Subgroups — C₂×C₆₇×#3517 #30742041

Order 17152.

7854 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3517	C₂	1	32	yes	maximal, direct product factor
2	C₂×#557×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#553×C₆₇	C₂	1	6	yes	maximal, complemented
4	C₂×#550×C₆₇	C₂	1	3	yes	maximal, complemented
5	C₂×#550×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#550×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#547×C₆₇	C₂	1	3	yes	maximal, complemented
8	C₂×#540×C₆₇	C₂	1	6	yes	maximal, complemented
9	C₂×#532×C₆₇	C₂	1	3	yes	maximal, complemented
10	C₂×#529×C₆₇	C₂	1	3	yes	maximal, complemented
11	C₂×#520×C₆₇	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×#41×C₆₇	C₂²	1	2	yes	—
15	C₂²×#41×C₆₇	C₂²	1	6	yes	—
16	#557×C₆₇	C₂²	1	16	yes	complemented
17	#553×C₆₇	C₂²	1	96	yes	complemented
18	#550×C₆₇	C₂²	1	48	yes	complemented
19	#550×C₆₇	C₂²	1	16	yes	complemented
20	#550×C₆₇	C₂²	1	32	yes	complemented
21	#547×C₆₇	C₂²	1	48	yes	complemented
22	#540×C₆₇	C₂²	1	96	yes	complemented
23	#532×C₆₇	C₂²	1	48	yes	complemented
24	C₂×C₄⋊Q₈×C₆₇	C₂²	1	3	yes	—
25	C₂×#127×C₆₇	C₂²	1	3	yes	—
26	C₂×#127×C₆₇	C₂²	1	6	yes	—
27	#529×C₆₇	C₂²	1	48	yes	complemented
28	C₂×#126×C₆₇	C₂²	1	6	yes	—
29	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#125×C₆₇	C₂²	1	6	yes	—
31	C₂×#124×C₆₇	C₂²	1	3	yes	—
32	C₂×#123×C₆₇	C₂²	1	6	yes	—
33	C₂×#123×C₆₇	C₂²	1	12	yes	—
34	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	3	yes	—
35	C₂×#121×C₆₇	C₂²	1	3	yes	—
36	C₂×#121×C₆₇	C₂²	1	6	yes	—
37	C₂×#121×C₆₇	C₂²	1	6	yes	—
38	C₂×#121×C₆₇	C₂²	1	6	yes	—
39	C₂×#121×C₆₇	C₂²	1	12	yes	—
40	C₂×#121×C₆₇	C₂²	1	6	yes	—
41	#520×C₆₇	C₂²	1	32	yes	complemented
42	C₄×#41×C₆₇	C₂²	1	16	yes	complemented
43	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
46	C₂×C₄×Q₈×C₆₇	C₂²	1	3	yes	—
47	C₂×C₄×D₄×C₆₇	C₂²	1	3	yes	—
48	C₂×C₄×D₄×C₆₇	C₂²	1	3	yes	—
49	C₂×C₄×D₄×C₆₇	C₂²	1	6	yes	—
50	C₂×C₄×D₄×C₆₇	C₂²	1	6	yes	—
51	C₂×C₄×D₄×C₆₇	C₂²	1	3	yes	—
52	C₂×C₄×D₄×C₆₇	C₂²	1	6	yes	—
53	C₂×C₄×D₄×C₆₇	C₂²	1	6	yes	—
54	C₂×#117×C₆₇	C₂²	1	3	yes	—
55	C₂×#117×C₆₇	C₂²	1	6	yes	—
56	C₂²×C₄⋊C₄×C₆₇	C₂²	1	6	yes	—
57	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
58	C₂×#41×C₆₇	—	2	8	no	—
59	C₂×#41×C₆₇	—	2	16	no	—
60	C₂×#41×C₆₇	—	4	24	no	—
61	C₂×#41×C₆₇	C₂³	1	8	yes	—
62	C₂×#41×C₆₇	C₂³	1	16	yes	—
63	C₂×#41×C₆₇	C₂³	1	48	yes	—
64	C₂²×Q₈×C₆₇	C₂³	1	3	yes	—
65	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₂²×D₄×C₆₇	C₂³	1	3	yes	—
67	C₂²×D₄×C₆₇	C₂³	1	6	yes	—
68	C₂²×D₄×C₆₇	C₂³	1	3	yes	—
69	C₂²×D₄×C₆₇	C₂³	1	6	yes	—
70	C₂²×D₄×C₆₇	C₂³	1	6	yes	—
71	C₂²×D₄×C₆₇	C₂³	1	12	yes	—
72	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
73	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
74	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
75	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
76	C₂³×C₄×C₆₇	C₂³	1	12	yes	—
77	C₄⋊Q₈×C₆₇	C₂³	1	24	yes	—
78	#127×C₆₇	C₂³	1	24	yes	—
79	#127×C₆₇	C₂³	1	48	yes	—
80	#126×C₆₇	C₂³	1	48	yes	—
81	#125×C₆₇	C₂³	1	8	yes	—
82	#125×C₆₇	C₂³	1	48	yes	—
83	#124×C₆₇	C₂³	1	24	yes	—
84	#123×C₆₇	C₂³	1	48	yes	—
85	#123×C₆₇	C₂³	1	96	yes	—
86	C₂²⋊Q₈×C₆₇	C₂³	1	24	yes	—
87	#121×C₆₇	C₂³	1	24	yes	—
88	#121×C₆₇	C₂³	1	48	yes	—
89	#121×C₆₇	C₂³	1	48	yes	—
90	#121×C₆₇	C₂³	1	48	yes	—
91	#121×C₆₇	C₂³	1	96	yes	—
92	#121×C₆₇	C₂³	1	48	yes	—
93	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
94	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
95	C₄×Q₈×C₆₇	C₂³	1	16	yes	—
96	C₄×Q₈×C₆₇	C₂³	1	24	yes	—
97	C₄×D₄×C₆₇	C₂³	1	24	yes	—
98	C₄×D₄×C₆₇	C₂³	1	24	yes	—
99	C₄×D₄×C₆₇	C₂³	1	48	yes	—
100	C₄×D₄×C₆₇	C₂³	1	48	yes	—
101	C₄×D₄×C₆₇	C₂³	1	24	yes	—
102	C₄×D₄×C₆₇	C₂³	1	48	yes	—
103	C₄×D₄×C₆₇	C₂³	1	48	yes	—
104	#117×C₆₇	C₂³	1	24	yes	—
105	#117×C₆₇	C₂³	1	48	yes	—
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	48	yes	—
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
114	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
115	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
116	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
117	C₂×C₂²⋊C₄×C₆₇	C₂³	1	6	yes	—
118	C₂×C₂²⋊C₄×C₆₇	C₂³	1	6	yes	—
119	C₂×C₂²⋊C₄×C₆₇	C₂³	1	6	yes	—
120	C₂×C₂²⋊C₄×C₆₇	C₂³	1	6	yes	—
121	C₂×C₂²⋊C₄×C₆₇	C₂³	1	6	yes	—
122	C₂×C₂²⋊C₄×C₆₇	C₂³	1	6	yes	—
123	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
124	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
125	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
126	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
127	C₂×C₄²×C₆₇	C₂³	1	24	yes	—
128	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
129	C₂⁴×C₆₇	C₂⁴	1	3	yes	—
130	C₂⁴×C₆₇	C₂⁴	1	6	yes	—
131	#41×C₆₇	—	2	32	no	—
132	#41×C₆₇	—	2	64	no	—
133	#41×C₆₇	—	4	96	no	—
134	C₂×Q₈×C₆₇	—	2	12	no	—
135	C₂×Q₈×C₆₇	#41	1	8	yes	complemented
136	C₂×Q₈×C₆₇	C₂⁴	1	12	yes	—
137	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
138	C₂×D₄×C₆₇	—	2	12	no	—
139	C₂×D₄×C₆₇	—	2	24	no	—
140	C₂×D₄×C₆₇	—	2	12	no	—
141	C₂×D₄×C₆₇	—	4	12	no	—
142	C₂×D₄×C₆₇	—	4	12	no	—
143	C₂×D₄×C₆₇	—	4	24	no	—
144	C₂×D₄×C₆₇	C₂⁴	1	12	yes	—
145	C₂×D₄×C₆₇	C₂⁴	1	24	yes	—
146	C₂×D₄×C₆₇	C₂⁴	1	12	yes	—
147	C₂×D₄×C₆₇	C₂⁴	1	24	yes	—
148	C₂×D₄×C₆₇	C₂⁴	1	24	yes	—
149	C₂×D₄×C₆₇	C₂⁴	1	48	yes	—
150	C₂²×C₄×C₆₇	—	2	12	no	—
151	C₂²×C₄×C₆₇	—	2	24	no	—
152	C₂²×C₄×C₆₇	—	2	12	no	—
153	C₂²×C₄×C₆₇	—	4	6	no	—
154	C₂²×C₄×C₆₇	—	4	24	no	—
155	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
156	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
157	C₂²×C₄×C₆₇	C₂⁴	1	24	yes	—
158	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
159	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
160	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
162	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
163	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
164	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₆₇	C₂⁴	1	12	yes	—
166	C₂²×C₄×C₆₇	C₂⁴	1	48	yes	—
167	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
168	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
169	C₄⋊C₄×C₆₇	C₂⁴	1	12	yes	—
170	C₄⋊C₄×C₆₇	C₂⁴	1	24	yes	—
171	C₄⋊C₄×C₆₇	C₂⁴	1	24	yes	—
172	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
173	C₄⋊C₄×C₆₇	C₂⁴	1	12	yes	—
174	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
175	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
176	C₄⋊C₄×C₆₇	C₂⁴	1	24	yes	—
177	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
178	C₄⋊C₄×C₆₇	C₂⁴	1	24	yes	—
179	C₂²⋊C₄×C₆₇	C₂⁴	1	24	yes	—
180	C₂²⋊C₄×C₆₇	C₂⁴	1	24	yes	—
181	C₂²⋊C₄×C₆₇	C₂⁴	1	24	yes	—
182	C₂²⋊C₄×C₆₇	C₂⁴	1	24	yes	—
183	C₂²⋊C₄×C₆₇	C₂⁴	1	24	yes	—
184	C₂²⋊C₄×C₆₇	C₂⁴	1	24	yes	—
185	C₄²×C₆₇	C₂⁴	1	4	yes	—
186	C₄²×C₆₇	C₂⁴	1	12	yes	—
187	C₄²×C₆₇	C₂⁴	1	12	yes	—
188	C₄²×C₆₇	C₂⁴	1	12	yes	—
189	C₄²×C₆₇	C₂⁴	1	24	yes	—
190	C₂³×C₆₇	—	2	6	no	—
191	C₂³×C₆₇	—	2	12	no	—
192	C₂³×C₆₇	—	2	6	no	—
193	C₂³×C₆₇	—	2	12	no	—
194	C₂³×C₆₇	—	2	6	no	—
195	C₂³×C₆₇	—	2	12	no	—
196	C₂³×C₆₇	C₂⁵	1	6	yes	—
197	C₂³×C₆₇	C₂⁵	1	12	yes	—
198	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
199	Q₈×C₆₇	—	2	24	no	—
200	Q₈×C₆₇	C₂×#41	1	16	yes	complemented
201	D₄×C₆₇	—	2	24	no	—
202	D₄×C₆₇	—	2	48	no	—
203	D₄×C₆₇	—	2	24	no	—
204	D₄×C₆₇	—	4	24	no	—
205	D₄×C₆₇	—	4	24	no	—
206	D₄×C₆₇	—	4	48	no	—
207	C₂×C₄×C₆₇	—	2	2	no	—
208	C₂×C₄×C₆₇	—	2	4	no	—
209	C₂×C₄×C₆₇	—	2	6	no	—
210	C₂×C₄×C₆₇	—	2	12	no	—
211	C₂×C₄×C₆₇	—	2	2	no	—
212	C₂×C₄×C₆₇	—	2	6	no	—
213	C₂×C₄×C₆₇	—	2	4	no	—
214	C₂×C₄×C₆₇	—	2	24	no	—
215	C₂×C₄×C₆₇	—	2	48	no	—
216	C₂×C₄×C₆₇	—	2	24	no	—
217	C₂×C₄×C₆₇	—	4	12	no	—
218	C₂×C₄×C₆₇	—	4	48	no	—
219	C₂×C₄×C₆₇	C₂×#41	1	12	yes	—
220	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
221	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
222	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
223	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
224	C₂×C₄×C₆₇	C₂⁵	1	6	yes	—
225	C₂×C₄×C₆₇	C₂⁵	1	6	yes	—
226	C₂×C₄×C₆₇	C₂⁵	1	12	yes	—
227	C₂×C₄×C₆₇	C₂⁵	1	4	yes	—
228	C₂×C₄×C₆₇	C₂⁵	1	6	yes	—
229	C₂×C₄×C₆₇	C₂⁵	1	2	yes	—
230	C₂²×C₆₇	—	2	6	no	—
231	C₂²×C₆₇	—	2	12	no	—
232	C₂²×C₆₇	—	2	6	no	—
233	C₂²×C₆₇	—	2	12	no	—
234	C₂²×C₆₇	—	2	6	no	—
235	C₂²×C₆₇	—	2	12	no	—
236	C₂²×C₆₇	—	4	12	no	—
237	C₂²×C₆₇	—	4	24	no	—
238	C₂²×C₆₇	C₂²×#41	1	2	yes	—
239	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
240	C₂²×C₆₇	#585	1	2	yes	—
241	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
242	C₄×C₆₇	—	2	2	no	—
243	C₄×C₆₇	—	2	4	no	—
244	C₄×C₆₇	—	2	6	no	—
245	C₄×C₆₇	—	2	12	no	—
246	C₄×C₆₇	—	2	2	no	—
247	C₄×C₆₇	—	2	6	no	—
248	C₄×C₆₇	—	2	4	no	—
249	C₄×C₆₇	C₂²×#41	1	12	yes	—
250	C₄×C₆₇	C₂×H(5,𝔽₂)	1	4	yes	—
251	C₂×#3517	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
252	C₂×C₆₇	—	4	6	no	—
253	C₂×C₆₇	—	4	12	no	—
254	C₂×C₆₇	#3517	1	4	yes	direct product factor
255	C₂×C₆₇	C₂³×#41	1	1	yes	characteristic
256	C₂×C₆₇	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
257	C₂×C₆₇	C₂×#585	1	1	yes	characteristic
258	#3517	C₂×C₆₇	1	32	yes	direct product factor
259	C₂×#557	C₂×C₆₇	1	1	yes	characteristic, complemented
260	C₂×#553	C₂×C₆₇	1	6	yes	complemented
261	C₂×#550	C₂×C₆₇	1	3	yes	complemented
262	C₂×#550	C₂×C₆₇	1	1	yes	characteristic, complemented
263	C₂×#550	C₂×C₆₇	1	2	yes	complemented
264	C₂×#547	C₂×C₆₇	1	3	yes	complemented
265	C₂×#540	C₂×C₆₇	1	6	yes	complemented
266	C₂×#532	C₂×C₆₇	1	3	yes	complemented
267	C₂×#529	C₂×C₆₇	1	3	yes	complemented
268	C₂×#520	C₂×C₆₇	1	2	yes	complemented
269	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic, complemented
270	C₆₇	C₂×#3517	1	1	yes	characteristic, direct product factor, Sylow
271	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
272	C₂²×#41	C₂²×C₆₇	1	2	yes	—
273	C₂²×#41	C₂²×C₆₇	1	6	yes	—
274	#557	C₂²×C₆₇	1	16	yes	complemented
275	#553	C₂²×C₆₇	1	96	yes	complemented
276	#550	C₂²×C₆₇	1	48	yes	complemented
277	#550	C₂²×C₆₇	1	16	yes	complemented
278	#550	C₂²×C₆₇	1	32	yes	complemented
279	#547	C₂²×C₆₇	1	48	yes	complemented
280	#540	C₂²×C₆₇	1	96	yes	complemented
281	#532	C₂²×C₆₇	1	48	yes	complemented
282	C₂×C₄⋊Q₈	C₂²×C₆₇	1	3	yes	—
283	C₂×#127	C₂²×C₆₇	1	3	yes	—
284	C₂×#127	C₂²×C₆₇	1	6	yes	—
285	#529	C₂²×C₆₇	1	48	yes	complemented
286	C₂×#126	C₂²×C₆₇	1	6	yes	—
287	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
288	C₂×#125	C₂²×C₆₇	1	6	yes	—
289	C₂×#124	C₂²×C₆₇	1	3	yes	—
290	C₂×#123	C₂²×C₆₇	1	6	yes	—
291	C₂×#123	C₂²×C₆₇	1	12	yes	—
292	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	3	yes	—
293	C₂×#121	C₂²×C₆₇	1	3	yes	—
294	C₂×#121	C₂²×C₆₇	1	6	yes	—
295	C₂×#121	C₂²×C₆₇	1	6	yes	—
296	C₂×#121	C₂²×C₆₇	1	6	yes	—
297	C₂×#121	C₂²×C₆₇	1	12	yes	—
298	C₂×#121	C₂²×C₆₇	1	6	yes	—
299	#520	C₂²×C₆₇	1	32	yes	complemented
300	C₄×#41	C₂²×C₆₇	1	16	yes	complemented
301	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
302	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
303	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
304	C₂×C₄×Q₈	C₂²×C₆₇	1	3	yes	—
305	C₂×C₄×D₄	C₂²×C₆₇	1	3	yes	—
306	C₂×C₄×D₄	C₂²×C₆₇	1	3	yes	—
307	C₂×C₄×D₄	C₂²×C₆₇	1	6	yes	—
308	C₂×C₄×D₄	C₂²×C₆₇	1	6	yes	—
309	C₂×C₄×D₄	C₂²×C₆₇	1	3	yes	—
310	C₂×C₄×D₄	C₂²×C₆₇	1	6	yes	—
311	C₂×C₄×D₄	C₂²×C₆₇	1	6	yes	—
312	C₂×#117	C₂²×C₆₇	1	3	yes	—
313	C₂×#117	C₂²×C₆₇	1	6	yes	—
314	C₂²×C₄⋊C₄	C₂²×C₆₇	1	6	yes	—
315	C₂²×C₄²	C₂²×C₆₇	1	3	yes	—
316	C₂×#41	—	2	8	no	—
317	C₂×#41	—	2	16	no	—
318	C₂×#41	—	4	24	no	—
319	C₂×#41	C₂³×C₆₇	1	8	yes	—
320	C₂×#41	C₂³×C₆₇	1	16	yes	—
321	C₂×#41	C₂³×C₆₇	1	48	yes	—
322	C₂²×Q₈	C₂³×C₆₇	1	3	yes	—
323	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
324	C₂²×D₄	C₂³×C₆₇	1	3	yes	—
325	C₂²×D₄	C₂³×C₆₇	1	6	yes	—
326	C₂²×D₄	C₂³×C₆₇	1	3	yes	—
327	C₂²×D₄	C₂³×C₆₇	1	6	yes	—
328	C₂²×D₄	C₂³×C₆₇	1	6	yes	—
329	C₂²×D₄	C₂³×C₆₇	1	12	yes	—
330	C₂³×C₄	C₂³×C₆₇	1	6	yes	—
331	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
332	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
333	C₂³×C₄	C₂³×C₆₇	1	12	yes	—
334	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
335	C₄⋊Q₈	C₂³×C₆₇	1	24	yes	—
336	#127	C₂³×C₆₇	1	24	yes	—
337	#127	C₂³×C₆₇	1	48	yes	—
338	#126	C₂³×C₆₇	1	48	yes	—
339	#125	C₂³×C₆₇	1	8	yes	—
340	#125	C₂³×C₆₇	1	48	yes	—
341	#124	C₂³×C₆₇	1	24	yes	—
342	#123	C₂³×C₆₇	1	48	yes	—
343	#123	C₂³×C₆₇	1	96	yes	—
344	C₂²⋊Q₈	C₂³×C₆₇	1	24	yes	—
345	#121	C₂³×C₆₇	1	24	yes	—
346	#121	C₂³×C₆₇	1	48	yes	—
347	#121	C₂³×C₆₇	1	48	yes	—
348	#121	C₂³×C₆₇	1	48	yes	—
349	#121	C₂³×C₆₇	1	96	yes	—
350	#121	C₂³×C₆₇	1	48	yes	—
351	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
352	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
353	C₄×Q₈	C₂³×C₆₇	1	16	yes	—
354	C₄×Q₈	C₂³×C₆₇	1	24	yes	—
355	C₄×D₄	C₂³×C₆₇	1	24	yes	—
356	C₄×D₄	C₂³×C₆₇	1	24	yes	—
357	C₄×D₄	C₂³×C₆₇	1	48	yes	—
358	C₄×D₄	C₂³×C₆₇	1	48	yes	—
359	C₄×D₄	C₂³×C₆₇	1	24	yes	—
360	C₄×D₄	C₂³×C₆₇	1	48	yes	—
361	C₄×D₄	C₂³×C₆₇	1	48	yes	—
362	#117	C₂³×C₆₇	1	24	yes	—
363	#117	C₂³×C₆₇	1	48	yes	—
364	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
365	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
366	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
367	C₂×C₄⋊C₄	C₂³×C₆₇	1	48	yes	—
368	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
369	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
370	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
371	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
372	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
373	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
374	C₂×C₄⋊C₄	C₂³×C₆₇	1	6	yes	—
375	C₂×C₂²⋊C₄	C₂³×C₆₇	1	6	yes	—
376	C₂×C₂²⋊C₄	C₂³×C₆₇	1	6	yes	—
377	C₂×C₂²⋊C₄	C₂³×C₆₇	1	6	yes	—
378	C₂×C₂²⋊C₄	C₂³×C₆₇	1	6	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₆₇	1	6	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₆₇	1	6	yes	—
381	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
382	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
383	C₂×C₄²	C₂³×C₆₇	1	6	yes	—
384	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
385	C₂×C₄²	C₂³×C₆₇	1	24	yes	—
386	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
387	C₂⁴	C₂⁴×C₆₇	1	3	yes	—
388	C₂⁴	C₂⁴×C₆₇	1	6	yes	—
389	#41	—	2	32	no	—
390	#41	—	2	64	no	—
391	#41	—	4	96	no	—
392	C₂×Q₈	—	2	12	no	—
393	C₂×Q₈	#41×C₆₇	1	8	yes	complemented
394	C₂×Q₈	C₂⁴×C₆₇	1	12	yes	—
395	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
396	C₂×D₄	—	2	12	no	—
397	C₂×D₄	—	2	24	no	—
398	C₂×D₄	—	2	12	no	—
399	C₂×D₄	—	4	24	no	—
400	C₂×D₄	—	4	12	no	—
401	C₂×D₄	—	4	12	no	—
402	C₂×D₄	C₂⁴×C₆₇	1	12	yes	—
403	C₂×D₄	C₂⁴×C₆₇	1	24	yes	—
404	C₂×D₄	C₂⁴×C₆₇	1	12	yes	—
405	C₂×D₄	C₂⁴×C₆₇	1	24	yes	—
406	C₂×D₄	C₂⁴×C₆₇	1	24	yes	—
407	C₂×D₄	C₂⁴×C₆₇	1	48	yes	—
408	C₂²×C₄	—	2	24	no	—
409	C₂²×C₄	—	2	12	no	—
410	C₂²×C₄	—	2	12	no	—
411	C₂²×C₄	—	4	24	no	—
412	C₂²×C₄	—	4	6	no	—
413	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
414	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
415	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
416	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
417	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
418	C₂²×C₄	C₂⁴×C₆₇	1	12	yes	—
419	C₂²×C₄	C₂⁴×C₆₇	1	12	yes	—
420	C₂²×C₄	C₂⁴×C₆₇	1	6	yes	—
421	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
422	C₂²×C₄	C₂⁴×C₆₇	1	24	yes	—
423	C₂²×C₄	C₂⁴×C₆₇	1	12	yes	—
424	C₂²×C₄	C₂⁴×C₆₇	1	48	yes	—
425	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
426	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
427	C₄⋊C₄	C₂⁴×C₆₇	1	12	yes	—
428	C₄⋊C₄	C₂⁴×C₆₇	1	24	yes	—
429	C₄⋊C₄	C₂⁴×C₆₇	1	24	yes	—
430	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
431	C₄⋊C₄	C₂⁴×C₆₇	1	12	yes	—
432	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
433	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
434	C₄⋊C₄	C₂⁴×C₆₇	1	24	yes	—
435	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
436	C₄⋊C₄	C₂⁴×C₆₇	1	24	yes	—
437	C₂²⋊C₄	C₂⁴×C₆₇	1	24	yes	—
438	C₂²⋊C₄	C₂⁴×C₆₇	1	24	yes	—
439	C₂²⋊C₄	C₂⁴×C₆₇	1	24	yes	—
440	C₂²⋊C₄	C₂⁴×C₆₇	1	24	yes	—
441	C₂²⋊C₄	C₂⁴×C₆₇	1	24	yes	—
442	C₂²⋊C₄	C₂⁴×C₆₇	1	24	yes	—
443	C₄²	C₂⁴×C₆₇	1	4	yes	—
444	C₄²	C₂⁴×C₆₇	1	12	yes	—
445	C₄²	C₂⁴×C₆₇	1	12	yes	—
446	C₄²	C₂⁴×C₆₇	1	12	yes	—
447	C₄²	C₂⁴×C₆₇	1	24	yes	—
448	C₂³	—	2	6	no	—
449	C₂³	—	2	12	no	—
450	C₂³	—	2	6	no	—
451	C₂³	—	2	12	no	—
452	C₂³	—	2	6	no	—
453	C₂³	—	2	12	no	—
454	C₂³	C₂⁵×C₆₇	1	6	yes	—
455	C₂³	C₂⁵×C₆₇	1	12	yes	—
456	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
457	Q₈	—	2	24	no	—
458	Q₈	C₂×#41×C₆₇	1	16	yes	complemented
459	D₄	—	2	24	no	—
460	D₄	—	2	48	no	—
461	D₄	—	2	24	no	—
462	D₄	—	4	48	no	—
463	D₄	—	4	24	no	—
464	D₄	—	4	24	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	6	no	—
468	C₂×C₄	—	2	12	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	6	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	24	no	—
473	C₂×C₄	—	2	48	no	—
474	C₂×C₄	—	2	24	no	—
475	C₂×C₄	—	4	48	no	—
476	C₂×C₄	—	4	12	no	—
477	C₂×C₄	C₂×#41×C₆₇	1	12	yes	—
478	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
479	C₂×C₄	C₂⁵×C₆₇	1	2	yes	—
480	C₂×C₄	C₂⁵×C₆₇	1	2	yes	—
481	C₂×C₄	C₂⁵×C₆₇	1	4	yes	—
482	C₂×C₄	C₂⁵×C₆₇	1	6	yes	—
483	C₂×C₄	C₂⁵×C₆₇	1	6	yes	—
484	C₂×C₄	C₂⁵×C₆₇	1	12	yes	—
485	C₂×C₄	C₂⁵×C₆₇	1	4	yes	—
486	C₂×C₄	C₂⁵×C₆₇	1	6	yes	—
487	C₂×C₄	C₂⁵×C₆₇	1	2	yes	—
488	C₂²	—	2	6	no	—
489	C₂²	—	2	12	no	—
490	C₂²	—	2	6	no	—
491	C₂²	—	2	12	no	—
492	C₂²	—	2	6	no	—
493	C₂²	—	2	12	no	—
494	C₂²	—	4	12	no	—
495	C₂²	—	4	24	no	—
496	C₂²	C₂²×#41×C₆₇	1	2	yes	—
497	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
498	C₂²	#585×C₆₇	1	2	yes	—
499	C₂²	C₂⁶×C₆₇	1	1	yes	characteristic
500	C₄	—	2	2	no	—
501	C₄	—	2	4	no	—
502	C₄	—	2	6	no	—
503	C₄	—	2	12	no	—
504	C₄	—	2	2	no	—
505	C₄	—	2	6	no	—
506	C₄	—	2	4	no	—
507	C₄	C₂²×#41×C₆₇	1	12	yes	—
508	C₄	C₂×H(5,𝔽₂)×C₆₇	1	4	yes	—
509	C₂	—	4	6	no	—
510	C₂	—	4	12	no	—
511	C₂	C₆₇×#3517	1	4	yes	direct product factor
512	C₂	C₂³×#41×C₆₇	1	1	yes	characteristic
513	C₂	C₂²×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
514	C₂	C₂×#585×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×C₆₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3517	C₂	direct (N × H)
2. C₂×#557×C₆₇	C₂	semidirect
3. C₂×#553×C₆₇	C₂	semidirect
4. C₂×#550×C₆₇	C₂	semidirect
5. C₂×#550×C₆₇	C₂	semidirect
6. C₂×#550×C₆₇	C₂	semidirect
7. C₂×#547×C₆₇	C₂	semidirect
8. C₂×#540×C₆₇	C₂	semidirect
9. C₂×#532×C₆₇	C₂	semidirect
10. C₂×#529×C₆₇	C₂	semidirect
11. C₂×#520×C₆₇	C₂	semidirect
12. C₂×C₄×#41×C₆₇	C₂	semidirect
16. #557×C₆₇	C₂²	semidirect
17. #553×C₆₇	C₂²	semidirect
18. #550×C₆₇	C₂²	semidirect
19. #550×C₆₇	C₂²	semidirect
20. #550×C₆₇	C₂²	semidirect
21. #547×C₆₇	C₂²	semidirect
22. #540×C₆₇	C₂²	semidirect
23. #532×C₆₇	C₂²	semidirect
27. #529×C₆₇	C₂²	semidirect
41. #520×C₆₇	C₂²	semidirect
42. C₄×#41×C₆₇	C₂²	semidirect
135. C₂×Q₈×C₆₇	#41	semidirect
200. Q₈×C₆₇	C₂×#41	semidirect
251. C₂×#3517	C₆₇	direct (N × H)
254. C₂×C₆₇	#3517	direct (N × H)
258. #3517	C₂×C₆₇	direct (N × H)
259. C₂×#557	C₂×C₆₇	semidirect
260. C₂×#553	C₂×C₆₇	semidirect
261. C₂×#550	C₂×C₆₇	semidirect
262. C₂×#550	C₂×C₆₇	semidirect
263. C₂×#550	C₂×C₆₇	semidirect
264. C₂×#547	C₂×C₆₇	semidirect
265. C₂×#540	C₂×C₆₇	semidirect
266. C₂×#532	C₂×C₆₇	semidirect
267. C₂×#529	C₂×C₆₇	semidirect
268. C₂×#520	C₂×C₆₇	semidirect
269. C₂×C₄×#41	C₂×C₆₇	semidirect
270. C₆₇	C₂×#3517	direct (N × H)
274. #557	C₂²×C₆₇	semidirect
275. #553	C₂²×C₆₇	semidirect
276. #550	C₂²×C₆₇	semidirect
277. #550	C₂²×C₆₇	semidirect
278. #550	C₂²×C₆₇	semidirect
279. #547	C₂²×C₆₇	semidirect
280. #540	C₂²×C₆₇	semidirect
281. #532	C₂²×C₆₇	semidirect
285. #529	C₂²×C₆₇	semidirect
299. #520	C₂²×C₆₇	semidirect
300. C₄×#41	C₂²×C₆₇	semidirect
393. C₂×Q₈	#41×C₆₇	semidirect
458. Q₈	C₂×#41×C₆₇	semidirect
511. C₂	C₆₇×#3517	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3517	2
C₂×#557×C₆₇	2
C₂×#553×C₆₇	2
C₂×#550×C₆₇	2
C₂×#547×C₆₇	2
C₂×#540×C₆₇	2
C₂×#532×C₆₇	2
C₂×#529×C₆₇	2
C₂×#520×C₆₇	2
C₂×C₄×#41×C₆₇	2
C₂×#3517	67

Subgroups — C2×C67×#3517 #30742041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3517 #30742041