Subgroups — C₆₇×#21062 #30700313

Order 17152.

1126 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#428×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#417×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#416×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2311	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2274	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1928	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1927	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1926	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1925	C₂	1	2	yes	maximal, complemented
10	C₂×#340×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1729	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
19	#428×C₆₇	—	2	4	no	—
20	C₂×#107×C₆₇	C₂²	1	2	yes	—
21	C₂×#106×C₆₇	C₂²	1	2	yes	—
22	#417×C₆₇	C₄	1	8	yes	complemented
23	#416×C₆₇	—	2	4	no	—
24	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#103×C₆₇	C₂²	1	2	yes	—
27	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#102×C₆₇	C₂²	1	2	yes	—
30	#391×C₆₇	C₂²	1	2	yes	—
31	#390×C₆₇	C₂²	1	2	yes	—
32	#386×C₆₇	—	2	2	no	—
33	#380×C₆₇	—	2	1	no	—
34	#380×C₆₇	—	2	1	no	—
35	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
36	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
37	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
38	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
39	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
40	#340×C₆₇	C₂²	1	2	yes	—
41	#340×C₆₇	C₂²	1	2	yes	—
42	#336×C₆₇	C₂²	1	2	yes	—
43	#336×C₆₇	C₂²	1	2	yes	—
44	C₂×#41×C₆₇	—	2	2	no	—
45	C₂×#41×C₆₇	—	2	2	no	—
46	C₂×#41×C₆₇	—	2	2	no	—
47	C₂×#41×C₆₇	—	2	2	no	—
48	C₂×#41×C₆₇	—	2	2	no	—
49	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
50	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×D₄×C₆₇	—	2	1	no	—
52	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
53	C₂³×C₄×C₆₇	—	2	1	no	—
54	C₂³×C₄×C₆₇	—	2	1	no	—
55	C₂³×C₄×C₆₇	—	2	1	no	—
56	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂×M₁₆×C₆₇	—	2	2	no	—
58	C₂×M₁₆×C₆₇	—	2	2	no	—
59	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
60	C₂²×C₈×C₆₇	—	2	2	no	—
61	C₂²×C₈×C₆₇	—	2	2	no	—
62	C₂²×C₈×C₆₇	—	2	2	no	—
63	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
66	#117×C₆₇	—	2	2	no	—
67	#117×C₆₇	—	2	2	no	—
68	#117×C₆₇	—	2	2	no	—
69	#117×C₆₇	D₄	1	4	yes	—
70	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
71	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
72	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
73	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
75	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
76	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
83	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
84	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
86	C₂×C₄²×C₆₇	—	2	1	no	—
87	C₂×C₄²×C₆₇	—	2	2	no	—
88	C₂×C₄²×C₆₇	—	2	2	no	—
89	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
90	#107×C₆₇	—	2	4	no	—
91	#106×C₆₇	—	2	4	no	—
92	#103×C₆₇	—	2	2	no	—
93	#103×C₆₇	—	2	2	no	—
94	#103×C₆₇	C₂×C₄	1	8	yes	complemented
95	#102×C₆₇	—	2	2	no	—
96	#102×C₆₇	—	2	2	no	—
97	#102×C₆₇	C₂×C₄	1	8	yes	complemented
98	#95×C₆₇	—	2	2	no	—
99	#95×C₆₇	—	2	2	no	—
100	#95×C₆₇	—	2	2	no	—
101	#95×C₆₇	—	2	2	no	—
102	C₂⁴×C₆₇	—	2	1	no	—
103	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
104	#41×C₆₇	—	2	8	no	—
105	#41×C₆₇	—	2	8	no	—
106	#41×C₆₇	—	4	4	no	—
107	#41×C₆₇	—	4	4	no	—
108	C₂×Q₈×C₆₇	—	2	1	no	—
109	C₂×Q₈×C₆₇	—	2	1	no	—
110	C₂×Q₈×C₆₇	—	2	1	no	—
111	C₂×Q₈×C₆₇	—	2	1	no	—
112	C₂×Q₈×C₆₇	—	2	1	no	—
113	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
114	C₂×D₄×C₆₇	—	2	4	no	—
115	C₂×D₄×C₆₇	—	2	1	no	—
116	C₂×D₄×C₆₇	—	2	1	no	—
117	C₂×D₄×C₆₇	—	2	1	no	—
118	C₂×D₄×C₆₇	—	2	1	no	—
119	C₂×D₄×C₆₇	—	2	1	no	—
120	C₂×D₄×C₆₇	—	4	2	no	—
121	C₂×D₄×C₆₇	—	4	2	no	—
122	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
123	C₂²×C₄×C₆₇	—	2	1	no	—
124	C₂²×C₄×C₆₇	—	2	1	no	—
125	C₂²×C₄×C₆₇	—	2	2	no	—
126	C₂²×C₄×C₆₇	—	2	1	no	—
127	C₂²×C₄×C₆₇	—	2	4	no	—
128	C₂²×C₄×C₆₇	—	2	4	no	—
129	C₂²×C₄×C₆₇	—	2	2	no	—
130	C₂²×C₄×C₆₇	—	4	2	no	—
131	C₂²×C₄×C₆₇	—	4	2	no	—
132	C₂²×C₄×C₆₇	—	4	2	no	—
133	C₂²×C₄×C₆₇	—	8	1	no	—
134	C₂²×C₄×C₆₇	—	8	1	no	—
135	C₂²×C₄×C₆₇	—	8	1	no	—
136	C₂²×C₄×C₆₇	—	8	1	no	—
137	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
138	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
139	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
140	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₆₇	#41	1	2	yes	—
143	C₂²×C₄×C₆₇	#41	1	2	yes	—
144	C₂²×C₄×C₆₇	#41	1	2	yes	—
145	C₂²×C₄×C₆₇	#41	1	2	yes	—
146	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
147	M₁₆×C₆₇	—	2	8	no	—
148	C₂×C₈×C₆₇	—	2	2	no	—
149	C₂×C₈×C₆₇	—	2	2	no	—
150	C₂×C₈×C₆₇	—	2	1	no	—
151	C₂×C₈×C₆₇	—	2	1	no	—
152	C₂×C₈×C₆₇	—	2	1	no	—
153	C₂×C₈×C₆₇	—	2	1	no	—
154	C₂×C₈×C₆₇	—	2	1	no	—
155	C₂×C₈×C₆₇	—	2	1	no	—
156	C₂×C₈×C₆₇	—	4	4	no	—
157	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
158	C₄⋊C₄×C₆₇	—	2	4	no	—
159	C₄⋊C₄×C₆₇	—	2	2	no	—
160	C₄⋊C₄×C₆₇	—	2	2	no	—
161	C₄⋊C₄×C₆₇	—	2	2	no	—
162	C₄⋊C₄×C₆₇	—	2	1	no	—
163	C₄⋊C₄×C₆₇	—	2	1	no	—
164	C₄⋊C₄×C₆₇	—	2	2	no	—
165	C₄⋊C₄×C₆₇	—	2	2	no	—
166	C₄⋊C₄×C₆₇	—	4	2	no	—
167	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
168	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
169	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
170	C₂²⋊C₄×C₆₇	—	2	4	no	—
171	C₂²⋊C₄×C₆₇	—	2	4	no	—
172	C₂²⋊C₄×C₆₇	—	8	2	no	—
173	C₂²⋊C₄×C₆₇	—	8	2	no	—
174	C₄²×C₆₇	—	2	4	no	—
175	C₄²×C₆₇	—	2	2	no	—
176	C₄²×C₆₇	—	2	2	no	—
177	C₄²×C₆₇	—	8	2	no	—
178	C₄²×C₆₇	—	8	2	no	—
179	C₂³×C₆₇	—	2	1	no	—
180	C₂³×C₆₇	—	2	1	no	—
181	C₂³×C₆₇	—	2	1	no	—
182	C₂³×C₆₇	—	2	1	no	—
183	C₂³×C₆₇	—	2	2	no	—
184	C₂³×C₆₇	—	4	1	no	—
185	C₂³×C₆₇	—	4	1	no	—
186	C₂³×C₆₇	—	4	1	no	—
187	C₂³×C₆₇	—	4	1	no	—
188	C₂³×C₆₇	—	4	1	no	—
189	C₂³×C₆₇	—	4	1	no	—
190	C₂³×C₆₇	C₄×D₄	1	2	yes	—
191	C₂³×C₆₇	#121	1	2	yes	—
192	C₂³×C₆₇	#123	1	2	yes	—
193	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
194	Q₈×C₆₇	—	2	2	no	—
195	Q₈×C₆₇	—	2	2	no	—
196	Q₈×C₆₇	—	4	1	no	—
197	Q₈×C₆₇	—	4	1	no	—
198	D₄×C₆₇	—	2	2	no	—
199	D₄×C₆₇	—	2	2	no	—
200	D₄×C₆₇	—	4	8	no	—
201	D₄×C₆₇	—	4	1	no	—
202	D₄×C₆₇	—	4	1	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	2	no	—
205	C₂×C₄×C₆₇	—	2	2	no	—
206	C₂×C₄×C₆₇	—	2	2	no	—
207	C₂×C₄×C₆₇	—	2	2	no	—
208	C₂×C₄×C₆₇	—	2	2	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	2	no	—
211	C₂×C₄×C₆₇	—	2	2	no	—
212	C₂×C₄×C₆₇	—	2	2	no	—
213	C₂×C₄×C₆₇	—	4	1	no	—
214	C₂×C₄×C₆₇	—	4	1	no	—
215	C₂×C₄×C₆₇	—	4	1	no	—
216	C₂×C₄×C₆₇	—	4	2	no	—
217	C₂×C₄×C₆₇	—	4	1	no	—
218	C₂×C₄×C₆₇	—	4	2	no	—
219	C₂×C₄×C₆₇	—	4	1	no	—
220	C₂×C₄×C₆₇	—	4	1	no	—
221	C₂×C₄×C₆₇	—	4	8	no	—
222	C₂×C₄×C₆₇	—	4	2	no	—
223	C₂×C₄×C₆₇	—	4	2	no	—
224	C₂×C₄×C₆₇	—	4	2	no	—
225	C₂×C₄×C₆₇	—	8	4	no	—
226	C₂×C₄×C₆₇	—	8	4	no	—
227	C₂×C₄×C₆₇	#117	1	4	yes	—
228	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
229	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
230	C₂×C₄×C₆₇	#121	1	2	yes	—
231	C₂×C₄×C₆₇	#123	1	2	yes	—
232	C₂×C₄×C₆₇	#124	1	4	yes	—
233	C₂×C₄×C₆₇	#126	1	4	yes	—
234	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
235	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
237	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
238	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
239	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
240	C₈×C₆₇	—	2	4	no	—
241	C₈×C₆₇	—	4	1	no	—
242	C₈×C₆₇	—	4	1	no	—
243	C₂²×C₆₇	—	2	2	no	—
244	C₂²×C₆₇	—	2	2	no	—
245	C₂²×C₆₇	—	2	2	no	—
246	C₂²×C₆₇	—	2	2	no	—
247	C₂²×C₆₇	—	2	2	no	—
248	C₂²×C₆₇	—	2	2	no	—
249	C₂²×C₆₇	—	4	2	no	—
250	C₂²×C₆₇	—	4	2	no	—
251	C₂²×C₆₇	—	4	2	no	—
252	C₂²×C₆₇	—	8	1	no	—
253	C₂²×C₆₇	—	8	1	no	—
254	C₂²×C₆₇	—	8	1	no	—
255	C₂²×C₆₇	—	8	1	no	—
256	C₂²×C₆₇	#388	1	4	yes	—
257	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
258	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
259	C₂²×C₆₇	C₂×#123	1	1	yes	characteristic
260	C₂²×C₆₇	#576	1	1	yes	characteristic
261	C₂²×C₆₇	#577	1	1	yes	characteristic
262	C₂²×C₆₇	#577	1	1	yes	characteristic
263	C₂²×C₆₇	#578	1	1	yes	characteristic
264	C₄×C₆₇	—	4	2	no	—
265	C₄×C₆₇	—	4	2	no	—
266	C₄×C₆₇	—	4	4	no	—
267	C₄×C₆₇	—	4	2	no	—
268	C₄×C₆₇	—	4	2	no	—
269	C₄×C₆₇	—	4	2	no	—
270	C₄×C₆₇	—	8	2	no	—
271	C₄×C₆₇	#388	1	4	yes	—
272	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
273	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
274	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
275	C₄×C₆₇	C₂×#126	1	1	yes	characteristic
276	#21062	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
277	C₂×C₆₇	—	2	4	no	—
278	C₂×C₆₇	—	8	2	no	—
279	C₂×C₆₇	C₂×#388	1	1	yes	characteristic
280	C₂×C₆₇	#2946	1	1	yes	characteristic
281	C₂×C₆₇	#2947	1	1	yes	characteristic
282	C₂×C₆₇	#3045	1	1	yes	characteristic
283	C₂×C₆₇	#3046	1	1	yes	characteristic
284	C₂×C₆₇	#3098	1	1	yes	characteristic
285	C₂×C₆₇	#3099	1	1	yes	characteristic
286	C₂×#428	C₂×C₆₇	1	1	yes	characteristic, complemented
287	C₂×#417	C₂×C₆₇	1	1	yes	characteristic
288	C₂×#416	C₂×C₆₇	1	1	yes	characteristic
289	#2311	C₂×C₆₇	1	1	yes	characteristic
290	#2274	C₂×C₆₇	1	1	yes	characteristic, complemented
291	#1928	C₂×C₆₇	1	2	yes	complemented
292	#1927	C₂×C₆₇	1	2	yes	complemented
293	#1926	C₂×C₆₇	1	2	yes	complemented
294	#1925	C₂×C₆₇	1	2	yes	complemented
295	C₂×#340	C₂×C₆₇	1	1	yes	characteristic, complemented
296	#1729	C₂×C₆₇	1	1	yes	characteristic, complemented
297	C₆₇	#21062	1	1	yes	characteristic, direct product factor, Sylow
298	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
299	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
300	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
301	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
302	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
303	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
304	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
305	#428	—	2	4	no	—
306	C₂×#107	C₂²×C₆₇	1	2	yes	—
307	C₂×#106	C₂²×C₆₇	1	2	yes	—
308	#417	C₄×C₆₇	1	8	yes	complemented
309	#416	—	2	4	no	—
310	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
311	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
312	C₂×#103	C₂²×C₆₇	1	2	yes	—
313	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
314	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
315	C₂×#102	C₂²×C₆₇	1	2	yes	—
316	#391	C₂²×C₆₇	1	2	yes	—
317	#390	C₂²×C₆₇	1	2	yes	—
318	#386	—	2	2	no	—
319	#380	—	2	1	no	—
320	#380	—	2	1	no	—
321	C₄×C₄⋊C₄	—	2	1	no	—
322	C₄×C₄⋊C₄	—	2	1	no	—
323	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
324	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
325	C₄×C₂²⋊C₄	—	2	2	no	—
326	#340	C₂²×C₆₇	1	2	yes	—
327	#340	C₂²×C₆₇	1	2	yes	—
328	#336	C₂²×C₆₇	1	2	yes	—
329	#336	C₂²×C₆₇	1	2	yes	—
330	C₂×#41	—	2	2	no	—
331	C₂×#41	—	2	2	no	—
332	C₂×#41	—	2	2	no	—
333	C₂×#41	—	2	2	no	—
334	C₂×#41	—	2	2	no	—
335	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
336	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
337	C₂²×D₄	—	2	1	no	—
338	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
339	C₂³×C₄	—	2	1	no	—
340	C₂³×C₄	—	2	1	no	—
341	C₂³×C₄	—	2	1	no	—
342	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
343	C₂×M₁₆	—	2	2	no	—
344	C₂×M₁₆	—	2	2	no	—
345	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
346	C₂²×C₈	—	2	2	no	—
347	C₂²×C₈	—	2	2	no	—
348	C₂²×C₈	—	2	2	no	—
349	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
350	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
351	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
352	#117	—	2	2	no	—
353	#117	—	2	2	no	—
354	#117	—	2	2	no	—
355	#117	D₄×C₆₇	1	4	yes	—
356	C₂×C₄⋊C₄	—	2	1	no	—
357	C₂×C₄⋊C₄	—	2	2	no	—
358	C₂×C₄⋊C₄	—	2	2	no	—
359	C₂×C₄⋊C₄	—	2	2	no	—
360	C₂×C₄⋊C₄	—	2	1	no	—
361	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
362	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
363	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
364	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
365	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
366	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
367	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
368	C₂×C₂²⋊C₄	—	2	1	no	—
369	C₂×C₂²⋊C₄	—	2	2	no	—
370	C₂×C₂²⋊C₄	—	2	2	no	—
371	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
372	C₂×C₄²	—	2	1	no	—
373	C₂×C₄²	—	2	2	no	—
374	C₂×C₄²	—	2	2	no	—
375	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
376	#107	—	2	4	no	—
377	#106	—	2	4	no	—
378	#103	—	2	2	no	—
379	#103	—	2	2	no	—
380	#103	C₂×C₄×C₆₇	1	8	yes	complemented
381	#102	—	2	2	no	—
382	#102	—	2	2	no	—
383	#102	C₂×C₄×C₆₇	1	8	yes	complemented
384	#95	—	2	2	no	—
385	#95	—	2	2	no	—
386	#95	—	2	2	no	—
387	#95	—	2	2	no	—
388	C₂⁴	—	2	1	no	—
389	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
390	#41	—	2	8	no	—
391	#41	—	2	8	no	—
392	#41	—	4	4	no	—
393	#41	—	4	4	no	—
394	C₂×Q₈	—	2	1	no	—
395	C₂×Q₈	—	2	1	no	—
396	C₂×Q₈	—	2	1	no	—
397	C₂×Q₈	—	2	1	no	—
398	C₂×Q₈	—	2	1	no	—
399	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
400	C₂×D₄	—	2	4	no	—
401	C₂×D₄	—	2	1	no	—
402	C₂×D₄	—	2	1	no	—
403	C₂×D₄	—	2	1	no	—
404	C₂×D₄	—	2	1	no	—
405	C₂×D₄	—	2	1	no	—
406	C₂×D₄	—	4	2	no	—
407	C₂×D₄	—	4	2	no	—
408	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
409	C₂²×C₄	—	2	1	no	—
410	C₂²×C₄	—	2	1	no	—
411	C₂²×C₄	—	2	1	no	—
412	C₂²×C₄	—	2	2	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	4	2	no	—
417	C₂²×C₄	—	4	2	no	—
418	C₂²×C₄	—	4	2	no	—
419	C₂²×C₄	—	8	1	no	—
420	C₂²×C₄	—	8	1	no	—
421	C₂²×C₄	—	8	1	no	—
422	C₂²×C₄	—	8	1	no	—
423	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
424	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
425	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
426	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
427	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
428	C₂²×C₄	#41×C₆₇	1	2	yes	—
429	C₂²×C₄	#41×C₆₇	1	2	yes	—
430	C₂²×C₄	#41×C₆₇	1	2	yes	—
431	C₂²×C₄	#41×C₆₇	1	2	yes	—
432	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
433	M₁₆	—	2	8	no	—
434	C₂×C₈	—	2	2	no	—
435	C₂×C₈	—	2	2	no	—
436	C₂×C₈	—	2	1	no	—
437	C₂×C₈	—	2	1	no	—
438	C₂×C₈	—	2	1	no	—
439	C₂×C₈	—	2	1	no	—
440	C₂×C₈	—	2	1	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	4	4	no	—
443	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
444	C₄⋊C₄	—	2	4	no	—
445	C₄⋊C₄	—	2	2	no	—
446	C₄⋊C₄	—	2	2	no	—
447	C₄⋊C₄	—	2	2	no	—
448	C₄⋊C₄	—	2	1	no	—
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	—	2	2	no	—
451	C₄⋊C₄	—	2	2	no	—
452	C₄⋊C₄	—	4	2	no	—
453	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
454	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
455	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
456	C₂²⋊C₄	—	2	4	no	—
457	C₂²⋊C₄	—	2	4	no	—
458	C₂²⋊C₄	—	8	2	no	—
459	C₂²⋊C₄	—	8	2	no	—
460	C₄²	—	2	4	no	—
461	C₄²	—	2	2	no	—
462	C₄²	—	2	2	no	—
463	C₄²	—	8	2	no	—
464	C₄²	—	8	2	no	—
465	C₂³	—	2	1	no	—
466	C₂³	—	2	1	no	—
467	C₂³	—	2	1	no	—
468	C₂³	—	2	1	no	—
469	C₂³	—	2	2	no	—
470	C₂³	—	4	1	no	—
471	C₂³	—	4	1	no	—
472	C₂³	—	4	1	no	—
473	C₂³	—	4	1	no	—
474	C₂³	—	4	1	no	—
475	C₂³	—	4	1	no	—
476	C₂³	C₄×D₄×C₆₇	1	2	yes	—
477	C₂³	#121×C₆₇	1	2	yes	—
478	C₂³	#123×C₆₇	1	2	yes	—
479	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
480	Q₈	—	2	2	no	—
481	Q₈	—	2	2	no	—
482	Q₈	—	4	1	no	—
483	Q₈	—	4	1	no	—
484	D₄	—	2	2	no	—
485	D₄	—	2	2	no	—
486	D₄	—	4	8	no	—
487	D₄	—	4	1	no	—
488	D₄	—	4	1	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	4	1	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	—	4	1	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	—	4	1	no	—
506	C₂×C₄	—	4	1	no	—
507	C₂×C₄	—	4	8	no	—
508	C₂×C₄	—	4	2	no	—
509	C₂×C₄	—	4	2	no	—
510	C₂×C₄	—	4	2	no	—
511	C₂×C₄	—	8	4	no	—
512	C₂×C₄	—	8	4	no	—
513	C₂×C₄	#117×C₆₇	1	4	yes	—
514	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
515	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
516	C₂×C₄	#121×C₆₇	1	2	yes	—
517	C₂×C₄	#123×C₆₇	1	2	yes	—
518	C₂×C₄	#124×C₆₇	1	4	yes	—
519	C₂×C₄	#126×C₆₇	1	4	yes	—
520	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
521	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
522	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
523	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
524	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
525	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
526	C₈	—	2	4	no	—
527	C₈	—	4	1	no	—
528	C₈	—	4	1	no	—
529	C₂²	—	2	2	no	—
530	C₂²	—	2	2	no	—
531	C₂²	—	2	2	no	—
532	C₂²	—	2	2	no	—
533	C₂²	—	2	2	no	—
534	C₂²	—	2	2	no	—
535	C₂²	—	4	2	no	—
536	C₂²	—	4	2	no	—
537	C₂²	—	4	2	no	—
538	C₂²	—	8	1	no	—
539	C₂²	—	8	1	no	—
540	C₂²	—	8	1	no	—
541	C₂²	—	8	1	no	—
542	C₂²	#388×C₆₇	1	4	yes	—
543	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
544	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
545	C₂²	C₂×#123×C₆₇	1	1	yes	characteristic
546	C₂²	#576×C₆₇	1	1	yes	characteristic
547	C₂²	#577×C₆₇	1	1	yes	characteristic
548	C₂²	#577×C₆₇	1	1	yes	characteristic
549	C₂²	#578×C₆₇	1	1	yes	characteristic
550	C₄	—	4	2	no	—
551	C₄	—	4	2	no	—
552	C₄	—	4	2	no	—
553	C₄	—	4	2	no	—
554	C₄	—	4	2	no	—
555	C₄	—	4	4	no	—
556	C₄	—	8	2	no	—
557	C₄	#388×C₆₇	1	4	yes	—
558	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
559	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
560	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
561	C₄	C₂×#126×C₆₇	1	1	yes	characteristic
562	C₂	—	2	4	no	—
563	C₂	—	8	2	no	—
564	C₂	C₂×#388×C₆₇	1	1	yes	characteristic
565	C₂	C₆₇×#2946	1	1	yes	characteristic
566	C₂	C₆₇×#2947	1	1	yes	characteristic
567	C₂	C₆₇×#3045	1	1	yes	characteristic
568	C₂	C₆₇×#3046	1	1	yes	characteristic
569	C₂	C₆₇×#3098	1	1	yes	characteristic
570	C₂	C₆₇×#3099	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#428×C₆₇	C₂	semidirect
5. C₆₇×#2274	C₂	semidirect
6. C₆₇×#1928	C₂	semidirect
7. C₆₇×#1927	C₂	semidirect
8. C₆₇×#1926	C₂	semidirect
9. C₆₇×#1925	C₂	semidirect
10. C₂×#340×C₆₇	C₂	semidirect
11. C₆₇×#1729	C₂	semidirect
22. #417×C₆₇	C₄	semidirect
94. #103×C₆₇	C₂×C₄	semidirect
97. #102×C₆₇	C₂×C₄	semidirect
276. #21062	C₆₇	direct (N × H)
286. C₂×#428	C₂×C₆₇	semidirect
290. #2274	C₂×C₆₇	semidirect
291. #1928	C₂×C₆₇	semidirect
292. #1927	C₂×C₆₇	semidirect
293. #1926	C₂×C₆₇	semidirect
294. #1925	C₂×C₆₇	semidirect
295. C₂×#340	C₂×C₆₇	semidirect
296. #1729	C₂×C₆₇	semidirect
297. C₆₇	#21062	direct (N × H)
308. #417	C₄×C₆₇	semidirect
380. #103	C₂×C₄×C₆₇	semidirect
383. #102	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#428×C₆₇	2
C₂×#417×C₆₇	2
C₂×#416×C₆₇	2
C₆₇×#2311	2
C₆₇×#2274	2
C₆₇×#1928	2
C₆₇×#1927	2
C₆₇×#1926	2
C₆₇×#1925	2
C₂×#340×C₆₇	2
C₆₇×#1729	2
#21062	67

Subgroups — C67×#21062 #30700313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21062 #30700313