Subgroups — C₆₇×#22896 #30702147

Order 17152.

1016 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#455×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2947	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2897	C₂	1	2	yes	maximal
4	C₆₇×#1958	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1931	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1908	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1888	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1860	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1764	C₂	1	2	yes	maximal, complemented
10	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#137×C₆₇	C₂²	1	2	yes	—
12	C₂×#131×C₆₇	C₂²	1	2	yes	—
13	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
14	#520×C₆₇	C₂²	1	2	yes	—
15	#519×C₆₇	C₂²	1	1	yes	characteristic, complemented
16	#455×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	complemented
22	C₂×#104×C₆₇	C₂²	1	2	yes	—
23	#418×C₆₇	—	2	2	no	—
24	#418×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	C₂²	1	2	yes	complemented
30	C₂×#102×C₆₇	—	2	2	no	—
31	C₂×#102×C₆₇	C₂²	1	2	yes	—
32	C₂×#101×C₆₇	C₂²	1	1	yes	characteristic, complemented
33	C₂×#100×C₆₇	C₂²	1	1	yes	characteristic
34	#405×C₆₇	C₂²	1	2	yes	complemented
35	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	#382×C₆₇	C₂²	1	1	yes	characteristic, complemented
37	#343×C₆₇	C₂²	1	2	yes	complemented
38	#341×C₆₇	C₂²	1	2	yes	complemented
39	#339×C₆₇	C₂²	1	2	yes	complemented
40	H(5,𝔽₂)×C₆₇	—	2	4	no	—
41	H(5,𝔽₂)×C₆₇	—	4	1	no	—
42	H(5,𝔽₂)×C₆₇	C₂×C₄	1	4	yes	—
43	C₂×#41×C₆₇	—	2	2	no	—
44	C₂×#41×C₆₇	C₂³	1	2	yes	—
45	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₆₇	—	2	2	no	—
47	C₂²×D₄×C₆₇	—	4	1	no	—
48	C₂²×D₄×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	#124×C₆₇	—	4	1	no	—
61	#124×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	C₂³	1	2	yes	—
72	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	#107×C₆₇	—	2	2	no	—
75	#106×C₆₇	—	2	2	no	—
76	#104×C₆₇	C₂×C₄	1	8	yes	complemented
77	#103×C₆₇	—	2	2	no	—
78	#103×C₆₇	—	2	2	no	—
79	#103×C₆₇	—	2	2	no	—
80	#103×C₆₇	D₄	1	2	yes	—
81	#103×C₆₇	D₄	1	2	yes	—
82	#102×C₆₇	—	2	2	no	—
83	#102×C₆₇	—	2	2	no	—
84	#102×C₆₇	—	2	2	no	—
85	#102×C₆₇	—	2	2	no	—
86	#102×C₆₇	—	2	2	no	—
87	#102×C₆₇	D₄	1	2	yes	—
88	#102×C₆₇	D₄	1	2	yes	—
89	#101×C₆₇	—	4	1	no	—
90	#100×C₆₇	C₂×C₄	1	4	yes	complemented
91	#97×C₆₇	—	2	2	no	—
92	C₄×C₈×C₆₇	—	2	2	no	—
93	C₄×C₈×C₆₇	—	4	1	no	—
94	#95×C₆₇	—	4	1	no	—
95	C₂⁴×C₆₇	—	2	1	no	—
96	C₂⁴×C₆₇	—	4	1	no	—
97	#41×C₆₇	—	2	2	no	—
98	#41×C₆₇	—	2	2	no	—
99	#41×C₆₇	—	2	8	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	2	no	—
105	C₂×Q₈×C₆₇	—	2	2	no	—
106	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
107	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
108	C₂×D₄×C₆₇	—	2	2	no	—
109	C₂×D₄×C₆₇	—	2	8	no	—
110	C₂×D₄×C₆₇	—	2	2	no	—
111	C₂×D₄×C₆₇	—	2	2	no	—
112	C₂×D₄×C₆₇	—	2	1	no	—
113	C₂×D₄×C₆₇	—	4	4	no	—
114	C₂×D₄×C₆₇	—	4	2	no	—
115	C₂×D₄×C₆₇	—	4	4	no	—
116	C₂×D₄×C₆₇	—	4	4	no	—
117	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
118	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
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	1	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	2	no	—
143	C₄⋊C₄×C₆₇	—	4	2	no	—
144	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
145	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
146	C₂²⋊C₄×C₆₇	—	2	2	no	—
147	C₂²⋊C₄×C₆₇	—	4	4	no	—
148	C₂²⋊C₄×C₆₇	—	4	2	no	—
149	C₄²×C₆₇	—	2	2	no	—
150	C₄²×C₆₇	—	2	1	no	—
151	C₄²×C₆₇	—	4	2	no	—
152	C₄²×C₆₇	—	4	1	no	—
153	C₄²×C₆₇	—	4	1	no	—
154	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
155	C₂³×C₆₇	—	2	2	no	—
156	C₂³×C₆₇	—	2	2	no	—
157	C₂³×C₆₇	—	4	4	no	—
158	C₂³×C₆₇	—	4	1	no	—
159	C₂³×C₆₇	—	4	1	no	—
160	C₂³×C₆₇	—	8	2	no	—
161	C₂³×C₆₇	—	16	2	no	—
162	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
163	Q₈×C₆₇	—	2	2	no	—
164	Q₈×C₆₇	—	2	4	no	—
165	Q₈×C₆₇	—	4	2	no	—
166	Q₈×C₆₇	C₄×D₄	1	4	yes	—
167	D₄×C₆₇	—	2	4	no	—
168	D₄×C₆₇	—	2	4	no	—
169	D₄×C₆₇	—	2	4	no	—
170	D₄×C₆₇	—	2	2	no	—
171	D₄×C₆₇	—	4	8	no	—
172	D₄×C₆₇	—	4	2	no	—
173	D₄×C₆₇	C₄×D₄	1	4	yes	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	4	no	—
176	C₂×C₄×C₆₇	—	2	4	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	—	2	2	no	—
181	C₂×C₄×C₆₇	—	4	4	no	—
182	C₂×C₄×C₆₇	—	4	2	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	2	no	—
197	C₂²×C₆₇	—	2	4	no	—
198	C₂²×C₆₇	—	4	2	no	—
199	C₂²×C₆₇	—	4	2	no	—
200	C₂²×C₆₇	—	4	2	no	—
201	C₂²×C₆₇	—	8	2	no	—
202	C₂²×C₆₇	—	8	2	no	—
203	C₂²×C₆₇	—	16	1	no	—
204	C₂²×C₆₇	—	16	4	no	—
205	C₂²×C₆₇	#518	1	1	yes	characteristic
206	C₂²×C₆₇	D₄²	1	1	yes	characteristic
207	C₂²×C₆₇	#551	1	1	yes	characteristic
208	C₄×C₆₇	—	2	4	no	—
209	C₄×C₆₇	—	4	2	no	—
210	C₄×C₆₇	—	4	2	no	—
211	C₄×C₆₇	—	4	2	no	—
212	C₄×C₆₇	—	4	2	no	—
213	C₄×C₆₇	—	8	2	no	—
214	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
215	C₄×C₆₇	#546	1	2	yes	—
216	#22896	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
217	C₂×C₆₇	—	2	1	no	—
218	C₂×C₆₇	—	2	1	no	—
219	C₂×C₆₇	—	4	4	no	—
220	C₂×C₆₇	—	8	2	no	—
221	C₂×C₆₇	#2358	1	1	yes	characteristic
222	C₂×C₆₇	#3290	1	2	yes	—
223	C₂×#455	C₂×C₆₇	1	1	yes	characteristic
224	#2947	C₂×C₆₇	1	2	yes	complemented
225	#2897	C₂×C₆₇	1	2	yes	—
226	#1958	C₂×C₆₇	1	1	yes	characteristic, complemented
227	#1931	C₂×C₆₇	1	2	yes	complemented
228	#1908	C₂×C₆₇	1	2	yes	complemented
229	#1888	C₂×C₆₇	1	1	yes	characteristic, complemented
230	#1860	C₂×C₆₇	1	2	yes	complemented
231	#1764	C₂×C₆₇	1	2	yes	complemented
232	C₆₇	#22896	1	1	yes	characteristic, direct product factor, Sylow
233	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	1	yes	characteristic
234	C₂×#137	C₂²×C₆₇	1	2	yes	—
235	C₂×#131	C₂²×C₆₇	1	2	yes	—
236	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
237	#520	C₂²×C₆₇	1	2	yes	—
238	#519	C₂²×C₆₇	1	1	yes	characteristic, complemented
239	#455	C₄×C₆₇	1	8	yes	complemented
240	#441	—	2	2	no	—
241	#440	—	2	2	no	—
242	C₄×Q₁₆	—	2	2	no	—
243	C₄×SD₁₆	—	2	2	no	—
244	#428	C₂²×C₆₇	1	2	yes	complemented
245	C₂×#104	C₂²×C₆₇	1	2	yes	—
246	#418	—	2	2	no	—
247	#418	C₂²×C₆₇	1	2	yes	—
248	#417	—	2	2	no	—
249	#417	C₂²×C₆₇	1	2	yes	—
250	#416	—	2	2	no	—
251	#416	C₂²×C₆₇	1	2	yes	—
252	C₂×#103	C₂²×C₆₇	1	2	yes	complemented
253	C₂×#102	—	2	2	no	—
254	C₂×#102	C₂²×C₆₇	1	2	yes	—
255	C₂×#101	C₂²×C₆₇	1	1	yes	characteristic, complemented
256	C₂×#100	C₂²×C₆₇	1	1	yes	characteristic
257	#405	C₂²×C₆₇	1	2	yes	complemented
258	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
259	#382	C₂²×C₆₇	1	1	yes	characteristic, complemented
260	#343	C₂²×C₆₇	1	2	yes	complemented
261	#341	C₂²×C₆₇	1	2	yes	complemented
262	#339	C₂²×C₆₇	1	2	yes	complemented
263	H(5,𝔽₂)	—	2	4	no	—
264	H(5,𝔽₂)	—	4	1	no	—
265	H(5,𝔽₂)	C₂×C₄×C₆₇	1	4	yes	—
266	C₂×#41	—	2	2	no	—
267	C₂×#41	C₂³×C₆₇	1	2	yes	—
268	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
269	C₂²×D₄	—	2	2	no	—
270	C₂²×D₄	—	4	1	no	—
271	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
272	#137	—	2	4	no	—
273	#137	C₂×C₄×C₆₇	1	8	yes	complemented
274	C₂×Q₁₆	—	2	2	no	—
275	C₂×SD₁₆	—	2	2	no	—
276	#131	—	2	4	no	—
277	#131	—	4	2	no	—
278	C₂×M₁₆	—	2	2	no	—
279	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
280	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
281	C₂²×C₈	—	2	2	no	—
282	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
283	#124	—	4	1	no	—
284	#124	C₂×C₄×C₆₇	1	4	yes	complemented
285	C₄×Q₈	—	2	2	no	—
286	C₄×Q₈	—	2	2	no	—
287	C₄×Q₈	—	2	2	no	—
288	C₄×D₄	—	2	2	no	—
289	C₄×D₄	—	4	2	no	—
290	#117	—	2	2	no	—
291	#117	—	4	1	no	—
292	#117	C₂³×C₆₇	1	2	yes	—
293	C₂×C₄⋊C₄	—	2	2	no	—
294	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
295	C₂×C₂²⋊C₄	—	4	1	no	—
296	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
297	#107	—	2	2	no	—
298	#106	—	2	2	no	—
299	#104	C₂×C₄×C₆₇	1	8	yes	complemented
300	#103	—	2	2	no	—
301	#103	—	2	2	no	—
302	#103	—	2	2	no	—
303	#103	D₄×C₆₇	1	2	yes	—
304	#103	D₄×C₆₇	1	2	yes	—
305	#102	—	2	2	no	—
306	#102	—	2	2	no	—
307	#102	—	2	2	no	—
308	#102	—	2	2	no	—
309	#102	—	2	2	no	—
310	#102	D₄×C₆₇	1	2	yes	—
311	#102	D₄×C₆₇	1	2	yes	—
312	#101	—	4	1	no	—
313	#100	C₂×C₄×C₆₇	1	4	yes	complemented
314	#97	—	2	2	no	—
315	C₄×C₈	—	2	2	no	—
316	C₄×C₈	—	4	1	no	—
317	#95	—	4	1	no	—
318	C₂⁴	—	2	1	no	—
319	C₂⁴	—	4	1	no	—
320	#41	—	2	2	no	—
321	#41	—	2	2	no	—
322	#41	—	2	8	no	—
323	#41	—	4	4	no	—
324	#41	C₂²×C₄×C₆₇	1	4	yes	—
325	#41	#41×C₆₇	1	4	yes	—
326	C₂×Q₈	—	2	1	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	—	2	2	no	—
329	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
330	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	8	no	—
333	C₂×D₄	—	2	2	no	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	—	2	1	no	—
336	C₂×D₄	—	4	4	no	—
337	C₂×D₄	—	4	2	no	—
338	C₂×D₄	—	4	4	no	—
339	C₂×D₄	—	4	4	no	—
340	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
341	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	4	1	no	—
347	C₂²×C₄	—	4	2	no	—
348	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
349	Q₁₆	—	2	8	no	—
350	SD₁₆	—	2	8	no	—
351	M₁₆	—	2	2	no	—
352	M₁₆	—	2	2	no	—
353	M₁₆	—	2	2	no	—
354	M₁₆	—	4	4	no	—
355	M₁₆	C₂²×C₄×C₆₇	1	4	yes	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	4	4	no	—
361	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
362	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	4	2	no	—
366	C₄⋊C₄	—	4	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	4	no	—
371	C₂²⋊C₄	—	4	2	no	—
372	C₄²	—	2	2	no	—
373	C₄²	—	2	1	no	—
374	C₄²	—	4	2	no	—
375	C₄²	—	4	1	no	—
376	C₄²	—	4	1	no	—
377	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
378	C₂³	—	2	2	no	—
379	C₂³	—	2	2	no	—
380	C₂³	—	4	4	no	—
381	C₂³	—	4	1	no	—
382	C₂³	—	4	1	no	—
383	C₂³	—	8	2	no	—
384	C₂³	—	16	2	no	—
385	C₂³	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
386	Q₈	—	2	2	no	—
387	Q₈	—	2	4	no	—
388	Q₈	—	4	2	no	—
389	Q₈	C₄×D₄×C₆₇	1	4	yes	—
390	D₄	—	2	4	no	—
391	D₄	—	2	4	no	—
392	D₄	—	2	4	no	—
393	D₄	—	2	2	no	—
394	D₄	—	4	8	no	—
395	D₄	—	4	2	no	—
396	D₄	C₄×D₄×C₆₇	1	4	yes	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	4	4	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	8	2	no	—
410	C₂×C₄	—	16	2	no	—
411	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
412	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
413	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
414	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
415	C₈	—	2	4	no	—
416	C₈	—	2	2	no	—
417	C₈	—	2	2	no	—
418	C₈	—	4	2	no	—
419	C₂²	—	2	2	no	—
420	C₂²	—	2	4	no	—
421	C₂²	—	4	2	no	—
422	C₂²	—	4	2	no	—
423	C₂²	—	4	2	no	—
424	C₂²	—	8	2	no	—
425	C₂²	—	8	2	no	—
426	C₂²	—	16	1	no	—
427	C₂²	—	16	4	no	—
428	C₂²	#518×C₆₇	1	1	yes	characteristic
429	C₂²	D₄²×C₆₇	1	1	yes	characteristic
430	C₂²	#551×C₆₇	1	1	yes	characteristic
431	C₄	—	2	4	no	—
432	C₄	—	4	2	no	—
433	C₄	—	4	2	no	—
434	C₄	—	4	2	no	—
435	C₄	—	4	2	no	—
436	C₄	—	8	2	no	—
437	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
438	C₄	#546×C₆₇	1	2	yes	—
439	C₂	—	2	1	no	—
440	C₂	—	2	1	no	—
441	C₂	—	4	4	no	—
442	C₂	—	8	2	no	—
443	C₂	C₆₇×#2358	1	1	yes	characteristic
444	C₂	C₆₇×#3290	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×H(5,𝔽₂)×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×#1958	C₂	semidirect
5. C₆₇×#1931	C₂	semidirect
6. C₆₇×#1908	C₂	semidirect
7. C₆₇×#1888	C₂	semidirect
8. C₆₇×#1860	C₂	semidirect
9. C₆₇×#1764	C₂	semidirect
15. #519×C₆₇	C₂²	semidirect
16. #455×C₆₇	C₄	semidirect
21. #428×C₆₇	C₂²	semidirect
29. C₂×#103×C₆₇	C₂²	semidirect
32. C₂×#101×C₆₇	C₂²	semidirect
34. #405×C₆₇	C₂²	semidirect
35. C₂×C₄×C₈×C₆₇	C₂²	semidirect
36. #382×C₆₇	C₂²	semidirect
37. #343×C₆₇	C₂²	semidirect
38. #341×C₆₇	C₂²	semidirect
39. #339×C₆₇	C₂²	semidirect
50. #137×C₆₇	C₂×C₄	semidirect
61. #124×C₆₇	C₂×C₄	semidirect
76. #104×C₆₇	C₂×C₄	semidirect
90. #100×C₆₇	C₂×C₄	semidirect
216. #22896	C₆₇	direct (N × H)
224. #2947	C₂×C₆₇	semidirect
226. #1958	C₂×C₆₇	semidirect
227. #1931	C₂×C₆₇	semidirect
228. #1908	C₂×C₆₇	semidirect
229. #1888	C₂×C₆₇	semidirect
230. #1860	C₂×C₆₇	semidirect
231. #1764	C₂×C₆₇	semidirect
232. C₆₇	#22896	direct (N × H)
238. #519	C₂²×C₆₇	semidirect
239. #455	C₄×C₆₇	semidirect
244. #428	C₂²×C₆₇	semidirect
252. C₂×#103	C₂²×C₆₇	semidirect
255. C₂×#101	C₂²×C₆₇	semidirect
257. #405	C₂²×C₆₇	semidirect
258. C₂×C₄×C₈	C₂²×C₆₇	semidirect
259. #382	C₂²×C₆₇	semidirect
260. #343	C₂²×C₆₇	semidirect
261. #341	C₂²×C₆₇	semidirect
262. #339	C₂²×C₆₇	semidirect
273. #137	C₂×C₄×C₆₇	semidirect
284. #124	C₂×C₄×C₆₇	semidirect
299. #104	C₂×C₄×C₆₇	semidirect
313. #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×#455×C₆₇	2
C₆₇×#2947	2
C₆₇×#2897	2
C₆₇×#1958	2
C₆₇×#1931	2
C₆₇×#1908	2
C₆₇×#1888	2
C₆₇×#1860	2
C₆₇×#1764	2
#22896	67

Subgroups — C67×#22896 #30702147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22896 #30702147