Subgroups — C₄×#467×C₆₇ #30700773

Order 17152.

1054 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#467×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₄×C₂²⋊Q₈×C₆₇	C₂	1	2	yes	maximal
5	C₆₇×#1969	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1942	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1933	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1892	C₂	1	2	yes	maximal
9	C₆₇×#1808	C₂	1	1	yes	maximal, characteristic
10	C₄×#107×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#103×C₆₇	C₂	1	2	yes	maximal, complemented
12	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	#467×C₆₇	C₄	1	8	yes	direct product factor
19	C₄×Q₁₆×C₆₇	—	2	4	no	—
20	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#103×C₆₇	C₂²	1	2	yes	—
24	C₂×#103×C₆₇	C₂²	1	2	yes	—
25	C₂×#103×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₆₇	—	2	4	no	—
27	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
31	#391×C₆₇	C₂²	1	2	yes	—
32	#389×C₆₇	C₂²	1	2	yes	—
33	#387×C₆₇	—	2	2	no	—
34	#385×C₆₇	—	2	2	no	—
35	#380×C₆₇	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
38	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
39	#340×C₆₇	C₂²	1	2	yes	—
40	#340×C₆₇	C₂²	1	2	yes	—
41	#340×C₆₇	C₂²	1	2	yes	—
42	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
43	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×Q₁₆×C₆₇	—	2	2	no	—
47	C₂×Q₁₆×C₆₇	—	2	2	no	—
48	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
49	C₂²×C₈×C₆₇	—	2	2	no	—
50	C₂²×C₈×C₆₇	—	2	2	no	—
51	C₂²×C₈×C₆₇	—	2	2	no	—
52	C₂²×C₈×C₆₇	—	2	2	no	—
53	C₂²×C₈×C₆₇	—	2	2	no	—
54	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²⋊Q₈×C₆₇	—	2	4	no	—
60	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
61	C₄×Q₈×C₆₇	—	2	4	no	—
62	C₄×Q₈×C₆₇	—	2	4	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
71	C₂×C₄²×C₆₇	—	2	2	no	—
72	C₂×C₄²×C₆₇	—	2	2	no	—
73	C₂×C₄²×C₆₇	—	2	2	no	—
74	C₂×C₄²×C₆₇	—	2	2	no	—
75	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
79	#107×C₆₇	—	2	2	no	—
80	#107×C₆₇	—	2	2	no	—
81	#107×C₆₇	C₂×C₄	1	4	yes	complemented
82	#103×C₆₇	—	2	4	no	—
83	#103×C₆₇	—	2	4	no	—
84	#103×C₆₇	C₂×C₄	1	8	yes	complemented
85	C₄×C₈×C₆₇	—	2	2	no	—
86	C₄×C₈×C₆₇	—	2	2	no	—
87	C₄×C₈×C₆₇	—	2	2	no	—
88	C₄×C₈×C₆₇	D₄	1	4	yes	—
89	#95×C₆₇	—	2	2	no	—
90	#95×C₆₇	—	2	2	no	—
91	#95×C₆₇	—	2	2	no	—
92	#95×C₆₇	—	2	2	no	—
93	#95×C₆₇	—	2	2	no	—
94	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂×Q₈×C₆₇	—	2	2	no	—
96	C₂×Q₈×C₆₇	—	2	2	no	—
97	C₂×Q₈×C₆₇	—	2	2	no	—
98	C₂×Q₈×C₆₇	—	2	2	no	—
99	C₂×Q₈×C₆₇	—	2	2	no	—
100	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
101	C₂²×C₄×C₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	1	no	—
107	C₂²×C₄×C₆₇	—	2	1	no	—
108	C₂²×C₄×C₆₇	—	2	1	no	—
109	C₂²×C₄×C₆₇	—	2	1	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	2	2	no	—
112	C₂²×C₄×C₆₇	—	2	2	no	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	complemented
115	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
116	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	#41	1	2	yes	—
122	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
123	Q₁₆×C₆₇	—	2	8	no	—
124	C₂×C₈×C₆₇	—	2	8	no	—
125	C₂×C₈×C₆₇	—	2	1	no	—
126	C₂×C₈×C₆₇	—	2	1	no	—
127	C₂×C₈×C₆₇	—	2	1	no	—
128	C₂×C₈×C₆₇	—	2	1	no	—
129	C₂×C₈×C₆₇	—	2	1	no	—
130	C₂×C₈×C₆₇	—	2	1	no	—
131	C₂×C₈×C₆₇	—	2	8	no	—
132	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
133	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
134	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
135	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
136	C₂×C₈×C₆₇	#41	1	2	yes	—
137	C₂×C₈×C₆₇	#41	1	2	yes	—
138	C₄⋊C₄×C₆₇	—	2	8	no	—
139	C₄⋊C₄×C₆₇	—	2	2	no	—
140	C₄⋊C₄×C₆₇	—	2	2	no	—
141	C₄⋊C₄×C₆₇	—	2	2	no	—
142	C₄⋊C₄×C₆₇	—	2	2	no	—
143	C₄⋊C₄×C₆₇	—	2	2	no	—
144	C₄⋊C₄×C₆₇	—	4	4	no	—
145	C₄⋊C₄×C₆₇	—	4	4	no	—
146	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
147	C₂²⋊C₄×C₆₇	—	2	8	no	—
148	C₂²⋊C₄×C₆₇	—	4	4	no	—
149	C₄²×C₆₇	—	2	2	no	—
150	C₄²×C₆₇	—	2	2	no	—
151	C₄²×C₆₇	—	2	1	no	—
152	C₄²×C₆₇	—	2	1	no	—
153	C₄²×C₆₇	—	4	4	no	—
154	C₄²×C₆₇	—	4	4	no	—
155	C₄²×C₆₇	C₂×D₄	1	2	yes	—
156	C₄²×C₆₇	#41	1	2	yes	—
157	C₂³×C₆₇	—	2	1	no	—
158	C₂³×C₆₇	—	2	1	no	—
159	C₂³×C₆₇	—	2	1	no	—
160	C₂³×C₆₇	—	2	1	no	—
161	C₂³×C₆₇	C₄×D₄	1	2	yes	—
162	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
163	C₂³×C₆₇	#135	1	2	yes	—
164	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
165	Q₈×C₆₇	—	2	4	no	—
166	Q₈×C₆₇	—	2	4	no	—
167	Q₈×C₆₇	—	4	2	no	—
168	Q₈×C₆₇	—	4	2	no	—
169	C₂×C₄×C₆₇	—	2	8	no	—
170	C₂×C₄×C₆₇	—	2	1	no	—
171	C₂×C₄×C₆₇	—	2	1	no	—
172	C₂×C₄×C₆₇	—	2	1	no	—
173	C₂×C₄×C₆₇	—	2	1	no	—
174	C₂×C₄×C₆₇	—	2	1	no	—
175	C₂×C₄×C₆₇	—	2	1	no	—
176	C₂×C₄×C₆₇	—	2	1	no	—
177	C₂×C₄×C₆₇	—	2	1	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	2	2	no	—
181	C₂×C₄×C₆₇	—	2	2	no	—
182	C₂×C₄×C₆₇	—	2	8	no	—
183	C₂×C₄×C₆₇	—	2	4	no	—
184	C₂×C₄×C₆₇	—	2	4	no	—
185	C₂×C₄×C₆₇	—	4	2	no	—
186	C₂×C₄×C₆₇	—	4	2	no	—
187	C₂×C₄×C₆₇	—	4	2	no	—
188	C₂×C₄×C₆₇	—	4	2	no	—
189	C₂×C₄×C₆₇	—	4	2	no	—
190	C₂×C₄×C₆₇	—	4	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	C₄×D₄	1	2	yes	—
196	C₂×C₄×C₆₇	#121	1	2	yes	—
197	C₂×C₄×C₆₇	#121	1	2	yes	—
198	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
199	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
200	C₂×C₄×C₆₇	#135	1	2	yes	—
201	C₂×C₄×C₆₇	#135	1	2	yes	—
202	C₂×C₄×C₆₇	C₂³×C₄	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	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
208	C₈×C₆₇	—	2	2	no	—
209	C₈×C₆₇	—	2	2	no	—
210	C₈×C₆₇	—	2	2	no	—
211	C₈×C₆₇	C₄×D₄	1	4	yes	—
212	C₂²×C₆₇	—	2	2	no	—
213	C₂²×C₆₇	—	2	2	no	—
214	C₂²×C₆₇	—	2	2	no	—
215	C₂²×C₆₇	—	2	2	no	—
216	C₂²×C₆₇	—	2	2	no	—
217	C₂²×C₆₇	—	2	2	no	—
218	C₂²×C₆₇	C₄×Q₁₆	1	4	yes	complemented
219	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
220	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
221	C₂²×C₆₇	#532	1	1	yes	characteristic
222	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
223	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
224	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
225	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
226	C₄×C₆₇	—	2	2	no	—
227	C₄×C₆₇	—	2	2	no	—
228	C₄×C₆₇	—	2	2	no	—
229	C₄×C₆₇	—	2	2	no	—
230	C₄×C₆₇	—	4	4	no	—
231	C₄×C₆₇	—	4	4	no	—
232	C₄×C₆₇	—	8	2	no	—
233	C₄×C₆₇	—	8	2	no	—
234	C₄×C₆₇	#467	1	4	yes	direct product factor
235	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
236	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
237	C₄×C₆₇	#525	1	1	yes	characteristic
238	C₄×C₆₇	#529	1	1	yes	characteristic
239	C₄×#467	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
240	C₂×C₆₇	—	2	4	no	—
241	C₂×C₆₇	C₄×#121	1	1	yes	characteristic
242	C₂×C₆₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
243	C₂×C₆₇	C₄×#135	1	1	yes	characteristic
244	C₂×C₆₇	C₂×#467	1	1	yes	characteristic
245	C₂×C₆₇	#3050	1	1	yes	characteristic
246	C₂×C₆₇	#3168	1	1	yes	characteristic
247	C₂×C₆₇	#3170	1	1	yes	characteristic
248	C₂×#467	C₂×C₆₇	1	1	yes	characteristic
249	C₂×C₄×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
250	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic
251	C₄×C₂²⋊Q₈	C₂×C₆₇	1	2	yes	—
252	#1969	C₂×C₆₇	1	1	yes	characteristic, complemented
253	#1942	C₂×C₆₇	1	1	yes	characteristic, complemented
254	#1933	C₂×C₆₇	1	2	yes	complemented
255	#1892	C₂×C₆₇	1	2	yes	—
256	#1808	C₂×C₆₇	1	1	yes	characteristic
257	C₄×#107	C₂×C₆₇	1	1	yes	characteristic, complemented
258	C₄×#103	C₂×C₆₇	1	2	yes	complemented
259	C₆₇	C₄×#467	1	1	yes	characteristic, direct product factor, Sylow
260	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
261	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
262	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
263	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
264	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
265	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
266	#467	C₄×C₆₇	1	8	yes	direct product factor
267	C₄×Q₁₆	—	2	4	no	—
268	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
269	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
270	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
271	C₂×#103	C₂²×C₆₇	1	2	yes	—
272	C₂×#103	C₂²×C₆₇	1	2	yes	—
273	C₂×#103	C₂²×C₆₇	1	2	yes	—
274	C₂×C₄×C₈	—	2	4	no	—
275	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
276	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
277	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
278	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
279	#391	C₂²×C₆₇	1	2	yes	—
280	#389	C₂²×C₆₇	1	2	yes	—
281	#387	—	2	2	no	—
282	#385	—	2	2	no	—
283	#380	C₂²×C₆₇	1	2	yes	—
284	C₄×C₄⋊C₄	—	2	2	no	—
285	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
286	C₄×C₂²⋊C₄	—	2	2	no	—
287	#340	C₂²×C₆₇	1	2	yes	—
288	#340	C₂²×C₆₇	1	2	yes	—
289	#340	C₂²×C₆₇	1	2	yes	—
290	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
291	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
292	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
293	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
294	C₂×Q₁₆	—	2	2	no	—
295	C₂×Q₁₆	—	2	2	no	—
296	C₂×Q₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
297	C₂²×C₈	—	2	2	no	—
298	C₂²×C₈	—	2	2	no	—
299	C₂²×C₈	—	2	2	no	—
300	C₂²×C₈	—	2	2	no	—
301	C₂²×C₈	—	2	2	no	—
302	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
303	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
304	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
305	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
306	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
307	C₂²⋊Q₈	—	2	4	no	—
308	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
309	C₄×Q₈	—	2	4	no	—
310	C₄×Q₈	—	2	4	no	—
311	C₂×C₄⋊C₄	—	2	2	no	—
312	C₂×C₄⋊C₄	—	2	2	no	—
313	C₂×C₄⋊C₄	—	2	2	no	—
314	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
315	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
316	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
317	C₂×C₂²⋊C₄	—	2	2	no	—
318	C₂×C₂²⋊C₄	—	2	2	no	—
319	C₂×C₄²	—	2	2	no	—
320	C₂×C₄²	—	2	2	no	—
321	C₂×C₄²	—	2	2	no	—
322	C₂×C₄²	—	2	2	no	—
323	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
324	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
325	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
326	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
327	#107	—	2	2	no	—
328	#107	—	2	2	no	—
329	#107	C₂×C₄×C₆₇	1	4	yes	complemented
330	#103	—	2	4	no	—
331	#103	—	2	4	no	—
332	#103	C₂×C₄×C₆₇	1	8	yes	complemented
333	C₄×C₈	—	2	2	no	—
334	C₄×C₈	—	2	2	no	—
335	C₄×C₈	—	2	2	no	—
336	C₄×C₈	D₄×C₆₇	1	4	yes	—
337	#95	—	2	2	no	—
338	#95	—	2	2	no	—
339	#95	—	2	2	no	—
340	#95	—	2	2	no	—
341	#95	—	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	C₂²×C₄×C₆₇	1	4	yes	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	Q₁₆×C₆₇	1	4	yes	complemented
363	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
364	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
365	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
366	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
367	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
368	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
369	C₂²×C₄	#41×C₆₇	1	2	yes	—
370	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
371	Q₁₆	—	2	8	no	—
372	C₂×C₈	—	2	8	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	8	no	—
380	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
381	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
382	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
383	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
384	C₂×C₈	#41×C₆₇	1	2	yes	—
385	C₂×C₈	#41×C₆₇	1	2	yes	—
386	C₄⋊C₄	—	2	8	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	4	4	no	—
393	C₄⋊C₄	—	4	4	no	—
394	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
395	C₂²⋊C₄	—	2	8	no	—
396	C₂²⋊C₄	—	4	4	no	—
397	C₄²	—	2	2	no	—
398	C₄²	—	2	2	no	—
399	C₄²	—	2	1	no	—
400	C₄²	—	2	1	no	—
401	C₄²	—	4	4	no	—
402	C₄²	—	4	4	no	—
403	C₄²	C₂×D₄×C₆₇	1	2	yes	—
404	C₄²	#41×C₆₇	1	2	yes	—
405	C₂³	—	2	1	no	—
406	C₂³	—	2	1	no	—
407	C₂³	—	2	1	no	—
408	C₂³	—	2	1	no	—
409	C₂³	C₄×D₄×C₆₇	1	2	yes	—
410	C₂³	C₂×Q₁₆×C₆₇	1	2	yes	—
411	C₂³	#135×C₆₇	1	2	yes	—
412	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
413	Q₈	—	2	4	no	—
414	Q₈	—	2	4	no	—
415	Q₈	—	4	2	no	—
416	Q₈	—	4	2	no	—
417	C₂×C₄	—	2	8	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	8	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	4	no	—
432	C₂×C₄	—	2	4	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	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	2	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
444	C₂×C₄	#121×C₆₇	1	2	yes	—
445	C₂×C₄	#121×C₆₇	1	2	yes	—
446	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
447	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
448	C₂×C₄	#135×C₆₇	1	2	yes	—
449	C₂×C₄	#135×C₆₇	1	2	yes	—
450	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
451	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
452	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
453	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
454	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
455	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
456	C₈	—	2	2	no	—
457	C₈	—	2	2	no	—
458	C₈	—	2	2	no	—
459	C₈	C₄×D₄×C₆₇	1	4	yes	—
460	C₂²	—	2	2	no	—
461	C₂²	—	2	2	no	—
462	C₂²	—	2	2	no	—
463	C₂²	—	2	2	no	—
464	C₂²	—	2	2	no	—
465	C₂²	—	2	2	no	—
466	C₂²	C₄×Q₁₆×C₆₇	1	4	yes	complemented
467	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
468	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
469	C₂²	#532×C₆₇	1	1	yes	characteristic
470	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
471	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
472	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
473	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
474	C₄	—	2	2	no	—
475	C₄	—	2	2	no	—
476	C₄	—	2	2	no	—
477	C₄	—	2	2	no	—
478	C₄	—	4	4	no	—
479	C₄	—	4	4	no	—
480	C₄	—	8	2	no	—
481	C₄	—	8	2	no	—
482	C₄	#467×C₆₇	1	4	yes	direct product factor
483	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
484	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
485	C₄	#525×C₆₇	1	1	yes	characteristic
486	C₄	#529×C₆₇	1	1	yes	characteristic
487	C₂	—	2	4	no	—
488	C₂	C₄×#121×C₆₇	1	1	yes	characteristic
489	C₂	C₂×C₄×Q₁₆×C₆₇	1	1	yes	characteristic
490	C₂	C₄×#135×C₆₇	1	1	yes	characteristic
491	C₂	C₂×#467×C₆₇	1	1	yes	characteristic
492	C₂	C₆₇×#3050	1	1	yes	characteristic
493	C₂	C₆₇×#3168	1	1	yes	characteristic
494	C₂	C₆₇×#3170	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₁₆×C₆₇	C₂	semidirect
5. C₆₇×#1969	C₂	semidirect
6. C₆₇×#1942	C₂	semidirect
7. C₆₇×#1933	C₂	semidirect
10. C₄×#107×C₆₇	C₂	semidirect
11. C₄×#103×C₆₇	C₂	semidirect
18. #467×C₆₇	C₄	direct (N × H)
48. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
81. #107×C₆₇	C₂×C₄	semidirect
84. #103×C₆₇	C₂×C₄	semidirect
114. C₂²×C₄×C₆₇	Q₁₆	semidirect
218. C₂²×C₆₇	C₄×Q₁₆	semidirect
234. C₄×C₆₇	#467	direct (N × H)
239. C₄×#467	C₆₇	direct (N × H)
249. C₂×C₄×Q₁₆	C₂×C₆₇	semidirect
252. #1969	C₂×C₆₇	semidirect
253. #1942	C₂×C₆₇	semidirect
254. #1933	C₂×C₆₇	semidirect
257. C₄×#107	C₂×C₆₇	semidirect
258. C₄×#103	C₂×C₆₇	semidirect
259. C₆₇	C₄×#467	direct (N × H)
266. #467	C₄×C₆₇	direct (N × H)
296. C₂×Q₁₆	C₂×C₄×C₆₇	semidirect
329. #107	C₂×C₄×C₆₇	semidirect
332. #103	C₂×C₄×C₆₇	semidirect
362. C₂²×C₄	Q₁₆×C₆₇	semidirect
466. C₂²	C₄×Q₁₆×C₆₇	semidirect
482. C₄	#467×C₆₇	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×#467×C₆₇	2
C₂×C₄×Q₁₆×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₄×C₂²⋊Q₈×C₆₇	2
C₆₇×#1969	2
C₆₇×#1942	2
C₆₇×#1933	2
C₆₇×#1892	2
C₆₇×#1808	2
C₄×#107×C₆₇	2
C₄×#103×C₆₇	2
C₄×#467	67

Subgroups — C4×#467×C67 #30700773

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#467×C₆₇ #30700773