Subgroups — C₆₇×#20401 #30699652

Order 17152.

1444 conjugacy classes of subgroups.

The subgroup lattice has 434 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×#413×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#410×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2873	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1757	C₂	1	8	yes	maximal, complemented
6	C₂×#344×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
8	#568×C₆₇	—	2	4	no	—
9	#568×C₆₇	—	2	4	no	—
10	C₂×#131×C₆₇	C₄	1	4	yes	complemented
11	C₂×#131×C₆₇	C₂²	1	4	yes	—
12	C₂²×M₁₆×C₆₇	C₄	1	4	yes	complemented
13	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#117×C₆₇	C₂²	1	2	yes	—
17	#413×C₆₇	C₄	1	4	yes	complemented
18	#413×C₆₇	C₂²	1	4	yes	—
19	C₂×#101×C₆₇	C₄	1	2	yes	complemented
20	C₂×#100×C₆₇	C₄	1	2	yes	complemented
21	#410×C₆₇	C₄	1	4	yes	—
22	#410×C₆₇	C₂²	1	4	yes	—
23	C₂×#99×C₆₇	C₄	1	4	yes	—
24	#405×C₆₇	—	2	4	no	—
25	#405×C₆₇	C₂²	1	8	yes	complemented
26	C₄×M₁₆×C₆₇	—	2	4	no	—
27	#344×C₆₇	C₂²	1	8	yes	complemented
28	C₂×#41×C₆₇	—	2	2	no	—
29	C₂×#41×C₆₇	—	2	2	no	—
30	C₂×#41×C₆₇	—	2	2	no	—
31	C₂×#41×C₆₇	—	2	2	no	—
32	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
34	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
35	C₂×#41×C₆₇	C₂³	1	2	yes	—
36	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
38	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
40	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	#131×C₆₇	—	2	16	no	—
42	#131×C₆₇	—	2	16	no	—
43	C₂×M₁₆×C₆₇	—	2	2	no	—
44	C₂×M₁₆×C₆₇	—	2	2	no	—
45	C₂×M₁₆×C₆₇	—	2	2	no	—
46	C₂×M₁₆×C₆₇	—	2	2	no	—
47	C₂×M₁₆×C₆₇	—	2	2	no	—
48	C₂×M₁₆×C₆₇	—	2	8	no	—
49	C₂×M₁₆×C₆₇	—	2	8	no	—
50	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
51	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
52	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
53	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
54	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
55	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
56	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
57	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
58	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
59	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
60	C₂²×C₈×C₆₇	—	2	2	no	—
61	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
62	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
63	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
64	#117×C₆₇	—	2	2	no	—
65	#117×C₆₇	—	2	2	no	—
66	#117×C₆₇	C₂×C₄	1	4	yes	—
67	#117×C₆₇	C₂³	1	4	yes	—
68	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
70	C₂×C₄²×C₆₇	—	2	2	no	—
71	#101×C₆₇	C₂×C₄	1	8	yes	complemented
72	#100×C₆₇	C₂×C₄	1	8	yes	complemented
73	#99×C₆₇	C₂×C₄	1	16	yes	—
74	#97×C₆₇	—	2	8	no	—
75	C₄×C₈×C₆₇	—	2	8	no	—
76	C₂⁴×C₆₇	C₄²	1	2	yes	—
77	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	#41×C₆₇	—	2	4	no	—
79	#41×C₆₇	—	2	4	no	—
80	#41×C₆₇	—	2	4	no	—
81	#41×C₆₇	—	2	4	no	—
82	#41×C₆₇	—	2	4	no	—
83	#41×C₆₇	—	2	4	no	—
84	#41×C₆₇	—	2	4	no	—
85	#41×C₆₇	—	2	4	no	—
86	C₂×Q₈×C₆₇	—	2	2	no	—
87	C₂×Q₈×C₆₇	—	2	2	no	—
88	C₂×Q₈×C₆₇	C₄²	1	2	yes	—
89	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
90	C₂×D₄×C₆₇	—	2	4	no	—
91	C₂×D₄×C₆₇	—	2	4	no	—
92	C₂×D₄×C₆₇	—	2	2	no	—
93	C₂×D₄×C₆₇	—	2	2	no	—
94	C₂×D₄×C₆₇	C₄²	1	4	yes	—
95	C₂×D₄×C₆₇	C₄²	1	2	yes	—
96	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
97	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	—	2	4	no	—
103	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
104	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
105	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
106	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₆₇	#41	1	2	yes	—
114	C₂²×C₄×C₆₇	#41	1	2	yes	—
115	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
116	M₁₆×C₆₇	—	2	2	no	—
117	M₁₆×C₆₇	—	2	2	no	—
118	M₁₆×C₆₇	—	2	4	no	—
119	M₁₆×C₆₇	—	2	4	no	—
120	M₁₆×C₆₇	—	2	4	no	—
121	M₁₆×C₆₇	—	2	4	no	—
122	M₁₆×C₆₇	—	2	16	no	—
123	M₁₆×C₆₇	C₂²⋊C₄	1	4	yes	—
124	M₁₆×C₆₇	C₂²⋊C₄	1	4	yes	—
125	M₁₆×C₆₇	C₂²⋊C₄	1	8	yes	—
126	M₁₆×C₆₇	C₂²×C₄	1	8	yes	—
127	C₂×C₈×C₆₇	—	2	4	no	—
128	C₂×C₈×C₆₇	—	2	4	no	—
129	C₂×C₈×C₆₇	—	2	4	no	—
130	C₂×C₈×C₆₇	—	2	4	no	—
131	C₂×C₈×C₆₇	—	2	16	no	—
132	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
133	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
134	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
135	C₄⋊C₄×C₆₇	—	2	4	no	—
136	C₄⋊C₄×C₆₇	—	2	4	no	—
137	C₂²⋊C₄×C₆₇	—	2	4	no	—
138	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
139	C₄²×C₆₇	—	2	4	no	—
140	C₄²×C₆₇	—	2	4	no	—
141	C₂³×C₆₇	—	2	2	no	—
142	C₂³×C₆₇	—	2	2	no	—
143	C₂³×C₆₇	—	8	1	no	—
144	C₂³×C₆₇	—	8	2	no	—
145	C₂³×C₆₇	C₂×C₄²	1	4	yes	—
146	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
147	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
148	C₂³×C₆₇	C₄×D₄	1	2	yes	—
149	C₂³×C₆₇	C₄×D₄	1	2	yes	—
150	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
151	Q₈×C₆₇	—	2	4	no	—
152	Q₈×C₆₇	—	2	4	no	—
153	D₄×C₆₇	—	2	8	no	—
154	D₄×C₆₇	—	2	8	no	—
155	D₄×C₆₇	—	2	4	no	—
156	D₄×C₆₇	—	2	4	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	2	no	—
159	C₂×C₄×C₆₇	—	2	4	no	—
160	C₂×C₄×C₆₇	—	2	4	no	—
161	C₂×C₄×C₆₇	—	2	4	no	—
162	C₂×C₄×C₆₇	—	2	4	no	—
163	C₂×C₄×C₆₇	—	2	8	no	—
164	C₂×C₄×C₆₇	—	8	4	no	—
165	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
166	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
168	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
169	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
170	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
172	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
173	C₂×C₄×C₆₇	#117	1	2	yes	—
174	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
175	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
176	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
177	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
178	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
179	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
180	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
183	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
184	C₈×C₆₇	—	2	4	no	—
185	C₈×C₆₇	—	2	8	no	—
186	C₈×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
187	C₂²×C₆₇	—	2	4	no	—
188	C₂²×C₆₇	—	4	2	no	—
189	C₂²×C₆₇	—	4	1	no	—
190	C₂²×C₆₇	—	4	1	no	—
191	C₂²×C₆₇	—	4	2	no	—
192	C₂²×C₆₇	—	4	1	no	—
193	C₂²×C₆₇	—	4	4	no	—
194	C₂²×C₆₇	—	8	2	no	—
195	C₂²×C₆₇	—	8	2	no	—
196	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
197	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
198	C₂²×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
199	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
200	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
201	C₄×C₆₇	—	2	4	no	—
202	C₄×C₆₇	—	4	8	no	—
203	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
204	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
205	C₄×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
206	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
207	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
208	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
209	#20401	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
210	C₂×C₆₇	—	2	2	no	—
211	C₂×C₆₇	—	2	1	no	—
212	C₂×C₆₇	—	2	1	no	—
213	C₂×C₆₇	—	2	1	no	—
214	C₂×C₆₇	—	2	2	no	—
215	C₂×C₆₇	—	4	4	no	—
216	C₂×C₆₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
217	C₂×#568	C₂×C₆₇	1	1	yes	characteristic
218	C₂×#413	C₂×C₆₇	1	1	yes	characteristic
219	C₂×#410	C₂×C₆₇	1	1	yes	characteristic
220	#2873	C₂×C₆₇	1	2	yes	complemented
221	#1757	C₂×C₆₇	1	8	yes	complemented
222	C₂×#344	C₂×C₆₇	1	2	yes	complemented
223	C₆₇	#20401	1	1	yes	characteristic, direct product factor, Sylow
224	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
225	#568	—	2	4	no	—
226	#568	—	2	4	no	—
227	C₂×#131	C₄×C₆₇	1	4	yes	complemented
228	C₂×#131	C₂²×C₆₇	1	4	yes	—
229	C₂²×M₁₆	C₄×C₆₇	1	4	yes	complemented
230	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
231	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
232	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
233	C₂×#117	C₂²×C₆₇	1	2	yes	—
234	#413	C₄×C₆₇	1	4	yes	complemented
235	#413	C₂²×C₆₇	1	4	yes	—
236	C₂×#101	C₄×C₆₇	1	2	yes	complemented
237	C₂×#100	C₄×C₆₇	1	2	yes	complemented
238	#410	C₄×C₆₇	1	4	yes	—
239	#410	C₂²×C₆₇	1	4	yes	—
240	C₂×#99	C₄×C₆₇	1	4	yes	—
241	#405	—	2	4	no	—
242	#405	C₂²×C₆₇	1	8	yes	complemented
243	C₄×M₁₆	—	2	4	no	—
244	#344	C₂²×C₆₇	1	8	yes	complemented
245	C₂×#41	—	2	2	no	—
246	C₂×#41	—	2	2	no	—
247	C₂×#41	—	2	2	no	—
248	C₂×#41	—	2	2	no	—
249	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
250	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
251	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
252	C₂×#41	C₂³×C₆₇	1	2	yes	—
253	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
254	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
255	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
256	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
257	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
258	#131	—	2	16	no	—
259	#131	—	2	16	no	—
260	C₂×M₁₆	—	2	2	no	—
261	C₂×M₁₆	—	2	2	no	—
262	C₂×M₁₆	—	2	2	no	—
263	C₂×M₁₆	—	2	2	no	—
264	C₂×M₁₆	—	2	2	no	—
265	C₂×M₁₆	—	2	8	no	—
266	C₂×M₁₆	—	2	8	no	—
267	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
268	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
269	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
270	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
271	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
272	C₂×M₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
273	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
274	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
275	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
276	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
277	C₂²×C₈	—	2	2	no	—
278	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
279	C₂²×C₈	C₂×C₄×C₆₇	1	8	yes	complemented
280	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
281	#117	—	2	2	no	—
282	#117	—	2	2	no	—
283	#117	C₂×C₄×C₆₇	1	4	yes	—
284	#117	C₂³×C₆₇	1	4	yes	—
285	C₂×C₄⋊C₄	—	2	2	no	—
286	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
287	C₂×C₄²	—	2	2	no	—
288	#101	C₂×C₄×C₆₇	1	8	yes	complemented
289	#100	C₂×C₄×C₆₇	1	8	yes	complemented
290	#99	C₂×C₄×C₆₇	1	16	yes	—
291	#97	—	2	8	no	—
292	C₄×C₈	—	2	8	no	—
293	C₂⁴	C₄²×C₆₇	1	2	yes	—
294	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
295	#41	—	2	4	no	—
296	#41	—	2	4	no	—
297	#41	—	2	4	no	—
298	#41	—	2	4	no	—
299	#41	—	2	4	no	—
300	#41	—	2	4	no	—
301	#41	—	2	4	no	—
302	#41	—	2	4	no	—
303	C₂×Q₈	—	2	2	no	—
304	C₂×Q₈	—	2	2	no	—
305	C₂×Q₈	C₄²×C₆₇	1	2	yes	—
306	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
307	C₂×D₄	—	2	4	no	—
308	C₂×D₄	—	2	4	no	—
309	C₂×D₄	—	2	2	no	—
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	C₄²×C₆₇	1	4	yes	—
312	C₂×D₄	C₄²×C₆₇	1	2	yes	—
313	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
314	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	—	2	4	no	—
320	C₂²×C₄	C₄²×C₆₇	1	2	yes	—
321	C₂²×C₄	C₄²×C₆₇	1	4	yes	—
322	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
323	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
324	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
325	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
326	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
327	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
328	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
329	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
330	C₂²×C₄	#41×C₆₇	1	2	yes	—
331	C₂²×C₄	#41×C₆₇	1	2	yes	—
332	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
333	M₁₆	—	2	2	no	—
334	M₁₆	—	2	2	no	—
335	M₁₆	—	2	4	no	—
336	M₁₆	—	2	4	no	—
337	M₁₆	—	2	4	no	—
338	M₁₆	—	2	4	no	—
339	M₁₆	—	2	16	no	—
340	M₁₆	C₂²⋊C₄×C₆₇	1	4	yes	—
341	M₁₆	C₂²⋊C₄×C₆₇	1	4	yes	—
342	M₁₆	C₂²⋊C₄×C₆₇	1	8	yes	—
343	M₁₆	C₂²×C₄×C₆₇	1	8	yes	—
344	C₂×C₈	—	2	4	no	—
345	C₂×C₈	—	2	4	no	—
346	C₂×C₈	—	2	4	no	—
347	C₂×C₈	—	2	4	no	—
348	C₂×C₈	—	2	16	no	—
349	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
350	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
351	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
352	C₄⋊C₄	—	2	4	no	—
353	C₄⋊C₄	—	2	4	no	—
354	C₂²⋊C₄	—	2	4	no	—
355	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
356	C₄²	—	2	4	no	—
357	C₄²	—	2	4	no	—
358	C₂³	—	2	2	no	—
359	C₂³	—	2	2	no	—
360	C₂³	—	8	1	no	—
361	C₂³	—	8	2	no	—
362	C₂³	C₂×C₄²×C₆₇	1	4	yes	—
363	C₂³	C₂×C₄²×C₆₇	1	1	yes	characteristic
364	C₂³	C₂×C₄²×C₆₇	1	1	yes	characteristic
365	C₂³	C₄×D₄×C₆₇	1	2	yes	—
366	C₂³	C₄×D₄×C₆₇	1	2	yes	—
367	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
368	Q₈	—	2	4	no	—
369	Q₈	—	2	4	no	—
370	D₄	—	2	8	no	—
371	D₄	—	2	8	no	—
372	D₄	—	2	4	no	—
373	D₄	—	2	4	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	4	no	—
377	C₂×C₄	—	2	4	no	—
378	C₂×C₄	—	2	4	no	—
379	C₂×C₄	—	2	4	no	—
380	C₂×C₄	—	2	8	no	—
381	C₂×C₄	—	8	4	no	—
382	C₂×C₄	C₂×C₄²×C₆₇	1	4	yes	—
383	C₂×C₄	C₂×C₄²×C₆₇	1	1	yes	characteristic
384	C₂×C₄	C₂×C₄²×C₆₇	1	1	yes	characteristic
385	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
386	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
387	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
388	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
389	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
390	C₂×C₄	#117×C₆₇	1	2	yes	—
391	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
392	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
393	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
394	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
395	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
396	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
397	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
398	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
399	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
400	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
401	C₈	—	2	4	no	—
402	C₈	—	2	8	no	—
403	C₈	C₂×C₂²⋊C₄×C₆₇	1	8	yes	—
404	C₂²	—	2	4	no	—
405	C₂²	—	4	2	no	—
406	C₂²	—	4	1	no	—
407	C₂²	—	4	1	no	—
408	C₂²	—	4	2	no	—
409	C₂²	—	4	1	no	—
410	C₂²	—	4	4	no	—
411	C₂²	—	8	2	no	—
412	C₂²	—	8	2	no	—
413	C₂²	C₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
414	C₂²	C₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
415	C₂²	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
416	C₂²	C₂²×C₄²×C₆₇	1	1	yes	characteristic
417	C₂²	C₂×C₄×D₄×C₆₇	1	2	yes	—
418	C₄	—	2	4	no	—
419	C₄	—	4	8	no	—
420	C₄	C₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
421	C₄	C₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
422	C₄	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
423	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
424	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
425	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
426	C₂	—	2	2	no	—
427	C₂	—	2	1	no	—
428	C₂	—	2	1	no	—
429	C₂	—	2	1	no	—
430	C₂	—	2	2	no	—
431	C₂	—	4	4	no	—
432	C₂	C₂×C₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₆₇	C₂²×C₂²⋊C₄
commutator	C₂²	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂×C₆₇	C₂×C₄×C₂²⋊C₄
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₆₇ ◁ C₂²×M₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#2873	C₂	semidirect
5. C₆₇×#1757	C₂	semidirect
6. C₂×#344×C₆₇	C₂	semidirect
10. C₂×#131×C₆₇	C₄	semidirect
12. C₂²×M₁₆×C₆₇	C₄	semidirect
17. #413×C₆₇	C₄	semidirect
19. C₂×#101×C₆₇	C₄	semidirect
20. C₂×#100×C₆₇	C₄	semidirect
25. #405×C₆₇	C₂²	semidirect
27. #344×C₆₇	C₂²	semidirect
55. C₂×M₁₆×C₆₇	C₂×C₄	semidirect
62. C₂²×C₈×C₆₇	C₂×C₄	semidirect
71. #101×C₆₇	C₂×C₄	semidirect
72. #100×C₆₇	C₂×C₄	semidirect
209. #20401	C₆₇	direct (N × H)
220. #2873	C₂×C₆₇	semidirect
221. #1757	C₂×C₆₇	semidirect
222. C₂×#344	C₂×C₆₇	semidirect
223. C₆₇	#20401	direct (N × H)
227. C₂×#131	C₄×C₆₇	semidirect
229. C₂²×M₁₆	C₄×C₆₇	semidirect
234. #413	C₄×C₆₇	semidirect
236. C₂×#101	C₄×C₆₇	semidirect
237. C₂×#100	C₄×C₆₇	semidirect
242. #405	C₂²×C₆₇	semidirect
244. #344	C₂²×C₆₇	semidirect
272. C₂×M₁₆	C₂×C₄×C₆₇	semidirect
279. C₂²×C₈	C₂×C₄×C₆₇	semidirect
288. #101	C₂×C₄×C₆₇	semidirect
289. #100	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×#413×C₆₇	2
C₂×#410×C₆₇	2
C₆₇×#2873	2
C₆₇×#1757	2
C₂×#344×C₆₇	2
#20401	67

Subgroups — C67×#20401 #30699652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20401 #30699652