Subgroups — C₆₇×#20885 #30700136

Order 17152.

1358 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#568×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#430×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2905	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#104×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2896	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1866	C₂	1	8	yes	maximal, complemented
7	C₂×#339×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1728	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
10	#568×C₆₇	—	2	4	no	—
11	#568×C₆₇	—	2	4	no	—
12	C₂×#131×C₆₇	—	2	2	no	—
13	C₂×#131×C₆₇	C₂²	1	4	yes	—
14	C₂²×M₁₆×C₆₇	—	2	2	no	—
15	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	#430×C₆₇	—	2	4	no	—
22	C₂×#108×C₆₇	C₂²	1	4	yes	complemented
23	#424×C₆₇	—	2	2	no	—
24	#424×C₆₇	C₂²	1	4	yes	complemented
25	#423×C₆₇	—	2	2	no	—
26	C₂×#104×C₆₇	—	2	4	no	—
27	C₂×#104×C₆₇	C₂²	1	4	yes	—
28	#419×C₆₇	C₄	1	2	yes	complemented
29	#418×C₆₇	C₄	1	2	yes	complemented
30	#417×C₆₇	C₄	1	4	yes	complemented
31	#416×C₆₇	C₂²	1	4	yes	—
32	#339×C₆₇	C₂²	1	4	yes	complemented
33	#336×C₆₇	C₂²	1	4	yes	complemented
34	C₂×#41×C₆₇	—	2	1	no	—
35	C₂×#41×C₆₇	—	2	2	no	—
36	C₂×#41×C₆₇	—	2	1	no	—
37	C₂×#41×C₆₇	—	2	2	no	—
38	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×#41×C₆₇	D₄	1	4	yes	—
40	C₂×#41×C₆₇	Q₈	1	4	yes	—
41	C₂×#41×C₆₇	C₂³	1	2	yes	—
42	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×D₄×C₆₇	—	2	1	no	—
44	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₆₇	—	2	1	no	—
46	C₂³×C₄×C₆₇	—	2	1	no	—
47	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
48	#131×C₆₇	—	2	16	no	—
49	#131×C₆₇	—	2	8	no	—
50	#131×C₆₇	—	2	8	no	—
51	C₂×M₁₆×C₆₇	—	2	16	no	—
52	C₂×M₁₆×C₆₇	—	2	1	no	—
53	C₂×M₁₆×C₆₇	—	2	2	no	—
54	C₂×M₁₆×C₆₇	—	2	2	no	—
55	C₂×M₁₆×C₆₇	—	2	2	no	—
56	C₂×M₁₆×C₆₇	—	2	2	no	—
57	C₂×M₁₆×C₆₇	—	2	2	no	—
58	C₂×M₁₆×C₆₇	—	2	1	no	—
59	C₂×M₁₆×C₆₇	—	2	4	no	—
60	C₂×M₁₆×C₆₇	—	2	2	no	—
61	C₂×M₁₆×C₆₇	—	2	2	no	—
62	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
63	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
64	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
65	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
66	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
67	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
68	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
69	C₂²×C₈×C₆₇	—	2	1	no	—
70	C₂²×C₈×C₆₇	—	2	4	no	—
71	C₂²×C₈×C₆₇	—	2	1	no	—
72	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
73	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
74	#117×C₆₇	—	2	1	no	—
75	#117×C₆₇	—	2	1	no	—
76	#117×C₆₇	C₂×C₄	1	2	yes	—
77	#117×C₆₇	C₂³	1	2	yes	—
78	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
79	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
80	C₂×C₄²×C₆₇	—	2	1	no	—
81	C₂×C₄²×C₆₇	—	2	2	no	—
82	C₂×C₄²×C₆₇	—	2	2	no	—
83	C₂×C₄²×C₆₇	—	2	1	no	—
84	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
85	#108×C₆₇	—	2	8	no	—
86	C₄⋊C₈×C₆₇	—	2	8	no	—
87	#104×C₆₇	—	2	8	no	—
88	#103×C₆₇	C₂×C₄	1	8	yes	complemented
89	#102×C₆₇	C₂×C₄	1	8	yes	complemented
90	C₂⁴×C₆₇	—	2	1	no	—
91	C₂⁴×C₆₇	#41	1	1	yes	characteristic
92	#41×C₆₇	—	2	4	no	—
93	#41×C₆₇	—	2	4	no	—
94	#41×C₆₇	—	2	4	no	—
95	#41×C₆₇	—	2	4	no	—
96	#41×C₆₇	—	2	4	no	—
97	#41×C₆₇	—	2	2	no	—
98	#41×C₆₇	—	2	2	no	—
99	#41×C₆₇	C₄⋊C₄	1	8	yes	—
100	#41×C₆₇	C₂×D₄	1	4	yes	—
101	#41×C₆₇	C₂×Q₈	1	4	yes	—
102	C₂×Q₈×C₆₇	—	2	1	no	—
103	C₂×Q₈×C₆₇	—	2	2	no	—
104	C₂×Q₈×C₆₇	C₄⋊C₄	1	4	yes	—
105	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
106	C₂×D₄×C₆₇	—	2	2	no	—
107	C₂×D₄×C₆₇	—	2	4	no	—
108	C₂×D₄×C₆₇	—	2	4	no	—
109	C₂×D₄×C₆₇	—	2	2	no	—
110	C₂×D₄×C₆₇	—	2	2	no	—
111	C₂×D₄×C₆₇	—	2	1	no	—
112	C₂×D₄×C₆₇	C₄⋊C₄	1	4	yes	—
113	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
114	C₂²×C₄×C₆₇	—	2	1	no	—
115	C₂²×C₄×C₆₇	—	2	2	no	—
116	C₂²×C₄×C₆₇	—	2	4	no	—
117	C₂²×C₄×C₆₇	—	2	4	no	—
118	C₂²×C₄×C₆₇	—	2	2	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	—	2	2	no	—
121	C₂²×C₄×C₆₇	—	4	4	no	—
122	C₂²×C₄×C₆₇	—	4	2	no	—
123	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	C₂²×C₄	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	1	yes	characteristic
128	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
130	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
131	C₂²×C₄×C₆₇	#41	1	2	yes	—
132	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
133	M₁₆×C₆₇	—	2	16	no	—
134	M₁₆×C₆₇	—	2	4	no	—
135	M₁₆×C₆₇	—	2	4	no	—
136	M₁₆×C₆₇	—	2	4	no	—
137	M₁₆×C₆₇	—	2	4	no	—
138	M₁₆×C₆₇	—	2	4	no	—
139	M₁₆×C₆₇	—	2	2	no	—
140	M₁₆×C₆₇	—	2	2	no	—
141	M₁₆×C₆₇	—	2	2	no	—
142	M₁₆×C₆₇	—	4	2	no	—
143	M₁₆×C₆₇	C₂×D₄	1	4	yes	—
144	C₂×C₈×C₆₇	—	2	16	no	—
145	C₂×C₈×C₆₇	—	2	4	no	—
146	C₂×C₈×C₆₇	—	2	2	no	—
147	C₂×C₈×C₆₇	—	2	2	no	—
148	C₂×C₈×C₆₇	—	2	2	no	—
149	C₂×C₈×C₆₇	—	2	2	no	—
150	C₂×C₈×C₆₇	—	2	2	no	—
151	C₂×C₈×C₆₇	—	2	2	no	—
152	C₂×C₈×C₆₇	—	4	2	no	—
153	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
154	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
155	C₄⋊C₄×C₆₇	—	2	2	no	—
156	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
157	C₂²⋊C₄×C₆₇	—	2	2	no	—
158	C₂²⋊C₄×C₆₇	—	2	2	no	—
159	C₄²×C₆₇	—	2	2	no	—
160	C₄²×C₆₇	—	2	2	no	—
161	C₄²×C₆₇	—	2	2	no	—
162	C₄²×C₆₇	—	2	2	no	—
163	C₄²×C₆₇	—	4	2	no	—
164	C₂³×C₆₇	—	2	2	no	—
165	C₂³×C₆₇	—	2	4	no	—
166	C₂³×C₆₇	—	4	2	no	—
167	C₂³×C₆₇	—	8	2	no	—
168	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
169	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
170	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
171	C₂³×C₆₇	#123	1	2	yes	—
172	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
173	Q₈×C₆₇	—	2	2	no	—
174	Q₈×C₆₇	—	2	4	no	—
175	Q₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
176	D₄×C₆₇	—	2	4	no	—
177	D₄×C₆₇	—	2	4	no	—
178	D₄×C₆₇	—	2	8	no	—
179	D₄×C₆₇	—	2	2	no	—
180	D₄×C₆₇	—	2	4	no	—
181	D₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
182	C₂×C₄×C₆₇	—	2	2	no	—
183	C₂×C₄×C₆₇	—	2	4	no	—
184	C₂×C₄×C₆₇	—	2	4	no	—
185	C₂×C₄×C₆₇	—	2	4	no	—
186	C₂×C₄×C₆₇	—	2	8	no	—
187	C₂×C₄×C₆₇	—	2	4	no	—
188	C₂×C₄×C₆₇	—	2	4	no	—
189	C₂×C₄×C₆₇	—	4	2	no	—
190	C₂×C₄×C₆₇	—	4	8	no	—
191	C₂×C₄×C₆₇	—	4	2	no	—
192	C₂×C₄×C₆₇	—	8	2	no	—
193	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
194	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
195	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
196	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
197	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
199	C₂×C₄×C₆₇	#120	1	2	yes	—
200	C₂×C₄×C₆₇	#120	1	2	yes	—
201	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
202	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
203	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
204	C₂×C₄×C₆₇	#123	1	2	yes	—
205	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
206	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
207	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
208	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
211	C₈×C₆₇	—	2	4	no	—
212	C₈×C₆₇	—	2	4	no	—
213	C₈×C₆₇	—	2	4	no	—
214	C₈×C₆₇	—	4	2	no	—
215	C₂²×C₆₇	—	2	2	no	—
216	C₂²×C₆₇	—	2	2	no	—
217	C₂²×C₆₇	—	2	2	no	—
218	C₂²×C₆₇	—	2	4	no	—
219	C₂²×C₆₇	—	4	2	no	—
220	C₂²×C₆₇	—	4	2	no	—
221	C₂²×C₆₇	—	4	4	no	—
222	C₂²×C₆₇	—	8	4	no	—
223	C₂²×C₆₇	#385	1	2	yes	—
224	C₂²×C₆₇	#385	1	2	yes	—
225	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
226	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
227	C₂²×C₆₇	C₂×#123	1	1	yes	characteristic
228	C₄×C₆₇	—	2	4	no	—
229	C₄×C₆₇	—	4	4	no	—
230	C₄×C₆₇	—	4	4	no	—
231	C₄×C₆₇	#385	1	2	yes	—
232	C₄×C₆₇	#385	1	2	yes	—
233	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
234	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
235	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
236	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
237	#20885	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
238	C₂×C₆₇	—	2	1	no	—
239	C₂×C₆₇	—	2	2	no	—
240	C₂×C₆₇	—	2	1	no	—
241	C₂×C₆₇	—	2	2	no	—
242	C₂×C₆₇	—	4	4	no	—
243	C₂×C₆₇	C₂×#385	1	1	yes	characteristic
244	C₂×C₆₇	#2956	1	2	yes	—
245	C₂×#568	C₂×C₆₇	1	1	yes	characteristic
246	C₂×#430	C₂×C₆₇	1	1	yes	characteristic, complemented
247	#2905	C₂×C₆₇	1	1	yes	characteristic, complemented
248	C₂²×#104	C₂×C₆₇	1	1	yes	characteristic
249	#2896	C₂×C₆₇	1	1	yes	characteristic
250	#1866	C₂×C₆₇	1	8	yes	complemented
251	C₂×#339	C₂×C₆₇	1	1	yes	characteristic, complemented
252	#1728	C₂×C₆₇	1	1	yes	characteristic, complemented
253	C₆₇	#20885	1	1	yes	characteristic, direct product factor, Sylow
254	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
255	#568	—	2	4	no	—
256	#568	—	2	4	no	—
257	C₂×#131	—	2	2	no	—
258	C₂×#131	C₂²×C₆₇	1	4	yes	—
259	C₂²×M₁₆	—	2	2	no	—
260	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
261	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
262	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
263	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
264	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
265	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
266	#430	—	2	4	no	—
267	C₂×#108	C₂²×C₆₇	1	4	yes	complemented
268	#424	—	2	2	no	—
269	#424	C₂²×C₆₇	1	4	yes	complemented
270	#423	—	2	2	no	—
271	C₂×#104	—	2	4	no	—
272	C₂×#104	C₂²×C₆₇	1	4	yes	—
273	#419	C₄×C₆₇	1	2	yes	complemented
274	#418	C₄×C₆₇	1	2	yes	complemented
275	#417	C₄×C₆₇	1	4	yes	complemented
276	#416	C₂²×C₆₇	1	4	yes	—
277	#339	C₂²×C₆₇	1	4	yes	complemented
278	#336	C₂²×C₆₇	1	4	yes	complemented
279	C₂×#41	—	2	1	no	—
280	C₂×#41	—	2	2	no	—
281	C₂×#41	—	2	1	no	—
282	C₂×#41	—	2	2	no	—
283	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
284	C₂×#41	D₄×C₆₇	1	4	yes	—
285	C₂×#41	Q₈×C₆₇	1	4	yes	—
286	C₂×#41	C₂³×C₆₇	1	2	yes	—
287	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
288	C₂²×D₄	—	2	1	no	—
289	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
290	C₂³×C₄	—	2	1	no	—
291	C₂³×C₄	—	2	1	no	—
292	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
293	#131	—	2	16	no	—
294	#131	—	2	8	no	—
295	#131	—	2	8	no	—
296	C₂×M₁₆	—	2	16	no	—
297	C₂×M₁₆	—	2	1	no	—
298	C₂×M₁₆	—	2	2	no	—
299	C₂×M₁₆	—	2	2	no	—
300	C₂×M₁₆	—	2	2	no	—
301	C₂×M₁₆	—	2	2	no	—
302	C₂×M₁₆	—	2	2	no	—
303	C₂×M₁₆	—	2	1	no	—
304	C₂×M₁₆	—	2	4	no	—
305	C₂×M₁₆	—	2	2	no	—
306	C₂×M₁₆	—	2	2	no	—
307	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
308	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
309	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
310	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
311	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
312	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
313	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
314	C₂²×C₈	—	2	1	no	—
315	C₂²×C₈	—	2	4	no	—
316	C₂²×C₈	—	2	1	no	—
317	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
318	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
319	#117	—	2	1	no	—
320	#117	—	2	1	no	—
321	#117	C₂×C₄×C₆₇	1	2	yes	—
322	#117	C₂³×C₆₇	1	2	yes	—
323	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
324	C₂×C₂²⋊C₄	—	2	1	no	—
325	C₂×C₄²	—	2	1	no	—
326	C₂×C₄²	—	2	2	no	—
327	C₂×C₄²	—	2	2	no	—
328	C₂×C₄²	—	2	1	no	—
329	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
330	#108	—	2	8	no	—
331	C₄⋊C₈	—	2	8	no	—
332	#104	—	2	8	no	—
333	#103	C₂×C₄×C₆₇	1	8	yes	complemented
334	#102	C₂×C₄×C₆₇	1	8	yes	complemented
335	C₂⁴	—	2	1	no	—
336	C₂⁴	#41×C₆₇	1	1	yes	characteristic
337	#41	—	2	4	no	—
338	#41	—	2	4	no	—
339	#41	—	2	4	no	—
340	#41	—	2	4	no	—
341	#41	—	2	4	no	—
342	#41	—	2	2	no	—
343	#41	—	2	2	no	—
344	#41	C₄⋊C₄×C₆₇	1	8	yes	—
345	#41	C₂×D₄×C₆₇	1	4	yes	—
346	#41	C₂×Q₈×C₆₇	1	4	yes	—
347	C₂×Q₈	—	2	1	no	—
348	C₂×Q₈	—	2	2	no	—
349	C₂×Q₈	C₄⋊C₄×C₆₇	1	4	yes	—
350	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
351	C₂×D₄	—	2	2	no	—
352	C₂×D₄	—	2	4	no	—
353	C₂×D₄	—	2	4	no	—
354	C₂×D₄	—	2	2	no	—
355	C₂×D₄	—	2	2	no	—
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	C₄⋊C₄×C₆₇	1	4	yes	—
358	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	4	no	—
362	C₂²×C₄	—	2	4	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	4	4	no	—
367	C₂²×C₄	—	4	2	no	—
368	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
369	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
370	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
371	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
372	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
373	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
374	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
375	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
376	C₂²×C₄	#41×C₆₇	1	2	yes	—
377	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
378	M₁₆	—	2	16	no	—
379	M₁₆	—	2	4	no	—
380	M₁₆	—	2	4	no	—
381	M₁₆	—	2	4	no	—
382	M₁₆	—	2	4	no	—
383	M₁₆	—	2	4	no	—
384	M₁₆	—	2	2	no	—
385	M₁₆	—	2	2	no	—
386	M₁₆	—	2	2	no	—
387	M₁₆	—	4	2	no	—
388	M₁₆	C₂×D₄×C₆₇	1	4	yes	—
389	C₂×C₈	—	2	16	no	—
390	C₂×C₈	—	2	4	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	4	2	no	—
398	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
399	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
402	C₂²⋊C₄	—	2	2	no	—
403	C₂²⋊C₄	—	2	2	no	—
404	C₄²	—	2	2	no	—
405	C₄²	—	2	2	no	—
406	C₄²	—	2	2	no	—
407	C₄²	—	2	2	no	—
408	C₄²	—	4	2	no	—
409	C₂³	—	2	2	no	—
410	C₂³	—	2	4	no	—
411	C₂³	—	4	2	no	—
412	C₂³	—	8	2	no	—
413	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
414	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
415	C₂³	C₂²⋊Q₈×C₆₇	1	2	yes	—
416	C₂³	#123×C₆₇	1	2	yes	—
417	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
418	Q₈	—	2	2	no	—
419	Q₈	—	2	4	no	—
420	Q₈	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
421	D₄	—	2	4	no	—
422	D₄	—	2	4	no	—
423	D₄	—	2	8	no	—
424	D₄	—	2	2	no	—
425	D₄	—	2	4	no	—
426	D₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	4	no	—
429	C₂×C₄	—	2	4	no	—
430	C₂×C₄	—	2	4	no	—
431	C₂×C₄	—	2	8	no	—
432	C₂×C₄	—	2	4	no	—
433	C₂×C₄	—	2	4	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	8	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	8	2	no	—
438	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
439	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
440	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
441	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
442	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
443	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
444	C₂×C₄	#120×C₆₇	1	2	yes	—
445	C₂×C₄	#120×C₆₇	1	2	yes	—
446	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
447	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
448	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
449	C₂×C₄	#123×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	C₂²×D₄×C₆₇	1	1	yes	characteristic
453	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
454	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
455	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
456	C₈	—	2	4	no	—
457	C₈	—	2	4	no	—
458	C₈	—	2	4	no	—
459	C₈	—	4	2	no	—
460	C₂²	—	2	2	no	—
461	C₂²	—	2	2	no	—
462	C₂²	—	2	2	no	—
463	C₂²	—	2	4	no	—
464	C₂²	—	4	2	no	—
465	C₂²	—	4	2	no	—
466	C₂²	—	4	4	no	—
467	C₂²	—	8	4	no	—
468	C₂²	#385×C₆₇	1	2	yes	—
469	C₂²	#385×C₆₇	1	2	yes	—
470	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
471	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
472	C₂²	C₂×#123×C₆₇	1	1	yes	characteristic
473	C₄	—	2	4	no	—
474	C₄	—	4	4	no	—
475	C₄	—	4	4	no	—
476	C₄	#385×C₆₇	1	2	yes	—
477	C₄	#385×C₆₇	1	2	yes	—
478	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
479	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
480	C₄	C₂×#120×C₆₇	1	1	yes	characteristic
481	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
482	C₂	—	2	1	no	—
483	C₂	—	2	2	no	—
484	C₂	—	2	1	no	—
485	C₂	—	2	2	no	—
486	C₂	—	4	4	no	—
487	C₂	C₂×#385×C₆₇	1	1	yes	characteristic
488	C₂	C₆₇×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×M₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#430×C₆₇	C₂	semidirect
3. C₆₇×#2905	C₂	semidirect
6. C₆₇×#1866	C₂	semidirect
7. C₂×#339×C₆₇	C₂	semidirect
8. C₆₇×#1728	C₂	semidirect
22. C₂×#108×C₆₇	C₂²	semidirect
24. #424×C₆₇	C₂²	semidirect
28. #419×C₆₇	C₄	semidirect
29. #418×C₆₇	C₄	semidirect
30. #417×C₆₇	C₄	semidirect
32. #339×C₆₇	C₂²	semidirect
33. #336×C₆₇	C₂²	semidirect
88. #103×C₆₇	C₂×C₄	semidirect
89. #102×C₆₇	C₂×C₄	semidirect
237. #20885	C₆₇	direct (N × H)
246. C₂×#430	C₂×C₆₇	semidirect
247. #2905	C₂×C₆₇	semidirect
250. #1866	C₂×C₆₇	semidirect
251. C₂×#339	C₂×C₆₇	semidirect
252. #1728	C₂×C₆₇	semidirect
253. C₆₇	#20885	direct (N × H)
267. C₂×#108	C₂²×C₆₇	semidirect
269. #424	C₂²×C₆₇	semidirect
273. #419	C₄×C₆₇	semidirect
274. #418	C₄×C₆₇	semidirect
275. #417	C₄×C₆₇	semidirect
277. #339	C₂²×C₆₇	semidirect
278. #336	C₂²×C₆₇	semidirect
333. #103	C₂×C₄×C₆₇	semidirect
334. #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×#568×C₆₇	2
C₂×#430×C₆₇	2
C₆₇×#2905	2
C₂²×#104×C₆₇	2
C₆₇×#2896	2
C₆₇×#1866	2
C₂×#339×C₆₇	2
C₆₇×#1728	2
#20885	67

Subgroups — C67×#20885 #30700136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20885 #30700136