Subgroups — C₆₇×#21237 #30700488

Order 17152.

1566 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#137×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#428×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#417×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#419×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2297	C₂	1	1	yes	maximal, characteristic
7	C₂×#391×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1973	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1971	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1969	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1968	C₂	1	2	yes	maximal, complemented
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#137×C₆₇	—	2	4	no	—
15	C₂×#137×C₆₇	—	2	4	no	—
16	C₂×#137×C₆₇	C₄	1	8	yes	complemented
17	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
18	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
19	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	#428×C₆₇	—	2	4	no	—
25	C₂×#107×C₆₇	C₂²	1	2	yes	—
26	C₂×#106×C₆₇	C₂²	1	2	yes	—
27	#419×C₆₇	—	2	4	no	—
28	#417×C₆₇	—	2	4	no	—
29	C₂×#103×C₆₇	C₂²	1	2	yes	—
30	C₂×#103×C₆₇	C₂²	1	2	yes	—
31	C₂×#103×C₆₇	C₂²	1	2	yes	—
32	C₂×#102×C₆₇	C₂²	1	2	yes	—
33	C₄×M₁₆×C₆₇	—	2	4	no	—
34	C₂×#97×C₆₇	C₂²	1	2	yes	—
35	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
36	#391×C₆₇	—	2	2	no	—
37	#391×C₆₇	C₂²	1	2	yes	—
38	#391×C₆₇	C₂²	1	2	yes	—
39	#391×C₆₇	C₂²	1	1	yes	characteristic
40	#391×C₆₇	C₂²	1	1	yes	characteristic
41	#390×C₆₇	C₂²	1	1	yes	characteristic
42	#390×C₆₇	C₂²	1	1	yes	characteristic
43	#386×C₆₇	—	2	2	no	—
44	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
45	C₂×#95×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	—	2	2	no	—
50	C₂×#41×C₆₇	—	2	2	no	—
51	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
52	C₂²×Q₈×C₆₇	—	2	1	no	—
53	C₂²×Q₈×C₆₇	—	2	2	no	—
54	C₂²×Q₈×C₆₇	—	2	2	no	—
55	C₂²×Q₈×C₆₇	—	2	2	no	—
56	C₂²×Q₈×C₆₇	—	2	2	no	—
57	C₂²×Q₈×C₆₇	—	2	2	no	—
58	C₂²×Q₈×C₆₇	C₂×C₄	1	4	yes	—
59	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
60	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
61	C₂²×D₄×C₆₇	—	2	1	no	—
62	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₆₇	—	2	1	no	—
64	C₂³×C₄×C₆₇	—	2	1	no	—
65	C₂³×C₄×C₆₇	—	2	1	no	—
66	C₂³×C₄×C₆₇	D₄	1	2	yes	—
67	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
68	#137×C₆₇	—	2	32	no	—
69	C₂×Q₁₆×C₆₇	—	2	4	no	—
70	C₂×Q₁₆×C₆₇	—	2	4	no	—
71	C₂×Q₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
72	C₂×SD₁₆×C₆₇	—	2	4	no	—
73	C₂×SD₁₆×C₆₇	—	2	4	no	—
74	C₂×SD₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
75	C₂×M₁₆×C₆₇	—	2	2	no	—
76	C₂×M₁₆×C₆₇	—	2	2	no	—
77	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
78	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
79	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
80	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
81	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
82	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
83	#117×C₆₇	—	2	2	no	—
84	#117×C₆₇	—	2	2	no	—
85	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
86	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
87	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
91	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
92	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
93	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
94	C₂×C₄²×C₆₇	—	2	2	no	—
95	C₂×C₄²×C₆₇	—	2	2	no	—
96	C₂×C₄²×C₆₇	—	2	1	no	—
97	C₂×C₄²×C₆₇	D₄	1	2	yes	—
98	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
100	#107×C₆₇	—	2	4	no	—
101	#106×C₆₇	—	2	4	no	—
102	#103×C₆₇	—	2	4	no	—
103	#103×C₆₇	—	2	4	no	—
104	#103×C₆₇	—	2	4	no	—
105	#102×C₆₇	—	2	4	no	—
106	#97×C₆₇	—	2	4	no	—
107	C₄×C₈×C₆₇	—	2	4	no	—
108	#95×C₆₇	—	2	4	no	—
109	#95×C₆₇	—	2	4	no	—
110	#95×C₆₇	—	2	2	no	—
111	#95×C₆₇	—	2	2	no	—
112	C₂⁴×C₆₇	—	2	1	no	—
113	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
114	#41×C₆₇	—	2	8	no	—
115	#41×C₆₇	—	2	8	no	—
116	#41×C₆₇	—	4	4	no	—
117	#41×C₆₇	—	4	4	no	—
118	C₂×Q₈×C₆₇	—	2	4	no	—
119	C₂×Q₈×C₆₇	—	2	2	no	—
120	C₂×Q₈×C₆₇	—	2	1	no	—
121	C₂×Q₈×C₆₇	—	2	8	no	—
122	C₂×Q₈×C₆₇	—	2	8	no	—
123	C₂×Q₈×C₆₇	—	2	2	no	—
124	C₂×Q₈×C₆₇	—	2	1	no	—
125	C₂×Q₈×C₆₇	—	2	2	no	—
126	C₂×Q₈×C₆₇	—	2	2	no	—
127	C₂×Q₈×C₆₇	—	2	1	no	—
128	C₂×Q₈×C₆₇	—	2	1	no	—
129	C₂×Q₈×C₆₇	—	2	2	no	—
130	C₂×Q₈×C₆₇	—	2	1	no	—
131	C₂×Q₈×C₆₇	—	4	2	no	—
132	C₂×Q₈×C₆₇	—	4	4	no	—
133	C₂×Q₈×C₆₇	—	4	4	no	—
134	C₂×Q₈×C₆₇	—	4	2	no	—
135	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
136	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
137	C₂×D₄×C₆₇	—	2	4	no	—
138	C₂×D₄×C₆₇	—	2	1	no	—
139	C₂×D₄×C₆₇	—	2	1	no	—
140	C₂×D₄×C₆₇	—	2	1	no	—
141	C₂×D₄×C₆₇	—	2	1	no	—
142	C₂×D₄×C₆₇	—	2	1	no	—
143	C₂×D₄×C₆₇	—	4	2	no	—
144	C₂×D₄×C₆₇	—	4	2	no	—
145	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
146	C₂²×C₄×C₆₇	—	2	4	no	—
147	C₂²×C₄×C₆₇	—	2	4	no	—
148	C₂²×C₄×C₆₇	—	2	1	no	—
149	C₂²×C₄×C₆₇	—	2	2	no	—
150	C₂²×C₄×C₆₇	—	2	4	no	—
151	C₂²×C₄×C₆₇	—	2	4	no	—
152	C₂²×C₄×C₆₇	—	2	1	no	—
153	C₂²×C₄×C₆₇	—	2	1	no	—
154	C₂²×C₄×C₆₇	—	2	2	no	—
155	C₂²×C₄×C₆₇	—	4	2	no	—
156	C₂²×C₄×C₆₇	—	4	2	no	—
157	C₂²×C₄×C₆₇	—	4	2	no	—
158	C₂²×C₄×C₆₇	—	4	2	no	—
159	C₂²×C₄×C₆₇	—	4	2	no	—
160	C₂²×C₄×C₆₇	—	4	4	no	—
161	C₂²×C₄×C₆₇	—	4	2	no	—
162	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
163	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
164	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
169	C₂²×C₄×C₆₇	#41	1	2	yes	—
170	C₂²×C₄×C₆₇	#41	1	2	yes	—
171	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
172	Q₁₆×C₆₇	—	2	16	no	—
173	SD₁₆×C₆₇	—	2	16	no	—
174	M₁₆×C₆₇	—	2	8	no	—
175	M₁₆×C₆₇	C₂²⋊C₄	1	16	yes	complemented
176	C₂×C₈×C₆₇	—	2	2	no	—
177	C₂×C₈×C₆₇	—	2	2	no	—
178	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
179	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
180	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
181	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
182	C₄⋊C₄×C₆₇	—	2	2	no	—
183	C₄⋊C₄×C₆₇	—	2	1	no	—
184	C₄⋊C₄×C₆₇	—	2	1	no	—
185	C₄⋊C₄×C₆₇	—	2	1	no	—
186	C₄⋊C₄×C₆₇	—	2	2	no	—
187	C₄⋊C₄×C₆₇	—	2	1	no	—
188	C₄⋊C₄×C₆₇	—	4	2	no	—
189	C₂²⋊C₄×C₆₇	—	4	4	no	—
190	C₂²⋊C₄×C₆₇	—	4	2	no	—
191	C₂²⋊C₄×C₆₇	—	4	4	no	—
192	C₄²×C₆₇	—	2	4	no	—
193	C₄²×C₆₇	—	2	1	no	—
194	C₄²×C₆₇	—	2	1	no	—
195	C₄²×C₆₇	—	2	1	no	—
196	C₄²×C₆₇	—	2	1	no	—
197	C₄²×C₆₇	—	4	2	no	—
198	C₂³×C₆₇	—	2	2	no	—
199	C₂³×C₆₇	—	2	2	no	—
200	C₂³×C₆₇	—	2	2	no	—
201	C₂³×C₆₇	—	4	1	no	—
202	C₂³×C₆₇	—	4	1	no	—
203	C₂³×C₆₇	—	4	2	no	—
204	C₂³×C₆₇	—	4	2	no	—
205	C₂³×C₆₇	C₄×D₄	1	2	yes	—
206	C₂³×C₆₇	#121	1	2	yes	—
207	C₂³×C₆₇	#127	1	2	yes	—
208	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
209	Q₈×C₆₇	—	2	4	no	—
210	Q₈×C₆₇	—	2	2	no	—
211	Q₈×C₆₇	—	2	4	no	—
212	Q₈×C₆₇	—	2	2	no	—
213	Q₈×C₆₇	—	4	2	no	—
214	Q₈×C₆₇	—	4	1	no	—
215	Q₈×C₆₇	—	4	2	no	—
216	Q₈×C₆₇	—	4	1	no	—
217	Q₈×C₆₇	—	4	8	no	—
218	D₄×C₆₇	—	2	2	no	—
219	D₄×C₆₇	—	2	2	no	—
220	D₄×C₆₇	—	4	8	no	—
221	D₄×C₆₇	—	4	1	no	—
222	D₄×C₆₇	—	4	1	no	—
223	C₂×C₄×C₆₇	—	2	2	no	—
224	C₂×C₄×C₆₇	—	2	2	no	—
225	C₂×C₄×C₆₇	—	2	2	no	—
226	C₂×C₄×C₆₇	—	2	4	no	—
227	C₂×C₄×C₆₇	—	2	2	no	—
228	C₂×C₄×C₆₇	—	2	2	no	—
229	C₂×C₄×C₆₇	—	2	2	no	—
230	C₂×C₄×C₆₇	—	4	8	no	—
231	C₂×C₄×C₆₇	—	4	1	no	—
232	C₂×C₄×C₆₇	—	4	2	no	—
233	C₂×C₄×C₆₇	—	4	1	no	—
234	C₂×C₄×C₆₇	—	4	2	no	—
235	C₂×C₄×C₆₇	—	4	8	no	—
236	C₂×C₄×C₆₇	—	4	8	no	—
237	C₂×C₄×C₆₇	—	4	1	no	—
238	C₂×C₄×C₆₇	—	4	1	no	—
239	C₂×C₄×C₆₇	—	4	2	no	—
240	C₂×C₄×C₆₇	—	4	2	no	—
241	C₂×C₄×C₆₇	—	4	2	no	—
242	C₂×C₄×C₆₇	—	4	4	no	—
243	C₂×C₄×C₆₇	—	8	4	no	—
244	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
245	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
246	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
247	C₂×C₄×C₆₇	#121	1	4	yes	—
248	C₂×C₄×C₆₇	#121	1	2	yes	—
249	C₂×C₄×C₆₇	#124	1	4	yes	—
250	C₂×C₄×C₆₇	#127	1	2	yes	—
251	C₂×C₄×C₆₇	#137	1	4	yes	complemented
252	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
253	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
254	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
255	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
256	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
257	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
258	C₈×C₆₇	—	2	4	no	—
259	C₈×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
260	C₂²×C₆₇	—	2	2	no	—
261	C₂²×C₆₇	—	2	2	no	—
262	C₂²×C₆₇	—	2	4	no	—
263	C₂²×C₆₇	—	2	4	no	—
264	C₂²×C₆₇	—	4	4	no	—
265	C₂²×C₆₇	—	4	2	no	—
266	C₂²×C₆₇	—	8	1	no	—
267	C₂²×C₆₇	—	8	1	no	—
268	C₂²×C₆₇	—	8	2	no	—
269	C₂²×C₆₇	#390	1	4	yes	—
270	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
271	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
272	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
273	C₂²×C₆₇	C₂×#137	1	2	yes	—
274	C₂²×C₆₇	#575	1	2	yes	—
275	C₄×C₆₇	—	2	4	no	—
276	C₄×C₆₇	—	4	2	no	—
277	C₄×C₆₇	—	4	2	no	—
278	C₄×C₆₇	—	4	4	no	—
279	C₄×C₆₇	—	8	1	no	—
280	C₄×C₆₇	—	8	1	no	—
281	C₄×C₆₇	—	8	2	no	—
282	C₄×C₆₇	#390	1	4	yes	—
283	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
284	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
285	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
286	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
287	#21237	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
288	C₂×C₆₇	—	2	4	no	—
289	C₂×C₆₇	—	8	2	no	—
290	C₂×C₆₇	C₂×#390	1	1	yes	characteristic
291	C₂×C₆₇	C₄×#137	1	2	yes	—
292	C₂×C₆₇	#3056	1	2	yes	—
293	C₂×C₆₇	#3152	1	1	yes	characteristic
294	C₂×C₆₇	#3153	1	1	yes	characteristic
295	C₂²×#137	C₂×C₆₇	1	1	yes	characteristic
296	C₂×#428	C₂×C₆₇	1	1	yes	characteristic, complemented
297	C₂×#417	C₂×C₆₇	1	1	yes	characteristic
298	C₂×#419	C₂×C₆₇	1	1	yes	characteristic, complemented
299	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
300	#2297	C₂×C₆₇	1	1	yes	characteristic
301	C₂×#391	C₂×C₆₇	1	1	yes	characteristic, complemented
302	#1973	C₂×C₆₇	1	2	yes	complemented
303	#1971	C₂×C₆₇	1	2	yes	complemented
304	#1969	C₂×C₆₇	1	2	yes	complemented
305	#1968	C₂×C₆₇	1	2	yes	complemented
306	C₆₇	#21237	1	1	yes	characteristic, direct product factor, Sylow
307	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
308	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
309	C₂×#137	—	2	4	no	—
310	C₂×#137	—	2	4	no	—
311	C₂×#137	C₄×C₆₇	1	8	yes	complemented
312	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
313	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
314	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
315	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
316	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
317	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
318	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
319	#428	—	2	4	no	—
320	C₂×#107	C₂²×C₆₇	1	2	yes	—
321	C₂×#106	C₂²×C₆₇	1	2	yes	—
322	#419	—	2	4	no	—
323	#417	—	2	4	no	—
324	C₂×#103	C₂²×C₆₇	1	2	yes	—
325	C₂×#103	C₂²×C₆₇	1	2	yes	—
326	C₂×#103	C₂²×C₆₇	1	2	yes	—
327	C₂×#102	C₂²×C₆₇	1	2	yes	—
328	C₄×M₁₆	—	2	4	no	—
329	C₂×#97	C₂²×C₆₇	1	2	yes	—
330	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
331	#391	—	2	2	no	—
332	#391	C₂²×C₆₇	1	2	yes	—
333	#391	C₂²×C₆₇	1	1	yes	characteristic
334	#391	C₂²×C₆₇	1	1	yes	characteristic
335	#391	C₂²×C₆₇	1	2	yes	—
336	#390	C₂²×C₆₇	1	1	yes	characteristic
337	#390	C₂²×C₆₇	1	1	yes	characteristic
338	#386	—	2	2	no	—
339	C₄×C₂²⋊C₄	—	2	2	no	—
340	C₂×#95	—	2	2	no	—
341	C₂×#41	—	2	2	no	—
342	C₂×#41	—	2	2	no	—
343	C₂×#41	—	2	2	no	—
344	C₂×#41	—	2	2	no	—
345	C₂×#41	—	2	2	no	—
346	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
347	C₂²×Q₈	—	2	1	no	—
348	C₂²×Q₈	—	2	2	no	—
349	C₂²×Q₈	—	2	2	no	—
350	C₂²×Q₈	—	2	2	no	—
351	C₂²×Q₈	—	2	2	no	—
352	C₂²×Q₈	—	2	2	no	—
353	C₂²×Q₈	C₂×C₄×C₆₇	1	4	yes	—
354	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
355	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
356	C₂²×D₄	—	2	1	no	—
357	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
358	C₂³×C₄	—	2	1	no	—
359	C₂³×C₄	—	2	1	no	—
360	C₂³×C₄	—	2	1	no	—
361	C₂³×C₄	D₄×C₆₇	1	2	yes	—
362	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
363	#137	—	2	32	no	—
364	C₂×Q₁₆	—	2	4	no	—
365	C₂×Q₁₆	—	2	4	no	—
366	C₂×Q₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
367	C₂×SD₁₆	—	2	4	no	—
368	C₂×SD₁₆	—	2	4	no	—
369	C₂×SD₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
370	C₂×M₁₆	—	2	2	no	—
371	C₂×M₁₆	—	2	2	no	—
372	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
373	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
374	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
375	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
376	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
377	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
378	#117	—	2	2	no	—
379	#117	—	2	2	no	—
380	C₂×C₄⋊C₄	—	2	2	no	—
381	C₂×C₄⋊C₄	—	2	2	no	—
382	C₂×C₄⋊C₄	—	2	1	no	—
383	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
384	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
385	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
386	C₂×C₂²⋊C₄	—	2	1	no	—
387	C₂×C₂²⋊C₄	—	2	2	no	—
388	C₂×C₂²⋊C₄	—	2	2	no	—
389	C₂×C₄²	—	2	2	no	—
390	C₂×C₄²	—	2	2	no	—
391	C₂×C₄²	—	2	1	no	—
392	C₂×C₄²	D₄×C₆₇	1	2	yes	—
393	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
394	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
395	#107	—	2	4	no	—
396	#106	—	2	4	no	—
397	#103	—	2	4	no	—
398	#103	—	2	4	no	—
399	#103	—	2	4	no	—
400	#102	—	2	4	no	—
401	#97	—	2	4	no	—
402	C₄×C₈	—	2	4	no	—
403	#95	—	2	4	no	—
404	#95	—	2	4	no	—
405	#95	—	2	2	no	—
406	#95	—	2	2	no	—
407	C₂⁴	—	2	1	no	—
408	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
409	#41	—	2	8	no	—
410	#41	—	2	8	no	—
411	#41	—	4	4	no	—
412	#41	—	4	4	no	—
413	C₂×Q₈	—	2	4	no	—
414	C₂×Q₈	—	2	2	no	—
415	C₂×Q₈	—	2	1	no	—
416	C₂×Q₈	—	2	8	no	—
417	C₂×Q₈	—	2	8	no	—
418	C₂×Q₈	—	2	2	no	—
419	C₂×Q₈	—	2	1	no	—
420	C₂×Q₈	—	2	2	no	—
421	C₂×Q₈	—	2	2	no	—
422	C₂×Q₈	—	2	1	no	—
423	C₂×Q₈	—	2	1	no	—
424	C₂×Q₈	—	2	2	no	—
425	C₂×Q₈	—	2	1	no	—
426	C₂×Q₈	—	4	2	no	—
427	C₂×Q₈	—	4	4	no	—
428	C₂×Q₈	—	4	4	no	—
429	C₂×Q₈	—	4	2	no	—
430	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
431	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
432	C₂×D₄	—	2	4	no	—
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	—	2	1	no	—
435	C₂×D₄	—	2	1	no	—
436	C₂×D₄	—	2	1	no	—
437	C₂×D₄	—	2	1	no	—
438	C₂×D₄	—	4	2	no	—
439	C₂×D₄	—	4	2	no	—
440	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
441	C₂²×C₄	—	2	4	no	—
442	C₂²×C₄	—	2	4	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	2	2	no	—
445	C₂²×C₄	—	2	4	no	—
446	C₂²×C₄	—	2	4	no	—
447	C₂²×C₄	—	2	2	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	4	2	no	—
451	C₂²×C₄	—	4	2	no	—
452	C₂²×C₄	—	4	2	no	—
453	C₂²×C₄	—	4	2	no	—
454	C₂²×C₄	—	4	2	no	—
455	C₂²×C₄	—	4	4	no	—
456	C₂²×C₄	—	4	2	no	—
457	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
458	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
459	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
460	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
461	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
462	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
463	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
464	C₂²×C₄	#41×C₆₇	1	2	yes	—
465	C₂²×C₄	#41×C₆₇	1	2	yes	—
466	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
467	Q₁₆	—	2	16	no	—
468	SD₁₆	—	2	16	no	—
469	M₁₆	—	2	8	no	—
470	M₁₆	C₂²⋊C₄×C₆₇	1	16	yes	complemented
471	C₂×C₈	—	2	2	no	—
472	C₂×C₈	—	2	2	no	—
473	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
474	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
475	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
476	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
477	C₄⋊C₄	—	2	2	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	4	2	no	—
484	C₂²⋊C₄	—	4	4	no	—
485	C₂²⋊C₄	—	4	2	no	—
486	C₂²⋊C₄	—	4	4	no	—
487	C₄²	—	2	4	no	—
488	C₄²	—	2	1	no	—
489	C₄²	—	2	1	no	—
490	C₄²	—	2	1	no	—
491	C₄²	—	2	1	no	—
492	C₄²	—	4	2	no	—
493	C₂³	—	2	2	no	—
494	C₂³	—	2	2	no	—
495	C₂³	—	2	2	no	—
496	C₂³	—	4	1	no	—
497	C₂³	—	4	1	no	—
498	C₂³	—	4	2	no	—
499	C₂³	—	4	2	no	—
500	C₂³	C₄×D₄×C₆₇	1	2	yes	—
501	C₂³	#121×C₆₇	1	2	yes	—
502	C₂³	#127×C₆₇	1	2	yes	—
503	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
504	Q₈	—	2	4	no	—
505	Q₈	—	2	2	no	—
506	Q₈	—	2	4	no	—
507	Q₈	—	2	2	no	—
508	Q₈	—	4	2	no	—
509	Q₈	—	4	1	no	—
510	Q₈	—	4	2	no	—
511	Q₈	—	4	1	no	—
512	Q₈	—	4	8	no	—
513	D₄	—	2	2	no	—
514	D₄	—	2	2	no	—
515	D₄	—	4	8	no	—
516	D₄	—	4	1	no	—
517	D₄	—	4	1	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	4	8	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	2	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	2	no	—
530	C₂×C₄	—	4	8	no	—
531	C₂×C₄	—	4	8	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	2	no	—
535	C₂×C₄	—	4	2	no	—
536	C₂×C₄	—	4	2	no	—
537	C₂×C₄	—	4	4	no	—
538	C₂×C₄	—	8	4	no	—
539	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
540	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
541	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
542	C₂×C₄	#121×C₆₇	1	4	yes	—
543	C₂×C₄	#121×C₆₇	1	2	yes	—
544	C₂×C₄	#124×C₆₇	1	4	yes	—
545	C₂×C₄	#127×C₆₇	1	2	yes	—
546	C₂×C₄	#137×C₆₇	1	4	yes	complemented
547	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
548	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
549	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
550	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
551	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
552	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
553	C₈	—	2	4	no	—
554	C₈	C₂×C₂²⋊C₄×C₆₇	1	8	yes	—
555	C₂²	—	2	4	no	—
556	C₂²	—	2	2	no	—
557	C₂²	—	2	2	no	—
558	C₂²	—	2	4	no	—
559	C₂²	—	4	2	no	—
560	C₂²	—	4	4	no	—
561	C₂²	—	8	2	no	—
562	C₂²	—	8	1	no	—
563	C₂²	—	8	1	no	—
564	C₂²	#390×C₆₇	1	4	yes	—
565	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
566	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
567	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
568	C₂²	C₂×#137×C₆₇	1	2	yes	—
569	C₂²	#575×C₆₇	1	2	yes	—
570	C₄	—	2	4	no	—
571	C₄	—	4	2	no	—
572	C₄	—	4	2	no	—
573	C₄	—	4	4	no	—
574	C₄	—	8	1	no	—
575	C₄	—	8	2	no	—
576	C₄	—	8	1	no	—
577	C₄	#390×C₆₇	1	4	yes	—
578	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
579	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
580	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
581	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
582	C₂	—	2	4	no	—
583	C₂	—	8	2	no	—
584	C₂	C₂×#390×C₆₇	1	1	yes	characteristic
585	C₂	C₄×#137×C₆₇	1	2	yes	—
586	C₂	C₆₇×#3056	1	2	yes	—
587	C₂	C₆₇×#3152	1	1	yes	characteristic
588	C₂	C₆₇×#3153	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 ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#428×C₆₇	C₂	semidirect
4. C₂×#419×C₆₇	C₂	semidirect
5. C₂×C₄×M₁₆×C₆₇	C₂	semidirect
7. C₂×#391×C₆₇	C₂	semidirect
8. C₆₇×#1973	C₂	semidirect
9. C₆₇×#1971	C₂	semidirect
10. C₆₇×#1969	C₂	semidirect
11. C₆₇×#1968	C₂	semidirect
16. C₂×#137×C₆₇	C₄	semidirect
71. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
74. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
175. M₁₆×C₆₇	C₂²⋊C₄	semidirect
251. C₂×C₄×C₆₇	#137	semidirect
287. #21237	C₆₇	direct (N × H)
296. C₂×#428	C₂×C₆₇	semidirect
298. C₂×#419	C₂×C₆₇	semidirect
299. C₂×C₄×M₁₆	C₂×C₆₇	semidirect
301. C₂×#391	C₂×C₆₇	semidirect
302. #1973	C₂×C₆₇	semidirect
303. #1971	C₂×C₆₇	semidirect
304. #1969	C₂×C₆₇	semidirect
305. #1968	C₂×C₆₇	semidirect
306. C₆₇	#21237	direct (N × H)
311. C₂×#137	C₄×C₆₇	semidirect
366. C₂×Q₁₆	C₂×C₄×C₆₇	semidirect
369. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
470. M₁₆	C₂²⋊C₄×C₆₇	semidirect
546. C₂×C₄	#137×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#137×C₆₇	2
C₂×#428×C₆₇	2
C₂×#417×C₆₇	2
C₂×#419×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₆₇×#2297	2
C₂×#391×C₆₇	2
C₆₇×#1973	2
C₆₇×#1971	2
C₆₇×#1969	2
C₆₇×#1968	2
#21237	67

Subgroups — C67×#21237 #30700488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21237 #30700488