Subgroups — C₂×C₆₇×#3336 #30740478

Order 17152.

1866 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#552×C₆₇	C₂	1	2	yes	maximal
2	C₆₇×#3336	C₂	1	16	yes	maximal, direct product factor
3	C₂×#499×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#485×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#483×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#479×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#467×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#467×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
11	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#107×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	#552×C₆₇	—	2	8	no	—
17	#552×C₆₇	C₂²	1	16	yes	—
18	C₂×#126×C₆₇	—	2	2	no	—
19	C₂×#125×C₆₇	—	2	2	no	—
20	C₂×#125×C₆₇	C₂²	1	2	yes	—
21	C₂×#123×C₆₇	—	2	2	no	—
22	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#117×C₆₇	—	2	2	no	—
27	C₂×#117×C₆₇	C₂²	1	2	yes	—
28	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
29	#499×C₆₇	C₂²	1	8	yes	complemented
30	#485×C₆₇	C₂²	1	16	yes	complemented
31	#483×C₆₇	C₂²	1	16	yes	complemented
32	#479×C₆₇	C₂²	1	16	yes	complemented
33	#467×C₆₇	C₂²	1	8	yes	complemented
34	#467×C₆₇	C₂²	1	8	yes	complemented
35	C₄×Q₁₆×C₆₇	C₂²	1	8	yes	complemented
36	C₈×D₄×C₆₇	C₂²	1	8	yes	—
37	#427×C₆₇	C₂²	1	8	yes	complemented
38	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#107×C₆₇	C₂²	1	2	yes	—
41	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×#107×C₆₇	C₂²	1	8	yes	complemented
43	C₂×#106×C₆₇	C₂²	1	2	yes	—
44	C₂×#106×C₆₇	C₂²	1	2	yes	—
45	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
46	C₂×#103×C₆₇	C₂²	1	2	yes	—
47	C₂×#103×C₆₇	C₂²	1	2	yes	—
48	C₂×#103×C₆₇	C₂²	1	2	yes	—
49	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
50	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
51	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
52	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
53	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
54	C₂³×C₄×C₆₇	—	2	2	no	—
55	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂×Q₁₆×C₆₇	—	2	4	no	—
58	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
59	C₂²×C₈×C₆₇	—	2	4	no	—
60	C₂²×C₈×C₆₇	—	2	4	no	—
61	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
62	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
63	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	#126×C₆₇	—	2	16	no	—
68	#125×C₆₇	—	2	16	no	—
69	#125×C₆₇	—	2	4	no	—
70	#125×C₆₇	C₂³	1	8	yes	—
71	#123×C₆₇	—	2	16	no	—
72	C₂²⋊Q₈×C₆₇	—	2	4	no	—
73	C₂²⋊Q₈×C₆₇	—	2	4	no	—
74	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
75	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
76	C₄×Q₈×C₆₇	—	2	4	no	—
77	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
78	C₄×D₄×C₆₇	D₄	1	4	yes	—
79	C₄×D₄×C₆₇	C₂³	1	4	yes	—
80	#117×C₆₇	—	2	4	no	—
81	#117×C₆₇	—	2	16	no	—
82	#117×C₆₇	C₂³	1	8	yes	—
83	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
84	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
85	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
86	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
87	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
88	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
89	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
90	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
92	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
93	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
95	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
96	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
97	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
98	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
99	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
100	C₂×C₄²×C₆₇	—	2	2	no	—
101	C₂×C₄²×C₆₇	—	2	2	no	—
102	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
103	#107×C₆₇	C₂³	1	4	yes	—
104	#107×C₆₇	C₂³	1	4	yes	—
105	#107×C₆₇	C₂³	1	8	yes	—
106	#107×C₆₇	C₂³	1	4	yes	—
107	#106×C₆₇	C₂³	1	8	yes	—
108	#106×C₆₇	C₂³	1	8	yes	—
109	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
110	#103×C₆₇	C₂³	1	8	yes	—
111	#103×C₆₇	C₂³	1	8	yes	—
112	#103×C₆₇	C₂³	1	8	yes	—
113	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
114	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
115	C₄×C₈×C₆₇	C₂³	1	4	yes	—
116	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂×Q₈×C₆₇	—	2	2	no	—
119	C₂×Q₈×C₆₇	—	2	4	no	—
120	C₂×Q₈×C₆₇	—	2	4	no	—
121	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
122	C₂×D₄×C₆₇	—	2	4	no	—
123	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
124	C₂²×C₄×C₆₇	—	2	2	no	—
125	C₂²×C₄×C₆₇	—	2	2	no	—
126	C₂²×C₄×C₆₇	—	2	2	no	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	—	2	8	no	—
129	C₂²×C₄×C₆₇	—	2	2	no	—
130	C₂²×C₄×C₆₇	—	2	2	no	—
131	C₂²×C₄×C₆₇	—	2	2	no	—
132	C₂²×C₄×C₆₇	—	2	2	no	—
133	C₂²×C₄×C₆₇	—	4	8	no	—
134	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
141	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
142	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
143	Q₁₆×C₆₇	—	2	8	no	—
144	C₂×C₈×C₆₇	—	2	2	no	—
145	C₂×C₈×C₆₇	—	2	2	no	—
146	C₂×C₈×C₆₇	—	2	2	no	—
147	C₂×C₈×C₆₇	—	2	8	no	—
148	C₂×C₈×C₆₇	—	2	8	no	—
149	C₂×C₈×C₆₇	#41	1	4	yes	—
150	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
151	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
152	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
153	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
154	C₄⋊C₄×C₆₇	—	2	2	no	—
155	C₄⋊C₄×C₆₇	—	2	2	no	—
156	C₄⋊C₄×C₆₇	—	2	8	no	—
157	C₄⋊C₄×C₆₇	—	2	8	no	—
158	C₄⋊C₄×C₆₇	—	2	8	no	—
159	C₄⋊C₄×C₆₇	—	2	8	no	—
160	C₄⋊C₄×C₆₇	—	2	8	no	—
161	C₄⋊C₄×C₆₇	—	2	8	no	—
162	C₄⋊C₄×C₆₇	—	2	2	no	—
163	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
164	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
165	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
166	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
167	C₂²⋊C₄×C₆₇	—	2	8	no	—
168	C₂²⋊C₄×C₆₇	—	2	8	no	—
169	C₂²⋊C₄×C₆₇	—	2	8	no	—
170	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
171	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
172	C₄²×C₆₇	—	2	8	no	—
173	C₄²×C₆₇	—	2	8	no	—
174	C₄²×C₆₇	C₂×D₄	1	2	yes	—
175	C₄²×C₆₇	C₂⁴	1	2	yes	—
176	C₂³×C₆₇	—	2	2	no	—
177	C₂³×C₆₇	—	2	2	no	—
178	C₂³×C₆₇	—	2	2	no	—
179	C₂³×C₆₇	—	2	2	no	—
180	C₂³×C₆₇	—	2	2	no	—
181	C₂³×C₆₇	#135	1	4	yes	—
182	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
183	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
184	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
185	Q₈×C₆₇	—	2	4	no	—
186	Q₈×C₆₇	—	2	4	no	—
187	Q₈×C₆₇	—	4	4	no	—
188	D₄×C₆₇	—	2	8	no	—
189	C₂×C₄×C₆₇	—	2	2	no	—
190	C₂×C₄×C₆₇	—	2	2	no	—
191	C₂×C₄×C₆₇	—	2	2	no	—
192	C₂×C₄×C₆₇	—	2	2	no	—
193	C₂×C₄×C₆₇	—	2	4	no	—
194	C₂×C₄×C₆₇	—	2	4	no	—
195	C₂×C₄×C₆₇	—	2	2	no	—
196	C₂×C₄×C₆₇	—	2	2	no	—
197	C₂×C₄×C₆₇	—	2	2	no	—
198	C₂×C₄×C₆₇	—	2	2	no	—
199	C₂×C₄×C₆₇	—	2	4	no	—
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	—	4	4	no	—
204	C₂×C₄×C₆₇	—	4	16	no	—
205	C₂×C₄×C₆₇	—	4	4	no	—
206	C₂×C₄×C₆₇	—	4	4	no	—
207	C₂×C₄×C₆₇	—	4	4	no	—
208	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
209	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
210	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
211	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
212	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
214	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
215	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
216	C₈×C₆₇	—	2	2	no	—
217	C₈×C₆₇	—	2	2	no	—
218	C₈×C₆₇	—	2	2	no	—
219	C₈×C₆₇	C₂×#41	1	4	yes	—
220	C₂²×C₆₇	—	2	8	no	—
221	C₂²×C₆₇	—	2	2	no	—
222	C₂²×C₆₇	—	2	2	no	—
223	C₂²×C₆₇	—	2	2	no	—
224	C₂²×C₆₇	—	2	2	no	—
225	C₂²×C₆₇	—	2	2	no	—
226	C₂²×C₆₇	—	4	4	no	—
227	C₂²×C₆₇	#547	1	2	yes	—
228	C₂²×C₆₇	C₂×#135	1	4	yes	—
229	C₂²×C₆₇	C₂×#135	1	2	yes	—
230	C₂²×C₆₇	#578	1	2	yes	—
231	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
232	C₄×C₆₇	—	2	2	no	—
233	C₄×C₆₇	—	2	2	no	—
234	C₄×C₆₇	—	2	2	no	—
235	C₄×C₆₇	—	2	2	no	—
236	C₄×C₆₇	—	4	4	no	—
237	C₄×C₆₇	—	4	4	no	—
238	C₄×C₆₇	—	4	4	no	—
239	C₄×C₆₇	—	4	4	no	—
240	C₄×C₆₇	#547	1	2	yes	—
241	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
242	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
243	C₂×#3336	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
244	C₂×C₆₇	—	2	4	no	—
245	C₂×C₆₇	—	4	2	no	—
246	C₂×C₆₇	#3336	1	4	yes	direct product factor
247	C₂×C₆₇	C₂×#547	1	1	yes	characteristic
248	C₂×C₆₇	C₂²×#135	1	1	yes	characteristic
249	C₂×C₆₇	C₂×#578	1	1	yes	characteristic
250	C₂×#552	C₂×C₆₇	1	2	yes	—
251	#3336	C₂×C₆₇	1	16	yes	direct product factor
252	C₂×#499	C₂×C₆₇	1	1	yes	characteristic, complemented
253	C₂×#485	C₂×C₆₇	1	2	yes	complemented
254	C₂×#483	C₂×C₆₇	1	2	yes	complemented
255	C₂×#479	C₂×C₆₇	1	2	yes	complemented
256	C₂×#467	C₂×C₆₇	1	1	yes	characteristic, complemented
257	C₂×#467	C₂×C₆₇	1	1	yes	characteristic, complemented
258	C₂×C₄×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
259	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic
260	C₂×#427	C₂×C₆₇	1	1	yes	characteristic, complemented
261	C₂²×#107	C₂×C₆₇	1	1	yes	characteristic, complemented
262	C₆₇	C₂×#3336	1	1	yes	characteristic, direct product factor, Sylow
263	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
264	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
265	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
266	#552	—	2	8	no	—
267	#552	C₂²×C₆₇	1	16	yes	—
268	C₂×#126	—	2	2	no	—
269	C₂×#125	—	2	2	no	—
270	C₂×#125	C₂²×C₆₇	1	2	yes	—
271	C₂×#123	—	2	2	no	—
272	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
273	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
274	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
275	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
276	C₂×#117	—	2	2	no	—
277	C₂×#117	C₂²×C₆₇	1	2	yes	—
278	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
279	#499	C₂²×C₆₇	1	8	yes	complemented
280	#485	C₂²×C₆₇	1	16	yes	complemented
281	#483	C₂²×C₆₇	1	16	yes	complemented
282	#479	C₂²×C₆₇	1	16	yes	complemented
283	#467	C₂²×C₆₇	1	8	yes	complemented
284	#467	C₂²×C₆₇	1	8	yes	complemented
285	C₄×Q₁₆	C₂²×C₆₇	1	8	yes	complemented
286	C₈×D₄	C₂²×C₆₇	1	8	yes	—
287	#427	C₂²×C₆₇	1	8	yes	complemented
288	C₂×#107	C₂²×C₆₇	1	2	yes	—
289	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
290	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
291	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
292	C₂×#107	C₂²×C₆₇	1	8	yes	complemented
293	C₂×#106	C₂²×C₆₇	1	2	yes	—
294	C₂×#106	C₂²×C₆₇	1	2	yes	—
295	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
296	C₂×#103	C₂²×C₆₇	1	2	yes	—
297	C₂×#103	C₂²×C₆₇	1	2	yes	—
298	C₂×#103	C₂²×C₆₇	1	2	yes	—
299	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
300	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
301	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
302	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
303	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
304	C₂³×C₄	—	2	2	no	—
305	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
306	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
307	C₂×Q₁₆	—	2	4	no	—
308	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
309	C₂²×C₈	—	2	4	no	—
310	C₂²×C₈	—	2	4	no	—
311	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
312	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
313	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
314	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
315	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
316	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
317	#126	—	2	16	no	—
318	#125	—	2	16	no	—
319	#125	—	2	4	no	—
320	#125	C₂³×C₆₇	1	8	yes	—
321	#123	—	2	16	no	—
322	C₂²⋊Q₈	—	2	4	no	—
323	C₂²⋊Q₈	—	2	4	no	—
324	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
325	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
326	C₄×Q₈	—	2	4	no	—
327	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
328	C₄×D₄	D₄×C₆₇	1	4	yes	—
329	C₄×D₄	C₂³×C₆₇	1	4	yes	—
330	#117	—	2	4	no	—
331	#117	—	2	16	no	—
332	#117	C₂³×C₆₇	1	8	yes	—
333	C₂×C₄⋊C₄	—	2	2	no	—
334	C₂×C₄⋊C₄	—	2	4	no	—
335	C₂×C₄⋊C₄	—	2	2	no	—
336	C₂×C₄⋊C₄	—	2	2	no	—
337	C₂×C₄⋊C₄	—	2	2	no	—
338	C₂×C₄⋊C₄	—	2	2	no	—
339	C₂×C₄⋊C₄	—	2	2	no	—
340	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
341	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
342	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
343	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
344	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
345	C₂×C₂²⋊C₄	—	2	2	no	—
346	C₂×C₂²⋊C₄	—	2	2	no	—
347	C₂×C₂²⋊C₄	—	2	2	no	—
348	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
349	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
350	C₂×C₄²	—	2	2	no	—
351	C₂×C₄²	—	2	2	no	—
352	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
353	#107	C₂³×C₆₇	1	4	yes	—
354	#107	C₂³×C₆₇	1	4	yes	—
355	#107	C₂³×C₆₇	1	8	yes	—
356	#107	C₂³×C₆₇	1	4	yes	—
357	#106	C₂³×C₆₇	1	8	yes	—
358	#106	C₂³×C₆₇	1	8	yes	—
359	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
360	#103	C₂³×C₆₇	1	8	yes	—
361	#103	C₂³×C₆₇	1	8	yes	—
362	#103	C₂³×C₆₇	1	8	yes	—
363	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
364	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
365	C₄×C₈	C₂³×C₆₇	1	4	yes	—
366	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
367	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
368	C₂×Q₈	—	2	2	no	—
369	C₂×Q₈	—	2	4	no	—
370	C₂×Q₈	—	2	4	no	—
371	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
372	C₂×D₄	—	2	4	no	—
373	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	8	no	—
383	C₂²×C₄	—	4	8	no	—
384	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
385	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
386	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
387	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
388	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
389	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
390	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
391	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
392	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
393	Q₁₆	—	2	8	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	2	8	no	—
398	C₂×C₈	—	2	8	no	—
399	C₂×C₈	#41×C₆₇	1	4	yes	—
400	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
401	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
402	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
403	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	8	no	—
407	C₄⋊C₄	—	2	8	no	—
408	C₄⋊C₄	—	2	8	no	—
409	C₄⋊C₄	—	2	8	no	—
410	C₄⋊C₄	—	2	8	no	—
411	C₄⋊C₄	—	2	8	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
414	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
415	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
416	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
417	C₂²⋊C₄	—	2	8	no	—
418	C₂²⋊C₄	—	2	8	no	—
419	C₂²⋊C₄	—	2	8	no	—
420	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
421	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
422	C₄²	—	2	8	no	—
423	C₄²	—	2	8	no	—
424	C₄²	C₂×D₄×C₆₇	1	2	yes	—
425	C₄²	C₂⁴×C₆₇	1	2	yes	—
426	C₂³	—	2	2	no	—
427	C₂³	—	2	2	no	—
428	C₂³	—	2	2	no	—
429	C₂³	—	2	2	no	—
430	C₂³	—	2	2	no	—
431	C₂³	#135×C₆₇	1	4	yes	—
432	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
433	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
434	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
435	Q₈	—	2	4	no	—
436	Q₈	—	2	4	no	—
437	Q₈	—	4	4	no	—
438	D₄	—	2	8	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	4	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	4	4	no	—
454	C₂×C₄	—	4	16	no	—
455	C₂×C₄	—	4	4	no	—
456	C₂×C₄	—	4	4	no	—
457	C₂×C₄	—	4	4	no	—
458	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
459	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
460	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
461	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
462	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
463	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
464	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
465	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
466	C₈	—	2	2	no	—
467	C₈	—	2	2	no	—
468	C₈	—	2	2	no	—
469	C₈	C₂×#41×C₆₇	1	4	yes	—
470	C₂²	—	2	2	no	—
471	C₂²	—	2	8	no	—
472	C₂²	—	2	2	no	—
473	C₂²	—	2	2	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	4	4	no	—
477	C₂²	#547×C₆₇	1	2	yes	—
478	C₂²	C₂×#135×C₆₇	1	4	yes	—
479	C₂²	C₂×#135×C₆₇	1	2	yes	—
480	C₂²	#578×C₆₇	1	2	yes	—
481	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
482	C₄	—	2	2	no	—
483	C₄	—	2	2	no	—
484	C₄	—	2	2	no	—
485	C₄	—	2	2	no	—
486	C₄	—	4	4	no	—
487	C₄	—	4	4	no	—
488	C₄	—	4	4	no	—
489	C₄	—	4	4	no	—
490	C₄	#547×C₆₇	1	2	yes	—
491	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
492	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
493	C₂	—	2	4	no	—
494	C₂	—	4	2	no	—
495	C₂	C₆₇×#3336	1	4	yes	direct product factor
496	C₂	C₂×#547×C₆₇	1	1	yes	characteristic
497	C₂	C₂²×#135×C₆₇	1	1	yes	characteristic
498	C₂	C₂×#578×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	D₄
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
2. C₆₇×#3336	C₂	direct (N × H)
3. C₂×#499×C₆₇	C₂	semidirect
4. C₂×#485×C₆₇	C₂	semidirect
5. C₂×#483×C₆₇	C₂	semidirect
6. C₂×#479×C₆₇	C₂	semidirect
7. C₂×#467×C₆₇	C₂	semidirect
8. C₂×#467×C₆₇	C₂	semidirect
9. C₂×C₄×Q₁₆×C₆₇	C₂	semidirect
11. C₂×#427×C₆₇	C₂	semidirect
12. C₂²×#107×C₆₇	C₂	semidirect
29. #499×C₆₇	C₂²	semidirect
30. #485×C₆₇	C₂²	semidirect
31. #483×C₆₇	C₂²	semidirect
32. #479×C₆₇	C₂²	semidirect
33. #467×C₆₇	C₂²	semidirect
34. #467×C₆₇	C₂²	semidirect
35. C₄×Q₁₆×C₆₇	C₂²	semidirect
37. #427×C₆₇	C₂²	semidirect
42. C₂×#107×C₆₇	C₂²	semidirect
243. C₂×#3336	C₆₇	direct (N × H)
246. C₂×C₆₇	#3336	direct (N × H)
251. #3336	C₂×C₆₇	direct (N × H)
252. C₂×#499	C₂×C₆₇	semidirect
253. C₂×#485	C₂×C₆₇	semidirect
254. C₂×#483	C₂×C₆₇	semidirect
255. C₂×#479	C₂×C₆₇	semidirect
256. C₂×#467	C₂×C₆₇	semidirect
257. C₂×#467	C₂×C₆₇	semidirect
258. C₂×C₄×Q₁₆	C₂×C₆₇	semidirect
260. C₂×#427	C₂×C₆₇	semidirect
261. C₂²×#107	C₂×C₆₇	semidirect
262. C₆₇	C₂×#3336	direct (N × H)
279. #499	C₂²×C₆₇	semidirect
280. #485	C₂²×C₆₇	semidirect
281. #483	C₂²×C₆₇	semidirect
282. #479	C₂²×C₆₇	semidirect
283. #467	C₂²×C₆₇	semidirect
284. #467	C₂²×C₆₇	semidirect
285. C₄×Q₁₆	C₂²×C₆₇	semidirect
287. #427	C₂²×C₆₇	semidirect
292. C₂×#107	C₂²×C₆₇	semidirect
495. C₂	C₆₇×#3336	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×#552×C₆₇	2
C₆₇×#3336	2
C₂×#499×C₆₇	2
C₂×#485×C₆₇	2
C₂×#483×C₆₇	2
C₂×#479×C₆₇	2
C₂×#467×C₆₇	2
C₂×C₄×Q₁₆×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₂×#427×C₆₇	2
C₂²×#107×C₆₇	2
C₂×#3336	67

Subgroups — C2×C67×#3336 #30740478

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3336 #30740478