Subgroups — C₆₇×#22894 #30702145

Order 17152.

998 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#456×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2947	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2898	C₂	1	2	yes	maximal
4	C₆₇×#1956	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1930	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1908	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1886	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1859	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1764	C₂	1	2	yes	maximal, complemented
10	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#137×C₆₇	C₂²	1	2	yes	—
12	C₂×#131×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	#520×C₆₇	C₂²	1	2	yes	—
15	#519×C₆₇	C₂²	1	1	yes	characteristic
16	#456×C₆₇	C₄	1	8	yes	complemented
17	#441×C₆₇	—	2	2	no	—
18	#440×C₆₇	—	2	2	no	—
19	C₄×Q₁₆×C₆₇	—	2	2	no	—
20	C₄×SD₁₆×C₆₇	—	2	2	no	—
21	#428×C₆₇	C₂²	1	2	yes	—
22	C₂×#104×C₆₇	C₂²	1	2	yes	—
23	#419×C₆₇	—	2	2	no	—
24	#419×C₆₇	C₂²	1	2	yes	—
25	#417×C₆₇	—	2	2	no	—
26	#417×C₆₇	C₂²	1	2	yes	—
27	#416×C₆₇	—	2	2	no	—
28	#416×C₆₇	C₂²	1	2	yes	—
29	C₂×#103×C₆₇	—	2	2	no	—
30	C₂×#103×C₆₇	C₂²	1	2	yes	—
31	C₂×#103×C₆₇	C₂²	1	2	yes	—
32	C₂×#101×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×#101×C₆₇	C₂²	1	1	yes	characteristic
34	#405×C₆₇	C₂²	1	2	yes	—
35	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
36	#384×C₆₇	C₂²	1	1	yes	characteristic
37	#343×C₆₇	C₂²	1	2	yes	—
38	#341×C₆₇	C₂²	1	2	yes	—
39	#339×C₆₇	C₂²	1	2	yes	—
40	2₋¹⁺⁴×C₆₇	—	2	4	no	—
41	2₋¹⁺⁴×C₆₇	—	4	1	no	—
42	2₋¹⁺⁴×C₆₇	C₂×C₄	1	4	yes	—
43	C₂×#41×C₆₇	—	2	2	no	—
44	C₂×#41×C₆₇	—	4	1	no	—
45	C₂×#41×C₆₇	C₂³	1	2	yes	—
46	C₂²×Q₈×C₆₇	—	2	2	no	—
47	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
49	#137×C₆₇	—	2	4	no	—
50	#137×C₆₇	C₂×C₄	1	8	yes	complemented
51	C₂×Q₁₆×C₆₇	—	2	2	no	—
52	C₂×SD₁₆×C₆₇	—	2	2	no	—
53	#131×C₆₇	—	2	4	no	—
54	#131×C₆₇	—	4	2	no	—
55	C₂×M₁₆×C₆₇	—	2	2	no	—
56	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
57	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
58	C₂²×C₈×C₆₇	—	2	2	no	—
59	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
60	C₄⋊Q₈×C₆₇	—	4	1	no	—
61	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
62	C₄×Q₈×C₆₇	—	2	2	no	—
63	C₄×Q₈×C₆₇	—	2	2	no	—
64	C₄×Q₈×C₆₇	—	2	2	no	—
65	C₄×D₄×C₆₇	—	2	2	no	—
66	C₄×D₄×C₆₇	—	4	2	no	—
67	#117×C₆₇	—	2	2	no	—
68	#117×C₆₇	—	4	1	no	—
69	#117×C₆₇	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
71	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
72	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
73	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	#107×C₆₇	—	2	2	no	—
76	#106×C₆₇	—	2	2	no	—
77	#104×C₆₇	C₂×C₄	1	8	yes	complemented
78	#103×C₆₇	—	2	2	no	—
79	#103×C₆₇	—	2	2	no	—
80	#103×C₆₇	—	2	2	no	—
81	#103×C₆₇	—	2	2	no	—
82	#103×C₆₇	—	2	2	no	—
83	#103×C₆₇	—	2	2	no	—
84	#103×C₆₇	—	2	2	no	—
85	#103×C₆₇	D₄	1	2	yes	—
86	#103×C₆₇	D₄	1	2	yes	—
87	#102×C₆₇	—	2	2	no	—
88	#102×C₆₇	D₄	1	2	yes	—
89	#102×C₆₇	D₄	1	2	yes	—
90	#101×C₆₇	—	4	1	no	—
91	#101×C₆₇	C₂×C₄	1	4	yes	complemented
92	#97×C₆₇	—	2	2	no	—
93	C₄×C₈×C₆₇	—	2	2	no	—
94	C₄×C₈×C₆₇	—	4	1	no	—
95	#41×C₆₇	—	2	2	no	—
96	#41×C₆₇	—	2	2	no	—
97	#41×C₆₇	—	2	8	no	—
98	#41×C₆₇	—	4	4	no	—
99	#41×C₆₇	—	4	4	no	—
100	#41×C₆₇	—	4	4	no	—
101	#41×C₆₇	C₂²×C₄	1	4	yes	—
102	#41×C₆₇	#41	1	4	yes	—
103	C₂×Q₈×C₆₇	—	2	1	no	—
104	C₂×Q₈×C₆₇	—	2	1	no	—
105	C₂×Q₈×C₆₇	—	2	8	no	—
106	C₂×Q₈×C₆₇	—	2	2	no	—
107	C₂×Q₈×C₆₇	—	2	2	no	—
108	C₂×Q₈×C₆₇	—	2	2	no	—
109	C₂×Q₈×C₆₇	—	2	2	no	—
110	C₂×Q₈×C₆₇	—	4	1	no	—
111	C₂×Q₈×C₆₇	—	4	4	no	—
112	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
113	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
114	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
115	C₂×D₄×C₆₇	—	2	2	no	—
116	C₂×D₄×C₆₇	—	4	1	no	—
117	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
118	C₂²×C₄×C₆₇	—	2	1	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	—	2	1	no	—
121	C₂²×C₄×C₆₇	—	2	1	no	—
122	C₂²×C₄×C₆₇	—	2	2	no	—
123	C₂²×C₄×C₆₇	—	4	2	no	—
124	C₂²×C₄×C₆₇	—	4	2	no	—
125	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
126	Q₁₆×C₆₇	—	2	8	no	—
127	SD₁₆×C₆₇	—	2	8	no	—
128	M₁₆×C₆₇	—	2	2	no	—
129	M₁₆×C₆₇	—	2	2	no	—
130	M₁₆×C₆₇	—	2	2	no	—
131	M₁₆×C₆₇	—	4	4	no	—
132	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
133	C₂×C₈×C₆₇	—	2	2	no	—
134	C₂×C₈×C₆₇	—	2	2	no	—
135	C₂×C₈×C₆₇	—	2	2	no	—
136	C₂×C₈×C₆₇	—	2	2	no	—
137	C₂×C₈×C₆₇	—	4	4	no	—
138	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
139	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
140	C₄⋊C₄×C₆₇	—	2	2	no	—
141	C₄⋊C₄×C₆₇	—	2	2	no	—
142	C₄⋊C₄×C₆₇	—	4	4	no	—
143	C₄⋊C₄×C₆₇	—	4	2	no	—
144	C₄⋊C₄×C₆₇	—	4	1	no	—
145	C₄⋊C₄×C₆₇	—	4	1	no	—
146	C₄⋊C₄×C₆₇	—	8	2	no	—
147	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
148	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
149	C₂²⋊C₄×C₆₇	—	2	2	no	—
150	C₂²⋊C₄×C₆₇	—	4	2	no	—
151	C₄²×C₆₇	—	2	2	no	—
152	C₄²×C₆₇	—	2	1	no	—
153	C₄²×C₆₇	—	4	2	no	—
154	C₄²×C₆₇	—	4	2	no	—
155	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
156	C₂³×C₆₇	—	2	2	no	—
157	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
158	Q₈×C₆₇	—	2	2	no	—
159	Q₈×C₆₇	—	2	4	no	—
160	Q₈×C₆₇	—	2	4	no	—
161	Q₈×C₆₇	—	4	4	no	—
162	Q₈×C₆₇	—	4	2	no	—
163	Q₈×C₆₇	—	4	2	no	—
164	Q₈×C₆₇	C₄×D₄	1	4	yes	—
165	D₄×C₆₇	—	2	4	no	—
166	D₄×C₆₇	—	2	4	no	—
167	D₄×C₆₇	—	2	2	no	—
168	D₄×C₆₇	—	4	4	no	—
169	D₄×C₆₇	C₄×D₄	1	4	yes	—
170	C₂×C₄×C₆₇	—	2	2	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	2	2	no	—
174	C₂×C₄×C₆₇	—	2	4	no	—
175	C₂×C₄×C₆₇	—	2	4	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	4	8	no	—
181	C₂×C₄×C₆₇	—	4	1	no	—
182	C₂×C₄×C₆₇	—	4	1	no	—
183	C₂×C₄×C₆₇	—	4	1	no	—
184	C₂×C₄×C₆₇	—	4	1	no	—
185	C₂×C₄×C₆₇	—	4	2	no	—
186	C₂×C₄×C₆₇	—	8	2	no	—
187	C₂×C₄×C₆₇	—	16	2	no	—
188	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
189	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
190	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
191	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
192	C₈×C₆₇	—	2	4	no	—
193	C₈×C₆₇	—	2	2	no	—
194	C₈×C₆₇	—	2	2	no	—
195	C₈×C₆₇	—	4	2	no	—
196	C₂²×C₆₇	—	2	1	no	—
197	C₂²×C₆₇	—	2	1	no	—
198	C₂²×C₆₇	—	2	4	no	—
199	C₂²×C₆₇	—	4	2	no	—
200	C₂²×C₆₇	—	4	2	no	—
201	C₂²×C₆₇	#518	1	1	yes	characteristic
202	C₂²×C₆₇	D₄²	1	1	yes	characteristic
203	C₂²×C₆₇	#551	1	1	yes	characteristic
204	C₄×C₆₇	—	2	4	no	—
205	C₄×C₆₇	—	2	4	no	—
206	C₄×C₆₇	—	4	2	no	—
207	C₄×C₆₇	—	4	2	no	—
208	C₄×C₆₇	—	4	2	no	—
209	C₄×C₆₇	—	4	2	no	—
210	C₄×C₆₇	—	8	2	no	—
211	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
212	C₄×C₆₇	#546	1	2	yes	—
213	#22894	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
214	C₂×C₆₇	—	2	1	no	—
215	C₂×C₆₇	—	2	1	no	—
216	C₂×C₆₇	—	4	4	no	—
217	C₂×C₆₇	#2358	1	1	yes	characteristic
218	C₂×C₆₇	#3291	1	1	yes	characteristic
219	C₂×C₆₇	#3293	1	1	yes	characteristic
220	C₂×#456	C₂×C₆₇	1	1	yes	characteristic
221	#2947	C₂×C₆₇	1	2	yes	complemented
222	#2898	C₂×C₆₇	1	2	yes	—
223	#1956	C₂×C₆₇	1	1	yes	characteristic, complemented
224	#1930	C₂×C₆₇	1	2	yes	complemented
225	#1908	C₂×C₆₇	1	2	yes	complemented
226	#1886	C₂×C₆₇	1	1	yes	characteristic, complemented
227	#1859	C₂×C₆₇	1	2	yes	complemented
228	#1764	C₂×C₆₇	1	2	yes	complemented
229	C₆₇	#22894	1	1	yes	characteristic, direct product factor, Sylow
230	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	1	yes	characteristic
231	C₂×#137	C₂²×C₆₇	1	2	yes	—
232	C₂×#131	C₂²×C₆₇	1	2	yes	—
233	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
234	#520	C₂²×C₆₇	1	2	yes	—
235	#519	C₂²×C₆₇	1	1	yes	characteristic
236	#456	C₄×C₆₇	1	8	yes	complemented
237	#441	—	2	2	no	—
238	#440	—	2	2	no	—
239	C₄×Q₁₆	—	2	2	no	—
240	C₄×SD₁₆	—	2	2	no	—
241	#428	C₂²×C₆₇	1	2	yes	—
242	C₂×#104	C₂²×C₆₇	1	2	yes	—
243	#419	—	2	2	no	—
244	#419	C₂²×C₆₇	1	2	yes	—
245	#417	—	2	2	no	—
246	#417	C₂²×C₆₇	1	2	yes	—
247	#416	—	2	2	no	—
248	#416	C₂²×C₆₇	1	2	yes	—
249	C₂×#103	—	2	2	no	—
250	C₂×#103	C₂²×C₆₇	1	2	yes	—
251	C₂×#103	C₂²×C₆₇	1	2	yes	—
252	C₂×#101	C₂²×C₆₇	1	1	yes	characteristic
253	C₂×#101	C₂²×C₆₇	1	1	yes	characteristic
254	#405	C₂²×C₆₇	1	2	yes	—
255	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
256	#384	C₂²×C₆₇	1	1	yes	characteristic
257	#343	C₂²×C₆₇	1	2	yes	—
258	#341	C₂²×C₆₇	1	2	yes	—
259	#339	C₂²×C₆₇	1	2	yes	—
260	2₋¹⁺⁴	—	2	4	no	—
261	2₋¹⁺⁴	—	4	1	no	—
262	2₋¹⁺⁴	C₂×C₄×C₆₇	1	4	yes	—
263	C₂×#41	—	2	2	no	—
264	C₂×#41	—	4	1	no	—
265	C₂×#41	C₂³×C₆₇	1	2	yes	—
266	C₂²×Q₈	—	2	2	no	—
267	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
268	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
269	#137	—	2	4	no	—
270	#137	C₂×C₄×C₆₇	1	8	yes	complemented
271	C₂×Q₁₆	—	2	2	no	—
272	C₂×SD₁₆	—	2	2	no	—
273	#131	—	2	4	no	—
274	#131	—	4	2	no	—
275	C₂×M₁₆	—	2	2	no	—
276	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
277	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
278	C₂²×C₈	—	2	2	no	—
279	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
280	C₄⋊Q₈	—	4	1	no	—
281	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
282	C₄×Q₈	—	2	2	no	—
283	C₄×Q₈	—	2	2	no	—
284	C₄×Q₈	—	2	2	no	—
285	C₄×D₄	—	2	2	no	—
286	C₄×D₄	—	4	2	no	—
287	#117	—	2	2	no	—
288	#117	—	4	1	no	—
289	#117	C₂³×C₆₇	1	2	yes	—
290	C₂×C₄⋊C₄	—	2	2	no	—
291	C₂×C₄⋊C₄	—	4	1	no	—
292	C₂×C₄⋊C₄	—	4	1	no	—
293	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
294	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
295	#107	—	2	2	no	—
296	#106	—	2	2	no	—
297	#104	C₂×C₄×C₆₇	1	8	yes	complemented
298	#103	—	2	2	no	—
299	#103	—	2	2	no	—
300	#103	—	2	2	no	—
301	#103	—	2	2	no	—
302	#103	—	2	2	no	—
303	#103	—	2	2	no	—
304	#103	—	2	2	no	—
305	#103	D₄×C₆₇	1	2	yes	—
306	#103	D₄×C₆₇	1	2	yes	—
307	#102	—	2	2	no	—
308	#102	D₄×C₆₇	1	2	yes	—
309	#102	D₄×C₆₇	1	2	yes	—
310	#101	—	4	1	no	—
311	#101	C₂×C₄×C₆₇	1	4	yes	complemented
312	#97	—	2	2	no	—
313	C₄×C₈	—	2	2	no	—
314	C₄×C₈	—	4	1	no	—
315	#41	—	2	2	no	—
316	#41	—	2	2	no	—
317	#41	—	2	8	no	—
318	#41	—	4	4	no	—
319	#41	—	4	4	no	—
320	#41	—	4	4	no	—
321	#41	C₂²×C₄×C₆₇	1	4	yes	—
322	#41	#41×C₆₇	1	4	yes	—
323	C₂×Q₈	—	2	1	no	—
324	C₂×Q₈	—	2	1	no	—
325	C₂×Q₈	—	2	8	no	—
326	C₂×Q₈	—	2	2	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	—	2	2	no	—
329	C₂×Q₈	—	2	2	no	—
330	C₂×Q₈	—	4	1	no	—
331	C₂×Q₈	—	4	4	no	—
332	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
333	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
334	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	4	1	no	—
337	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	4	2	no	—
344	C₂²×C₄	—	4	2	no	—
345	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
346	Q₁₆	—	2	8	no	—
347	SD₁₆	—	2	8	no	—
348	M₁₆	—	2	2	no	—
349	M₁₆	—	2	2	no	—
350	M₁₆	—	2	2	no	—
351	M₁₆	—	4	4	no	—
352	M₁₆	C₂²×C₄×C₆₇	1	4	yes	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	4	4	no	—
358	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
359	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	4	4	no	—
363	C₄⋊C₄	—	4	2	no	—
364	C₄⋊C₄	—	4	1	no	—
365	C₄⋊C₄	—	4	1	no	—
366	C₄⋊C₄	—	8	2	no	—
367	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
368	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
369	C₂²⋊C₄	—	2	2	no	—
370	C₂²⋊C₄	—	4	2	no	—
371	C₄²	—	2	2	no	—
372	C₄²	—	2	1	no	—
373	C₄²	—	4	2	no	—
374	C₄²	—	4	2	no	—
375	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
376	C₂³	—	2	2	no	—
377	C₂³	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
378	Q₈	—	2	2	no	—
379	Q₈	—	2	4	no	—
380	Q₈	—	2	4	no	—
381	Q₈	—	4	4	no	—
382	Q₈	—	4	2	no	—
383	Q₈	—	4	2	no	—
384	Q₈	C₄×D₄×C₆₇	1	4	yes	—
385	D₄	—	2	4	no	—
386	D₄	—	2	4	no	—
387	D₄	—	2	2	no	—
388	D₄	—	4	4	no	—
389	D₄	C₄×D₄×C₆₇	1	4	yes	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	4	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	4	8	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	1	no	—
403	C₂×C₄	—	4	1	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	8	2	no	—
407	C₂×C₄	—	16	2	no	—
408	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
409	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
410	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
411	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
412	C₈	—	2	4	no	—
413	C₈	—	2	2	no	—
414	C₈	—	2	2	no	—
415	C₈	—	4	2	no	—
416	C₂²	—	2	1	no	—
417	C₂²	—	2	1	no	—
418	C₂²	—	2	4	no	—
419	C₂²	—	4	2	no	—
420	C₂²	—	4	2	no	—
421	C₂²	#518×C₆₇	1	1	yes	characteristic
422	C₂²	D₄²×C₆₇	1	1	yes	characteristic
423	C₂²	#551×C₆₇	1	1	yes	characteristic
424	C₄	—	2	4	no	—
425	C₄	—	2	4	no	—
426	C₄	—	4	2	no	—
427	C₄	—	4	2	no	—
428	C₄	—	4	2	no	—
429	C₄	—	4	2	no	—
430	C₄	—	8	2	no	—
431	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
432	C₄	#546×C₆₇	1	2	yes	—
433	C₂	—	2	1	no	—
434	C₂	—	2	1	no	—
435	C₂	—	4	4	no	—
436	C₂	C₆₇×#2358	1	1	yes	characteristic
437	C₂	C₆₇×#3291	1	1	yes	characteristic
438	C₂	C₆₇×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×2₋¹⁺⁴×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2947	C₂	semidirect
4. C₆₇×#1956	C₂	semidirect
5. C₆₇×#1930	C₂	semidirect
6. C₆₇×#1908	C₂	semidirect
7. C₆₇×#1886	C₂	semidirect
8. C₆₇×#1859	C₂	semidirect
9. C₆₇×#1764	C₂	semidirect
16. #456×C₆₇	C₄	semidirect
50. #137×C₆₇	C₂×C₄	semidirect
61. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
77. #104×C₆₇	C₂×C₄	semidirect
91. #101×C₆₇	C₂×C₄	semidirect
213. #22894	C₆₇	direct (N × H)
221. #2947	C₂×C₆₇	semidirect
223. #1956	C₂×C₆₇	semidirect
224. #1930	C₂×C₆₇	semidirect
225. #1908	C₂×C₆₇	semidirect
226. #1886	C₂×C₆₇	semidirect
227. #1859	C₂×C₆₇	semidirect
228. #1764	C₂×C₆₇	semidirect
229. C₆₇	#22894	direct (N × H)
236. #456	C₄×C₆₇	semidirect
270. #137	C₂×C₄×C₆₇	semidirect
281. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
297. #104	C₂×C₄×C₆₇	semidirect
311. #101	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×#456×C₆₇	2
C₆₇×#2947	2
C₆₇×#2898	2
C₆₇×#1956	2
C₆₇×#1930	2
C₆₇×#1908	2
C₆₇×#1886	2
C₆₇×#1859	2
C₆₇×#1764	2
#22894	67

Subgroups — C67×#22894 #30702145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22894 #30702145