Subgroups — C₆₇×#25982 #30705233

Order 17152.

884 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3083	C₂	1	2	yes	maximal, complemented
2	C₆₇×#2897	C₂	1	2	yes	maximal
3	C₆₇×#2093	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2063	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2027	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1866	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1859	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1816	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1792	C₂	1	1	yes	maximal, characteristic
10	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#131×C₆₇	C₂²	1	2	yes	—
12	#544×C₆₇	C₂²	1	1	yes	characteristic, complemented
13	#520×C₆₇	C₂²	1	2	yes	—
14	#479×C₆₇	—	2	2	no	—
15	#478×C₆₇	—	2	2	no	—
16	#477×C₆₇	—	2	2	no	—
17	#474×C₆₇	—	2	2	no	—
18	C₂×#108×C₆₇	C₂²	1	2	yes	complemented
19	#428×C₆₇	C₂²	1	2	yes	complemented
20	C₂×#107×C₆₇	C₂²	1	2	yes	complemented
21	#424×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	#417×C₆₇	C₂²	1	2	yes	—
28	#416×C₆₇	—	2	2	no	—
29	#416×C₆₇	C₂²	1	2	yes	—
30	C₂×#102×C₆₇	—	2	2	no	—
31	C₂×#102×C₆₇	C₂²	1	2	yes	—
32	C₂×#100×C₆₇	C₂²	1	1	yes	characteristic
33	#390×C₆₇	C₂²	1	1	yes	characteristic
34	#384×C₆₇	C₂²	1	1	yes	characteristic, complemented
35	#343×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	#343×C₆₇	C₂²	1	2	yes	complemented
37	#339×C₆₇	C₂²	1	2	yes	complemented
38	#336×C₆₇	C₂²	1	1	yes	characteristic, complemented
39	H(5,𝔽₂)×C₆₇	—	2	4	no	—
40	H(5,𝔽₂)×C₆₇	—	4	1	no	—
41	H(5,𝔽₂)×C₆₇	Q₈	1	4	yes	complemented
42	C₂×#41×C₆₇	—	2	2	no	—
43	C₂×#41×C₆₇	C₂³	1	2	yes	—
44	C₂²×D₄×C₆₇	—	2	2	no	—
45	C₂²×D₄×C₆₇	—	4	1	no	—
46	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
47	#131×C₆₇	—	2	4	no	—
48	#131×C₆₇	—	4	2	no	—
49	C₂×M₁₆×C₆₇	—	2	2	no	—
50	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
51	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
52	C₂²×C₈×C₆₇	—	2	2	no	—
53	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
54	C₄⋊Q₈×C₆₇	—	2	2	no	—
55	#125×C₆₇	—	2	2	no	—
56	C₂²⋊Q₈×C₆₇	—	4	1	no	—
57	C₄×Q₈×C₆₇	—	2	2	no	—
58	C₄×D₄×C₆₇	—	2	2	no	—
59	C₄×D₄×C₆₇	—	4	2	no	—
60	#117×C₆₇	—	2	2	no	—
61	#117×C₆₇	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
67	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
68	#108×C₆₇	—	4	2	no	—
69	#107×C₆₇	—	2	2	no	—
70	#107×C₆₇	—	2	2	no	—
71	#107×C₆₇	—	2	2	no	—
72	#106×C₆₇	—	2	2	no	—
73	C₄⋊C₈×C₆₇	—	2	2	no	—
74	C₄⋊C₈×C₆₇	—	2	2	no	—
75	#104×C₆₇	—	2	4	no	—
76	#103×C₆₇	—	2	2	no	—
77	#103×C₆₇	—	2	2	no	—
78	#103×C₆₇	D₄	1	2	yes	—
79	#103×C₆₇	D₄	1	2	yes	—
80	#102×C₆₇	—	2	2	no	—
81	#102×C₆₇	—	2	2	no	—
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	D₄	1	2	yes	—
87	#102×C₆₇	D₄	1	2	yes	—
88	#100×C₆₇	—	4	1	no	—
89	C₂⁴×C₆₇	—	2	1	no	—
90	C₂⁴×C₆₇	—	4	1	no	—
91	#41×C₆₇	—	2	2	no	—
92	#41×C₆₇	—	2	2	no	—
93	#41×C₆₇	—	2	8	no	—
94	#41×C₆₇	—	4	4	no	—
95	#41×C₆₇	C₂×Q₈	1	4	yes	—
96	#41×C₆₇	#41	1	4	yes	—
97	C₂×Q₈×C₆₇	—	4	1	no	—
98	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
99	C₂×D₄×C₆₇	—	2	2	no	—
100	C₂×D₄×C₆₇	—	2	8	no	—
101	C₂×D₄×C₆₇	—	2	2	no	—
102	C₂×D₄×C₆₇	—	2	2	no	—
103	C₂×D₄×C₆₇	—	2	1	no	—
104	C₂×D₄×C₆₇	—	4	4	no	—
105	C₂×D₄×C₆₇	—	4	2	no	—
106	C₂×D₄×C₆₇	—	4	4	no	—
107	C₂×D₄×C₆₇	—	4	4	no	—
108	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
109	C₂×D₄×C₆₇	C₂×Q₈	1	2	yes	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	2	1	no	—
112	C₂²×C₄×C₆₇	—	2	1	no	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	—	4	1	no	—
115	C₂²×C₄×C₆₇	—	4	2	no	—
116	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
117	M₁₆×C₆₇	—	2	2	no	—
118	M₁₆×C₆₇	—	2	2	no	—
119	M₁₆×C₆₇	—	2	2	no	—
120	M₁₆×C₆₇	—	2	2	no	—
121	M₁₆×C₆₇	—	4	4	no	—
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	4	4	no	—
127	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
128	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
129	C₄⋊C₄×C₆₇	—	2	2	no	—
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	4	2	no	—
133	C₄⋊C₄×C₆₇	—	4	1	no	—
134	C₄⋊C₄×C₆₇	—	4	2	no	—
135	C₄⋊C₄×C₆₇	—	8	2	no	—
136	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
137	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
138	C₂²⋊C₄×C₆₇	—	2	2	no	—
139	C₂²⋊C₄×C₆₇	—	4	2	no	—
140	C₂²⋊C₄×C₆₇	—	8	2	no	—
141	C₄²×C₆₇	—	2	2	no	—
142	C₄²×C₆₇	—	2	2	no	—
143	C₄²×C₆₇	—	4	2	no	—
144	C₂³×C₆₇	—	2	2	no	—
145	C₂³×C₆₇	—	2	2	no	—
146	C₂³×C₆₇	—	4	4	no	—
147	C₂³×C₆₇	—	4	1	no	—
148	C₂³×C₆₇	—	4	1	no	—
149	C₂³×C₆₇	—	8	1	no	—
150	C₂³×C₆₇	—	8	1	no	—
151	C₂³×C₆₇	—	16	1	no	—
152	C₂³×C₆₇	—	16	1	no	—
153	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
154	Q₈×C₆₇	—	2	2	no	—
155	Q₈×C₆₇	—	16	1	no	—
156	Q₈×C₆₇	C₂²⋊Q₈	1	4	yes	—
157	D₄×C₆₇	—	2	4	no	—
158	D₄×C₆₇	—	2	4	no	—
159	D₄×C₆₇	—	2	4	no	—
160	D₄×C₆₇	—	2	2	no	—
161	D₄×C₆₇	—	4	8	no	—
162	D₄×C₆₇	—	4	2	no	—
163	D₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	4	no	—
166	C₂×C₄×C₆₇	—	2	4	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	2	2	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	4	4	no	—
171	C₂×C₄×C₆₇	—	4	1	no	—
172	C₂×C₄×C₆₇	—	4	1	no	—
173	C₂×C₄×C₆₇	—	4	2	no	—
174	C₂×C₄×C₆₇	—	4	1	no	—
175	C₂×C₄×C₆₇	—	8	1	no	—
176	C₂×C₄×C₆₇	—	8	2	no	—
177	C₂×C₄×C₆₇	—	16	2	no	—
178	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
179	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
181	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
182	C₈×C₆₇	—	2	2	no	—
183	C₈×C₆₇	—	2	2	no	—
184	C₈×C₆₇	—	4	2	no	—
185	C₈×C₆₇	—	4	2	no	—
186	C₂²×C₆₇	—	2	1	no	—
187	C₂²×C₆₇	—	2	1	no	—
188	C₂²×C₆₇	—	2	4	no	—
189	C₂²×C₆₇	—	4	2	no	—
190	C₂²×C₆₇	—	4	2	no	—
191	C₂²×C₆₇	—	4	2	no	—
192	C₂²×C₆₇	—	8	2	no	—
193	C₂²×C₆₇	—	8	1	no	—
194	C₂²×C₆₇	—	8	1	no	—
195	C₂²×C₆₇	—	16	1	no	—
196	C₂²×C₆₇	—	16	4	no	—
197	C₂²×C₆₇	#543	1	1	yes	characteristic
198	C₂²×C₆₇	D₄²	1	1	yes	characteristic
199	C₂²×C₆₇	#551	1	1	yes	characteristic
200	C₄×C₆₇	—	2	4	no	—
201	C₄×C₆₇	—	4	2	no	—
202	C₄×C₆₇	—	4	2	no	—
203	C₄×C₆₇	—	4	2	no	—
204	C₄×C₆₇	—	8	2	no	—
205	C₄×C₆₇	—	8	1	no	—
206	C₄×C₆₇	C₂×C₂²⋊Q₈	1	2	yes	—
207	C₄×C₆₇	#546	1	2	yes	—
208	#25982	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
209	C₂×C₆₇	—	2	1	no	—
210	C₂×C₆₇	—	2	1	no	—
211	C₂×C₆₇	—	4	4	no	—
212	C₂×C₆₇	—	8	2	no	—
213	C₂×C₆₇	#2437	1	1	yes	characteristic
214	C₂×C₆₇	#3289	1	1	yes	characteristic
215	C₂×C₆₇	#3292	1	1	yes	characteristic
216	#3083	C₂×C₆₇	1	2	yes	complemented
217	#2897	C₂×C₆₇	1	2	yes	—
218	#2093	C₂×C₆₇	1	1	yes	characteristic, complemented
219	#2063	C₂×C₆₇	1	2	yes	complemented
220	#2027	C₂×C₆₇	1	1	yes	characteristic, complemented
221	#1866	C₂×C₆₇	1	2	yes	complemented
222	#1859	C₂×C₆₇	1	2	yes	complemented
223	#1816	C₂×C₆₇	1	2	yes	complemented
224	#1792	C₂×C₆₇	1	1	yes	characteristic
225	C₆₇	#25982	1	1	yes	characteristic, direct product factor, Sylow
226	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	1	yes	characteristic
227	C₂×#131	C₂²×C₆₇	1	2	yes	—
228	#544	C₂²×C₆₇	1	1	yes	characteristic, complemented
229	#520	C₂²×C₆₇	1	2	yes	—
230	#479	—	2	2	no	—
231	#478	—	2	2	no	—
232	#477	—	2	2	no	—
233	#474	—	2	2	no	—
234	C₂×#108	C₂²×C₆₇	1	2	yes	complemented
235	#428	C₂²×C₆₇	1	2	yes	complemented
236	C₂×#107	C₂²×C₆₇	1	2	yes	complemented
237	#424	C₂²×C₆₇	1	2	yes	complemented
238	C₂×#104	C₂²×C₆₇	1	2	yes	—
239	#418	—	2	2	no	—
240	#418	C₂²×C₆₇	1	2	yes	—
241	#417	—	2	2	no	—
242	#417	C₂²×C₆₇	1	2	yes	—
243	#417	C₂²×C₆₇	1	2	yes	—
244	#416	—	2	2	no	—
245	#416	C₂²×C₆₇	1	2	yes	—
246	C₂×#102	—	2	2	no	—
247	C₂×#102	C₂²×C₆₇	1	2	yes	—
248	C₂×#100	C₂²×C₆₇	1	1	yes	characteristic
249	#390	C₂²×C₆₇	1	1	yes	characteristic
250	#384	C₂²×C₆₇	1	1	yes	characteristic, complemented
251	#343	C₂²×C₆₇	1	2	yes	complemented
252	#343	C₂²×C₆₇	1	1	yes	characteristic, complemented
253	#339	C₂²×C₆₇	1	2	yes	complemented
254	#336	C₂²×C₆₇	1	1	yes	characteristic, complemented
255	H(5,𝔽₂)	—	2	4	no	—
256	H(5,𝔽₂)	—	4	1	no	—
257	H(5,𝔽₂)	Q₈×C₆₇	1	4	yes	complemented
258	C₂×#41	—	2	2	no	—
259	C₂×#41	C₂³×C₆₇	1	2	yes	—
260	C₂²×D₄	—	2	2	no	—
261	C₂²×D₄	—	4	1	no	—
262	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
263	#131	—	2	4	no	—
264	#131	—	4	2	no	—
265	C₂×M₁₆	—	2	2	no	—
266	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
267	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
268	C₂²×C₈	—	2	2	no	—
269	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
270	C₄⋊Q₈	—	2	2	no	—
271	#125	—	2	2	no	—
272	C₂²⋊Q₈	—	4	1	no	—
273	C₄×Q₈	—	2	2	no	—
274	C₄×D₄	—	2	2	no	—
275	C₄×D₄	—	4	2	no	—
276	#117	—	2	2	no	—
277	#117	C₂³×C₆₇	1	2	yes	—
278	C₂×C₄⋊C₄	—	2	2	no	—
279	C₂×C₄⋊C₄	—	4	1	no	—
280	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
281	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
282	C₂×C₂²⋊C₄	—	4	1	no	—
283	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
284	#108	—	4	2	no	—
285	#107	—	2	2	no	—
286	#107	—	2	2	no	—
287	#107	—	2	2	no	—
288	#106	—	2	2	no	—
289	C₄⋊C₈	—	2	2	no	—
290	C₄⋊C₈	—	2	2	no	—
291	#104	—	2	4	no	—
292	#103	—	2	2	no	—
293	#103	—	2	2	no	—
294	#103	D₄×C₆₇	1	2	yes	—
295	#103	D₄×C₆₇	1	2	yes	—
296	#102	—	2	2	no	—
297	#102	—	2	2	no	—
298	#102	—	2	2	no	—
299	#102	—	2	2	no	—
300	#102	—	2	2	no	—
301	#102	—	2	2	no	—
302	#102	D₄×C₆₇	1	2	yes	—
303	#102	D₄×C₆₇	1	2	yes	—
304	#100	—	4	1	no	—
305	C₂⁴	—	2	1	no	—
306	C₂⁴	—	4	1	no	—
307	#41	—	2	2	no	—
308	#41	—	2	2	no	—
309	#41	—	2	8	no	—
310	#41	—	4	4	no	—
311	#41	C₂×Q₈×C₆₇	1	4	yes	—
312	#41	#41×C₆₇	1	4	yes	—
313	C₂×Q₈	—	4	1	no	—
314	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	8	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	2	2	no	—
319	C₂×D₄	—	2	1	no	—
320	C₂×D₄	—	4	4	no	—
321	C₂×D₄	—	4	2	no	—
322	C₂×D₄	—	4	4	no	—
323	C₂×D₄	—	4	4	no	—
324	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
325	C₂×D₄	C₂×Q₈×C₆₇	1	2	yes	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	4	1	no	—
331	C₂²×C₄	—	4	2	no	—
332	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
333	M₁₆	—	2	2	no	—
334	M₁₆	—	2	2	no	—
335	M₁₆	—	2	2	no	—
336	M₁₆	—	2	2	no	—
337	M₁₆	—	4	4	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	4	4	no	—
343	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
344	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	4	2	no	—
349	C₄⋊C₄	—	4	1	no	—
350	C₄⋊C₄	—	4	2	no	—
351	C₄⋊C₄	—	8	2	no	—
352	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
353	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
354	C₂²⋊C₄	—	2	2	no	—
355	C₂²⋊C₄	—	4	2	no	—
356	C₂²⋊C₄	—	8	2	no	—
357	C₄²	—	2	2	no	—
358	C₄²	—	2	2	no	—
359	C₄²	—	4	2	no	—
360	C₂³	—	2	2	no	—
361	C₂³	—	2	2	no	—
362	C₂³	—	4	4	no	—
363	C₂³	—	4	1	no	—
364	C₂³	—	4	1	no	—
365	C₂³	—	8	1	no	—
366	C₂³	—	8	1	no	—
367	C₂³	—	16	1	no	—
368	C₂³	—	16	1	no	—
369	C₂³	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
370	Q₈	—	2	2	no	—
371	Q₈	—	16	1	no	—
372	Q₈	C₂²⋊Q₈×C₆₇	1	4	yes	—
373	D₄	—	2	4	no	—
374	D₄	—	2	4	no	—
375	D₄	—	2	4	no	—
376	D₄	—	2	2	no	—
377	D₄	—	4	8	no	—
378	D₄	—	4	2	no	—
379	D₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	4	no	—
382	C₂×C₄	—	2	4	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	4	4	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	—	8	1	no	—
392	C₂×C₄	—	8	2	no	—
393	C₂×C₄	—	16	2	no	—
394	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
395	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
396	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
397	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
398	C₈	—	2	2	no	—
399	C₈	—	2	2	no	—
400	C₈	—	4	2	no	—
401	C₈	—	4	2	no	—
402	C₂²	—	2	1	no	—
403	C₂²	—	2	1	no	—
404	C₂²	—	2	4	no	—
405	C₂²	—	4	2	no	—
406	C₂²	—	4	2	no	—
407	C₂²	—	4	2	no	—
408	C₂²	—	8	2	no	—
409	C₂²	—	8	1	no	—
410	C₂²	—	8	1	no	—
411	C₂²	—	16	1	no	—
412	C₂²	—	16	4	no	—
413	C₂²	#543×C₆₇	1	1	yes	characteristic
414	C₂²	D₄²×C₆₇	1	1	yes	characteristic
415	C₂²	#551×C₆₇	1	1	yes	characteristic
416	C₄	—	2	4	no	—
417	C₄	—	4	2	no	—
418	C₄	—	4	2	no	—
419	C₄	—	4	2	no	—
420	C₄	—	8	2	no	—
421	C₄	—	8	1	no	—
422	C₄	C₂×C₂²⋊Q₈×C₆₇	1	2	yes	—
423	C₄	#546×C₆₇	1	2	yes	—
424	C₂	—	2	1	no	—
425	C₂	—	2	1	no	—
426	C₂	—	4	4	no	—
427	C₂	—	8	2	no	—
428	C₂	C₆₇×#2437	1	1	yes	characteristic
429	C₂	C₆₇×#3289	1	1	yes	characteristic
430	C₂	C₆₇×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×H(5,𝔽₂)×C₆₇	C₂²
LCS	G ▷ C₂²×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
1. C₆₇×#3083	C₂	semidirect
3. C₆₇×#2093	C₂	semidirect
4. C₆₇×#2063	C₂	semidirect
5. C₆₇×#2027	C₂	semidirect
6. C₆₇×#1866	C₂	semidirect
7. C₆₇×#1859	C₂	semidirect
8. C₆₇×#1816	C₂	semidirect
12. #544×C₆₇	C₂²	semidirect
18. C₂×#108×C₆₇	C₂²	semidirect
19. #428×C₆₇	C₂²	semidirect
20. C₂×#107×C₆₇	C₂²	semidirect
21. #424×C₆₇	C₂²	semidirect
34. #384×C₆₇	C₂²	semidirect
35. #343×C₆₇	C₂²	semidirect
36. #343×C₆₇	C₂²	semidirect
37. #339×C₆₇	C₂²	semidirect
38. #336×C₆₇	C₂²	semidirect
41. H(5,𝔽₂)×C₆₇	Q₈	semidirect
208. #25982	C₆₇	direct (N × H)
216. #3083	C₂×C₆₇	semidirect
218. #2093	C₂×C₆₇	semidirect
219. #2063	C₂×C₆₇	semidirect
220. #2027	C₂×C₆₇	semidirect
221. #1866	C₂×C₆₇	semidirect
222. #1859	C₂×C₆₇	semidirect
223. #1816	C₂×C₆₇	semidirect
225. C₆₇	#25982	direct (N × H)
228. #544	C₂²×C₆₇	semidirect
234. C₂×#108	C₂²×C₆₇	semidirect
235. #428	C₂²×C₆₇	semidirect
236. C₂×#107	C₂²×C₆₇	semidirect
237. #424	C₂²×C₆₇	semidirect
250. #384	C₂²×C₆₇	semidirect
251. #343	C₂²×C₆₇	semidirect
252. #343	C₂²×C₆₇	semidirect
253. #339	C₂²×C₆₇	semidirect
254. #336	C₂²×C₆₇	semidirect
257. H(5,𝔽₂)	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3083	2
C₆₇×#2897	2
C₆₇×#2093	2
C₆₇×#2063	2
C₆₇×#2027	2
C₆₇×#1866	2
C₆₇×#1859	2
C₆₇×#1816	2
C₆₇×#1792	2
#25982	67

Subgroups — C67×#25982 #30705233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25982 #30705233