Subgroups — C₆₇×#33192 #30712443

Order 17152.

998 conjugacy classes of subgroups.

The subgroup lattice has 514 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×#473×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#472×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#471×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3040	C₂	1	2	yes	maximal, complemented
6	C₄×#131×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1979	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1978	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1977	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1967	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1848	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₆₇	C₂²	1	2	yes	—
13	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	2	yes	—
14	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
16	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#124×C₆₇	C₂²	1	2	yes	—
20	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
21	#495×C₆₇	—	2	1	no	—
22	#495×C₆₇	—	2	1	no	—
23	#495×C₆₇	—	2	1	no	—
24	#495×C₆₇	—	2	1	no	—
25	#473×C₆₇	—	2	4	no	—
26	#472×C₆₇	—	2	4	no	—
27	#471×C₆₇	—	2	8	no	—
28	#464×C₆₇	—	2	4	no	—
29	#463×C₆₇	—	2	4	no	—
30	C₂×#108×C₆₇	C₂²	1	2	yes	—
31	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
33	#423×C₆₇	C₂²	1	2	yes	—
34	#419×C₆₇	C₂²	1	2	yes	—
35	#418×C₆₇	C₂²	1	2	yes	—
36	C₂×#101×C₆₇	C₂²	1	2	yes	—
37	C₂×#101×C₆₇	C₂²	1	2	yes	—
38	C₂×#100×C₆₇	C₂²	1	2	yes	—
39	C₂×#100×C₆₇	C₂²	1	2	yes	—
40	#405×C₆₇	—	2	1	no	—
41	#405×C₆₇	—	2	1	no	—
42	#405×C₆₇	C₂²	1	1	yes	characteristic
43	#405×C₆₇	C₂²	1	1	yes	characteristic
44	C₄×M₁₆×C₆₇	—	2	1	no	—
45	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₆₇	—	2	1	no	—
47	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
48	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
50	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
51	#137×C₆₇	—	2	4	no	—
52	#137×C₆₇	—	2	4	no	—
53	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
54	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
55	C₂×Q₁₆×C₆₇	—	2	2	no	—
56	C₂×Q₁₆×C₆₇	—	2	2	no	—
57	C₂×Q₁₆×C₆₇	—	2	1	no	—
58	C₂×Q₁₆×C₆₇	—	2	1	no	—
59	C₂×Q₁₆×C₆₇	—	2	2	no	—
60	C₂×SD₁₆×C₆₇	—	2	2	no	—
61	C₂×SD₁₆×C₆₇	—	2	2	no	—
62	C₂×SD₁₆×C₆₇	—	2	4	no	—
63	C₂×SD₁₆×C₆₇	—	2	2	no	—
64	C₂×SD₁₆×C₆₇	—	2	2	no	—
65	C₂×SD₁₆×C₆₇	—	2	4	no	—
66	C₂×D₈×C₆₇	—	2	2	no	—
67	C₂×D₈×C₆₇	—	2	2	no	—
68	C₂×D₈×C₆₇	—	2	1	no	—
69	C₂×D₈×C₆₇	—	2	1	no	—
70	C₂×D₈×C₆₇	—	2	2	no	—
71	#131×C₆₇	—	2	2	no	—
72	#131×C₆₇	—	2	2	no	—
73	#131×C₆₇	D₄	1	4	yes	—
74	#131×C₆₇	D₄	1	4	yes	—
75	C₂×M₁₆×C₆₇	—	2	1	no	—
76	C₂×M₁₆×C₆₇	—	2	1	no	—
77	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
78	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
79	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
80	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₆₇	—	2	1	no	—
82	C₂²×C₈×C₆₇	—	2	1	no	—
83	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
84	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
85	#124×C₆₇	—	2	4	no	—
86	#124×C₆₇	—	2	2	no	—
87	#124×C₆₇	—	2	2	no	—
88	C₄×Q₈×C₆₇	D₄	1	2	yes	—
89	C₄×D₄×C₆₇	—	2	2	no	—
90	C₄×D₄×C₆₇	D₄	1	2	yes	—
91	#117×C₆₇	—	2	1	no	—
92	#117×C₆₇	C₂³	1	1	yes	characteristic
93	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
94	C₂×C₄²×C₆₇	—	2	1	no	—
95	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
96	#108×C₆₇	—	2	4	no	—
97	#108×C₆₇	—	2	2	no	—
98	#108×C₆₇	—	2	2	no	—
99	C₄⋊C₈×C₆₇	—	2	4	no	—
100	#103×C₆₇	—	2	4	no	—
101	#102×C₆₇	—	2	4	no	—
102	#101×C₆₇	—	2	4	no	—
103	#101×C₆₇	—	2	4	no	—
104	#100×C₆₇	—	2	4	no	—
105	#100×C₆₇	—	2	4	no	—
106	#97×C₆₇	—	2	1	no	—
107	#97×C₆₇	—	2	1	no	—
108	#97×C₆₇	—	2	1	no	—
109	C₄×C₈×C₆₇	—	2	1	no	—
110	C₄×C₈×C₆₇	—	2	1	no	—
111	C₄×C₈×C₆₇	—	2	1	no	—
112	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
113	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
114	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
115	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
116	C₂⁴×C₆₇	—	2	2	no	—
117	#41×C₆₇	—	2	1	no	—
118	#41×C₆₇	—	2	1	no	—
119	#41×C₆₇	C₂×D₄	1	2	yes	—
120	#41×C₆₇	C₂×D₄	1	2	yes	—
121	C₂×Q₈×C₆₇	—	2	2	no	—
122	C₂×Q₈×C₆₇	—	2	2	no	—
123	C₂×Q₈×C₆₇	—	2	2	no	—
124	C₂×Q₈×C₆₇	—	2	2	no	—
125	C₂×Q₈×C₆₇	—	2	2	no	—
126	C₂×Q₈×C₆₇	—	2	2	no	—
127	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
128	C₂×D₄×C₆₇	—	2	1	no	—
129	C₂×D₄×C₆₇	—	2	2	no	—
130	C₂×D₄×C₆₇	—	2	2	no	—
131	C₂×D₄×C₆₇	—	2	2	no	—
132	C₂×D₄×C₆₇	—	2	2	no	—
133	C₂×D₄×C₆₇	—	2	2	no	—
134	C₂×D₄×C₆₇	—	2	2	no	—
135	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₆₇	—	2	1	no	—
137	C₂²×C₄×C₆₇	—	2	2	no	—
138	C₂²×C₄×C₆₇	—	2	2	no	—
139	C₂²×C₄×C₆₇	#41	1	2	yes	—
140	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
141	Q₁₆×C₆₇	—	4	4	no	—
142	Q₁₆×C₆₇	—	4	2	no	—
143	Q₁₆×C₆₇	—	4	2	no	—
144	SD₁₆×C₆₇	—	4	4	no	—
145	SD₁₆×C₆₇	—	4	4	no	—
146	SD₁₆×C₆₇	—	4	4	no	—
147	SD₁₆×C₆₇	—	4	4	no	—
148	D₈×C₆₇	—	4	4	no	—
149	D₈×C₆₇	—	4	2	no	—
150	D₈×C₆₇	—	4	2	no	—
151	M₁₆×C₆₇	—	2	1	no	—
152	M₁₆×C₆₇	—	2	1	no	—
153	M₁₆×C₆₇	—	2	4	no	—
154	M₁₆×C₆₇	—	2	4	no	—
155	M₁₆×C₆₇	—	2	2	no	—
156	M₁₆×C₆₇	—	2	2	no	—
157	M₁₆×C₆₇	—	2	2	no	—
158	M₁₆×C₆₇	—	2	2	no	—
159	M₁₆×C₆₇	C₂×D₄	1	2	yes	—
160	M₁₆×C₆₇	C₂×D₄	1	2	yes	—
161	C₂×C₈×C₆₇	—	2	1	no	—
162	C₂×C₈×C₆₇	—	2	1	no	—
163	C₂×C₈×C₆₇	—	2	4	no	—
164	C₂×C₈×C₆₇	—	2	4	no	—
165	C₂×C₈×C₆₇	—	2	2	no	—
166	C₂×C₈×C₆₇	—	2	2	no	—
167	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
168	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
169	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
170	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
171	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
172	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
173	C₂×C₈×C₆₇	#41	1	2	yes	—
174	C₂×C₈×C₆₇	#41	1	2	yes	—
175	C₄⋊C₄×C₆₇	—	2	2	no	—
176	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
177	C₂²⋊C₄×C₆₇	—	2	1	no	—
178	C₂²⋊C₄×C₆₇	—	2	2	no	—
179	C₂²⋊C₄×C₆₇	—	4	4	no	—
180	C₂²⋊C₄×C₆₇	—	4	4	no	—
181	C₄²×C₆₇	—	2	2	no	—
182	C₄²×C₆₇	—	2	1	no	—
183	C₄²×C₆₇	C₂×D₄	1	2	yes	—
184	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
185	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
186	C₂³×C₆₇	—	2	1	no	—
187	C₂³×C₆₇	—	4	2	no	—
188	C₂³×C₆₇	—	4	2	no	—
189	C₂³×C₆₇	—	4	2	no	—
190	C₂³×C₆₇	—	8	4	no	—
191	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
192	Q₈×C₆₇	—	2	1	no	—
193	Q₈×C₆₇	—	4	2	no	—
194	Q₈×C₆₇	—	4	2	no	—
195	Q₈×C₆₇	—	4	2	no	—
196	Q₈×C₆₇	—	4	2	no	—
197	Q₈×C₆₇	#127	1	2	yes	—
198	D₄×C₆₇	—	2	2	no	—
199	D₄×C₆₇	—	2	2	no	—
200	D₄×C₆₇	—	2	1	no	—
201	D₄×C₆₇	—	4	2	no	—
202	D₄×C₆₇	—	4	2	no	—
203	D₄×C₆₇	—	4	2	no	—
204	D₄×C₆₇	—	4	2	no	—
205	D₄×C₆₇	#127	1	2	yes	—
206	C₂×C₄×C₆₇	—	2	1	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	—	4	2	no	—
211	C₂×C₄×C₆₇	—	4	4	no	—
212	C₂×C₄×C₆₇	—	4	2	no	—
213	C₂×C₄×C₆₇	—	4	2	no	—
214	C₂×C₄×C₆₇	—	4	2	no	—
215	C₂×C₄×C₆₇	#121	1	2	yes	—
216	C₂×C₄×C₆₇	#121	1	2	yes	—
217	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
222	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
223	C₈×C₆₇	—	2	2	no	—
224	C₈×C₆₇	—	2	2	no	—
225	C₈×C₆₇	—	4	2	no	—
226	C₈×C₆₇	—	4	2	no	—
227	C₈×C₆₇	#121	1	2	yes	—
228	C₈×C₆₇	#121	1	2	yes	—
229	C₈×C₆₇	#121	1	2	yes	—
230	C₈×C₆₇	#121	1	2	yes	—
231	C₂²×C₆₇	—	2	2	no	—
232	C₂²×C₆₇	—	2	2	no	—
233	C₂²×C₆₇	—	4	2	no	—
234	C₂²×C₆₇	—	8	2	no	—
235	C₂²×C₆₇	—	8	4	no	—
236	C₂²×C₆₇	—	16	2	no	—
237	C₂²×C₆₇	—	16	2	no	—
238	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
239	C₂²×C₆₇	#532	1	1	yes	characteristic
240	C₂²×C₆₇	#532	1	1	yes	characteristic
241	C₄×C₆₇	—	2	2	no	—
242	C₄×C₆₇	—	2	2	no	—
243	C₄×C₆₇	—	2	2	no	—
244	C₄×C₆₇	—	4	2	no	—
245	C₄×C₆₇	—	8	2	no	—
246	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
247	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
248	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
249	C₄×C₆₇	#532	1	1	yes	characteristic
250	#33192	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
251	C₂×C₆₇	—	2	1	no	—
252	C₂×C₆₇	—	2	1	no	—
253	C₂×C₆₇	—	4	2	no	—
254	C₂×C₆₇	—	16	2	no	—
255	C₂×C₆₇	#2856	1	1	yes	characteristic
256	C₂×C₆₇	#3067	1	2	yes	—
257	C₂×#495	C₂×C₆₇	1	1	yes	characteristic, complemented
258	C₂×#473	C₂×C₆₇	1	1	yes	characteristic, complemented
259	C₂×#472	C₂×C₆₇	1	1	yes	characteristic
260	C₂×#471	C₂×C₆₇	1	2	yes	complemented
261	#3040	C₂×C₆₇	1	2	yes	complemented
262	C₄×#131	C₂×C₆₇	1	1	yes	characteristic, complemented
263	#1979	C₂×C₆₇	1	1	yes	characteristic, complemented
264	#1978	C₂×C₆₇	1	1	yes	characteristic, complemented
265	#1977	C₂×C₆₇	1	2	yes	complemented
266	#1967	C₂×C₆₇	1	2	yes	complemented
267	#1848	C₂×C₆₇	1	1	yes	characteristic, complemented
268	C₆₇	#33192	1	1	yes	characteristic, direct product factor, Sylow
269	C₂×#137	C₂²×C₆₇	1	2	yes	—
270	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	2	yes	—
271	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
272	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
273	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
274	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
275	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
276	C₂×#124	C₂²×C₆₇	1	2	yes	—
277	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
278	#495	—	2	1	no	—
279	#495	—	2	1	no	—
280	#495	—	2	1	no	—
281	#495	—	2	1	no	—
282	#473	—	2	4	no	—
283	#472	—	2	4	no	—
284	#471	—	2	8	no	—
285	#464	—	2	4	no	—
286	#463	—	2	4	no	—
287	C₂×#108	C₂²×C₆₇	1	2	yes	—
288	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
289	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic
290	#423	C₂²×C₆₇	1	2	yes	—
291	#419	C₂²×C₆₇	1	2	yes	—
292	#418	C₂²×C₆₇	1	2	yes	—
293	C₂×#101	C₂²×C₆₇	1	2	yes	—
294	C₂×#101	C₂²×C₆₇	1	2	yes	—
295	C₂×#100	C₂²×C₆₇	1	2	yes	—
296	C₂×#100	C₂²×C₆₇	1	2	yes	—
297	#405	—	2	1	no	—
298	#405	—	2	1	no	—
299	#405	C₂²×C₆₇	1	1	yes	characteristic
300	#405	C₂²×C₆₇	1	1	yes	characteristic
301	C₄×M₁₆	—	2	1	no	—
302	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
303	C₂×C₄×C₈	—	2	1	no	—
304	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
305	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
306	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
307	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
308	#137	—	2	4	no	—
309	#137	—	2	4	no	—
310	AGL(1,ℤ/8)	—	2	4	no	—
311	AGL(1,ℤ/8)	—	2	4	no	—
312	C₂×Q₁₆	—	2	2	no	—
313	C₂×Q₁₆	—	2	2	no	—
314	C₂×Q₁₆	—	2	1	no	—
315	C₂×Q₁₆	—	2	1	no	—
316	C₂×Q₁₆	—	2	2	no	—
317	C₂×SD₁₆	—	2	2	no	—
318	C₂×SD₁₆	—	2	2	no	—
319	C₂×SD₁₆	—	2	4	no	—
320	C₂×SD₁₆	—	2	2	no	—
321	C₂×SD₁₆	—	2	2	no	—
322	C₂×SD₁₆	—	2	4	no	—
323	C₂×D₈	—	2	2	no	—
324	C₂×D₈	—	2	2	no	—
325	C₂×D₈	—	2	1	no	—
326	C₂×D₈	—	2	1	no	—
327	C₂×D₈	—	2	2	no	—
328	#131	—	2	2	no	—
329	#131	—	2	2	no	—
330	#131	D₄×C₆₇	1	4	yes	—
331	#131	D₄×C₆₇	1	4	yes	—
332	C₂×M₁₆	—	2	1	no	—
333	C₂×M₁₆	—	2	1	no	—
334	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
335	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
336	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
337	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
338	C₂²×C₈	—	2	1	no	—
339	C₂²×C₈	—	2	1	no	—
340	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
341	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
342	#124	—	2	4	no	—
343	#124	—	2	2	no	—
344	#124	—	2	2	no	—
345	C₄×Q₈	D₄×C₆₇	1	2	yes	—
346	C₄×D₄	—	2	2	no	—
347	C₄×D₄	D₄×C₆₇	1	2	yes	—
348	#117	—	2	1	no	—
349	#117	C₂³×C₆₇	1	1	yes	characteristic
350	C₂×C₂²⋊C₄	—	2	4	no	—
351	C₂×C₄²	—	2	1	no	—
352	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
353	#108	—	2	4	no	—
354	#108	—	2	2	no	—
355	#108	—	2	2	no	—
356	C₄⋊C₈	—	2	4	no	—
357	#103	—	2	4	no	—
358	#102	—	2	4	no	—
359	#101	—	2	4	no	—
360	#101	—	2	4	no	—
361	#100	—	2	4	no	—
362	#100	—	2	4	no	—
363	#97	—	2	1	no	—
364	#97	—	2	1	no	—
365	#97	—	2	1	no	—
366	C₄×C₈	—	2	1	no	—
367	C₄×C₈	—	2	1	no	—
368	C₄×C₈	—	2	1	no	—
369	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
370	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
371	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
372	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
373	C₂⁴	—	2	2	no	—
374	#41	—	2	1	no	—
375	#41	—	2	1	no	—
376	#41	C₂×D₄×C₆₇	1	2	yes	—
377	#41	C₂×D₄×C₆₇	1	2	yes	—
378	C₂×Q₈	—	2	2	no	—
379	C₂×Q₈	—	2	2	no	—
380	C₂×Q₈	—	2	2	no	—
381	C₂×Q₈	—	2	2	no	—
382	C₂×Q₈	—	2	2	no	—
383	C₂×Q₈	—	2	2	no	—
384	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
385	C₂×D₄	—	2	1	no	—
386	C₂×D₄	—	2	2	no	—
387	C₂×D₄	—	2	2	no	—
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	—	2	2	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	2	2	no	—
396	C₂²×C₄	#41×C₆₇	1	2	yes	—
397	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
398	Q₁₆	—	4	4	no	—
399	Q₁₆	—	4	2	no	—
400	Q₁₆	—	4	2	no	—
401	SD₁₆	—	4	4	no	—
402	SD₁₆	—	4	4	no	—
403	SD₁₆	—	4	4	no	—
404	SD₁₆	—	4	4	no	—
405	D₈	—	4	4	no	—
406	D₈	—	4	2	no	—
407	D₈	—	4	2	no	—
408	M₁₆	—	2	1	no	—
409	M₁₆	—	2	1	no	—
410	M₁₆	—	2	4	no	—
411	M₁₆	—	2	4	no	—
412	M₁₆	—	2	2	no	—
413	M₁₆	—	2	2	no	—
414	M₁₆	—	2	2	no	—
415	M₁₆	—	2	2	no	—
416	M₁₆	C₂×D₄×C₆₇	1	2	yes	—
417	M₁₆	C₂×D₄×C₆₇	1	2	yes	—
418	C₂×C₈	—	2	1	no	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	2	4	no	—
421	C₂×C₈	—	2	4	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
425	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
426	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
427	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
428	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
429	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
430	C₂×C₈	#41×C₆₇	1	2	yes	—
431	C₂×C₈	#41×C₆₇	1	2	yes	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
434	C₂²⋊C₄	—	2	1	no	—
435	C₂²⋊C₄	—	2	2	no	—
436	C₂²⋊C₄	—	4	4	no	—
437	C₂²⋊C₄	—	4	4	no	—
438	C₄²	—	2	2	no	—
439	C₄²	—	2	1	no	—
440	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
441	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
442	C₄²	C₂×D₄×C₆₇	1	2	yes	—
443	C₂³	—	2	1	no	—
444	C₂³	—	4	2	no	—
445	C₂³	—	4	2	no	—
446	C₂³	—	4	2	no	—
447	C₂³	—	8	4	no	—
448	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
449	Q₈	—	2	1	no	—
450	Q₈	—	4	2	no	—
451	Q₈	—	4	2	no	—
452	Q₈	—	4	2	no	—
453	Q₈	—	4	2	no	—
454	Q₈	#127×C₆₇	1	2	yes	—
455	D₄	—	2	2	no	—
456	D₄	—	2	2	no	—
457	D₄	—	2	1	no	—
458	D₄	—	4	2	no	—
459	D₄	—	4	2	no	—
460	D₄	—	4	2	no	—
461	D₄	—	4	2	no	—
462	D₄	#127×C₆₇	1	2	yes	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	4	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	#121×C₆₇	1	2	yes	—
473	C₂×C₄	#121×C₆₇	1	2	yes	—
474	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
475	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
476	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
477	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
478	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
479	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
480	C₈	—	2	2	no	—
481	C₈	—	2	2	no	—
482	C₈	—	4	2	no	—
483	C₈	—	4	2	no	—
484	C₈	#121×C₆₇	1	2	yes	—
485	C₈	#121×C₆₇	1	2	yes	—
486	C₈	#121×C₆₇	1	2	yes	—
487	C₈	#121×C₆₇	1	2	yes	—
488	C₂²	—	2	2	no	—
489	C₂²	—	2	2	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	8	4	no	—
492	C₂²	—	8	2	no	—
493	C₂²	—	16	2	no	—
494	C₂²	—	16	2	no	—
495	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
496	C₂²	#532×C₆₇	1	1	yes	characteristic
497	C₂²	#532×C₆₇	1	1	yes	characteristic
498	C₄	—	2	2	no	—
499	C₄	—	2	2	no	—
500	C₄	—	2	2	no	—
501	C₄	—	4	2	no	—
502	C₄	—	8	2	no	—
503	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
504	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
505	C₄	C₂×#127×C₆₇	1	1	yes	characteristic
506	C₄	#532×C₆₇	1	1	yes	characteristic
507	C₂	—	2	1	no	—
508	C₂	—	2	1	no	—
509	C₂	—	4	2	no	—
510	C₂	—	16	2	no	—
511	C₂	C₆₇×#2856	1	1	yes	characteristic
512	C₂	C₆₇×#3067	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#472×C₆₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×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×#473×C₆₇	C₂	semidirect
4. C₂×#471×C₆₇	C₂	semidirect
5. C₆₇×#3040	C₂	semidirect
6. C₄×#131×C₆₇	C₂	semidirect
7. C₆₇×#1979	C₂	semidirect
8. C₆₇×#1978	C₂	semidirect
9. C₆₇×#1977	C₂	semidirect
10. C₆₇×#1967	C₂	semidirect
11. C₆₇×#1848	C₂	semidirect
250. #33192	C₆₇	direct (N × H)
257. C₂×#495	C₂×C₆₇	semidirect
258. C₂×#473	C₂×C₆₇	semidirect
260. C₂×#471	C₂×C₆₇	semidirect
261. #3040	C₂×C₆₇	semidirect
262. C₄×#131	C₂×C₆₇	semidirect
263. #1979	C₂×C₆₇	semidirect
264. #1978	C₂×C₆₇	semidirect
265. #1977	C₂×C₆₇	semidirect
266. #1967	C₂×C₆₇	semidirect
267. #1848	C₂×C₆₇	semidirect
268. C₆₇	#33192	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×#495×C₆₇	2
C₂×#473×C₆₇	2
C₂×#472×C₆₇	2
C₂×#471×C₆₇	2
C₆₇×#3040	2
C₄×#131×C₆₇	2
C₆₇×#1979	2
C₆₇×#1978	2
C₆₇×#1977	2
C₆₇×#1967	2
C₆₇×#1848	2
#33192	67

Subgroups — C67×#33192 #30712443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33192 #30712443