8	yes	—
20	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#125×C₆₇	—	2	1	no	—
23	C₂×#123×C₆₇	—	2	2	no	—
24	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
25	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
26	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
27	C₂×#121×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₄×C₆₇	—	2	1	no	—
30	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
33	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
34	#498×C₆₇	C₂²	1	8	yes	complemented
35	#483×C₆₇	C₂²	1	16	yes	complemented
36	#481×C₆₇	C₂²	1	16	yes	complemented
37	#476×C₆₇	C₂²	1	8	yes	complemented
38	#475×C₆₇	C₂²	1	8	yes	complemented
39	#465×C₆₇	C₂²	1	16	yes	complemented
40	C₄×SD₁₆×C₆₇	C₂²	1	8	yes	complemented
41	C₈×D₄×C₆₇	C₂²	1	8	yes	complemented
42	C₂×#106×C₆₇	C₂²	1	2	yes	—
43	C₂×#106×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×#106×C₆₇	C₂²	1	16	yes	complemented
48	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
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×C₂²⋊C₈×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	—	2	1	no	—
59	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₂²×D₄×C₆₇	—	2	2	no	—
61	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
62	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₆₇	—	2	2	no	—
64	C₂³×C₄×C₆₇	—	2	2	no	—
65	C₂³×C₄×C₆₇	—	2	1	no	—
66	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
67	C₂×SD₁₆×C₆₇	—	2	4	no	—
68	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
69	C₂²×C₈×C₆₇	—	2	8	no	—
70	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
71	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
72	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₄⋊Q₈×C₆₇	—	2	2	no	—
75	C₄⋊Q₈×C₆₇	—	2	2	no	—
76	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
77	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
78	#125×C₆₇	—	2	8	no	—
79	#123×C₆₇	—	2	16	no	—
80	C₂²⋊Q₈×C₆₇	—	2	4	no	—
81	C₂²⋊Q₈×C₆₇	—	2	8	no	—
82	C₂²⋊Q₈×C₆₇	—	2	16	no	—
83	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
84	#121×C₆₇	—	2	4	no	—
85	#121×C₆₇	C₂³	1	8	yes	—
86	C₄×Q₈×C₆₇	—	2	2	no	—
87	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
88	C₄×D₄×C₆₇	—	2	8	no	—
89	C₄×D₄×C₆₇	—	2	2	no	—
90	C₄×D₄×C₆₇	D₄	1	4	yes	—
91	C₄×D₄×C₆₇	C₂³	1	4	yes	—
92	C₄×D₄×C₆₇	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
94	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
95	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
96	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
97	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
98	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
99	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
100	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
101	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
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×C₂²⋊C₄×C₆₇	—	2	2	no	—
111	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
112	C₂×C₄²×C₆₇	—	2	1	no	—
113	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
114	#106×C₆₇	C₂³	1	8	yes	—
115	#106×C₆₇	C₂³	1	8	yes	—
116	#106×C₆₇	C₂³	1	8	yes	—
117	#106×C₆₇	C₂³	1	8	yes	—
118	#106×C₆₇	C₂³	1	4	yes	—
119	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
120	#103×C₆₇	C₂³	1	8	yes	—
121	#103×C₆₇	C₂³	1	4	yes	—
122	#102×C₆₇	C₂³	1	8	yes	—
123	#102×C₆₇	C₂³	1	4	yes	—
124	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
125	C₄×C₈×C₆₇	C₂³	1	4	yes	—
126	C₂⁴×C₆₇	—	2	1	no	—
127	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
128	#41×C₆₇	—	4	32	no	—
129	C₂×Q₈×C₆₇	—	2	1	no	—
130	C₂×Q₈×C₆₇	—	2	4	no	—
131	C₂×Q₈×C₆₇	—	2	4	no	—
132	C₂×Q₈×C₆₇	—	2	2	no	—
133	C₂×Q₈×C₆₇	—	2	2	no	—
134	C₂×Q₈×C₆₇	—	4	4	no	—
135	C₂×Q₈×C₆₇	—	8	2	no	—
136	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
137	C₂×D₄×C₆₇	—	2	8	no	—
138	C₂×D₄×C₆₇	—	2	1	no	—
139	C₂×D₄×C₆₇	—	2	2	no	—
140	C₂×D₄×C₆₇	—	2	2	no	—
141	C₂×D₄×C₆₇	—	4	8	no	—
142	C₂×D₄×C₆₇	SD₁₆	1	8	yes	complemented
143	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
144	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
145	C₂²×C₄×C₆₇	—	2	4	no	—
146	C₂²×C₄×C₆₇	—	2	4	no	—
147	C₂²×C₄×C₆₇	—	2	8	no	—
148	C₂²×C₄×C₆₇	—	2	1	no	—
149	C₂²×C₄×C₆₇	—	2	1	no	—
150	C₂²×C₄×C₆₇	—	2	2	no	—
151	C₂²×C₄×C₆₇	—	2	1	no	—
152	C₂²×C₄×C₆₇	—	2	2	no	—
153	C₂²×C₄×C₆₇	—	2	8	no	—
154	C₂²×C₄×C₆₇	—	2	4	no	—
155	C₂²×C₄×C₆₇	—	4	8	no	—
156	C₂²×C₄×C₆₇	—	4	4	no	—
157	C₂²×C₄×C₆₇	—	8	4	no	—
158	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
159	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
163	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
164	SD₁₆×C₆₇	—	2	8	no	—
165	C₂×C₈×C₆₇	—	2	4	no	—
166	C₂×C₈×C₆₇	—	2	2	no	—
167	C₂×C₈×C₆₇	—	2	16	no	—
168	C₂×C₈×C₆₇	#41	1	4	yes	complemented
169	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
170	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
171	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
172	C₄⋊C₄×C₆₇	—	2	1	no	—
173	C₄⋊C₄×C₆₇	—	2	1	no	—
174	C₄⋊C₄×C₆₇	—	2	2	no	—
175	C₄⋊C₄×C₆₇	—	2	8	no	—
176	C₄⋊C₄×C₆₇	—	2	8	no	—
177	C₄⋊C₄×C₆₇	—	2	2	no	—
178	C₄⋊C₄×C₆₇	—	2	4	no	—
179	C₄⋊C₄×C₆₇	—	2	4	no	—
180	C₄⋊C₄×C₆₇	—	2	8	no	—
181	C₄⋊C₄×C₆₇	—	2	4	no	—
182	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
183	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
184	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
185	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
186	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
187	C₂²⋊C₄×C₆₇	—	2	8	no	—
188	C₂²⋊C₄×C₆₇	—	2	4	no	—
189	C₂²⋊C₄×C₆₇	—	2	8	no	—
190	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
191	C₄²×C₆₇	—	2	4	no	—
192	C₄²×C₆₇	C₂×D₄	1	2	yes	—
193	C₄²×C₆₇	C₂⁴	1	2	yes	—
194	C₂³×C₆₇	—	2	4	no	—
195	C₂³×C₆₇	—	2	4	no	—
196	C₂³×C₆₇	—	2	2	no	—
197	C₂³×C₆₇	—	4	2	no	—
198	C₂³×C₆₇	—	4	2	no	—
199	C₂³×C₆₇	—	4	2	no	—
200	C₂³×C₆₇	C₂×SD₁₆	1	8	yes	—
201	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
202	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
203	Q₈×C₆₇	—	2	2	no	—
204	Q₈×C₆₇	—	2	2	no	—
205	Q₈×C₆₇	—	4	2	no	—
206	Q₈×C₆₇	—	4	8	no	—
207	Q₈×C₆₇	—	8	4	no	—
208	D₄×C₆₇	—	2	2	no	—
209	D₄×C₆₇	—	2	2	no	—
210	D₄×C₆₇	—	4	16	no	—
211	D₄×C₆₇	—	4	2	no	—
212	D₄×C₆₇	C₂×SD₁₆	1	16	yes	complemented
213	C₂×C₄×C₆₇	—	2	4	no	—
214	C₂×C₄×C₆₇	—	2	2	no	—
215	C₂×C₄×C₆₇	—	2	8	no	—
216	C₂×C₄×C₆₇	—	2	4	no	—
217	C₂×C₄×C₆₇	—	2	4	no	—
218	C₂×C₄×C₆₇	—	2	2	no	—
219	C₂×C₄×C₆₇	—	2	2	no	—
220	C₂×C₄×C₆₇	—	2	4	no	—
221	C₂×C₄×C₆₇	—	2	2	no	—
222	C₂×C₄×C₆₇	—	2	4	no	—
223	C₂×C₄×C₆₇	—	4	2	no	—
224	C₂×C₄×C₆₇	—	4	2	no	—
225	C₂×C₄×C₆₇	—	4	4	no	—
226	C₂×C₄×C₆₇	—	4	16	no	—
227	C₂×C₄×C₆₇	—	4	2	no	—
228	C₂×C₄×C₆₇	—	4	4	no	—
229	C₂×C₄×C₆₇	—	4	8	no	—
230	C₂×C₄×C₆₇	—	8	8	no	—
231	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
232	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
233	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
234	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
236	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
237	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
238	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
239	C₈×C₆₇	—	2	4	no	—
240	C₈×C₆₇	—	2	2	no	—
241	C₈×C₆₇	C₂×#41	1	4	yes	complemented
242	C₂²×C₆₇	—	2	16	no	—
243	C₂²×C₆₇	—	2	4	no	—
244	C₂²×C₆₇	—	2	4	no	—
245	C₂²×C₆₇	—	4	2	no	—
246	C₂²×C₆₇	—	4	2	no	—
247	C₂²×C₆₇	—	4	2	no	—
248	C₂²×C₆₇	—	8	4	no	—
249	C₂²×C₆₇	#547	1	2	yes	—
250	C₂²×C₆₇	C₂²×SD₁₆	1	8	yes	—
251	C₂²×C₆₇	C₂²×SD₁₆	1	2	yes	—
252	C₂²×C₆₇	#577	1	2	yes	—
253	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
254	C₄×C₆₇	—	2	4	no	—
255	C₄×C₆₇	—	2	2	no	—
256	C₄×C₆₇	—	4	2	no	—
257	C₄×C₆₇	—	4	2	no	—
258	C₄×C₆₇	—	4	4	no	—
259	C₄×C₆₇	—	4	2	no	—
260	C₄×C₆₇	—	4	4	no	—
261	C₄×C₆₇	#547	1	2	yes	—
262	C₄×C₆₇	C₂²×SD₁₆	1	4	yes	—
263	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
264	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
265	C₂×#3335	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
266	C₂×C₆₇	—	2	8	no	—
267	C₂×C₆₇	—	8	2	no	—
268	C₂×C₆₇	#3335	1	4	yes	direct product factor
269	C₂×C₆₇	C₂×#547	1	1	yes	characteristic
270	C₂×C₆₇	C₂³×SD₁₆	1	1	yes	characteristic
271	C₂×C₆₇	C₂×#577	1	1	yes	characteristic
272	C₂×#554	C₂×C₆₇	1	1	yes	characteristic, complemented
273	C₂×#547	C₂×C₆₇	1	1	yes	characteristic
274	#3335	C₂×C₆₇	1	16	yes	direct product factor
275	C₂×#498	C₂×C₆₇	1	1	yes	characteristic, complemented
276	C₂×#481	C₂×C₆₇	1	2	yes	complemented
277	C₂×#483	C₂×C₆₇	1	2	yes	complemented
278	C₂×#475	C₂×C₆₇	1	1	yes	characteristic, complemented
279	C₂×#476	C₂×C₆₇	1	1	yes	characteristic, complemented
280	C₂×#465	C₂×C₆₇	1	2	yes	complemented
281	C₂×C₄×SD₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
282	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
283	C₂²×#106	C₂×C₆₇	1	2	yes	complemented
284	C₆₇	C₂×#3335	1	1	yes	characteristic, direct product factor, Sylow
285	C₂²×#41	—	2	1	no	—
286	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
287	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
288	#554	—	2	4	no	—
289	#554	C₂²×C₆₇	1	8	yes	complemented
290	#547	—	2	4	no	—
291	#547	C₂²×C₆₇	1	8	yes	—
292	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
293	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
294	C₂×#125	—	2	1	no	—
295	C₂×#123	—	2	2	no	—
296	C₂×C₂²⋊Q₈	—	2	1	no	—
297	C₂×C₂²⋊Q₈	—	2	2	no	—
298	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
299	C₂×#121	C₂²×C₆₇	1	2	yes	—
300	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
301	C₂×C₄×D₄	—	2	1	no	—
302	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
303	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
304	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
305	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
306	#498	C₂²×C₆₇	1	8	yes	complemented
307	#483	C₂²×C₆₇	1	16	yes	complemented
308	#481	C₂²×C₆₇	1	16	yes	complemented
309	#476	C₂²×C₆₇	1	8	yes	complemented
310	#475	C₂²×C₆₇	1	8	yes	complemented
311	#465	C₂²×C₆₇	1	16	yes	complemented
312	C₄×SD₁₆	C₂²×C₆₇	1	8	yes	complemented
313	C₈×D₄	C₂²×C₆₇	1	8	yes	complemented
314	C₂×#106	C₂²×C₆₇	1	2	yes	—
315	C₂×#106	C₂²×C₆₇	1	2	yes	—
316	C₂×#106	C₂²×C₆₇	1	2	yes	—
317	C₂×#106	C₂²×C₆₇	1	2	yes	—
318	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
319	C₂×#106	C₂²×C₆₇	1	16	yes	complemented
320	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
321	C₂×#103	C₂²×C₆₇	1	2	yes	—
322	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
323	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
324	C₂×#102	C₂²×C₆₇	1	2	yes	—
325	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
326	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
327	C₂×#41	—	2	8	no	—
328	C₂×#41	—	4	8	no	—
329	C₂²×Q₈	—	2	1	no	—
330	C₂²×Q₈	—	2	1	no	—
331	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
332	C₂²×D₄	—	2	2	no	—
333	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
334	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
335	C₂³×C₄	—	2	2	no	—
336	C₂³×C₄	—	2	2	no	—
337	C₂³×C₄	—	2	1	no	—
338	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
339	C₂×SD₁₆	—	2	4	no	—
340	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
341	C₂²×C₈	—	2	8	no	—
342	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
343	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
344	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
345	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
346	C₄⋊Q₈	—	2	2	no	—
347	C₄⋊Q₈	—	2	2	no	—
348	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
349	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
350	#125	—	2	8	no	—
351	#123	—	2	16	no	—
352	C₂²⋊Q₈	—	2	4	no	—
353	C₂²⋊Q₈	—	2	8	no	—
354	C₂²⋊Q₈	—	2	16	no	—
355	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
356	#121	—	2	4	no	—
357	#121	C₂³×C₆₇	1	8	yes	—
358	C₄×Q₈	—	2	2	no	—
359	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
360	C₄×D₄	—	2	8	no	—
361	C₄×D₄	—	2	2	no	—
362	C₄×D₄	D₄×C₆₇	1	4	yes	—
363	C₄×D₄	C₂³×C₆₇	1	4	yes	—
364	C₄×D₄	C₂³×C₆₇	1	4	yes	—
365	C₂×C₄⋊C₄	—	2	2	no	—
366	C₂×C₄⋊C₄	—	2	2	no	—
367	C₂×C₄⋊C₄	—	2	4	no	—
368	C₂×C₄⋊C₄	—	2	4	no	—
369	C₂×C₄⋊C₄	—	2	1	no	—
370	C₂×C₄⋊C₄	—	2	1	no	—
371	C₂×C₄⋊C₄	—	2	2	no	—
372	C₂×C₄⋊C₄	—	2	1	no	—
373	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
374	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
375	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
376	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
377	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
378	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
379	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
380	C₂×C₂²⋊C₄	—	2	2	no	—
381	C₂×C₂²⋊C₄	—	2	1	no	—
382	C₂×C₂²⋊C₄	—	2	2	no	—
383	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
384	C₂×C₄²	—	2	1	no	—
385	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
386	#106	C₂³×C₆₇	1	4	yes	—
387	#106	C₂³×C₆₇	1	8	yes	—
388	#106	C₂³×C₆₇	1	8	yes	—
389	#106	C₂³×C₆₇	1	8	yes	—
390	#106	C₂³×C₆₇	1	8	yes	—
391	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
392	#103	C₂³×C₆₇	1	4	yes	—
393	#103	C₂³×C₆₇	1	8	yes	—
394	#102	C₂³×C₆₇	1	4	yes	—
395	#102	C₂³×C₆₇	1	8	yes	—
396	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
397	C₄×C₈	C₂³×C₆₇	1	4	yes	—
398	C₂⁴	—	2	1	no	—
399	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
400	#41	—	4	32	no	—
401	C₂×Q₈	—	2	1	no	—
402	C₂×Q₈	—	2	4	no	—
403	C₂×Q₈	—	2	4	no	—
404	C₂×Q₈	—	2	2	no	—
405	C₂×Q₈	—	2	2	no	—
406	C₂×Q₈	—	4	4	no	—
407	C₂×Q₈	—	8	2	no	—
408	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
409	C₂×D₄	—	2	8	no	—
410	C₂×D₄	—	2	1	no	—
411	C₂×D₄	—	2	2	no	—
412	C₂×D₄	—	2	2	no	—
413	C₂×D₄	—	4	8	no	—
414	C₂×D₄	SD₁₆×C₆₇	1	8	yes	complemented
415	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
416	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	1	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	2	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	—	2	2	no	—
424	C₂²×C₄	—	2	8	no	—
425	C₂²×C₄	—	2	8	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	4	8	no	—
428	C₂²×C₄	—	4	4	no	—
429	C₂²×C₄	—	8	4	no	—
430	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
431	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
432	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
433	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
434	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
435	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
436	SD₁₆	—	2	8	no	—
437	C₂×C₈	—	2	4	no	—
438	C₂×C₈	—	2	2	no	—
439	C₂×C₈	—	2	16	no	—
440	C₂×C₈	#41×C₆₇	1	4	yes	complemented
441	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
442	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
443	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
444	C₄⋊C₄	—	2	1	no	—
445	C₄⋊C₄	—	2	1	no	—
446	C₄⋊C₄	—	2	2	no	—
447	C₄⋊C₄	—	2	8	no	—
448	C₄⋊C₄	—	2	8	no	—
449	C₄⋊C₄	—	2	2	no	—
450	C₄⋊C₄	—	2	4	no	—
451	C₄⋊C₄	—	2	4	no	—
452	C₄⋊C₄	—	2	8	no	—
453	C₄⋊C₄	—	2	4	no	—
454	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
455	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
456	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
457	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
458	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
459	C₂²⋊C₄	—	2	8	no	—
460	C₂²⋊C₄	—	2	4	no	—
461	C₂²⋊C₄	—	2	8	no	—
462	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
463	C₄²	—	2	4	no	—
464	C₄²	C₂×D₄×C₆₇	1	2	yes	—
465	C₄²	C₂⁴×C₆₇	1	2	yes	—
466	C₂³	—	2	4	no	—
467	C₂³	—	2	4	no	—
468	C₂³	—	2	2	no	—
469	C₂³	—	4	2	no	—
470	C₂³	—	4	2	no	—
471	C₂³	—	4	2	no	—
472	C₂³	C₂×SD₁₆×C₆₇	1	8	yes	—
473	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
474	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
475	Q₈	—	2	2	no	—
476	Q₈	—	2	2	no	—
477	Q₈	—	4	2	no	—
478	Q₈	—	4	8	no	—
479	Q₈	—	8	4	no	—
480	D₄	—	2	2	no	—
481	D₄	—	2	2	no	—
482	D₄	—	4	16	no	—
483	D₄	—	4	2	no	—
484	D₄	C₂×SD₁₆×C₆₇	1	16	yes	complemented
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	8	no	—
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	4	2	no	—
496	C₂×C₄	—	4	2	no	—
497	C₂×C₄	—	4	4	no	—
498	C₂×C₄	—	4	16	no	—
499	C₂×C₄	—	4	2	no	—
500	C₂×C₄	—	4	4	no	—
501	C₂×C₄	—	4	8	no	—
502	C₂×C₄	—	8	8	no	—
503	C₂×C₄	C₂×SD₁₆×C₆₇	1	4	yes	—
504	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
505	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
506	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
507	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
508	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
509	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
510	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
511	C₈	—	2	4	no	—
512	C₈	—	2	2	no	—
513	C₈	C₂×#41×C₆₇	1	4	yes	complemented
514	C₂²	—	2	4	no	—
515	C₂²	—	2	16	no	—
516	C₂²	—	2	4	no	—
517	C₂²	—	4	2	no	—
518	C₂²	—	4	2	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	8	4	no	—
521	C₂²	#547×C₆₇	1	2	yes	—
522	C₂²	C₂²×SD₁₆×C₆₇	1	8	yes	—
523	C₂²	C₂²×SD₁₆×C₆₇	1	2	yes	—
524	C₂²	#577×C₆₇	1	2	yes	—
525	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
526	C₄	—	2	2	no	—
527	C₄	—	2	4	no	—
528	C₄	—	4	2	no	—
529	C₄	—	4	4	no	—
530	C₄	—	4	2	no	—
531	C₄	—	4	4	no	—
532	C₄	—	4	2	no	—
533	C₄	#547×C₆₇	1	2	yes	—
534	C₄	C₂²×SD₁₆×C₆₇	1	4	yes	—
535	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
536	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
537	C₂	—	2	8	no	—
538	C₂	—	8	2	no	—
539	C₂	C₆₇×#3335	1	4	yes	direct product factor
540	C₂	C₂×#547×C₆₇	1	1	yes	characteristic
541	C₂	C₂³×SD₁₆×C₆₇	1	1	yes	characteristic
542	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×#547×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#554×C₆₇	C₂	semidirect
3. C₆₇×#3335	C₂	direct (N × H)
4. C₂×#498×C₆₇	C₂	semidirect
5. C₂×#481×C₆₇	C₂	semidirect
6. C₂×#483×C₆₇	C₂	semidirect
7. C₂×#475×C₆₇	C₂	semidirect
8. C₂×#476×C₆₇	C₂	semidirect
9. C₂×#465×C₆₇	C₂	semidirect
10. C₂×C₄×SD₁₆×C₆₇	C₂	semidirect
11. C₂×C₈×D₄×C₆₇	C₂	semidirect
12. C₂²×#106×C₆₇	C₂	semidirect
17. #554×C₆₇	C₂²	semidirect
34. #498×C₆₇	C₂²	semidirect
35. #483×C₆₇	C₂²	semidirect
36. #481×C₆₇	C₂²	semidirect
37. #476×C₆₇	C₂²	semidirect
38. #475×C₆₇	C₂²	semidirect
39. #465×C₆₇	C₂²	semidirect
40. C₄×SD₁₆×C₆₇	C₂²	semidirect
41. C₈×D₄×C₆₇	C₂²	semidirect
47. C₂×#106×C₆₇	C₂²	semidirect
142. C₂×D₄×C₆₇	SD₁₆	semidirect
168. C₂×C₈×C₆₇	#41	semidirect
212. D₄×C₆₇	C₂×SD₁₆	semidirect
241. C₈×C₆₇	C₂×#41	semidirect
265. C₂×#3335	C₆₇	direct (N × H)
268. C₂×C₆₇	#3335	direct (N × H)
272. C₂×#554	C₂×C₆₇	semidirect
274. #3335	C₂×C₆₇	direct (N × H)
275. C₂×#498	C₂×C₆₇	semidirect
276. C₂×#481	C₂×C₆₇	semidirect
277. C₂×#483	C₂×C₆₇	semidirect
278. C₂×#475	C₂×C₆₇	semidirect
279. C₂×#476	C₂×C₆₇	semidirect
280. C₂×#465	C₂×C₆₇	semidirect
281. C₂×C₄×SD₁₆	C₂×C₆₇	semidirect
282. C₂×C₈×D₄	C₂×C₆₇	semidirect
283. C₂²×#106	C₂×C₆₇	semidirect
284. C₆₇	C₂×#3335	direct (N × H)
289. #554	C₂²×C₆₇	semidirect
306. #498	C₂²×C₆₇	semidirect
307. #483	C₂²×C₆₇	semidirect
308. #481	C₂²×C₆₇	semidirect
309. #476	C₂²×C₆₇	semidirect
310. #475	C₂²×C₆₇	semidirect
311. #465	C₂²×C₆₇	semidirect
312. C₄×SD₁₆	C₂²×C₆₇	semidirect
313. C₈×D₄	C₂²×C₆₇	semidirect
319. C₂×#106	C₂²×C₆₇	semidirect
414. C₂×D₄	SD₁₆×C₆₇	semidirect
440. C₂×C₈	#41×C₆₇	semidirect
484. D₄	C₂×SD₁₆×C₆₇	semidirect
513. C₈	C₂×#41×C₆₇	semidirect
539. C₂	C₆₇×#3335	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×#554×C₆₇	2
C₂×#547×C₆₇	2
C₆₇×#3335	2
C₂×#498×C₆₇	2
C₂×#481×C₆₇	2
C₂×#483×C₆₇	2
C₂×#475×C₆₇	2
C₂×#476×C₆₇	2
C₂×#465×C₆₇	2
C₂×C₄×SD₁₆×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₂²×#106×C₆₇	2
C₂×#3335	67

Subgroups — C2×C67×#3335 #30740477

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3335 #30740477