Subgroups — C₆₇×#33173 #30712424

Order 17152.

1194 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#495×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#467×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#466×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#465×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2850	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1970	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1969	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1968	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1958	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#124×C₆₇	C₂²	1	2	yes	—
18	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
19	C₂×#121×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	#495×C₆₇	—	2	4	no	—
22	#467×C₆₇	—	2	4	no	—
23	#466×C₆₇	—	2	4	no	—
24	#465×C₆₇	—	2	8	no	—
25	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#106×C₆₇	C₂²	1	2	yes	—
28	C₂×#103×C₆₇	C₂²	1	2	yes	—
29	C₂×#103×C₆₇	C₂²	1	2	yes	—
30	C₂×#102×C₆₇	C₂²	1	2	yes	—
31	C₂×#102×C₆₇	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₆₇	—	2	4	no	—
33	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	#391×C₆₇	C₂²	1	2	yes	—
38	#390×C₆₇	C₂²	1	2	yes	—
39	#388×C₆₇	—	2	4	no	—
40	#386×C₆₇	—	2	4	no	—
41	#382×C₆₇	C₂²	1	2	yes	—
42	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
43	C₂²×D₄×C₆₇	—	2	2	no	—
44	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
45	C₂³×C₄×C₆₇	D₄	1	2	yes	—
46	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
47	C₂×Q₁₆×C₆₇	—	2	2	no	—
48	C₂×Q₁₆×C₆₇	—	2	2	no	—
49	C₂×Q₁₆×C₆₇	—	2	2	no	—
50	C₂×SD₁₆×C₆₇	—	2	4	no	—
51	C₂×SD₁₆×C₆₇	—	2	4	no	—
52	C₂×SD₁₆×C₆₇	—	2	4	no	—
53	C₂×D₈×C₆₇	—	2	2	no	—
54	C₂×D₈×C₆₇	—	2	2	no	—
55	C₂×D₈×C₆₇	—	2	2	no	—
56	C₂²×C₈×C₆₇	—	2	2	no	—
57	C₂²×C₈×C₆₇	—	2	2	no	—
58	C₂²×C₈×C₆₇	—	2	2	no	—
59	C₂²×C₈×C₆₇	—	2	2	no	—
60	C₂²×C₈×C₆₇	D₄	1	4	yes	—
61	C₂²×C₈×C₆₇	D₄	1	4	yes	—
62	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
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	#124×C₆₇	—	2	4	no	—
67	#124×C₆₇	—	2	4	no	—
68	C₂²⋊Q₈×C₆₇	—	2	4	no	—
69	C₂²⋊Q₈×C₆₇	—	2	4	no	—
70	#121×C₆₇	—	2	4	no	—
71	#121×C₆₇	—	2	4	no	—
72	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
73	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
75	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
78	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
79	C₂×C₄²×C₆₇	—	2	2	no	—
80	C₂×C₄²×C₆₇	—	2	2	no	—
81	C₂×C₄²×C₆₇	D₄	1	2	yes	—
82	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
84	#107×C₆₇	—	2	2	no	—
85	#107×C₆₇	—	2	2	no	—
86	#106×C₆₇	—	2	4	no	—
87	#103×C₆₇	—	2	4	no	—
88	#103×C₆₇	—	2	4	no	—
89	#102×C₆₇	—	2	4	no	—
90	#102×C₆₇	—	2	4	no	—
91	C₄×C₈×C₆₇	—	2	2	no	—
92	C₄×C₈×C₆₇	—	2	2	no	—
93	C₄×C₈×C₆₇	—	2	2	no	—
94	C₄×C₈×C₆₇	D₄	1	4	yes	complemented
95	#95×C₆₇	—	2	4	no	—
96	#95×C₆₇	—	2	4	no	—
97	C₂⁴×C₆₇	—	2	2	no	—
98	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂×Q₈×C₆₇	—	2	2	no	—
100	C₂×Q₈×C₆₇	—	2	2	no	—
101	C₂×Q₈×C₆₇	—	2	2	no	—
102	C₂×Q₈×C₆₇	—	2	2	no	—
103	C₂×Q₈×C₆₇	—	2	2	no	—
104	C₂×Q₈×C₆₇	—	2	2	no	—
105	C₂×D₄×C₆₇	—	2	2	no	—
106	C₂×D₄×C₆₇	—	2	2	no	—
107	C₂×D₄×C₆₇	—	2	2	no	—
108	C₂×D₄×C₆₇	—	2	2	no	—
109	C₂×D₄×C₆₇	—	2	2	no	—
110	C₂×D₄×C₆₇	—	2	2	no	—
111	C₂×D₄×C₆₇	—	4	4	no	—
112	C₂×D₄×C₆₇	—	4	8	no	—
113	C₂²×C₄×C₆₇	—	2	4	no	—
114	C₂²×C₄×C₆₇	—	2	4	no	—
115	C₂²×C₄×C₆₇	—	2	4	no	—
116	C₂²×C₄×C₆₇	—	2	1	no	—
117	C₂²×C₄×C₆₇	—	2	1	no	—
118	C₂²×C₄×C₆₇	—	2	1	no	—
119	C₂²×C₄×C₆₇	—	2	1	no	—
120	C₂²×C₄×C₆₇	—	2	2	no	—
121	C₂²×C₄×C₆₇	—	2	2	no	—
122	C₂²×C₄×C₆₇	—	2	2	no	—
123	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
124	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
129	Q₁₆×C₆₇	—	4	4	no	—
130	SD₁₆×C₆₇	—	4	8	no	—
131	D₈×C₆₇	—	4	4	no	—
132	C₂×C₈×C₆₇	—	2	1	no	—
133	C₂×C₈×C₆₇	—	2	1	no	—
134	C₂×C₈×C₆₇	—	2	1	no	—
135	C₂×C₈×C₆₇	—	2	1	no	—
136	C₂×C₈×C₆₇	—	2	8	no	—
137	C₂×C₈×C₆₇	—	2	8	no	—
138	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
139	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
140	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
141	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
142	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
143	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
144	C₂×C₈×C₆₇	#41	1	2	yes	—
145	C₂×C₈×C₆₇	#41	1	2	yes	—
146	C₄⋊C₄×C₆₇	—	2	2	no	—
147	C₄⋊C₄×C₆₇	—	2	2	no	—
148	C₄⋊C₄×C₆₇	—	2	2	no	—
149	C₄⋊C₄×C₆₇	—	2	2	no	—
150	C₄⋊C₄×C₆₇	—	4	4	no	—
151	C₂²⋊C₄×C₆₇	—	4	4	no	—
152	C₂²⋊C₄×C₆₇	—	4	8	no	—
153	C₂²⋊C₄×C₆₇	—	4	4	no	—
154	C₂²⋊C₄×C₆₇	—	4	8	no	—
155	C₄²×C₆₇	—	2	4	no	—
156	C₄²×C₆₇	—	2	1	no	—
157	C₄²×C₆₇	—	2	1	no	—
158	C₄²×C₆₇	C₂×D₄	1	2	yes	—
159	C₄²×C₆₇	#41	1	2	yes	—
160	C₂³×C₆₇	—	2	1	no	—
161	C₂³×C₆₇	—	2	1	no	—
162	C₂³×C₆₇	—	2	2	no	—
163	C₂³×C₆₇	—	4	2	no	—
164	C₂³×C₆₇	—	4	2	no	—
165	C₂³×C₆₇	—	4	2	no	—
166	C₂³×C₆₇	—	4	4	no	—
167	C₂³×C₆₇	—	4	4	no	—
168	C₂³×C₆₇	#127	1	2	yes	—
169	C₂³×C₆₇	#135	1	4	yes	—
170	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
171	Q₈×C₆₇	—	4	2	no	—
172	Q₈×C₆₇	—	4	2	no	—
173	Q₈×C₆₇	—	4	2	no	—
174	Q₈×C₆₇	—	4	2	no	—
175	D₄×C₆₇	—	4	2	no	—
176	D₄×C₆₇	—	4	2	no	—
177	D₄×C₆₇	—	4	2	no	—
178	D₄×C₆₇	—	4	2	no	—
179	D₄×C₆₇	—	8	8	no	—
180	C₂×C₄×C₆₇	—	2	16	no	—
181	C₂×C₄×C₆₇	—	2	1	no	—
182	C₂×C₄×C₆₇	—	2	2	no	—
183	C₂×C₄×C₆₇	—	2	1	no	—
184	C₂×C₄×C₆₇	—	2	2	no	—
185	C₂×C₄×C₆₇	—	2	2	no	—
186	C₂×C₄×C₆₇	—	2	2	no	—
187	C₂×C₄×C₆₇	—	2	2	no	—
188	C₂×C₄×C₆₇	—	2	2	no	—
189	C₂×C₄×C₆₇	—	2	2	no	—
190	C₂×C₄×C₆₇	—	2	2	no	—
191	C₂×C₄×C₆₇	—	4	2	no	—
192	C₂×C₄×C₆₇	—	4	2	no	—
193	C₂×C₄×C₆₇	—	4	2	no	—
194	C₂×C₄×C₆₇	—	4	2	no	—
195	C₂×C₄×C₆₇	—	4	2	no	—
196	C₂×C₄×C₆₇	—	4	4	no	—
197	C₂×C₄×C₆₇	—	4	4	no	—
198	C₂×C₄×C₆₇	#121	1	4	yes	—
199	C₂×C₄×C₆₇	#127	1	2	yes	—
200	C₂×C₄×C₆₇	#135	1	4	yes	—
201	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
203	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
204	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
205	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
206	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
207	C₈×C₆₇	—	2	2	no	—
208	C₈×C₆₇	—	2	2	no	—
209	C₈×C₆₇	#121	1	4	yes	—
210	C₈×C₆₇	#121	1	4	yes	—
211	C₂²×C₆₇	—	2	2	no	—
212	C₂²×C₆₇	—	2	2	no	—
213	C₂²×C₆₇	—	2	4	no	—
214	C₂²×C₆₇	—	2	4	no	—
215	C₂²×C₆₇	—	8	2	no	—
216	C₂²×C₆₇	—	8	2	no	—
217	C₂²×C₆₇	—	8	2	no	—
218	C₂²×C₆₇	—	8	4	no	—
219	C₂²×C₆₇	—	8	4	no	—
220	C₂²×C₆₇	#495	1	4	yes	complemented
221	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
222	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
223	C₂²×C₆₇	#532	1	1	yes	characteristic
224	C₂²×C₆₇	C₂×#135	1	2	yes	—
225	C₂²×C₆₇	C₂×#135	1	2	yes	—
226	C₄×C₆₇	—	2	2	no	—
227	C₄×C₆₇	—	2	4	no	—
228	C₄×C₆₇	—	2	4	no	—
229	C₄×C₆₇	—	8	2	no	—
230	C₄×C₆₇	—	8	2	no	—
231	C₄×C₆₇	—	8	2	no	—
232	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
233	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
234	C₄×C₆₇	#532	1	1	yes	characteristic
235	C₄×C₆₇	#532	1	1	yes	characteristic
236	#33173	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
237	C₂×C₆₇	—	2	4	no	—
238	C₂×C₆₇	—	16	2	no	—
239	C₂×C₆₇	#2856	1	1	yes	characteristic
240	C₂×C₆₇	#3050	1	2	yes	—
241	C₂×C₆₇	C₂×#495	1	1	yes	characteristic
242	C₂×C₆₇	#3147	1	1	yes	characteristic
243	C₂×C₆₇	#3168	1	2	yes	—
244	C₂×#495	C₂×C₆₇	1	1	yes	characteristic, complemented
245	C₂×#467	C₂×C₆₇	1	1	yes	characteristic, complemented
246	C₂×#466	C₂×C₆₇	1	1	yes	characteristic
247	C₂×#465	C₂×C₆₇	1	2	yes	complemented
248	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
249	#2850	C₂×C₆₇	1	2	yes	complemented
250	#1970	C₂×C₆₇	1	1	yes	characteristic, complemented
251	#1969	C₂×C₆₇	1	1	yes	characteristic, complemented
252	#1968	C₂×C₆₇	1	2	yes	complemented
253	#1958	C₂×C₆₇	1	2	yes	complemented
254	#1846	C₂×C₆₇	1	1	yes	characteristic, complemented
255	C₆₇	#33173	1	1	yes	characteristic, direct product factor, Sylow
256	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
257	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
258	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
259	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
260	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
261	C₂×#124	C₂²×C₆₇	1	2	yes	—
262	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
263	C₂×#121	C₂²×C₆₇	1	2	yes	—
264	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
265	#495	—	2	4	no	—
266	#467	—	2	4	no	—
267	#466	—	2	4	no	—
268	#465	—	2	8	no	—
269	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
270	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
271	C₂×#106	C₂²×C₆₇	1	2	yes	—
272	C₂×#103	C₂²×C₆₇	1	2	yes	—
273	C₂×#103	C₂²×C₆₇	1	2	yes	—
274	C₂×#102	C₂²×C₆₇	1	2	yes	—
275	C₂×#102	C₂²×C₆₇	1	2	yes	—
276	C₂×C₄×C₈	—	2	4	no	—
277	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
278	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
279	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
280	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
281	#391	C₂²×C₆₇	1	2	yes	—
282	#390	C₂²×C₆₇	1	2	yes	—
283	#388	—	2	4	no	—
284	#386	—	2	4	no	—
285	#382	C₂²×C₆₇	1	2	yes	—
286	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
287	C₂²×D₄	—	2	2	no	—
288	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
289	C₂³×C₄	D₄×C₆₇	1	2	yes	—
290	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
291	C₂×Q₁₆	—	2	2	no	—
292	C₂×Q₁₆	—	2	2	no	—
293	C₂×Q₁₆	—	2	2	no	—
294	C₂×SD₁₆	—	2	4	no	—
295	C₂×SD₁₆	—	2	4	no	—
296	C₂×SD₁₆	—	2	4	no	—
297	C₂×D₈	—	2	2	no	—
298	C₂×D₈	—	2	2	no	—
299	C₂×D₈	—	2	2	no	—
300	C₂²×C₈	—	2	2	no	—
301	C₂²×C₈	—	2	2	no	—
302	C₂²×C₈	—	2	2	no	—
303	C₂²×C₈	—	2	2	no	—
304	C₂²×C₈	D₄×C₆₇	1	4	yes	—
305	C₂²×C₈	D₄×C₆₇	1	4	yes	—
306	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
307	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
308	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
309	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
310	#124	—	2	4	no	—
311	#124	—	2	4	no	—
312	C₂²⋊Q₈	—	2	4	no	—
313	C₂²⋊Q₈	—	2	4	no	—
314	#121	—	2	4	no	—
315	#121	—	2	4	no	—
316	C₂×C₄⋊C₄	—	2	2	no	—
317	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
318	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
319	C₂×C₂²⋊C₄	—	2	2	no	—
320	C₂×C₂²⋊C₄	—	2	2	no	—
321	C₂×C₂²⋊C₄	—	2	4	no	—
322	C₂×C₂²⋊C₄	—	2	4	no	—
323	C₂×C₄²	—	2	2	no	—
324	C₂×C₄²	—	2	2	no	—
325	C₂×C₄²	D₄×C₆₇	1	2	yes	—
326	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
327	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
328	#107	—	2	2	no	—
329	#107	—	2	2	no	—
330	#106	—	2	4	no	—
331	#103	—	2	4	no	—
332	#103	—	2	4	no	—
333	#102	—	2	4	no	—
334	#102	—	2	4	no	—
335	C₄×C₈	—	2	2	no	—
336	C₄×C₈	—	2	2	no	—
337	C₄×C₈	—	2	2	no	—
338	C₄×C₈	D₄×C₆₇	1	4	yes	complemented
339	#95	—	2	4	no	—
340	#95	—	2	4	no	—
341	C₂⁴	—	2	2	no	—
342	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
343	C₂×Q₈	—	2	2	no	—
344	C₂×Q₈	—	2	2	no	—
345	C₂×Q₈	—	2	2	no	—
346	C₂×Q₈	—	2	2	no	—
347	C₂×Q₈	—	2	2	no	—
348	C₂×Q₈	—	2	2	no	—
349	C₂×D₄	—	2	2	no	—
350	C₂×D₄	—	2	2	no	—
351	C₂×D₄	—	2	2	no	—
352	C₂×D₄	—	2	2	no	—
353	C₂×D₄	—	2	2	no	—
354	C₂×D₄	—	2	2	no	—
355	C₂×D₄	—	4	4	no	—
356	C₂×D₄	—	4	8	no	—
357	C₂²×C₄	—	2	4	no	—
358	C₂²×C₄	—	2	4	no	—
359	C₂²×C₄	—	2	4	no	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
368	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
369	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
370	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
371	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
372	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
373	Q₁₆	—	4	4	no	—
374	SD₁₆	—	4	8	no	—
375	D₈	—	4	4	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	—	2	8	no	—
381	C₂×C₈	—	2	8	no	—
382	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
383	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
384	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
385	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
386	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
387	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
388	C₂×C₈	#41×C₆₇	1	2	yes	—
389	C₂×C₈	#41×C₆₇	1	2	yes	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	2	2	no	—
393	C₄⋊C₄	—	2	2	no	—
394	C₄⋊C₄	—	4	4	no	—
395	C₂²⋊C₄	—	4	4	no	—
396	C₂²⋊C₄	—	4	8	no	—
397	C₂²⋊C₄	—	4	4	no	—
398	C₂²⋊C₄	—	4	8	no	—
399	C₄²	—	2	4	no	—
400	C₄²	—	2	1	no	—
401	C₄²	—	2	1	no	—
402	C₄²	C₂×D₄×C₆₇	1	2	yes	—
403	C₄²	#41×C₆₇	1	2	yes	—
404	C₂³	—	2	1	no	—
405	C₂³	—	2	1	no	—
406	C₂³	—	2	2	no	—
407	C₂³	—	4	2	no	—
408	C₂³	—	4	2	no	—
409	C₂³	—	4	2	no	—
410	C₂³	—	4	4	no	—
411	C₂³	—	4	4	no	—
412	C₂³	#127×C₆₇	1	2	yes	—
413	C₂³	#135×C₆₇	1	4	yes	—
414	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
415	Q₈	—	4	2	no	—
416	Q₈	—	4	2	no	—
417	Q₈	—	4	2	no	—
418	Q₈	—	4	2	no	—
419	D₄	—	4	2	no	—
420	D₄	—	4	2	no	—
421	D₄	—	4	2	no	—
422	D₄	—	4	2	no	—
423	D₄	—	8	8	no	—
424	C₂×C₄	—	2	16	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	4	no	—
441	C₂×C₄	—	4	4	no	—
442	C₂×C₄	#121×C₆₇	1	4	yes	—
443	C₂×C₄	#127×C₆₇	1	2	yes	—
444	C₂×C₄	#135×C₆₇	1	4	yes	—
445	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
446	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
447	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
448	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
449	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
450	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
451	C₈	—	2	2	no	—
452	C₈	—	2	2	no	—
453	C₈	#121×C₆₇	1	4	yes	—
454	C₈	#121×C₆₇	1	4	yes	—
455	C₂²	—	2	4	no	—
456	C₂²	—	2	2	no	—
457	C₂²	—	2	2	no	—
458	C₂²	—	2	4	no	—
459	C₂²	—	8	4	no	—
460	C₂²	—	8	2	no	—
461	C₂²	—	8	2	no	—
462	C₂²	—	8	2	no	—
463	C₂²	—	8	4	no	—
464	C₂²	#495×C₆₇	1	4	yes	complemented
465	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
466	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
467	C₂²	#532×C₆₇	1	1	yes	characteristic
468	C₂²	C₂×#135×C₆₇	1	2	yes	—
469	C₂²	C₂×#135×C₆₇	1	2	yes	—
470	C₄	—	2	4	no	—
471	C₄	—	2	4	no	—
472	C₄	—	2	2	no	—
473	C₄	—	8	2	no	—
474	C₄	—	8	2	no	—
475	C₄	—	8	2	no	—
476	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
477	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
478	C₄	#532×C₆₇	1	1	yes	characteristic
479	C₄	#532×C₆₇	1	1	yes	characteristic
480	C₂	—	2	4	no	—
481	C₂	—	16	2	no	—
482	C₂	C₆₇×#2856	1	1	yes	characteristic
483	C₂	C₆₇×#3050	1	2	yes	—
484	C₂	C₂×#495×C₆₇	1	1	yes	characteristic
485	C₂	C₆₇×#3147	1	1	yes	characteristic
486	C₂	C₆₇×#3168	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#466×C₆₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#495×C₆₇	C₂	semidirect
2. C₂×#467×C₆₇	C₂	semidirect
4. C₂×#465×C₆₇	C₂	semidirect
5. C₂²×C₄×C₈×C₆₇	C₂	semidirect
6. C₆₇×#2850	C₂	semidirect
7. C₆₇×#1970	C₂	semidirect
8. C₆₇×#1969	C₂	semidirect
9. C₆₇×#1968	C₂	semidirect
10. C₆₇×#1958	C₂	semidirect
11. C₆₇×#1846	C₂	semidirect
94. C₄×C₈×C₆₇	D₄	semidirect
220. C₂²×C₆₇	#495	semidirect
236. #33173	C₆₇	direct (N × H)
244. C₂×#495	C₂×C₆₇	semidirect
245. C₂×#467	C₂×C₆₇	semidirect
247. C₂×#465	C₂×C₆₇	semidirect
248. C₂²×C₄×C₈	C₂×C₆₇	semidirect
249. #2850	C₂×C₆₇	semidirect
250. #1970	C₂×C₆₇	semidirect
251. #1969	C₂×C₆₇	semidirect
252. #1968	C₂×C₆₇	semidirect
253. #1958	C₂×C₆₇	semidirect
254. #1846	C₂×C₆₇	semidirect
255. C₆₇	#33173	direct (N × H)
338. C₄×C₈	D₄×C₆₇	semidirect
464. C₂²	#495×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#495×C₆₇	2
C₂×#467×C₆₇	2
C₂×#466×C₆₇	2
C₂×#465×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₆₇×#2850	2
C₆₇×#1970	2
C₆₇×#1969	2
C₆₇×#1968	2
C₆₇×#1958	2
C₆₇×#1846	2
#33173	67

Subgroups — C67×#33173 #30712424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33173 #30712424