Subgroups — C₆₇×#22136 #30701387

Order 17152.

896 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#433×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2392	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1955	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1955	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1940	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1924	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1869	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1850	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1798	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1770	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
15	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
16	C₈×D₄×C₆₇	—	2	4	no	—
17	#433×C₆₇	—	2	4	no	—
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
29	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#97×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#398×C₆₇	C₄	1	1	yes	characteristic
33	#397×C₆₇	C₄	1	1	yes	characteristic
34	#396×C₆₇	C₄	1	2	yes	—
35	#394×C₆₇	C₄	1	2	yes	—
36	#394×C₆₇	C₄	1	1	yes	characteristic
37	#392×C₆₇	C₄	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
40	#336×C₆₇	C₂²	1	2	yes	—
41	#336×C₆₇	C₂²	1	2	yes	—
42	#336×C₆₇	C₂²	1	2	yes	—
43	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
45	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₆₇	—	2	4	no	—
48	C₂²×C₈×C₆₇	—	4	2	no	—
49	C₂²×C₈×C₆₇	—	4	2	no	—
50	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₄×D₄×C₆₇	—	2	2	no	—
57	C₄×D₄×C₆₇	—	2	2	no	—
58	C₄×D₄×C₆₇	—	4	2	no	—
59	C₄×D₄×C₆₇	—	4	2	no	—
60	#117×C₆₇	—	2	2	no	—
61	#117×C₆₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
67	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
70	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
71	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
72	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
73	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₆₇	—	2	2	no	—
78	C₄⋊C₈×C₆₇	—	2	2	no	—
79	C₄⋊C₈×C₆₇	—	2	2	no	—
80	C₄⋊C₈×C₆₇	—	2	2	no	—
81	C₄⋊C₈×C₆₇	—	4	2	no	—
82	C₂²⋊C₈×C₆₇	—	2	2	no	—
83	C₂²⋊C₈×C₆₇	—	2	2	no	—
84	C₂²⋊C₈×C₆₇	—	2	4	no	—
85	C₂²⋊C₈×C₆₇	—	4	2	no	—
86	C₂²⋊C₈×C₆₇	—	4	2	no	—
87	C₂²⋊C₈×C₆₇	—	4	2	no	—
88	#97×C₆₇	—	2	2	no	—
89	#97×C₆₇	—	4	2	no	—
90	C₄×C₈×C₆₇	D₄	1	4	yes	complemented
91	#95×C₆₇	C₂×C₄	1	2	yes	—
92	#95×C₆₇	C₂×C₄	1	2	yes	—
93	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
94	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
95	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂×D₄×C₆₇	—	2	4	no	—
98	C₂×D₄×C₆₇	—	2	4	no	—
99	C₂×D₄×C₆₇	—	2	2	no	—
100	C₂×D₄×C₆₇	—	4	2	no	—
101	C₂×D₄×C₆₇	—	4	2	no	—
102	C₂²×C₄×C₆₇	—	2	1	no	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	2	1	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	1	no	—
107	C₂²×C₄×C₆₇	—	2	1	no	—
108	C₂²×C₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	2	1	no	—
112	C₂²×C₄×C₆₇	—	2	2	no	—
113	C₂²×C₄×C₆₇	—	2	1	no	—
114	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
118	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
120	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
123	C₂×C₈×C₆₇	—	2	1	no	—
124	C₂×C₈×C₆₇	—	2	1	no	—
125	C₂×C₈×C₆₇	—	2	1	no	—
126	C₂×C₈×C₆₇	—	2	1	no	—
127	C₂×C₈×C₆₇	—	2	2	no	—
128	C₂×C₈×C₆₇	—	2	1	no	—
129	C₂×C₈×C₆₇	—	2	1	no	—
130	C₂×C₈×C₆₇	—	2	2	no	—
131	C₂×C₈×C₆₇	—	2	1	no	—
132	C₂×C₈×C₆₇	—	2	2	no	—
133	C₂×C₈×C₆₇	—	2	2	no	—
134	C₂×C₈×C₆₇	—	2	2	no	—
135	C₂×C₈×C₆₇	—	2	1	no	—
136	C₂×C₈×C₆₇	—	2	1	no	—
137	C₂×C₈×C₆₇	—	2	2	no	—
138	C₂×C₈×C₆₇	—	2	1	no	—
139	C₂×C₈×C₆₇	—	4	8	no	—
140	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
141	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
142	C₄⋊C₄×C₆₇	—	2	2	no	—
143	C₄⋊C₄×C₆₇	—	2	2	no	—
144	C₄⋊C₄×C₆₇	—	2	2	no	—
145	C₄⋊C₄×C₆₇	—	4	2	no	—
146	C₄⋊C₄×C₆₇	—	4	2	no	—
147	C₂²⋊C₄×C₆₇	—	2	4	no	—
148	C₂²⋊C₄×C₆₇	—	2	2	no	—
149	C₂²⋊C₄×C₆₇	—	2	2	no	—
150	C₂²⋊C₄×C₆₇	—	2	4	no	—
151	C₂²⋊C₄×C₆₇	—	4	2	no	—
152	C₂²⋊C₄×C₆₇	—	4	2	no	—
153	C₂²⋊C₄×C₆₇	—	4	2	no	—
154	C₄²×C₆₇	—	2	1	no	—
155	C₄²×C₆₇	—	2	1	no	—
156	C₄²×C₆₇	—	2	2	no	—
157	C₄²×C₆₇	—	2	2	no	—
158	C₄²×C₆₇	C₂×D₄	1	2	yes	—
159	C₄²×C₆₇	#41	1	2	yes	—
160	C₂³×C₆₇	—	2	2	no	—
161	C₂³×C₆₇	—	2	2	no	—
162	C₂³×C₆₇	—	2	2	no	—
163	C₂³×C₆₇	—	2	2	no	—
164	C₂³×C₆₇	—	2	2	no	—
165	C₂³×C₆₇	—	2	2	no	—
166	C₂³×C₆₇	—	2	1	no	—
167	C₂³×C₆₇	—	2	1	no	—
168	C₂³×C₆₇	—	2	1	no	—
169	C₂³×C₆₇	—	2	2	no	—
170	C₂³×C₆₇	—	2	2	no	—
171	C₂³×C₆₇	#131	1	4	yes	—
172	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
173	D₄×C₆₇	—	2	8	no	—
174	D₄×C₆₇	—	4	8	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	1	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	2	1	no	—
181	C₂×C₄×C₆₇	—	2	2	no	—
182	C₂×C₄×C₆₇	—	2	2	no	—
183	C₂×C₄×C₆₇	—	2	1	no	—
184	C₂×C₄×C₆₇	—	2	2	no	—
185	C₂×C₄×C₆₇	—	2	2	no	—
186	C₂×C₄×C₆₇	—	2	1	no	—
187	C₂×C₄×C₆₇	—	2	2	no	—
188	C₂×C₄×C₆₇	—	2	2	no	—
189	C₂×C₄×C₆₇	—	2	2	no	—
190	C₂×C₄×C₆₇	—	2	2	no	—
191	C₂×C₄×C₆₇	—	2	2	no	—
192	C₂×C₄×C₆₇	—	2	2	no	—
193	C₂×C₄×C₆₇	—	4	2	no	—
194	C₂×C₄×C₆₇	—	4	1	no	—
195	C₂×C₄×C₆₇	—	4	2	no	—
196	C₂×C₄×C₆₇	—	4	1	no	—
197	C₂×C₄×C₆₇	—	4	2	no	—
198	C₂×C₄×C₆₇	—	4	1	no	—
199	C₂×C₄×C₆₇	—	4	2	no	—
200	C₂×C₄×C₆₇	—	4	1	no	—
201	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
202	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
203	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
204	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
205	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
206	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
207	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
208	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
209	C₈×C₆₇	—	2	2	no	—
210	C₈×C₆₇	—	2	2	no	—
211	C₈×C₆₇	—	4	1	no	—
212	C₈×C₆₇	—	4	2	no	—
213	C₈×C₆₇	—	4	1	no	—
214	C₈×C₆₇	—	4	2	no	—
215	C₂²×C₆₇	—	2	4	no	—
216	C₂²×C₆₇	—	2	4	no	—
217	C₂²×C₆₇	—	2	4	no	—
218	C₂²×C₆₇	—	4	2	no	—
219	C₂²×C₆₇	—	4	2	no	—
220	C₂²×C₆₇	—	4	2	no	—
221	C₂²×C₆₇	—	4	2	no	—
222	C₂²×C₆₇	—	4	1	no	—
223	C₂²×C₆₇	—	4	1	no	—
224	C₂²×C₆₇	—	4	1	no	—
225	C₂²×C₆₇	—	4	2	no	—
226	C₂²×C₆₇	—	4	2	no	—
227	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
228	C₂²×C₆₇	#518	1	1	yes	characteristic
229	C₂²×C₆₇	#520	1	1	yes	characteristic
230	C₂²×C₆₇	C₂×#131	1	2	yes	—
231	C₂²×C₆₇	#568	1	2	yes	—
232	C₄×C₆₇	—	2	2	no	—
233	C₄×C₆₇	—	2	2	no	—
234	C₄×C₆₇	—	4	2	no	—
235	C₄×C₆₇	—	4	1	no	—
236	C₄×C₆₇	—	4	1	no	—
237	C₄×C₆₇	—	4	1	no	—
238	C₄×C₆₇	—	4	2	no	—
239	C₄×C₆₇	—	4	2	no	—
240	C₄×C₆₇	#535	1	1	yes	characteristic
241	C₄×C₆₇	#537	1	1	yes	characteristic
242	C₄×C₆₇	#540	1	1	yes	characteristic
243	C₄×C₆₇	#542	1	1	yes	characteristic
244	#22136	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
245	C₂×C₆₇	—	4	4	no	—
246	C₂×C₆₇	—	8	1	no	—
247	C₂×C₆₇	#2324	1	1	yes	characteristic
248	C₂×C₆₇	#2929	1	1	yes	characteristic
249	C₂×C₆₇	#2932	1	1	yes	characteristic
250	C₂×C₆₇	#2976	1	2	yes	—
251	C₂×C₆₇	#2992	1	2	yes	—
252	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
253	C₂×#433	C₂×C₆₇	1	1	yes	characteristic, complemented
254	#2392	C₂×C₆₇	1	1	yes	characteristic
255	#1955	C₂×C₆₇	1	1	yes	characteristic, complemented
256	#1955	C₂×C₆₇	1	2	yes	complemented
257	#1940	C₂×C₆₇	1	1	yes	characteristic, complemented
258	#1924	C₂×C₆₇	1	2	yes	complemented
259	#1869	C₂×C₆₇	1	2	yes	complemented
260	#1850	C₂×C₆₇	1	2	yes	complemented
261	#1798	C₂×C₆₇	1	1	yes	characteristic, complemented
262	#1770	C₂×C₆₇	1	1	yes	characteristic, complemented
263	C₆₇	#22136	1	1	yes	characteristic, direct product factor, Sylow
264	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
265	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
266	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
267	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
268	C₈×D₄	—	2	4	no	—
269	#433	—	2	4	no	—
270	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
271	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
272	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
273	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
274	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
275	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
276	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
277	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
278	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
279	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
280	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
281	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
282	C₂×#97	C₂²×C₆₇	1	2	yes	—
283	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
284	#398	C₄×C₆₇	1	1	yes	characteristic
285	#397	C₄×C₆₇	1	1	yes	characteristic
286	#396	C₄×C₆₇	1	2	yes	—
287	#394	C₄×C₆₇	1	2	yes	—
288	#394	C₄×C₆₇	1	1	yes	characteristic
289	#392	C₄×C₆₇	1	1	yes	characteristic
290	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
291	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
292	#336	C₂²×C₆₇	1	2	yes	—
293	#336	C₂²×C₆₇	1	2	yes	—
294	#336	C₂²×C₆₇	1	2	yes	—
295	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
296	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
297	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
298	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
299	C₂²×C₈	—	2	4	no	—
300	C₂²×C₈	—	4	2	no	—
301	C₂²×C₈	—	4	2	no	—
302	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
303	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
304	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
305	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
306	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
307	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
308	C₄×D₄	—	2	2	no	—
309	C₄×D₄	—	2	2	no	—
310	C₄×D₄	—	4	2	no	—
311	C₄×D₄	—	4	2	no	—
312	#117	—	2	2	no	—
313	#117	—	2	2	no	—
314	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
315	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
316	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
317	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
318	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
319	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
320	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
321	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
322	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
323	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
324	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
325	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
326	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
327	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
328	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
329	C₄⋊C₈	—	2	2	no	—
330	C₄⋊C₈	—	2	2	no	—
331	C₄⋊C₈	—	2	2	no	—
332	C₄⋊C₈	—	2	2	no	—
333	C₄⋊C₈	—	4	2	no	—
334	C₂²⋊C₈	—	2	2	no	—
335	C₂²⋊C₈	—	2	2	no	—
336	C₂²⋊C₈	—	2	4	no	—
337	C₂²⋊C₈	—	4	2	no	—
338	C₂²⋊C₈	—	4	2	no	—
339	C₂²⋊C₈	—	4	2	no	—
340	#97	—	2	2	no	—
341	#97	—	4	2	no	—
342	C₄×C₈	D₄×C₆₇	1	4	yes	complemented
343	#95	C₂×C₄×C₆₇	1	2	yes	—
344	#95	C₂×C₄×C₆₇	1	2	yes	—
345	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
346	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
347	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
348	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
349	C₂×D₄	—	2	4	no	—
350	C₂×D₄	—	2	4	no	—
351	C₂×D₄	—	2	2	no	—
352	C₂×D₄	—	4	2	no	—
353	C₂×D₄	—	4	2	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
367	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
368	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
369	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
370	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
371	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
372	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
373	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
374	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	—	2	1	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	2	1	no	—
391	C₂×C₈	—	4	8	no	—
392	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
393	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
394	C₄⋊C₄	—	2	2	no	—
395	C₄⋊C₄	—	2	2	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	—	4	2	no	—
398	C₄⋊C₄	—	4	2	no	—
399	C₂²⋊C₄	—	2	4	no	—
400	C₂²⋊C₄	—	2	2	no	—
401	C₂²⋊C₄	—	2	2	no	—
402	C₂²⋊C₄	—	2	4	no	—
403	C₂²⋊C₄	—	4	2	no	—
404	C₂²⋊C₄	—	4	2	no	—
405	C₂²⋊C₄	—	4	2	no	—
406	C₄²	—	2	1	no	—
407	C₄²	—	2	1	no	—
408	C₄²	—	2	2	no	—
409	C₄²	—	2	2	no	—
410	C₄²	C₂×D₄×C₆₇	1	2	yes	—
411	C₄²	#41×C₆₇	1	2	yes	—
412	C₂³	—	2	2	no	—
413	C₂³	—	2	2	no	—
414	C₂³	—	2	2	no	—
415	C₂³	—	2	2	no	—
416	C₂³	—	2	2	no	—
417	C₂³	—	2	2	no	—
418	C₂³	—	2	1	no	—
419	C₂³	—	2	1	no	—
420	C₂³	—	2	1	no	—
421	C₂³	—	2	2	no	—
422	C₂³	—	2	2	no	—
423	C₂³	#131×C₆₇	1	4	yes	—
424	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
425	D₄	—	2	8	no	—
426	D₄	—	4	8	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
454	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
455	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
456	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
457	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
458	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
459	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
460	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
461	C₈	—	2	2	no	—
462	C₈	—	2	2	no	—
463	C₈	—	4	2	no	—
464	C₈	—	4	1	no	—
465	C₈	—	4	1	no	—
466	C₈	—	4	2	no	—
467	C₂²	—	2	4	no	—
468	C₂²	—	2	4	no	—
469	C₂²	—	2	4	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	4	2	no	—
473	C₂²	—	4	2	no	—
474	C₂²	—	4	1	no	—
475	C₂²	—	4	1	no	—
476	C₂²	—	4	1	no	—
477	C₂²	—	4	2	no	—
478	C₂²	—	4	2	no	—
479	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
480	C₂²	#518×C₆₇	1	1	yes	characteristic
481	C₂²	#520×C₆₇	1	1	yes	characteristic
482	C₂²	C₂×#131×C₆₇	1	2	yes	—
483	C₂²	#568×C₆₇	1	2	yes	—
484	C₄	—	2	2	no	—
485	C₄	—	2	2	no	—
486	C₄	—	4	2	no	—
487	C₄	—	4	1	no	—
488	C₄	—	4	1	no	—
489	C₄	—	4	1	no	—
490	C₄	—	4	2	no	—
491	C₄	—	4	2	no	—
492	C₄	#535×C₆₇	1	1	yes	characteristic
493	C₄	#537×C₆₇	1	1	yes	characteristic
494	C₄	#540×C₆₇	1	1	yes	characteristic
495	C₄	#542×C₆₇	1	1	yes	characteristic
496	C₂	—	4	4	no	—
497	C₂	—	8	1	no	—
498	C₂	C₆₇×#2324	1	1	yes	characteristic
499	C₂	C₆₇×#2929	1	1	yes	characteristic
500	C₂	C₆₇×#2932	1	1	yes	characteristic
501	C₂	C₆₇×#2976	1	2	yes	—
502	C₂	C₆₇×#2992	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	#392×C₆₇	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₈×D₄×C₆₇	C₂	semidirect
2. C₂×#433×C₆₇	C₂	semidirect
4. C₆₇×#1955	C₂	semidirect
5. C₆₇×#1955	C₂	semidirect
6. C₆₇×#1940	C₂	semidirect
7. C₆₇×#1924	C₂	semidirect
8. C₆₇×#1869	C₂	semidirect
9. C₆₇×#1850	C₂	semidirect
10. C₆₇×#1798	C₂	semidirect
11. C₆₇×#1770	C₂	semidirect
90. C₄×C₈×C₆₇	D₄	semidirect
244. #22136	C₆₇	direct (N × H)
252. C₂×C₈×D₄	C₂×C₆₇	semidirect
253. C₂×#433	C₂×C₆₇	semidirect
255. #1955	C₂×C₆₇	semidirect
256. #1955	C₂×C₆₇	semidirect
257. #1940	C₂×C₆₇	semidirect
258. #1924	C₂×C₆₇	semidirect
259. #1869	C₂×C₆₇	semidirect
260. #1850	C₂×C₆₇	semidirect
261. #1798	C₂×C₆₇	semidirect
262. #1770	C₂×C₆₇	semidirect
263. C₆₇	#22136	direct (N × H)
342. C₄×C₈	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₆₇	2
C₂×#433×C₆₇	2
C₆₇×#2392	2
C₆₇×#1955	2
C₆₇×#1940	2
C₆₇×#1924	2
C₆₇×#1869	2
C₆₇×#1850	2
C₆₇×#1798	2
C₆₇×#1770	2
#22136	67

Subgroups — C67×#22136 #30701387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22136 #30701387